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## Simulating Poisson Networks Efficiently

Published:

When I first started my Ph.D. I learned about modeling large wireless network using point processes. They allow you to obtain tractable mathematical models that scale well with the size of the network. The most commonly used model in wireless networks is the homogeneous Poisson point proces (PPP), because it has a reasonable assumption about the network geometry and leads to tractable models. To briefly introduce the concept, say we have a set of two dimensional points $\phi = \{u_1, \cdots\}$ randomly distributed in $\mathbb{R}^2$, such that for every compact set $B \in \mathbb{R}^2$. The set of points $\phi$ is a HPPP if:

## Portfolio item number 1

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Short description of portfolio item number 1

## Portfolio item number 2

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Short description of portfolio item number 2

## Uma metodologia para geraçao de sequências aleatórias usando mapas caóticos

Published in XXXIII Simpósio Brasileiro de Telecomunicaçoes (SBrT 2015), Juiz de Fora, 2015

Geradores de números aleatórios são amplamente utilizados em aplicações científicas e tecnológicas. Particularmente em criptografia, são empregados em sistemas de chave pública, assinatura digital. Neste trabalho, apresentamos uma metodologia para o projeto destes geradores a partir de mapas caóticos baseada em duas técnicas: salto de amostras e discretização codificada variante no tempo. Mostramos que o procedimento possui alta taxa de geração de bits, além de dispensar pósprocessamento para melhoria de suas propriedades aleatórias. Validamos o método proposto empregando o teste NIST. Por fim, apresentamos um circuito que implementa o mapa caótico analisado neste trabalho.

Recommended citation: Artiles, J.A., Chaves, D.P., Evangelista, J.V. and Pimentel, C., "Uma metodologia para geraçao de sequências aleatórias usando mapas caóticos." XXXIII Simpósio Brasileiro de Telecomunicaçoes (SBrT 2015), Juiz de Fora, pp.1-5. #

## Spectral analysis of a chaotic map based on the hyperbolic tangent function

Published in Journal of Communication and Information Systems, 2016

This work generalizes the chaotic map recently introduced by the authors with the addition of two new control parameters. The Lyapunov exponent is employed to analyze the chaotic dynamics of the proposed map. The spectral characteristics of chaotic signals generated by this map as a function of the control parameters are also investigated. The proposed map can be used in symbolic dynamics based communication systems with advantages over current approaches based on piecewise linear maps.

Recommended citation: Souza, C., Evangelista, J.V., Chaves, D.P. and Pimentel, C., 2016. "Spectral analysis of a chaotic map based on the hyperbolic tangent function." Journal of Communication and Information Systems, 31(1). https://jvce92.github.io/files/2016-spectral-analysis-of-a-chaotic.pdf

## Physical-layer authentication Using chaotic maps

Published in Journal of Communication and Information Systems, 2016

Message authentication, which ensures that a received message comes from its acclaimed sender, is of fundamental importance for secure communication systems. We consider in this work a physical layer authentication system employing tag signals embedded in the message to provide a robust au- thentication method. This work diverges from previous work in the area when it comes to the tag generation method. While the previous works use methods based on cryptographic hash functions or on the channel side information our system employs unidimensional chaotic maps to generate these tags. Due to the loss of information about the initial condition of chaotic maps, we show that they are strong candidates for the tag generation process. We prove that chaotic tags provide a positive lower bound on the unconditional security of the system. Additionally, we calculate the probability of suc- cess for three possible attacks to the authentication system: impersonation, substitution and replay. Finally, we analyze how the system parameters affect these probabilities and some performance met- rics (bit error rate, outage probability, probability of false negative) and explore the tradeoff between security and performance in order to provide guidelines to design the system.

Recommended citation: Evangelista, Joao V. C., 2016, "Physical-layer authentication Using chaotic maps." M.Sc. Thesis, Universidade Federal de Pernambuco. https://jvce92.github.io/files/msc-thesis.pdf

## Analysis of the cell association for decoupled wireless access in a two tier network

Published in IEEE 28th Annual International Symposium on Personal, Indoor, and Mobile Radio Communications (PIMRC), 2017

In this paper, we analyze the association of a user terminal in a two-tier network (ie, macrocells and millimeter wave small cells). We assumed a decoupled wireless access where a user terminal has the liberty to choose different base stations (BSs) in uplink and downlink based on the received power and the channel quality. A practical blockage model where a human body is a blockage to millimeter wave (mmW) is considered. An in-depth simulation study is done to explore the effectiveness of decoupled wireless access in a crowded environment. In addition to that, a detailed analysis on the intuitiveness and the mathematical tractability of the blockage model used is also provided. In the end, few research questions on the efficacy of decoupled wireless access are raised in this paper.

Recommended citation: Z. Sattar, J. V. C. Evangelista, G. Kaddoum and N. Batani, "Analysis of the cell association for decoupled wireless access in a two tier network," 2017 IEEE 28th Annual International Symposium on Personal, Indoor, and Mobile Radio Communications (PIMRC), Montreal, QC, 2017, pp. 1-6. https://jvce92.github.io/files/2017-analysis-of-the-cell-association.pdf

## Fairness and Sum-Rate Maximization via Joint Channel and Power Allocation in Uplink SCMA Networks

Published in arXiv preprint arXiv:1805.11722, 2018

In this work, we consider a sparse code multiple access uplink network, where J users simultaneously transmit data over K subcarriers, such that J > K , with a constraint on the power transmitted by each user. To jointly optimize the codebook assignment and the transmitted power per subcarrier, two new iterative algorithms are proposed, the first one aims to maximize the sum-rate (Max-SR) of the network, while the second aims to maximize the minimum rate (Max-Min). In both cases, the optimization problem is of the mixed-integer nonlinear programming (MINLP) type, with non-convex objective functions. To address this issues, we employ an alternating optimization framework, where the codebook allocation problem is decoupled from the power allocation into two subproblems, and they are solved in an iterative alternating fashion. These subproblems are non-convex, therefore, we employ a sequential convex programming (SCP) method to solve them. An analysis of the convergence of the algorithms is provided. A comparison of the sum-rate and Jain’s fairness index of the novel algorithms with three other algorithms proposed in the literature is presented for varying power budgets. The Max-SR algorithm outperforms the others in the sum-rate sense, while the Max-Min outperforms them in the fairness sense, achieving near perfect fairness.

Recommended citation: Evangelista, J.V., Sattar, Z., Kaddoum, G. and Chaaban, A., 2018. "Fairness and Sum-Rate Maximization via Joint Channel and Power Allocation in Uplink SCMA Networks." arXiv preprint arXiv:1805.11722. https://jvce92.github.io/files/2018-fairness-and-sum-rate-maximization.pdf

## Spectral Efficiency Analysis of the Decoupled Access for Downlink and Uplink in Two Tier Network

Published in arXiv preprint arXiv:1808.02523, 2018

This paper analyzes the efficacy of decoupled wireless access in a two-tier heterogeneous network. The decoupled wireless access and its performance benefits have been studied in different scenarios recently. In this paper, an in-depth analysis on its efficacy from spectral efficiency perspective is provided. To achieve this task, (i) new closed form expressions for probability of association of user equipment with different tiers employing different frequency bands (i.e., microwave and millimeter wave) with different pathloss exponents are derived using univariate Fox’s H-functions; (ii) Distributions of the distance to the serving base stations are also derived; (iii) Exact expressions of spectral efficiency for different association cases are further obtained in a closed form using bivariate Fox’s H-functions. Furthermore, rigorous simulation results are provided which validate the aforementioned analytical results. In addition to that, a detailed discussion on the decoupling gain of decoupled wireless access and its efficacy is also provided. Lastly, despite the improvement provided by the decoupled wireless access, which is evident from the results presented in this paper, few questions are raised on its pragmatic value.

Recommended citation: Z. Sattar, J. V. C. Evangelista, G. Kaddoum and N. Batani,, 2018. "Spectral Efficiency Analysis of the Decoupled Access for Downlink and Uplink in Two Tier Network." arXiv preprint arXiv:1808.02523 https://jvce92.github.io/files/2018-spectral-efficiency-analysis-of-the-decoupled-access.pdf

## Talk 1 on Relevant Topic in Your Field

Published:

This is a description of your talk, which is a markdown files that can be all markdown-ified like any other post. Yay markdown!

Published:

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## Conference Proceeding talk 3 on Relevant Topic in Your Field

Published:

This is a description of your conference proceedings talk, note the different field in type. You can put anything in this field.

## Digital Circuits Teaching Assistant

Undergraduate Course, Universidade Federal de Pernambuco, Department of Electronics and Systems, 2013

I’ve taught the digital circuits lab classes and graded the student participation and reports. The focus of the lab was on the design and synthesis of digital circuits using the VHDL description language targeting Altera FPGAs.

## Analog Circuits Teaching Assistant

Undergraduate Course, Universidade Federal de Pernambuco, Department of Electronics and Systems, 2014

I’ve taught the analog circuits lab classes and graded the student participation and reports. The focus of the lab was on the analysis, simulation and design of:

• Differential amplifiers
• Multistage BJT and CMOS amplifiers
• Stable current and voltage sources I’ve also introduced the students to SPICE simulation through the ngspice software.


text:
What is the new boiling temperature for bromine, if the pressure changes from atmospheric pressure to "76000 Pa"?

To what temperature do we need to heat up bromine in order for it to boil on 76 000 Pa. At atmospheric pressure bromine boils on 58 degrees. Enthalpy of vaporization for bromine is $\text{194.26 J/g}$.

Jan 1, 2017

I got $\textcolor{b l u e}{{T}_{b 2} \approx {50.5}^{\circ} \text{C}}$, when I used ${T}_{b 1} = {58.8}^{\circ} \text{C}$ and your $\Delta {H}_{\text{vap}}$ of $\text{194.26 J/g}$, or $\Delta {\overline{H}}_{\text{vap" = "31.04 kJ/mol}}$. But that's not the $\Delta {\overline{H}}_{\text{vap}}$ I see here.

• If you use ${T}_{b 1} = {58}^{\circ} \text{C}$ and $\Delta {\overline{H}}_{\text{vap" = "31.04 kJ/mol}}$, you should get ${T}_{b 2} \approx {49.8}^{\circ} \text{C}$.
• If you use ${T}_{b 1} = {58}^{\circ} \text{C}$ and $\Delta {\overline{H}}_{\text{vap" = "29.96 kJ/mol}}$, you would get ${T}_{b 2} \approx {49.5}^{\circ} \text{C}$.
• If you use ${T}_{b 1} = {58.8}^{\circ} \text{C}$ and $\Delta {\overline{H}}_{\text{vap" = "29.96 kJ/mol}}$, you would get ${T}_{b 2} \approx {50.2}^{\circ} \text{C}$.
This demonstrates how good the assumption is that $\Delta {H}_{\text{vap}}$ doesn't vary in this temperature range.

So basically, you should get somewhere between ${49.5}^{\circ} \text{C}$ and ${50.5}^{\circ} \text{C}$.

Since we need to find a new temperature given two pressures, one temperature, and an enthalpy of vaporization $\Delta {H}_{\text{vap}}$, we should turn to the Clausius-Clapeyron equation in its natural logarithm form.

$\boldsymbol{\ln \left({P}_{2} / {P}_{1}\right) = \ln \left(\frac{P \left({T}_{b 2}\right)}{P \left({T}_{b 1}\right)}\right) = - \frac{\Delta {\overline{H}}_{\text{vap}}}{R} \left[\frac{1}{T} _ \left(b 2\right) - \frac{1}{T} _ \left(b 1\right)\right]}$,

where $P \left({T}_{b i}\right)$ is the vapor pressure of the substance at some specified boiling point ${T}_{b i}$, $\Delta {\overline{H}}_{\text{vap}}$ is the molar enthalpy of vaporization, in $\text{kJ"/"mol}$, and $R = \text{0.008314472 kJ/mol"cdot"K}$ is the universal gas constant.

To use it, we're solving for ${T}_{b 2}$. It's generally easier to plug in numbers early, actually.

Recall that atmospheric pressure is $\cancel{\text{1 atm" xx "101325 Pa"/cancel"atm" = "101325 Pa}}$. So, ${P}_{1} = \text{101325 Pa}$, and:

ln("76000 Pa"/("101325 Pa")) = -(194.26 xx 10^(-3) "kJ"/cancel"g" xx (2xx79.904 cancel"g")/("mol Br"_2))/("0.008314472 kJ/mol"cdot"K")[1/T_(b2) - 1/(58.8 + "273.15 K")]

$\implies \ln \left(0.7501\right) = - \left(31.044 \cancel{\text{kJ/mol")/(0.008314472 cancel"kJ/mol"cdot"K")[1/T_(b2) - 1/"331.95 K}}\right]$

= -"3733.77 K"[1/T_(b2) - 0.0030125 "K"^(-1)]

$\implies - \ln \frac{0.7501}{\text{3733.77 K" = 1/T_(b2) - 0.0030125 "K}} ^ \left(- 1\right)$

$\implies - \ln \frac{0.7501}{\text{3733.77 K" + 0.0030125 "K}} ^ \left(- 1\right) = \frac{1}{T} _ \left(b 2\right)$

$\implies {T}_{b 2} \approx \text{323.68 K}$,

or about $\textcolor{b l u e}{{50.5}^{\circ} \text{C}}$.

This should make physical sense because we say that boiling is easier at higher altitudes, at which the pressure is lower.

Lower pressure makes it easier to achieve the required vapor pressure, and thus, it is easier to boil at higher altitudes.

I derive the Clausius-Clapeyron equation below, if you want to take a look.

DISCLAIMER: DERIVATION BELOW!

I never remember the Clausius-Clapeyron equation, but I do remember the Clapeyron equation, so we can derive it by comparing with the Clapeyron equation:

$\boldsymbol{\frac{\mathrm{dP}}{\mathrm{dT}} = \frac{\Delta {H}_{\text{vap}}}{T \Delta {V}^{\left(l\right) \to \left(g\right)}}}$

where $\Delta {V}^{\left(l\right) \to \left(g\right)}$ is the change in volume due to vaporization, and $T$ is temperature in $\text{K}$. This equation tells you the slope of the liquid-vapor coexistence curve on a phase diagram.

Since a gas is much more compressible than a liquid, we can say that $\Delta {V}^{\left(l\right) \to \left(g\right)} \approx {V}_{g a s}$. Therefore:

$\frac{\mathrm{dP}}{\mathrm{dT}} \approx \frac{\Delta {H}_{\text{vap}}}{T {V}_{g a s}}$

If the gas is assumed ideal, then we can further substitute the ideal gas law to get that ${V}_{g a s} = \frac{n R {T}_{b}}{P}$, and:

$\frac{\mathrm{dP}}{\mathrm{dT}} \approx \frac{\Delta {H}_{\text{vap}} P}{n R {T}_{b}^{2}}$

$= \frac{\Delta {\overline{H}}_{\text{vap}} P}{R {T}_{b}^{2}}$

Recall that the Clausius-Clapeyron equation has $\frac{\mathrm{dl} n P}{\mathrm{dT}}$ on the left side. From the chain rule, $\frac{\mathrm{dl} n P}{\mathrm{dT}} = \frac{\mathrm{dl} n P}{\mathrm{dP}} \frac{\mathrm{dP}}{\mathrm{dT}} = \frac{1}{P} \frac{\mathrm{dP}}{\mathrm{dT}}$.

So, we obtain the Clausius-Clapeyron equation:

$\frac{1}{P} \frac{\mathrm{dP}}{\mathrm{dT}} = \textcolor{g r e e n}{\frac{\mathrm{dl} n P}{\mathrm{dT}} = \frac{\Delta {\overline{H}}_{\text{vap}}}{R {T}_{b}^{2}}}$

where DeltabarH_"vap" = (DeltaH_"vap")/n is the molar enthalpy of vaporization of ${\text{Br}}_{2} \left(l\right)$.

Now, when we integrate both sides (the left side varies from ${P}_{1}$ to ${P}_{2}$, and the right side varies from ${T}_{b 1}$ to ${T}_{b 2}$), and assume that $\Delta \overline{H}$ varies little in this temperature range, we get:

${\int}_{{P}_{1}}^{{P}_{2}} \mathrm{dl} n P$

$= {\int}_{{T}_{b 1}}^{{T}_{b 2}} \frac{\Delta {\overline{H}}_{\text{vap")/(RT_b^2)dT_b = (DeltabarH_"vap}}}{R} {\int}_{{T}_{b 1}}^{{T}_{b 2}} \frac{1}{{T}_{b}^{2}} {\mathrm{dT}}_{b}$

$\implies \boldsymbol{\ln \left({P}_{2} / {P}_{1}\right) = - \frac{\Delta {\overline{H}}_{\text{vap}}}{R} \left[\frac{1}{T} _ \left(b 2\right) - \frac{1}{T} _ \left(b 1\right)\right]}$

This is the equation we'd use to solve this problem.

format: plain


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# Velocity And Angular Velocity Of A Sphere Rolling Down A Ramp

I was given a question on a recent test that asked me the following:

A sphere is allowed to roll down a smooth inclined plane (No Friction), as it rolls down does its velocity remain constant, and does its angular velocity remain constant?

Now obviously I answered that both the velocity and angular velocity increase as the sphere rolls down the ramp, but I was marked wrong, my teacher said that the angular velocity doesn't increase and remains constant, because there isn't a torque acting on the sphere, but the velocity does increase.

Now here's my question, even if there wasn't a torque acting on the sphere, the velocity was still increasing, and since $$v=r\omega$$, and $$r$$ stays constant shouldn't increasing $$v$$ also increase $$\omega$$.

• Did you mean "and $r$ stays constant shouldn't increasing $v$ also increase $\omega$?"
– user191954
Dec 3, 2018 at 12:03
• Your problem lies in the last sentence. Which $v$ do you mean?
– user197851
Dec 3, 2018 at 12:04
• I mean the $v$ with which the sphere rolls down the plane. Dec 3, 2018 at 12:14
• Yes, it was supposed to be " shouldn't increasing v also increase ω?" Dec 3, 2018 at 12:15
• But the formula $v=r\omega$ does not involve the speed with which the sphere is moving. It refers to a different $v$. Remember, in your case, there is no friction. (Also perhaps the word "rolls" is misleading, and the word "slides" would be more accurate.)
– user197851
Dec 3, 2018 at 12:29

Now here's my question, even if there wasn't a torque acting on the sphere, the velocity was still increasing, and since $$v=rω$$, and $$r$$ stays constant shouldn't increasing $$v$$ also increase $$ω$$.
As the sphere slides down the incline, its velocity $$v$$ increases due to the Earth's gravitational acceleration.
However, without friction (and thus without torque) $$\omega$$ remains constant. $$v=rω$$ only applies for rolling without sliding, where there is enough friction to prevent sliding altogether.


text:
# Matrix lookup table

Status
Not open for further replies.

#### Mosaic

##### Well-Known Member
Hi:
I need an approach to look up data in a x-y matrix based upon x,y coords.

The data on the x-axis is stored as sequential nybbles in sequential bytes....about 6 bytes = 12 nybbles of x -data
The y axis are just bytes.

Asm pls if u have any sample code....16F886 is my chosen mcu.

Thanks!

Last edited:

#### Pommie

##### Well-Known Member
If you x data is only a nibble then use it as an index into a 16 byte table.
Code:
	addwf	PCL,f
dt	5,10,12,16,18,2,28,32
dt	40,50,62,74,86,100,120,160
Or, are you saying it is a 12 by (num y values) matrix. If so how many y values? Also, is memory a problem?

Mike.

#### Mosaic

##### Well-Known Member
We're looking at 12 by 12 matrices. memory is a problem. I need at least need 4 of them.

So i compressed the x range into nybbles to halve the eeprom requirement.
The tables are eprom based, and are mapped to GPR individually as required.User editable so no computed gotos in code space.
Gotta do it with FSR/INDF in GPR space with writes back to eeprom after editing.

The way i look at it is i need to figure a way to address the x,y coords and display/update the individual memory locations.

#### Mike - K8LH

##### Well-Known Member
We're looking at 12 by 12 matrices. memory is a problem. I need at least need 4 of them.

So i compressed the x range into nybbles to halve the eeprom requirement.
The tables are eprom based, and are mapped to GPR individually as required.User editable so no computed gotos in code space.
Gotta do it with FSR/INDF in GPR space with writes back to eeprom after editing.

The way i look at it is i need to figure a way to address the x,y coords and display/update the individual memory locations.
ArrayIndex = (Y * 6) + (X / 2)

where X = 0..11, Y = 0..11, and ArrayIndex = 0..71 (the "byte" index into the array)

the result of X & 1 (bit test X,0) determines which nybble to use.

the formula assumes you're storing arrays in EEPROM in Array[Y][X] format;

Code:
        de      0x21,0x22,0x44,0x33,0x21,0x30    ; Y = 0, X = 0..11 (nybbles)
de      0x31,0x33,0x31,0x22,0x24,0x22    ; Y = 1, X = 0..11 (nybbles)
......
de      0x44,0x31,0x33,0x31,0x41,0x12    ; Y = 11, X = 0..11 (nybbles)
Example Index routine;

Code:
;
;  get EEPROM array 'byte' index, 0..71
;
clrc                    ;
rlf     y,W             ; wreg = Y * 2
movwf   index           ; index = Y * 2
addwf   index,F         ; index = Y * 4 (C=0)
addwf   index,F         ; index = Y * 6 (C=0)
rrf     x,W             ; wreg = (X / 2)
addwf   index,F         ; index = (Y * 6) + (X / 2)

Last edited:

#### Mosaic

##### Well-Known Member
yes, that's it...

#### Mike - K8LH

##### Well-Known Member
yes, that's it...

Have fun... Regards, Mike, K8LH

3v0

#### Mosaic

##### Well-Known Member
Great stuff there Mike....!!!

Status
Not open for further replies.


text:
# Java Program to Display Sum of Series X^1 + X^2 + X^3 + ……… X^N

In the previous article we have discussed about Java Program to print Series 6 14 36 98 … N

In this article we are going to see how to print sum of the series X^1 + X^2 + X^3 + … + X^N by using Java programming language

## Java Program to Display Sum of Series X^1 + X^2 + X^3 + ……… X^N

On observing the pattern carefully, we can see

X is fixed and can be anything.

Sum series = X^1 + X^2 + X^3 + … + X^n,

where 1,2,3,..,n are the powers of X in each term respectively.

Example:

Suppose the value of X = 2 and n= 3
Then Sum of series = 2^1 + 2^2 + 2^3 = 2+4+8 = 14

Let’s see different ways to print sum of the series X^1 + X^2 + X^3 + … + X^N

### Method-1: Java Program to Display Sum of Series X^1 + X^2 + X^3 + ……… X^N By Using For Loop

Approach:

• Create Scanner class object.
• Declare an integer variable say ‘x’, which holds a fixed value.
• Declare an integer variable say ‘n’, which holds the Nth term power of the series.
• Prompt the user to enter 2 number as values of ‘x’, ‘n’ respectively .
• Let declare another integer variable say ‘result’ and initialize it to 0.
• Use a for loop from i=1 to i<=n, where the loop is incremented by 1
• Inside the loop we will find the value of x^i and then add that value with the result.
• Print the result.

Program:

import java.util.*;
public class Main
{
public static void main(String [] args)
{
//creating object of Scanner class
Scanner s = new Scanner(System.in);
//Taking input of number of elements in the series
System.out.println("Enter the value of x ");
int x = s.nextInt();
//Taking input of number of elements in the series
System.out.println("Enter the value of N ");
int n = s.nextInt();
int result = 0;
//for loop to print the series
for (int i = 1; i <= n; i++)
{
result +=Math.pow(x,i);
}
System.out.println("Sum of the series is " + result);
}
}

Output:

Enter the value of x
2
Enter the value of N
3
Sum of the series is 14

### Method-2: Java Program to Display Sum of Series X^1 + X^2 + X^3 + ……… X^N By Using While Loop

Approach:

• Create Scanner class object.
• Declare an integer variable say ‘x’, which holds a fixed value.
• Declare an integer variable say ‘n’, which holds the Nth term power of the series.
• Prompt the user to enter 2 number as values of ‘x’, ‘n’ respectively .
• Let declare another integer variable say ‘result’ and initialize it to 0.
• Declare and initialize an integer variable i=1
• Continue a while loop till i<=n, where the loop is incremented by 1
• Inside the loop we will find the value of x^i and then add that value with the result.
• Print the result.

Program:

import java.util.*;
public class Main
{
public static void main(String [] args)
{
//creating object of Scanner class
Scanner s = new Scanner(System.in);
//Taking input of number of elements in the series
System.out.println("Enter the value of x ");
int x = s.nextInt();
//Taking input of number of elements in the series
System.out.println("Enter the value of N ");
int n = s.nextInt();
int result=0;
System.out.print(result);
int i=1;
while(i<=n)
{
result +=Math.pow(x,i);
i++;
}
System.out.println("Sum of the series is " + result);
}
}

Output:

Enter the value of x
2
Enter the value of N
9
0Sum of the series is 1022

### Method-3: Java Program to Display Sum of Series X^1 + X^2 + X^3 + ……… X^N By Using User Defined Method

Approach:

• Create Scanner class object.
• Declare an integer variable say ‘x’, which holds a fixed value.
• Declare an integer variable say ‘n’, which holds the Nth term power of the series.
• Prompt the user to enter 2 number as values of ‘x’, ‘n’ respectively .
• Let declare another integer variable say ‘result’ and initialize it to 0.
• Use a for loop from i=1 to i<=n, where the loop is incremented by 1
• Inside the loop we will find the value of x^i and then add that value with the result.
• Print the result.

Program:

import java.util.*;
public class Main
{
public static void main(String[] args)
{
// creating object of scanner class
Scanner s = new Scanner(System.in);
//Taking input of number of elements in the series
System.out.println("Enter the value of x ");
int x = s.nextInt();
//Taking input of number of elements in the series
System.out.println("Enter the value of N ");
int n = s.nextInt();
// calling printSeries method to print the series
printSeries(x,n);
}

//printSeries metthod to print the series
public static void printSeries(int x, int n)
{
int result = 0;
System.out.print(result);
//for loop to print the series
for (int i = 1; i <=n; i++)
{
result +=Math.pow(x,i);
}
System.out.println("Sum of the series is " + result);
}
}

Output:

Enter the value of x
5
Enter the value of N
10
0Sum of the series is 12207030

Practice Java programming from home without using any fancy software just by tapping on this Simple Java Programs for Beginners tutorial.

Applications On:

1. X+X^2+X^3+…+X^N Program In Java
2. 1+X+X^2+X^3+…+X^N Program In Java
3. Display The Sum Of The Series 1 + X + X^2/2! + X^3/3! + … In Java
4. Summation Of X^N
5. Sum Of Series In Java
6. 1+X+X^2+X^3+…+X^N Program In C
7. Sum Of Series Program In Java

Related Java Programs:


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## Profile Posts Latest Activity Postings About

• Thanks for the honorific. ^_^

-Duxwing
Almost have a KT huh. I remember when I got the king tiger way back when that was my first tier 8. Well you can still go for it I have to say the E100 as my first tier 10 was a HUGE let down.
And you didn't ask me how I was

That's like super rude, bro
You listed world of tanks as one of the good games out now on the thread. So then I take it you must play it? Also what have you done to Reimu and Marisa?!?!
OBLIVIOUS ! ! !

WHERe HAVE YOU BEEN !
Greetings comrade of same post count!

1165 is truly a fine number. (;
did you vote TKL too?

I did not regret my decision too as I would have felt immeasurably guilty had I switched to TCB and he lost his deposit (which, in the final result, he did). At least now I can say "wen xin wu kui".
I see disdain as a sign of an arrogance. His first TV broadcast felt big brotherish.

TJS is a complete idiot la. Sadly my effort to help TKL keep his election deposit failed.

Of course I know of the JPSDS and SAF-ITE scheme. It's not about having a good life, but surviving. It's like molecules in boiling water. Some get enough energy and escape. But the rest are stuck and keep bumping into the other molecules.

Everyday I fret about school fees, text book costs, thankfully not quite at the "where's my next meal" level yet. Sure, you can say "but people can always work harder, earn more " - how much harder do we have to work? And of course, I know my own life is not much of a sample. But still.

Lol CNA is super funny today. Keep getting rebuffed by the candidates. Did you see the part where they panned to Tony Tan, interrupting something to say "We've got a live interview...well, perhaps not"?

TJS is such a media whore. And didn't get his speech telecast live. Idiot.
I'm not anti-foreigner and neither is my party, really. There is nothing wrong with immigration per se but infrastructure must be improved enough to handle the inflow and the labour market must still have jobs for Singaporeans.

I'm not sure about us failing to compete in the global arena. Much anecdotal evidence suggests that Singaporeans find it easier to compete overseas.

You sound like someone who was at least born into a reasonably functional family. I've had teenaged staff who had rather sad backgrounds, end up in the ITE, don't care about studying - what's going to happen when all these people become even more disenfranchised than they already are?

I like TKL. Frankly, I feel he's the only real finance guy. Tony Tan's worth is insufficiently independently attested to. And I'm impressed by the four-language nomination speech too. But he seems the least likely to win.

TT is arrogant. Maybe you see it as confidence and strength, I don't know.
You're voting for Tony Tan, I guess?
Points to you for image-editing skills.
Your avatar is from Disgaea, right?
Hi, do i know you from somewhere or is this just a prompt to do the intro thingy?
You should go back and check out the guideline thread. See the troll that you so warmly welcomed! =)
The picture was actually drawn by a Myanmese.
They won't touch you? Why the sudden muscle? But okay, that wasn't my point though. I just didn't really want to fill a thread with a completely irrelevant argument.

Oh and hey, I thought you were too busy, you know precious time and all.
I've replied to your post. How about you answer me with a visitors/private message? I think your beef with me doesn't have anything to do with the thread, and I think the mods would agree with me.

Your choice though, I'm only making a suggestion, not telling you what to do.
Pfft, why are YOU so whipped up?

I'm basing this on our skype convos with fill (and latte I think?). And I'm not stalking you, you pickled gizzard! I just happened to be in the same thread, at the same time, staring at your underpants! So there!

*edit
And like I said, it's my opinion.
"For instance, I do not speak with a Singlish accent" --- YES YOU DO, though you may not perceive it. Your enunciation is singaporean, though your inflections may *occasionally* sound western. The overall sound to me is still of a singaporean speaking in a slightly western manner.

That's my opinion anyway.
Thanks. Oh, and about the last thing you said in that "response to pod'lair" about what type you are.

I don't type people through text alone, there was a point where I was doing this, so I assume that is why you think I will. But over time I recognized the inherent weaknesses and unreliability of trying to properly interpret the use of cognitive functions simply by analysing the way a person is typing. So it is video or nothing.
Have you ever been viewing who is online, and then noticed that at the exact same moment someone else is viewing who is online. And then you wonder if where ever they are on earth, they too are looking at your status of "Viewing whose online."

And for a brief moment, you feel like you made a connection?
For what it's worth, I enjoy your every participation: comments and ideas.
Mm. Yes.

Distinction should be drawn between validity and soundness, as I said. The whole purpose is to identify arguments which may be untrue (even if they may be true part of the time), so we will not be misled.

Anyway, it is your right to have an illogical sig

I have studied logic semi-formally, but knew it before I did that. A book called "How to win every argument: the use and abuse of logic" taught me much (not sure if you've read that before). And a book called "How to win any argument" helped with the emotional side.

Though considering my lack of success hitherto, they mightn't help. Though I suspect that has more to do with the ineptitude of the reader than the authors .

And your last post wouldn't have been unassailable (logic wise). I would say "no offence", but I'm not particularly enamoured of that cliche.

Nice avatar, anyway.
You persist in remaining Oblivious!

A true conclusion can follow from true premises even if there is a flawed argument. The contentious word is because. Affirming the consequent is a fallacy in deductive logic regardless of whether there are conditions under which the latter can be fulfilled while the former is not, because true conclusions do not invariably follow from true premises when you use this logical structure (I'm sure you know what a logical fallacy is). Granted, it is usually inductively cogent - but "because" differs from "because it is highly likely that" & friends.

I'm sure you know what would be a valid form of your sig (maybe use modus tollens or something).

Then again - I missed something else too: "cannot be" != "is not".
Your signature is the logical fallacy of affirming the consequent.
*makes note to self*

Never take shower at Ob's place....
lol, the internet is a strange place, it's totally acceptable to miss it.
It was something about: 2) People tend to forget things.

But I really can't remember it.... arg!
:P


text:
## Journal of Symplectic Geometry

### The group of contact diffeomorphisms for compact contact manifolds

#### Abstract

For a compact contact manifold $M^{2n + 1}$, it is shown that the anisotropic Folland-Stein function spaces $\Gamma^{s} (M), s \geq (2n + 4)$ form an algebra. The notion of anisotropic regularity is extended to define the space of $\Gamma^{s}$-contact diffeomorphisms, which is shown to be a topological group under composition and a smooth Hilbert manifold. These results are used in a subsequent paper to analyse the action of the group of contact diffeomorphisms on the space of CR structures on a compact, three-dimensional manifold.

#### Article information

Source
J. Symplectic Geom., Volume 12, Number 1 (2014), 49-104.

Dates
First available in Project Euclid: 29 August 2014


text:
# How do you solve  x/-3+3=7?

Jun 16, 2015

$x = - 12$

#### Explanation:

Given $\frac{x}{- 3} + 3 = 7$

Clear the fraction by multiplying all terms (on both sides) by $\left(- 3\right)$
$\textcolor{w h i t e}{\text{XXXX}}$$x - 9 = - 21$
Add $9$ to both sides
$\textcolor{w h i t e}{\text{XXXX}}$$x = - 12$


text:
finding a particular solution...

can you check to see if my work is correct? i am to find an diffy equation that satisfies the boundary condition y(4) = 2. thanks in advance....IMG_0001.pdf


text:
Tag Info

6

I explain the concept by mysteriously showing this image as a prop: It always gets a laugh, which is nice :) Though, really, this image (which I haven't found before) might lead to a richer discussion: What I follow with is examples with arrays, along the lines of the following, and have students discuss in small groups what will work and what won't. We ...

4

I'm adding in a second answer to look at a few heftier assignments, worthy of a lab. These were created by a truly fantastic colleague of mine. The game Nim is excellent for a loop with no arrays. You can use just three variables to keep track of the state of the game, and a variable and an if statement to deal with the player. (Kids at this level will ...

4

I expect to update this a few times and hope to eventually have a long answer. But let me start by explaining some things that you may already be aware of, but other readers may not. First, OO programming is not, fundamentally, about inheritance and too many books and authors fail to understand that. Moreover, they use inheritance in a dreadful way that ...

4

Note that the following are intended for the instructor, not always directly for the student. If used well they require some analysis, not just coding, from the student. Use the fact that the sum of the first n odd numbers is $n^2$ to compute the integer square root of an integer. Test data should include 63, 64, and 65 to catch edge cases. This can be ...

3

Option 1: Scores Arsenal is going to play Real Madrid tomorrow. What is their score today? Focus on the difference between a match which concluded 0-0 and a match which has not been played yet. For the latter, the score is non-existant, which is what null represents. In general, focus on examples where zero has meaning and is not synonymous with an absence. ...

3

When you fill a form, and in the AGE field, you leave it empty THAT IS NULL. If you instead of leaving it empty, you fill it with 0 (zero) THAT IS ZERO

3

Google Classroom is a clean and feature light interface for managing the distribution and collection of course materials of a wide media variety. I teach about 100 students across five courses each semester using this platform. If you need a grade book that is integrated as well. I link to repl.it and let students fork my code in the cloud for some ...

3

My first instinct is that any LMS is overkill. LMSs manage classrooms, it's true, but they also are designed to help manage schools, and most of the administrative end (and most of the features outside of the literal classroom aspects, and even many of the features within the classrooms) of any LMS would be completely useless to you. Busing, medical ...

2

Maybe a better idea could be a quiz with automatically generated questions, like a code snippet and the request to determine the output for a given input, with different input values and variable names for each student. I actually have successfully implemented this (in part) for my first-semester programming tests this year. For example: Using the ...

2

Maybe didn't attend an exam, or register in the course, as apposed to took zero? -Or the difference between not having an account in a certain bank & having a zero balance in ur account? (The students may feel bad luck about 1st example, so I suggest u skip it unless too necessary)

2

Machine Learning for Kids is a resource for teaching ML to school age students that I have recently been playing around with. https://machinelearningforkids.co.uk/ It includes a range of projects that students can complete in Scratch, Python or App Inventor which range in challenge level. There is also a book you can buy but it is possible to use the site ...

2

These "laws" mostly reflect how people learn, so they are still quite relevant. But going through: ("Know thoroughly and familiarly the lesson you wish to teach; or, in other words, teach from a full mind and a clear understanding") is interesting in a CS context, since we often don't have fantastic knowledge of the material, because it ...

2

This is a bit of a frame-challenge, but what I'm understanding from your question is that your goals here are to: 1) Not overwhelm your students and avoid them getting lost in a plethora of blocks they won't need, 2) Force students to understand how to implement more complex blocks themselves. Have you considered instead of providing a restricted version of ...

2

VisualMelon's answer is a good one; I wanted to extend the suggestion to start on solo projects. Every team sport consists of members who need to both be individually skilled and fit, and work well together as a team. All of these things are hard to learn, and therefore more often than not it is better to learn these one at a time. Therefore, a team is ...

2

I would suggest not rushing into trying to contribute unless you have a very willing mentor: I suspect it's relatively unusual for people to contribute to packages/programs that they themselves don't use (indeed, I would be worried if this wasn't the case). That is, if you are not actually consuming any open-source Java packages/programs, then you probably ...

1

In my experience, engineers, statisticians and programmers will not use spreadsheets, they will use programming in languages such as R, python or matlab instead. To me it makes sense to spend a few hours teaching the basics of a spreadsheet app in high school so that students will know that it exists and will feel confident to learn more advanced usage if ...

1

Some basic SQL is there. Look under 2.2.3 Additional programming techniques. They also need to know about SQL injection for the network security topic.

1

I found a Python class to be so simple to write (for "mundane" tasks), that they can grow naturally out of a desire to simplify and refactor code in development. It is very much a pragmatic bottom-up approach; you were expecting to hack some simple code for a particular purpose, then it grows a bit and you find you have several functions handling &...

1

Now I have another favorite example called "Buy Low, Sell High", all the credit go to this article https://realpython.com/numpy-array-programming/#buy-low-sell-high Given a stock’s price history as a sequence, and assuming that you are only allowed to make one purchase and one sale, what is the maximum profit that can be obtained? For example, ...

1

Write a loop that reverses the characters of a string.

1

Since we introduce loops before we introduce arrays, we will often do simple programs like ask a user how many numbers they want to enter, and then find the sum of the numbers entered. Similarly, they can practice these concepts a few times to find the maximum, minimum, and average of the numbers entered. Having a few very similar tiny projects like this ...

1

Basically some experience with Discrete Mathematics and Probability , elementary Calculus is also recommend. And students are expected to be familiar with mathematical proofs, proofs by induction and good knowledge of a programming language. An Excerpt From CLRS What are the prerequisites for reading this book? You should have some programming experience. ...

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


text:
# Known Bugs And Quirks In XS Scripting¶

Written by: Alian713

This section is a list of known bugs, quirks and weird behaviour in XS Scripting. All the shown code can be directly copy pasted into any XS file to reproduce the bugs/effects documented.

Any new bugs discovered will be added to the list. This list will also be updated with each AoEII:DE update, removing any bugs that get fixed.

If you know of a bug that is not documented here, or if a bug listed here is actually fixed in the indicated game version, reach out to the authors of this guide! Check the About page for relevant information about the authors.

Game Version: 101.102.5558.0 (#71094) 9717890

Last Updated: 08.11.2022

Game Platform: Steam

OS: Windows 10


text:
• •

### 社鼠种群生态研究

1. 中国科学院动物研究所
• 出版日期:2006-07-29 发布日期:2007-09-13

### STUDIES 0N THE P0PULATION ECOLOGY OF SULPHUR BELLIED RAT

ZHANG Jie

• Online:2006-07-29 Published:2007-09-13

Abstract: In the period of 1983- 1986，the author had carried out the study on the population ecology of Sulphur Bellied Rat Rattus niviventer in Xishan area of Beijing。A totalnumber of 870( 464,406)rats were captured as samples．These sample rats were divided into five age groups according to their body weight excluding viscera ． Results showed that there were obvious seasonall disparities in age composition． The changes of the age composition were mainly caused by the changes of the reproductive behavior．Studies on the sex ratio represented that the proportion of female rats were generally higher than that of male in young and sub—adult groups．
The sulphur bellied rats started to breed from March to April， and most of them finished the breeding in September．Occasionally some individuals were found to be pregnant in October to December．The weight of the testis of the matured male rats were more than 0．8g．Females became matured at the age of 3— 8 months after being born， and usually started to breed only after overwintering．Most of females bred two foetus，but some of them bred once ，only a little individuals were found to breed three foetus in a year． The average of litter sizes for one foetus were 5.20． The number of litter sizes varied between 4．78—5．55．
The ecological life of sulphur bellied rat was 1．5 years or more．Some individuals participated breeding activities in two years．There were obvious seasonal difference in the number of population owing to the effect of reproduction and weather．The pattern of the changes in population was basically“S" type．The sulphur bellied rats inhabit onlv in mountain areas．They had not been found in plain．


text:
I wish I could have used Theorem X…

 IISC centenary source

Vijaya Ramachandran is now the Blakemore Regents Professor of Computer Science at The University of Texas at Austin. Once upon a time she was my PhD student, at Princeton, working on the theory of VLSI chip design. Long ago she moved successfully into other areas of theory. She became a leader in cache oblivious algorithms, an area created by Charles Leiserson, whom I advised as an undergraduate many years ago. It’s a small world.

Today I want to talk about theorems, old theorems and new, that I wish I could have used in a paper.

Call the phenomenon theorem envy: There are theorems that I have long known, but have never been able to use in any actual paper, in any proof of my own. I find this curiously unsatisfying, to know a cool result and yet be unable to use it in one of my papers. Perhaps I am alone in having this feeling, perhaps you only care about solving your problem, not how you solve it. Mostly I would agree with that. But part of me would like to use Theorem X in one of my papers. It just seems like it would be cool to do that.

This isn’t about envying those who originally proved the theorems. It’s about those who actually use the big gear in their toolboxes. We may have lots of shiny gadgets in our boxes, but part reason they’re shiny is they haven’t been used.

## Stirring the Brew

I realized this years ago when Ramachandran was working on her thesis. One of her results solved a problem that most had ignored—could not solve—about the cost of resizing drivers to make a chip go faster. One way to make a signal propagate faster across a chip is to use more power: the larger the driver, the faster the signal can move. This remains as significant today as then. The issue is not the speed-of-light restriction, but is related to the time it takes to charge a capacitive load:more power makes it happen faster. The downside then, and much worse today, is the added heat that power causes.

There is a curious cyclic issue in resizing drivers. The approach was in two steps:

1. Look at a chip and determine the wires that were too long and were slowing down the chip.

2. Resize their respective drivers so that they would go faster.

In practice this worked quite well. Yet there is a subtle bug: the resizing of the drivers moves parts of the chip, and this can make some wires become new bottlenecks. The obvious answer is to do the following:

1. Look at a chip and determine the wires that were long and were slowing down the chip.

2. Resize their respective drivers so that they would go faster.

3. Look at the chip, and if it is still slow repeat step 1.

The theoretical question is, does this process actually ever stop? Is it possible that the chip just blows up and thus no resizing will make it go faster? Or is this unlikely or even impossible?

The answer is that using the famous Brouwer Fixed-Point Theorem, Ramachandran was able to prove that this cannot happen. She showed that given some reasonable restrictions on the chip topology, the process becomes a continuous function on a convex compact set, which by the theorem has a fixed point. Brouwer means “brewer” in Dutch, and one way to visualize is stirring a brew in a pot—as opposed to a torus where the flow can go around with no vortex. The analysis also needs and shows quick convergence, but the essence is that things stop because the resizing operation has a fixed point.

I like her result. In my opinion its practical impact was shedding light on CAD, but I liked it even more because it used the wonderful Brouwer Fixed-Point Theorem. I never have been able to use that in any of my papers. Oh well.

So let me give a few other examples of other theorems that I would still like to get a chance to use in some future paper.

## A List

Here is a partial list of my top few:

${\bullet }$ Regularity Lemma. The famous Szemerédi regularity lemma states that every large enough graph can be divided into subsets of about the same size so that the edges between different subsets behave almost randomly. It was proved in a weaker form by Endre Szemerédi and used to prove his famous theorem on arithmetic progressions.

I have known it for many years, but have never been able to use it in my work. Of course many others have used it in their work. I would still like to be able to use it.

${\bullet }$ Ax—Grothendieck theorem. This is a famous theorem about the relationship between injectivity and surjectivity of multi-variate polynomials. It was proved independently by James Ax and Alexander Grothendieck. One version of the theorem is that if ${F}$ is a polynomial function from ${\mathbb{C}^{n}}$ to ${\mathbb{C}^{n}}$ and ${F}$ is injective then ${F}$ is bijective.

The full theorem is more general, and what’s remarkable is that the basic mechanism of the proof is really quite simple. It relies on the fact that injective implies surjective for finite sets: a map that is one-to-one on a finite set must be onto. I believe this should be useful in problems that concern polynomials, which after all is one of our main objects of study in complexity theory.

${\bullet }$ Baire Category Theorem. This is the famous theorem of René-Louis Baire, who proved it is in his 1899 doctoral thesis. It is an existence theorem showing that any countable family of open and everywhere-dense sets in a complete metric space has a non-empty intersection. It has been used in topology and analysis to prove the existence if many wonderful and exotic objects.

In our world we often prove the existence of exotic objects via the probabilistic method, but the Baire Category Theorem is another different method. It has constructive versions in certain settings, and complexity theorists have used them to show that certain oracle constructions are generic. I believe there should be some further interesting applications of it, and still hope to use it someday.

${\bullet }$ Feit-Thompson theorem: this is the famous theorem that every odd order group is solvable. It was proved by Walter Feit and John Thompson in the early 1960’s. It is a cornerstone to the understanding and eventual classification of finite simple groups: it implies that all simple groups are even order.

I have always thought that there might be some deep connection between this theorem and our inability in complexity theory to understand the power of computations modulo composites, but have never been able to make this idea into a theorem.

${\bullet}$ Hilbert’s Nullstellensatz. This does get applied, as we noted here. The cool German name is enough to induce envy. Particularly interesting are various effective and combinatorial forms of it.

This also stands for theorems in polynomial ideal theory and algebraic geometry in general, as used e.g. by Ketan Mulmuley and Milind Sohoni in their attack on ${\mathsf{P \neq NP}}$ type questions, and by Craig Gentry in crypto for some schemes of fully homomorphic encryption.

## Open Problems

Do you ever feel theorem envy? Which theorems would you like to use? Am I unique and alone in this affliction? Is there a cure?

1. June 11, 2014 3:57 pm

all simple groups are even order
or prime order (Cyclic groups)

June 11, 2014 4:38 pm

A cure for theorem envy ? I guess it’s what they call an axiom, a hypothesis, a conjecture, a phenomenon… Just repeat after me: “that’s the way it is!” 🙂

3. June 11, 2014 5:02 pm

I thought the Brouwer Fixed-Point Theorem was proven unstable.

4. June 11, 2014 6:11 pm

it seems you may have omitted some youve already mentioned in your blog. for example gromovs thm relates group theory & polynomial growth in a rare structural result giving sufficient/necessary conditions for a polynomial growth (& great blog on that). defn have proof envy on this one & seems like it ought to tie into TCS in some big way… ❓ 💡 ❗

also szemeredi’s thm has interested me also. its a big deal given that he got the \$1M abel prize in part for it. this also caused some semimajor angst in mathematical circles because szemeredis work was more combinatorial than topological. (a possible paradigm shift/ phase transition going on in mathematics reaching the highest echelons.)

June 11, 2014 9:50 pm

I envy Chandra-Furst-Lipton for being able to use a corollary of the multi-dim Van Der Warden theorem in their paper on multiparty protocols (STOC 1983).

Joel Spencer said that there are some theorems that just like you’d use them all the time but
you never seem to. Example: Every group is a group of perms. Group theorists must use this all the time. Alas, they rarely do. Another Example: Every number is the sum of at most four squares. Number theorists must use that all the time. Alas, they rarely due.

June 15, 2014 12:01 am

Cayley’s theorem is the opposite of a useless result – it’s so basic and deeply ingrained in group theory that its use becomes invisible. It’s the first line in a whole list of analogies between abstract groups and groups of symmetries.

June 12, 2014 3:34 am

Could you please provide a link to Ramachandran’s paper that uses the Brouwer Fixed-Point Theorem (or at least the title of the paper)?

7. June 15, 2014 9:29 am

Pip asks  “Do you ever feel theorem-envy? Which theorems would you like to use?”

In engineering contexts, what Pip calls “theorem envy” commonly manifests itself as the Enchantment of Naturality: the desire and dream that complex systems can find elegant explanations within mathematically natural frameworks.

Our consideration of the enchantment of naturality will be guided by Twain’s maxim “the humorous story is told gravely” and its by dual “the grave story is told humorously.”

To start, the enchanting naturality of Vijaya Ramachandran’s computational power metric is evident in a recent story in Aviation Week & Space Technology

Looming Shift in Supercomputing
by Graham Warwick, 19 May, 2014

Exascale computing — machines 1,000 times faster than today’s fastest supercomputers — will be prohibitively power-hungry without a fundamental change in how they operate.

Today’s top supercomputers consume around a watt of electricity for every 2 gigaflops of computing power [such that] exascale computations will need at least 500 megawatts — the Energy Department estimates 1-2 gigawatts — which would cost \$300-500 million a year for power.

A Natural Proposition  Supposing that exascale computing comes to pass (as foreseen by research funding agencies), then each doubling of algorithmic efficiency will acquire a concrete monetary value of \$300M-500M … an amount that nicely doubles the \$237M budget of the NSF’s Division of Mathematical Sciences.

Thus it is naturally evident that even one doubling of exascale algorithmic efficiency has sufficient power-value to support 3000-5000 mathematical researchers in perpetuity.

We therefore wonder: How might the researches of next-generation exascale-enterprise mathematicians be coordinated with sufficient laxity as to allow ample scope for creativity, and yet sufficient discipline as to foster exascale enterprises?

The Enchanting Naturality of Dynamics  Exascale computations commonly are simulations broadly conceived, whose mathematical frameworks are presently set forth in magisterial textbooks such as Bird, Stewart, and Lightfoot’s Transport Phenomena, Frenkel and Smit’s Understanding Molecular Simulation, Nielsen and Chuang’s Quantum Computation and Quantum Information, and Kloeden and Platen’s Numerical Solution of Stochastic Differential Equations; such that the physical elements of dynamics are encompassed in (say) 10,000 pages of (say) ten-or-twenty graduate-level textbooks.

It has to be said, however, that these books aren’t individually easy for students to read. Moreover, students often find that understanding any one of them doesn’t help all that much with understanding the others.

Geometry helps greatly to naturalize and unify the disparate physical elements of these textbooks, and toward this end mathematicians have blessed us (already) with Euclidean, Riemmannian, affine, symplectic, and algebraic notions of geometry; moreover in each decade striking new geometric notions are synthesized (see for example Terry Tao’s survey “Algebraic combinatorial geometry …”, arXiv:1310.6482).

This accelerating appreciation of mathematical naturality is good news, because otherwise 21st century students will scarcely be able to assimilate the accelerating extensions of our already-vast physical understanding. The prospect of rewriting and unifying the disorganized corpus of 20th century textbooks is perhaps the chief ingredient in the enchantment of naturality. It’s true that a lot of 20th century textbooks need rewriting, but on the other hand, \$500M/year suffices to support a lot of textbook-writing.

In a nutshell, the mathematical beauty of the proofs in Erdős’ Book originates equally in their enchanting naturality, and in their pedagogic concision, and in their inspiration for practical enterprise.

Broadening Appreciations of Naturality  Our century’s sustained accelerations in computational capacity and in physical understanding are accompanied too by an broadening appreciation of mathematical naturality, that holds forth the promise of explaining what exascale computing is really all about … which for young students (especially), is the thrilling new enterprises and abundant family-supporting jobs that exascale computing promises to foster.

The Enchantment of Naturality  In aggregate, the preceding considerations suggest that mathematical naturality — and the theorems that are its distilled essence — is an enchantment both pragmatic and platonic; that provides a language not solely for proofs, but for education and enterprises too.

An Answer to Pip’s Question  Theorem envy is a natural and healthy expression of the Enchantment of Naturality.


text:
Quantum particle in a 2 dimensional box

1. Jun 5, 2007

Unskilled

1. The problem statement, all variables and given/known data
I need some help

http://www.fysik.uu.se/kurser/1tt306/filer/TentKF06+short-answers.pdf [Broken]

On task 4 a) i don't understand why they have two solutions, one for odd n and the other for even n.

Last edited by a moderator: May 2, 2017
2. Jun 5, 2007

malawi_glenn

Hi there, I also study @ Uppsala ;) physics..

It has to do with the parity of the wave functions. Look on the symmetries around x=0. Ground state has zero nodes, 1st excited has one etc. Skiss the potetial and search for symmetry points.

Last edited by a moderator: May 2, 2017
3. Jun 6, 2007

StatusX

Note there's no qualititative difference between the x and y directions. You could write the wavefunction more symmetrically as:

$$\psi_{nm} = C \sin \left( n \pi \frac{x-(-A)}{A-(-A)} \right) \sin \left( m\pi \frac{y-0}{B-0}\right)$$

They just used some trig identities to rewrite this.

Last edited: Jun 6, 2007
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text:
# High school statistics questions and answers

Recent questions in High school statistics
Raina Cole 2022-12-27

### What number is 12% of 45?

Keira Gibson 2022-12-19

### The probability that an automobile being filled with gasoline also needs an oil change is 0.30; the probability that it needs a new oil filter is 0.40; and the probability that both the oil and the filter need changing is 0.10. (a) If the oil has to be changed, what is the probability that a new oil filter is needed? (b) If a new oil filter is needed, what is the probability that the oil has to be changed?

Kolton Rush 2022-12-19

### Leasing a car. The price of the car is$45,000. You have $3000 for a down payment. The term of the lease is and the interest rate is 3.5% APR. The buyout on the lease is51% of its purchase price and it is due at the end of the term. What are the monthly lease payments (before tax)?

Charlie Glass 2022-12-19

### The mean of sample A is significantly different than the mean of sample B. Sample A: $59,33,74,62,87,73$ Sample B: $53,67,72,57,93,79$ Use a two-tailed $t$-test of independent samples for the above hypothesis and data. What is the $p$-value?

glucidessho 2022-12-11

### What is mean and its advantages?

Mollie Wise 2022-12-04

### Which statement is true?Group of answer choicesThe larger the standard deviation, the higher the total risk.The larger the standard deviation, the lower the total risk.The standard deviation is not an indication of total risk.None of these are correct.

Brenda Leach 2022-12-01

### Which among the following is not a fundamental force of nature? Gravitational ForceElectromagnetic ForceBuoyant ForceWeak Nuclear Force

Scott Valenzuela 2022-11-29

### Еhere is a question that worries me: Why is it correct to say a normal distribution and the standard normal distribution?

Marques Flynn 2022-11-27

### For the data in the table, does y vary directly with x? Write an equation for the direct variation.$x:8,16,24\phantom{\rule{0ex}{0ex}}y:11,22,33$a. yes; y = 2.75xb. yes; y = 0.6875xc. yes; y = 1.375xd. no; y does not vary directly with x

Talia Frederick 2022-11-26

### Which of the following is not a measure of central tendency? Mode Standard deviation Median Mean

modnimCYf 2022-11-25

### Find the equation of the line perpendicular to $y=-\frac{7}{16}x$ that passes through (5,4)

liachta6VW 2022-11-25

agrarismoN07 2022-11-24

### You collect data from some population with mean $\mu$ and standard deviation $\sigma$ to test the following hypotheses: . You obtain a p-value of 0.053. Which of the following is true? a. A 95% confidence interval for $\mu$ will include the value 14. b. A 95% confidence interval for $\mu$ will include the value 1. c. A 90% confidence interval for $\mu$ will include the value 1. d. A 90% confidence interval for $\mu$ will include the value 14.

Elisha Cancino 2022-11-23

### Scores for a common standardized college aptitude test are normally distributed with a mean of 490 and a standard deviation of 103. Randomly selected students are given a Test Preparation Course before taking this test. Assume, for sake of argument, that the preparation course has no effect. If 1 student is randomly selected, find the probability that their score is at least 527.3.P(X > 527.3) = If 11 students are randomly selected, find the probability that their mean score is at least 527.3.P(¯¯¯X$\overline{X}$ > 527.3) =

Goundoubuf 2022-11-23

### i'm seeking out thoughts for a 15-hour mathematical enrichment course in a chinese language high faculty. What (pretty) simple concern would you advocate as a subject for any such course?historical past/issues:My students are generally pretty good at math, but many of them have no longer been uncovered to rigorous or summary mathematical reasoning. an amazing topic would be one that could not be impossibly hard for students who have by no means written or study proofs in English.i have taught this magnificence three times earlier than. (a part of the purpose that i'm posting that is that i have used up all my thoughts!) the primary semester I taught an introductory range theory elegance (which meandered its way toward a proof of quadratic reciprocity, though I think this become in the end too advanced/abstract for some of the students). the second one semester I taught fundamental graph idea and packages (with a focal point on planarity and coloring). The 1/3 semester I taught a class at the Rubik's dice.the students' math backgrounds are pretty numerous: a number of them take part in contest math competitions, and so are familiar with IMO-fashion techniques, however many aren't. a number of them may additionally realize some calculus, however I cannot assume it. all of them are superb at what in the united states is on occasion termed "pre-calculus": trigonometry, conic sections, systems of linear equations (though, shockingly, no matrices), and the like. They realize what a binomial coefficient is.So, any ideas? preferably, i'd like to find some thing a bit "sexy" (like the Rubik's cube) -- tries to encourage wide variety theory through cryptography seemed to fall on deaf ears, however being capable of "see" institution idea on the cube became pretty popular.(Responses specifically welcome from folks who grew up in the percent -- any mathematical subjects you desire were protected within the excessive college curriculum?)

Alvin Parks 2022-11-21

### I'm a graduate student studying quantum mechanics/quantum information and would like to consolidate my understanding of linear algebra. What are some good math books for that purpose?

Jaiden Elliott 2022-11-19

### How to deal with lack of peers?I am pursuing a B.S in Mathematics(freshman) in a country where people rarely study math for the sake of it and in a new university with good professors. Yet, it is new and I suffer from the lack of peers and it is literally impossible to pick up mathematical conversations on ideas with students . As a result, I find it difficult to exchange ideas with others.Is there something which can be done to make sure that lack of peers or the pressure to push oneself doesn't harm me much?

Uroskopieulm 2022-11-19


text:
acsch

Inverse hyperbolic cosecant

Syntax

`Y = acsch(X)`

Description

`Y = acsch(X)` returns the inverse hyperbolic cosecant for each element of `X`.

The `acsch` function operates element-wise on arrays. The function's domains and ranges include complex values. All angles are in radians.

Examples

collapse all

Graph of Inverse Hyperbolic Cosecant Function

Graph the inverse hyperbolic cosecant function over the domains and .

```x1 = -20:0.01:-1; x2 = 1:0.01:20; plot(x1,acsch(x1),x2,acsch(x2)), grid on ```

collapse all

Inverse Hyperbolic Cosecant

The inverse hyperbolic cosecant can be defined as

${\text{csch}}^{-1}\left(z\right)={\mathrm{sinh}}^{-1}\left(\frac{1}{z}\right).$


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# 108 (number)

(Redirected from Number 108)
 ← 107 108 109 →
Cardinal one hundred and eight
Ordinal 108th
(one hundred and eighth)
Factorization 22· 33
Divisors 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108
Roman numeral CVIII
Binary 11011002
Ternary 110003
Quaternary 12304
Quinary 4135
Senary 3006
Octal 1548
Duodecimal 9012
Vigesimal 5820
Base 36 3036

108 (one hundred [and] eight) is the natural number following 107 and preceding 109.

## In mathematics

108 (or nine dozen) is an abundant number and a semiperfect number. It is a tetranacci number.

108 is the hyperfactorial of 3 since it is of the form $1^1 \cdot 2^2 \cdot 3^3$.

108 is divisible by the value of its φ function, which is 36. 108 is also divisible by the total number of its divisors (12), hence it is a refactorable number.

In Euclidean space, the interior angles of a regular pentagon measure 108 degrees each.

There are 108 free polyominoes of order 7.

In base 10, it is a Harshad number and a self number.

The equation $2\sin\left(\frac{108^\circ}{2}\right) = \phi$ results in the golden ratio.

## Religion and the arts

The number 108 is considered sacred by several Eastern religions, such as Hinduism, Buddhism, Jainism, connected yoga and dharma based practices.

### Hinduism

Mukhya Shivaganas are 108 in number and hence all Shaiva sects majorly Lingayats use 108 rudraksha beaded lace for japa. Also they recite supreme lord Shiva's 108(AshtaaShatanaamaavaLi) names daily during their morning Shivapuja.
In Gaudiya Vaishnavism, there are 108 gopis of Vrindavan. Recital of these names, often accompanied by the counting of a 108-beaded mala, is considered sacred and often done during religious ceremonies. The recital is called namajapa. Accordingly, a japa mala usually has beads for 108 repetitions of a mantra. Srivaishnavism has 108 Divya Kshetras of Lord Vishnu, called as 108 DivyaDesam.

The well known bas-relief carving at the famous Angkor Wat temple in Cambodia relates the Hindu story of a serpent being pulled back and forth by 108 gods and asuras (demons), 54 gods pulling one way, and 54 asuras pulling the other, to churn the ocean of milk in order to produce the elixir of immortality. According to the Oriental Architecture site there are 5 monumental guardian gates to the fortified temple city of Angkor Thom.[1] In front of each gate stand giant statues of 54 gods (to the left of the causeway) and 54 demons (to the right of the causeway) which represent the churning of the ocean.

### Buddhism

Likewise, Tibetan Buddhist malas or rosaries (Tib. ཕྲེང་བ Wyl. phreng ba, "Trengwa") are usually 108 beads;[2] sometimes 111 including the guru bead(s), reflecting the words of the Buddha called in Tibetan the Kangyur (Wylie: Bka'-'gyur) in 108 volumes. Zen priests wear juzu (a ring of prayer beads) around their wrists, which consists of 108 beads.[3]

The Lankavatara Sutra has a section where the Bodhisattva Mahamati asks Buddha 108 questions[4] and another section where Buddha lists 108 statements of negation in the form of "A statement concerning X is not statement concerning X".[5] In a footnote, D.T. Suzuki explains that the Sanskrit word translated as "statement" is pada which can also mean "foot-step" or "a position." This confusion over the word "pada" explains why some have mistakenly held that the reference to 108 statements in the Lankavatara refer to the 108 steps that many temples have.[6]

In some schools of Buddhism it is believed that there are 108 feelings. According to Bhante Gunaratana[7] this number is reached by multiplying the senses smell, touch, taste, hearing, sight, and consciousness by whether they are painful, pleasant or neutral, and then again by whether these are internally generated or externally occurring, and yet again by past, present and future, finally we get 108 feelings. 6 × 3 × 2 × 3 = 108. In Japan, at the end of the year, a bell is chimed 108 times in Buddhist temples to finish the old year and welcome the new one. Each ring represents one of 108 earthly temptations a person must overcome to achieve nirvana.

### In Jewish culture and numerology

Jews often give gifts and charitable donations in multiples of the number 18, associated with the Hebrew word 'chai(חי)', meaning 'alive', 'living', or 'life'. See Chai (symbol). The number 108 is both a multiple of 18 (6 times 18) and it contains the numbers 1 and 8 that comprise the number 18.

### Pop culture

The number is prominent in Anime. The number 108 also appears in Crying Freeman as the sinister society of 108 Dragons, and Devil Hunter Yohko is the 108th generation demon slayer in her family, in Sekirei there are 108 beings called Sekirei which bond with humans, and in Sands of Destruction there are 108 laws of robotics in clock town.

### Ancient artifacts

The pre-historic monument Stonehenge is about 108 feet in diameter.[8][further explanation needed]

### Other references

In the neo-Gnostic teachings of Samael Aun Weor, an individual has 108 chances (lifetimes) to eliminate his egos and transcend the material world before "devolving" and having the egos forcefully removed in the infradimensions.[9]

## Martial arts

Many East Asian martial arts trace their roots back to Buddhism, specifically, to the Buddhist Shaolin Temple. Because of their ties to Buddhism, 108 has become an important symbolic number in a number of martial arts styles.

• According to Marma Adi and Ayurveda, there are 108 pressure points in the body, where consciousness and flesh intersect to give life to the living being.[10]
• The Chinese school of martial arts agrees with the South Indian school of martial arts on the principle of 108 pressure points.[11][12]
• 108 number figures prominently in the symbolism associated with karate, particularly the Gōjū-ryū discipline. The ultimate Gōjū-ryū kata, Suparinpei, literally translates to 108. Suparinpei is the Chinese pronunciation of the number 108, while gojūshi of Gojūshiho is the Japanese pronunciation of the number 54. The other Gōjū-ryū kata, Sanseru (meaning "36") and Seipai ("18") are factors of the number 108.[3]
• The 108 moves of the Yang Taijiquan long form and 108 moves in the Wing Chun wooden dummy form, taught by Yip Man, are noted in this regard.[6]
• Paek Pal Ki Hyung, the 7th form taught in the art of Kuk Sool Won, translates literally to "108 technique" form. It is also frequently referred to as the "eliminate 108 torments" form. Each motion corresponds with one of the 108 Buddhist torments or defilements.

## In sports

108 is the number that the Belgian cyclist Wouter Weylandt wore when he crashed fatally in the Giro d'Italia on May 9, 2011. As a tribute, many supporters held replicas of his race number by the side of the road the next day. The organization of the Giro d'Italia decided not to issue race number 108 in future editions, to commemorate him.

## Notes

1. ^ http://www.orientalarchitecture.com/cambodia/angkor/angkorthom.php
2. ^ Chapter 5 of 'Generating the Deity' ISBN 1-55939-055-7
3. ^ a b Hyaku Hachi No Bonno: The Influence of The 108 Defilements and Other Buddhist Concepts on Karate Thought and Practice By Charles C. Goodin. The article has appeared in Issue #7, Winter 1996-97 of Furyu: The Budo Journal.
4. ^ The Lankavatara Sutra translated by Daisetz Teitaro Suzuki, Chapter Two, Section II,[1]
5. ^ The Lankavatara Sutra translated by Daisetz Teitaro Suzuki, Chapter Two, Section III,[2]
6. ^ a b 108 STEPS: The Sino-Indian Connection in the Martial Arts by Joyotpaul Chaudhuri....
7. ^ Bhante Gunaratana, Four Foundations of Mindfulness in Plain English 2012, Wisdom Publications page 86


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# Beginner: Simulating Gaussian Process on $\mathbb{R}$ [duplicate]

How can one simulate a Gaussian process with nonzero covariance?

I've never simulated a process. I have searched for some basic examples, but I cannot make sense of all of it. Can someone give an an algorithm (continuous and discrete) followed with a minor explanation of such a simulation? How can the Fourier transform be used?

• Closely related to that particular covariance function: stats.stackexchange.com/questions/29091, though this, too, is for a set of points. Can you clarify what you expect a "simulation over the real line" to output - it's (obviously?) not possible to actually either store or generate an infinite number of values (one $f(t)$ for every $t$) with a computer, but perhaps you are after something else – Juho Kokkala Oct 19 '16 at 5:43
• @ Juho Kokkala I expect to use the fourier transform. For the real line, I am looking or a more theoretical way to proceed rather than using a computer. Is there a theoretical way to simulate over the line? – Iced Palmer Oct 19 '16 at 5:48
• @ Juho Kokkala also, regarding that example you cited. That looks like a discrete version. Can you give a continuous version? – Iced Palmer Oct 19 '16 at 5:50
• I don't know what a "theoretical way to simulate" would be - or, what tools are available - apparently the definition of the process itself is not allowed as an oracle for sampling from the process, but what is? Besides sampling the process in a finite set of points, sampling from a truncated basis function decomposition comes into mind but I'm not sure if that, either, is what you are after (my intuition would anyway be that a stationary process cannot be very accurately sampled over the entire real line with any method based on a finite number of random numbers) – Juho Kokkala Oct 19 '16 at 6:06
• I've edited the question, can the hold be removed? – Iced Palmer Oct 20 '16 at 6:07


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# ME's question at Yahoo! Answers regarding pursuit curve

#### MarkFL

Staff member
Here is the question:

How do you solve this really hard calculus question?

Ship A and ship B start 100 km apart from each other (oriented so that a horizontal line will pass through both). As time passes, ship A moves straight down at a speed of 10 km/hr. Ship B moves towards ship A at all times so that the slope at which ship B is moving on is always one who's line points directly towards ship A at that instance. Ship B is always moving at 12 km/hr. How do I find the function which represents the path which ship B moves along? The function may have as many variables in it as you want. I just need to know the function, and how to find it.
Here is a link to the question:

How do you solve this really hard calculus question? - Yahoo! Answers

I have posted a link there to this topic so the OP can find my response.

#### MarkFL

Staff member
Hello ME,

An interesting geometric model arises when one tries to determine the path of a pursuer chasing its "prey." This path is called a curve of pursuit. These problems were analyzed using methods of calculus circa 1760 (more than two centuries after Leonardo da Vinci had considered them).

The simplest problem is to find the curve along which a vessel moves when pursuing another vessel that flees along a straight line, assuming the speeds of the two vessels are constant.

Let's assume ship $B$, traveling at speed $\beta$, is pursuing ship $A$, which is traveling at speed $\alpha$. In addition, assume that ship $B$ begins (at time $t=0$) at the origin and pursues ship $A$, which begins at the point $(b,0)$ where $0<b$ and travels down the line $x=b$.

After $t$ hours, ship $B$ is located at the point $P(x,y)$, and ship $A$ is located at the point $Q(b,-\alpha t)$. The goal is then to describe the locus of points $P$, that is, to find $y$ as a function of $x$.

(a) Since ship $B$ is pursuing ship $A$, then at time $t$, ship $B$ must be heading right at ship $A$. That is, the tangent line to the curve of pursuit at $P$ must pass through the point $Q$. Using the point-slope formula, this implies:

$$\displaystyle y+\alpha t=\frac{dy}{dx}(x-b)$$

(1) $$\displaystyle \frac{dy}{dx}=\frac{y+\alpha t}{x-b}$$

(b) Since we know the speed at which ship $B$ is traveling, we know the distance it travels is $\beta t$. This distance is also the length of the pursuit curve from $(0,0)$ to $(x,y)$. Using the arc length formula from calculus, we find that:

(2) $$\displaystyle \beta t=\int_0^x \sqrt{1+\left(y'(u) \right)^2}\,du$$

Solving for $t$ in equations (1) and (2), we find that:

(3) $$\displaystyle (x-b)\frac{dy}{dx}-y=\frac{\alpha}{\beta}\int_0^x \sqrt{1+\left(y'(u) \right)^2}\,du$$

(c) Differentiating both sides of (3) with respect to $x$, we find:

$$\displaystyle \left((x-b)\frac{d^2y}{dx^2}+\frac{dy}{dx} \right)-\frac{dy}{dx}=\frac{\alpha}{\beta}\sqrt{1+\left( \frac{dy}{dx} \right)^2}$$

Letting $$\displaystyle \omega=\frac{dy}{dx}$$ we obtain the first order IVP:

$$\displaystyle (x-b)\frac{d\omega}{dx}=\frac{\alpha}{\beta}\sqrt{1+ \omega^2}$$ where $$\displaystyle \omega(0)=0$$

(d) Using separation of variables, and switching the dummy variables of integration so that we may use the boundaries as the limits, we find:

$$\displaystyle \int_0^{\omega}\frac{1}{\sqrt{1+u^2}}\,du= \frac{\alpha}{\beta}\int_0^x\frac{1}{v-b}\,dv$$

Integrating, we find:

$$\displaystyle \left[\ln|u+\sqrt{1+u^2}| \right]_0^{\omega}=\frac{\alpha}{\beta}\left[\ln|v-b| \right]_0^x$$

$$\displaystyle \ln|\omega+\sqrt{1+\omega^2}|=\ln\left|\left(1-\frac{x}{b} \right)^{\frac{\alpha}{\beta}} \right|$$

Since $$\displaystyle -\infty\le \omega<0$$ and $$\displaystyle 0<1-\frac{x}{b}$$ we may state:

$$\displaystyle \omega+\sqrt{1+\omega^2}=-\left(1-\frac{x}{b} \right)^{\frac{\alpha}{\beta}}$$

$$\displaystyle \left(\sqrt{1+\omega^2} \right)^2=\left(\left(1-\frac{x}{b} \right)^{\frac{\alpha}{\beta}}+\omega \right)^2$$

$$\displaystyle 1+\omega^2=\left(1-\frac{x}{b} \right)^{\frac{2\alpha}{\beta}}+2\omega\left(1-\frac{x}{b} \right)^{\frac{\alpha}{\beta}}+\omega^2$$

$$\displaystyle 2\omega\left(1-\frac{x}{b} \right)^{\frac{\alpha}{\beta}}=1-\left(1-\frac{x}{b} \right)^{\frac{2\alpha}{\beta}}$$

Back-substituting for $\omega$, we have the IVP:

$$\displaystyle \frac{dy}{dx}=\frac{1}{2}\left(\left(1-\frac{x}{b} \right)^{-\frac{\alpha}{\beta}}-\left(1-\frac{x}{b} \right)^{\frac{\alpha}{\beta}} \right)$$ where $$\displaystyle y(0)=0$$

Switching the dummy variables of integration so we can use the boundaries, we have:

$$\displaystyle \int_0^y\,du=\frac{1}{2}\int_0^x\left(1-\frac{v}{b} \right)^{-\frac{\alpha}{\beta}}-\left(1-\frac{v}{b} \right)^{\frac{\alpha}{\beta}}\,dv$$

$$\displaystyle _0^y=\frac{b}{2}\left[\frac{\left(1+\frac{v}{b} \right)^{1+\frac{\alpha}{\beta}}}{1+\frac{\alpha}{\beta}}-\frac{\left(1-\frac{v}{b} \right)^{1-\frac{\alpha}{\beta}}}{1-\frac{\alpha}{\beta}} \right]_0^x$$

After simplifying, we obtain:

$$\displaystyle y=\frac{b\beta}{2(\alpha^2-\beta^2)}\left((\beta-\alpha)\left(1-\frac{x}{b} \right)^{\frac{\beta+\alpha}{\beta}}-(\beta+\alpha)\left(1-\frac{x}{b} \right)^{\frac{\beta-\alpha}{\beta}}+2\alpha \right)$$

Plugging in the given data $b=100,\,\alpha=10,\,\beta=12$ we find:

$$\displaystyle y=-\frac{300}{11}\left(\left(1-\frac{x}{100} \right)^{\frac{11}{6}}-11\left(1-\frac{x}{100} \right)^{\frac{1}{6}}+10 \right)$$

Here is a plot of the pursuit curve, i.e., the path of ship B:

To ME and any other guests viewing this topic, I invite and encourage you to post other calculus questions here in our Calculus forum.

Best Regards,

Mark.


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## Modified Chebyshev-Picard iteration methods for station-keeping of translunar halo orbits.(English)Zbl 1264.70032

Summary: The halo orbits around the Earth-Moon $$L_2$$ libration point provide a great candidate orbit for a lunar communication satellite, where the satellite remains above the horizon on the far side of the Moon being visible from the Earth at all times. Such orbits are generally unstable, and station-keeping strategies are required to control the satellite to remain close to the reference orbit. A recently developed Modified Chebyshev-Picard Iteration method is used to compute corrective maneuvers at discrete time intervals for station-keeping of halo orbit satellite, and several key parameters affecting the mission performance are analyzed through numerical simulations. Compared with previously published results, the presented method provides a computationally efficient station-keeping approach which has a simple control structure that does not require weight turning and, most importantly, does not need state transition matrix or gradient information computation. The performance of the presented approach is shown to be comparable with published methods.

### MSC:

 70F15 Celestial mechanics 70F07 Three-body problems
Full Text:

### References:

 [1] R. W. Farquhar, “Station-keeping in the vicinity of collinear libration points with an application to a lunar communications problem,” in Proceedings of the Space Flight Mechanics Specialist Symposium, AAS Science and Technology Series, pp. 519-535, Denver, Colo, USA, July 1966. [2] R. W. Farquhar and A. A. Kamel, “Quasi-periodic orbits about the translunar libration point,” Celestial Mechanics, vol. 7, no. 4, pp. 458-473, 1973. · Zbl 0258.70011 [3] R. W. Farquhar, D. P. Muhonen, C. Newman, and H. Heuberger, “Trajectories and orbital maneuvers for the first libration-point satellite,” Journal of Guidance, Control and Dynamicss, vol. 3, no. 6, pp. 549-554, 1980. [4] J. V. Breakwell, A. A. Kamel, and M. J. Ratner, “Station-keeping for a translunar communication station,” Celestial Mechanics, vol. 10, pp. 357-373, 1974. [5] M. Xin, M. W. Dancer, S. Balakrishnan, and H. J. Pernicka, “Stationkeeping of an L2 Libration point satellite with \theta -D technique,” in Proceedings of the American Control Conference, Boston, Mass, USA, 2004. [6] J. E. Kulkarni, M. E. Campbell, and G. E. Dullerud, “Stabilization of spacecraft flight in halo orbits: an H\infty approach,” IEEE Transactions on Control Systems Technology, vol. 14, no. 3, pp. 572-578, 2006. [7] D. Cielaszyk and B. Wie, “New approach to halo orbit determination and control,” Journal of Guidance, Control, and Dynamics, vol. 19, no. 2, pp. 266-273, 1996. · Zbl 0851.70010 [8] D. W. Dunham and C. E. Roberts, “Stationkeeping techniques for libration-point satellites,” Journal of the Astronautical Sciences, vol. 49, no. 1, pp. 127-144, 2001. [9] G. Gomez, K. Howell, K. Howell, C. Simo, and J. Masdemont, “Station-keeping strategies for translunar libration point orbits,” in Proceedings of the AAS/AIAA Space Mechanics Meeting, Monterey, Calif, USA, 1998. [10] K. C. Howell and H. J. Pernicka, “Stationkeeping method for libration point trajectories,” Journal of Guidance, Control, and Dynamics, vol. 16, no. 1, pp. 151-159, 1993. [11] C. A. Renault and D. J. Scheeres, “Statistical analysis of control maneuvers in unstable orbital environments,” Journal of Guidance, Control, and Dynamics, vol. 26, no. 5, pp. 758-769, 2003. [12] E. D. Gustafson and D. J. Scheeres, “Optimal timing of control law updates for unstable systems with continuous control,” in Proceedings of the American Control Conference, Seattle, Wash, USA, June 2008. [13] X. Bai and J. L. Junkins, “Modified Chebyshev-Picard iteration methods for solution of initial value problems,” Advances in the Astronautical Sciences, vol. 139, pp. 345-362, 2011. [14] X. Bai and J. L. Junkins, “Modified Chebyshev-Picard iteration methods for solution of boundary value problems,” Advances in the Astronautical Sciences, vol. 140, pp. 381-400, 2011. [15] X. Bai, Modified Chebyshev-Picard iteration methods for solution of initial value and boundary value problems, Ph.D. dissertation, Texas A&M University, College Station, Tex, USA, 2010. [16] D. L. Richardson, “Analytic construction of periodic orbits about the collinear points,” Celestial Mechanics, vol. 22, no. 3, pp. 241-253, 1980. · Zbl 0465.34028 [17] H. Schaub and J. L. Junkins, Analytical Mechanics of Space Systems, American Institute of Aeronautics and Astronautics, Reston, Va, USA, 1st edition, 2003. · Zbl 1194.70001 [18] J. V. Breakwell and J. V. Brown, “The “Halo” family of 3-dimensional periodic orbits in the Earth-Moon restricted 3-body problem,” Celestial Mechanics, vol. 20, no. 4, pp. 389-404, 1979. · Zbl 0415.70020 [19] X. Bai, J. D. Turner, and J. L. Junkins, “Optimal thrust design of a mission to apophis based on a homotopy method,” in Proceedings of the AAS/AIAA Spaceflight Mechanics Meeting, Savannah, Ga, USA, 2009.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


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# How do you verify (1 + sec^2 x) / (1 + tan^2 x) = 1 + cos^2 x?

Jul 3, 2016

Use the Pythagorean Identity $1 + {\tan}^{2} x = {\sec}^{2} x$.

#### Explanation:

Recall the Pythagorean Identity $1 + {\tan}^{2} x = {\sec}^{2} x$ (this can be derived by dividing the identity ${\sin}^{2} x + {\cos}^{2} x = 1$ by ${\cos}^{2} x$). The key to this problem is applying this identity.

Since $1 + {\tan}^{2} x = {\sec}^{2} x$, we can replace the $1 + {\tan}^{2} x$ in the denominator with ${\sec}^{2} x$:
$\frac{1 + {\sec}^{2} x}{{\sec}^{2} x} = 1 + {\cos}^{2} x$

Now we can break the fraction up in two:
$\frac{1}{\sec} ^ 2 x + {\sec}^{2} \frac{x}{\sec} ^ 2 x = 1 + {\cos}^{2} x$

Since $\frac{1}{\sec} x = \cos x$, $\frac{1}{\sec} ^ 2 x = {\cos}^{2} x$; and ${\sec}^{2} \frac{x}{\sec} ^ 2 x = 1$. So:
${\cos}^{2} x + 1 = 1 + {\cos}^{2} x$

Using the commutative property of addition we can rearrange the left side of the equation to match the right:
$1 + {\cos}^{2} x = 1 + {\cos}^{2} x$


text:
# Disable/Enable Windows Lock Screen (Windows + L)

Although Windows is not my primary operating system, I do need to use it in order to get my day job done. Therefore I run Windows in a VirtualBox VM. A good habit to have, and I often use is to lock my computer whenever I walk away from it. Unfortunately, this can have some unexpected effects on some desktops. When the focus is on the VM and using the hotkeys (Windows + L) the VM will lock leaving the workstation unlocked. A quick web search didn't reveal the correct solution so I figured I would post it.

## Registry Edit

Open up the regedit.exe by typing it in the Windows menu search box or the run dialog. Then locate the following key. If it does not exist then create it.

HKEY_CURRENT_USER\Software\Microsoft\Windows\CurrentVersion\Policies\System

Create a new DWORD 32-bit and name it 'DisableLockWorkstation'. Then double click on it to give it the value on '1' for on. This change should work with out a restart.

I have not tried this and all Windows versions but at the time of this write-up, it works on Windows 7 & 10.


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# Prove that $\sum_{k=1}^{\infty} \large\frac{k}{\text{e}^{2\pi k}-1}=\frac{1}{24}-\frac{1}{8\pi}$

Prove that $$\sum_{k=1}^{\infty} \frac{k}{\text{e}^{2\pi k}-1}=\frac{1}{24}-\frac{1}{8\pi}$$

• $\large\frac{2\pi k}{\text{e}^{2\pi k}-1}=k\pi (\coth(k\pi)-1)$ – mrs Jan 8 '13 at 17:09
• I can transform the sum to $\sum_{m=1}^{\infty} (-1)^{m+1} \mathrm{csch}^2{(\pi m)}$. Unfortunately, I have to get back to work. I will come back if nobody has solved by then. – Ron Gordon Jan 8 '13 at 17:20
• @Chris'ssister Have you tried using the Laplace transform as a tool to find the limit of this series using this paper? See mathdl.maa.org/images/cms_upload/A_Laplace_Transform18380.pdf. – MathOverview Jan 8 '13 at 18:25
• Interestingly, if one blindly applies the Plana summation formula$$\sum_{k=1}^\infty f(k) = -\frac{1}{2} f(0) + \int_0^\infty f(t) \mathrm{d}t + \int_0^\infty i \frac{f(i t)-f(-i t)}{\\exp(2 \pi t)-1} \mathrm{d}t$$ although it is not applicable because $f(t)$ is not bounded in the right half complex plane, one gets $\frac{1}{12} - \frac{1}{4 \pi}$ which twice the exact answer. This hints that the sum may be amendable to complex analysis techniques. Here $f(k)$ refers to the summand. – user40314 Feb 9 '13 at 13:49
• It is amusing since, "essentially", the sum is the energy of a photon gas with discrete wave numbers. I guess @Fabian answer exploits this fact. – Felix Marin May 3 '14 at 23:02

## 3 Answers

Rewrite $$\frac{1}{e^{2\pi k} -1} = \sum_{n=1}^\infty e^{-2\pi k n}.$$

So we need to evaluate $$\sum_{n,k=1}^\infty k e^{-2\pi k n}.$$

Summing first over $k$, we have $$\sum_{k=1}^\infty k e^{-2\pi k n} = \frac1{2\pi}\frac{\partial}{\partial n} \sum_{k=1}^\infty e^{-2\pi k n} = \frac1{2\pi}\frac{\partial}{\partial n} \frac{1}{e^{2\pi n} -1} =\frac{e^{2\pi n}}{(e^{2\pi n}-1)^2} = \frac{1}{4 \sinh^2(\pi n)} .$$

The sum $$\sum_{n=1}^\infty \frac{1}{\sinh^2(\pi n)} =\frac{1}{6} - \frac{1}{2\pi}$$ is evaluated here, see also page 3 here, and the quoted result follows.

• beautiful answer! – nbubis Jan 8 '13 at 18:38
• Better yet, here's a link to the paper in which the result is derived. math.uiuc.edu/~berndt/articles/venkalatex.pdf – Ron Gordon Jan 8 '13 at 18:40
• @nbubis: thanks, I was trying to evaluate $\sum_n \text{csch}^2 (\pi n)$ from first principles but it seems not so easy. The second reference needs two pages to prove it (ok, he has a more general result). – Fabian Jan 8 '13 at 18:40
• @fabian: apologies, didn't see that. Very nontrivial exercise indeed. – Ron Gordon Jan 8 '13 at 18:41
• @Chris'ssister: I wondered a long time myself (tried it very hard). However, the statement which was evidently first proved by T. S. Nanjundiah [11] in 1951 in the paper made me decide that there is none. – Fabian Jan 8 '13 at 19:15

Consider the function complex valued function

$$f(z)=\frac{\cot(\pi z)}{\sinh^2(z \pi)}$$

and integrate it along a quadratic contour $\mathcal{C}$ with verticies $\{( N/2,i N/2),(- N/2,i N/2),(- N/2,-i N/2),( N/2,-i N/2)\}$ where $N$ is a (big) odd natural number. Note that the choice of such of kind verticies is necessary to bypass the poles of $f(z)$.

The result is (we travel counter-clockwise)

$$\oint_{\mathcal{C}}f(z)dz=\int_{N/2}^{-N/2}dxf(x+i N/2)+i\int_{N/2}^{-N/2}dyf(i y+ N/2)+\\\int_{-N/2}^{N/2}dxf(x-i N/2)+i\int_{-N/2}^{N/2}dyf(i y+ N/2)$$

we now have (this is an exercise for the reader)

$\lim_{N\rightarrow\infty}f(x\pm i N/2)=\frac{\mp i}{\cosh^2(\pi x)}$ and that $\lim_{N\rightarrow\infty}f(i y\pm N/2)=0$.

Furthermore it is elementary to show (the integrand is the derivative of $\tanh$) that $\int_{\mathbb{R}}\frac{1}{\cosh^2(\pi x)}dx=\frac{2}{\pi}$ which means that

$$\oint_{\mathcal{C}}f(z)dz=-2\frac{2i}{\pi}+2\cdot0=-\frac{4i}{\pi} \quad (\spadesuit)$$

in the limit $N\rightarrow \infty$. On the other hand we know by the residue theorem that (note that $(N-1)/2$ is an integer by construction)

$$\oint_{\mathcal{C}}f(z)dz=2 \pi i\sum_{-(N-1)/2\leq k\leq (N-1)/2}\text{Res}(f(z),z=k)+\text{Res}(f(z),z=i k)$$

it is now straightforward to show that

$$\text{Res}(f(z),z=k)=\text{Res}(f(z),z=i k) = \frac{1}{\pi\sinh^2(\pi k)} \quad \text{if}\,\,k\neq0$$

$$\text{Res}(f(z),z=0) =-\frac{2}{3 \pi}$$

which allows us to write that

$$\oint_{\mathcal{C}} f(z)dz = 8 i\sum_{k=1}^{\infty}\frac{1}{\sinh^2(\pi k)}-\frac{4 i}{3} \quad (\heartsuit)$$

In the limit of $N\rightarrow\infty$.

Now putting togehter $(\heartsuit)$ and $(\spadesuit)$ it follows that

$$\sum_{k=1}^{\infty}\frac{1}{\sinh^2(\pi k)}=\frac{1}{6}-\frac{1}{2\pi} \quad\blacksquare$$

this proves the result used by @Fabian

Appendix

if we set $f_a(z)=\frac{\cot(\pi z)}{\sinh^2(a z \pi)}$ and follow the same steps as above we can show that on the one hand the sum of residues equals

$$\oint_{\mathcal{C}} f_a(z)dz=4i\sum_{k\geq1}\frac{1}{\sinh^2(k \pi a)}+\frac{4i}{a^2}\sum_{k\geq1}\frac{1}{\sinh^2(k \pi/ a)}-\frac {2 i}{3}\frac{1+a^2}{a^2}\quad (\heartsuit \heartsuit)$$

and on the other hand, by parametrizing the contour of integration as shown above, we get that $$\oint_{\mathcal{C}} f_a(z)dz=-\frac{4i}{\pi a} \quad (\spadesuit \spadesuit)$$

Now putting togehter $(\heartsuit\heartsuit)$ and $(\spadesuit\spadesuit)$ it follows that

$$\sum_{k\geq1}\frac{1}{\sinh^2(k \pi a)}+\frac{1}{a^2}\sum_{k\geq1}\frac{1}{\sinh^2(k \pi/ a)}=\frac{1+a^2}{6 a^2}-\frac{1}{\pi a}\quad\blacksquare\quad\blacksquare$$

in accordance with Berndt et al.

• @r9m thanks, that is very kind of you – tired May 21 '17 at 16:56
• It is interesting to prove that some parts of the contour don't vanish. – Zaid Alyafeai May 22 '17 at 13:03
• Nice answer. For some reasons I simply missed this post. – user 1357113 Apr 29 '18 at 20:41

If we define Dedekind eta function $\eta(q)$ via the equation $$\eta(q) = q^{1/12}\prod_{i = 1}^{\infty}(1 - q^{2i})\tag{1}$$ then it can be proved via functional equation of Dedekind eta function that $$n^{1/4}\eta(e^{-\pi\sqrt{n}}) = \eta(e^{-\pi/\sqrt{n}})\tag{2}$$ Let's define function $P(q)$ as $$P(q) = 1 - 24\sum_{i = 1}^{\infty}\frac{iq^{2i}}{1 - q^{2i}} = 12q\frac{d}{dq}\log\eta(q)\tag{3}$$ Logarithmic differentiation of equation $(2)$ with respect to $n$ gives us $$nP(e^{-\pi\sqrt{n}}) + P(e^{-\pi/\sqrt{n}}) = \frac{6\sqrt{n}}{\pi}\tag{4}$$ and putting $n = 1$ we get $$P(e^{-\pi}) = \frac{3}{\pi}\tag{5}$$ and looking at the definition of $P(q)$ we see that the above equation is exactly what is asked in the question.

• this made short work of it (+1) – tired May 17 '17 at 21:59
• @tired: Ramanujan dealt with such sums of hyperbolic functions in great detail and I learnt most of his technique from his collected papers. The beauty of his approach is the use of elementary tools (algebra and calculus of single real variable) and it shows that these basic tools (available to any high schooler) can prove very deep results. – Paramanand Singh May 17 '17 at 22:02
• it is somehow logical that Ramanujan chosed this appproach because his lack of a formal mathematical education (as Euler did in a time where formal mathematics wasn't developed). And it is absolutely incredible how far he came... – tired May 17 '17 at 22:12
• @tired : fully agree. By the way in case you have not noticed the last equation in your answer dealing with variable $a$ is same as the equation $(4)$ of my answer (just put $a=\sqrt{n}$). – Paramanand Singh May 18 '17 at 3:28
• @tired: there are identities to handle the powers of hyperbolic functions also. I will extract some info and post after sometime. – Paramanand Singh May 18 '17 at 17:54


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# How can I write roman numbers in MathJax?

I need to write roman numbers in MathJax. I've tried almost every way. But, then I decided to copy the symbol itself. But it looks very ugly.

For example,

$$\displaystyle \mathcal {D}^{{\left(II \right)}}_n$$

I tried LaTex syntax. \rom. But, unfortunately it doesn$$$$t work.

Do you mean \mathrm{IX} perhaps? $$\mathrm{IX}$$?

There's a reason why the font family is called "Roman".

• Or the alternative but more archaic {\rm IX}. – ja72 Dec 23 '19 at 20:23

$$\mathcal{D}_n^{\tiny\mathrm{(IX)}}$$ might work ? put \tiny\mathrm{(IX)} in the exponent. That at least makes it look similar size.

• might bold it to make it look better. – Roddy MacPhee Dec 20 '19 at 0:59
• $ \texttt{\tiny} "$ may be a bit too small; if the size'd be reduced, probably just $ \texttt{\small} ".$ $$\begin{array}{|c|c|} \hline \left[\text{none}\right] & \mathcal{D}_{n}^{\left(\mathrm{IX}\right)} \\ \hline \texttt{\small} & \mathcal{D}_{n}^{\small{\left(\mathrm{IX}\right)}} \\ \hline \texttt{\Tiny} &\mathcal{D}_{n}^{\Tiny{\left(\mathrm{IX}\right)}} \\ \hline \texttt{\tiny} &\mathcal{D}_{n}^{\tiny{\left(\mathrm{IX}\right)}} \\ \hline \end{array}$$ – Nat Dec 20 '19 at 9:11
• I was simply making a size comparison with the n . ${}^{\boldsymbol\tiny\mathrm{(IX)}}$ looks closer to the boldness probably – Roddy MacPhee Dec 20 '19 at 9:38
• or resize the n – Roddy MacPhee Dec 20 '19 at 11:46
• So you were trying to get the subscript and superscripts to match in size? I can appreciate the aesthetic motivation behind that, though the superscript's and subscript's sizes differ by so much that it's probably not reasonably viable to make them both about the same size. – Nat Dec 21 '19 at 3:09
• @Nat: \scriptsize works, and would be the semantically correct choice here: $\mathcal{D}_n^{\scriptsize\mathrm{(IX)}}$. In this case, it appears to render slightly smaller than \small ($\mathcal{D}_n^{\small\mathrm{(IX)}}$) but slightly bigger than \Tiny` ($\mathcal{D}_n^{\Tiny\mathrm{(IX)}}$). – Ilmari Karonen Dec 23 '19 at 1:05
• You can also adjust the subscript writing placement with a forced space. – Roddy MacPhee Dec 23 '19 at 11:59
• $\mathcal{D}_{\;n}^{\scriptsize\mathrm{(IX)}}$ – Roddy MacPhee Dec 27 '19 at 17:42