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# zbMATH — the first resource for mathematics

## Algebraic Structures and their Applications

 Short Title: Algebr. Struct. Appl. Publisher: Yazd University, Yazd ISSN: 2382-9761; 2423-3447/e Online: http://as.yazd.ac.ir/browse?_action=issue Comments: Indexed cover-to-cover; This journal is available open access.
 Documents Indexed: 105 Publications (since 2014) References Indexed: 32 Publications with 337 References.
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#### Latest Issues

 7, No. 2 (2020) 7, No. 1 (2020) 6, No. 2 (2019) 6, No. 1 (2019) 5, No. 2 (2018) 5, No. 1 (2018) 4, No. 2 (2017) 4, No. 1 (2017) 3, No. 2 (2016) 3, No. 1 (2016) 2, No. 2 (2015) 2, No. 1 (2015) 1, No. 2 (2014) 1, No. 1 (2014)
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#### Fields

 39 Combinatorics (05-XX) 28 Group theory and generalizations (20-XX) 22 Commutative algebra (13-XX) 14 Associative rings and algebras (16-XX) 13 Order, lattices, ordered algebraic structures (06-XX) 6 Information and communication theory, circuits (94-XX) 5 Mathematical logic and foundations (03-XX) 5 Algebraic geometry (14-XX) 5 Linear and multilinear algebra; matrix theory (15-XX) 4 General topology (54-XX) 3 Category theory; homological algebra (18-XX) 3 Topological groups, Lie groups (22-XX) 3 Computer science (68-XX) 2 General algebraic systems (08-XX) 2 Nonassociative rings and algebras (17-XX) 2 Functional analysis (46-XX) 1 Number theory (11-XX) 1 Dynamical systems and ergodic theory (37-XX) 1 Abstract harmonic analysis (43-XX) 1 Operator theory (47-XX) 1 Differential geometry (53-XX) 1 Manifolds and cell complexes (57-XX) 1 Probability theory and stochastic processes (60-XX) 1 Numerical analysis (65-XX)

#### Citations contained in zbMATH Open

9 Publications have been cited 11 times in 11 Documents Cited by Year
Automorphism group of groups of order $$pqr$$. Zbl 1463.20037
Ghorbani, Modjtaba; Larki, Farzaneh Nowroozi
2014
Signless Laplacian spectral moments of graphs and ordering some graphs with respect to them. Zbl 1463.05358
Taghvaee, Fatemeh; Fath-Tabar, Gholam Hossein
2014
On new classes of multicone graphs determined by their spectrums. Zbl 1463.05368
Zeydi Abdian, Ali; Mirafzal, Seyed Morteza
2015
When does the complement of the annihilating-ideal graph of a commutative ring admit a cut vertex? Zbl 1463.13010
Visweswaran, S.; Parmar, Anirudhdha
2015
On the eigenvalues of non-commuting graphs. Zbl 1463.22009
Ghorbani, Modjtaba; Gharavi-Alkhansari, Zahra; Zaeem-Bashi, Ali
2017
Finite groups admitting a connected cubic integral bi-Cayley graph. Zbl 1463.05246
Arezoomand, Majid; Taeri, Bijan
2018
On medial filters of BE-algebras. Zbl 1463.06107
Rezaei, Akbar; Radfar, Akefe; Pourabdollah, Amir
2020
An upper bound on the distinguishing index of graphs with minimum degree at least two. Zbl 1463.05244
Alikhani, Saeid; Soltani, Samaneh
2020
Free ideals and real ideals of the ring of frame maps from $$\mathcal{P}(\mathbb{R})$$ to a frame. Zbl 1463.06088
2020
On medial filters of BE-algebras. Zbl 1463.06107
Rezaei, Akbar; Radfar, Akefe; Pourabdollah, Amir
2020
An upper bound on the distinguishing index of graphs with minimum degree at least two. Zbl 1463.05244
Alikhani, Saeid; Soltani, Samaneh
2020
Free ideals and real ideals of the ring of frame maps from $$\mathcal{P}(\mathbb{R})$$ to a frame. Zbl 1463.06088
2020
Finite groups admitting a connected cubic integral bi-Cayley graph. Zbl 1463.05246
Arezoomand, Majid; Taeri, Bijan
2018
On the eigenvalues of non-commuting graphs. Zbl 1463.22009
Ghorbani, Modjtaba; Gharavi-Alkhansari, Zahra; Zaeem-Bashi, Ali
2017
On new classes of multicone graphs determined by their spectrums. Zbl 1463.05368
Zeydi Abdian, Ali; Mirafzal, Seyed Morteza
2015
When does the complement of the annihilating-ideal graph of a commutative ring admit a cut vertex? Zbl 1463.13010
Visweswaran, S.; Parmar, Anirudhdha
2015
Automorphism group of groups of order $$pqr$$. Zbl 1463.20037
Ghorbani, Modjtaba; Larki, Farzaneh Nowroozi
2014
Signless Laplacian spectral moments of graphs and ordering some graphs with respect to them. Zbl 1463.05358
Taghvaee, Fatemeh; Fath-Tabar, Gholam Hossein
2014
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#### Cited by 19 Authors

 3 Ghorbani, Modjtaba 2 Fath-Tabar, Gholam Hossein 2 Rajabi Parsa, Mina 1 Aijaz, Malla 1 Arezoomand, Majid 1 Gharavi-Alkhansari, Zahra 1 Imrich, Wilfried 1 Kalinowski, Rafał 1 Mahmoudi, Darghadam A. 1 Nowroozi Larki, F. 1 Pilśniak, Monika 1 Pirzada, Shariefuddin 1 Rahmani, Moghaddam M. 1 Sadeghi, Ghadir 1 Seyyed-Hadi, A. 1 Taghvaee, Fatemeh 1 Woźniak, Mariusz 1 Yousefpour, H. 1 Zeydi, Abdian A.
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#### Cited in 7 Journals

 5 Journal of Algebraic Systems 1 Kragujevac Journal of Mathematics 1 Journal of the Indonesian Mathematical Society 1 Asian-European Journal of Mathematics 1 Ars Mathematica Contemporanea 1 Transactions on Combinatorics 1 Electronic Journal of Graph Theory and Applications
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#### Cited in 9 Fields

 9 Combinatorics (05-XX) 3 Group theory and generalizations (20-XX) 1 Mathematical logic and foundations (03-XX) 1 Order, lattices, ordered algebraic structures (06-XX) 1 Number theory (11-XX) 1 Commutative algebra (13-XX) 1 Linear and multilinear algebra; matrix theory (15-XX) 1 Measure and integration (28-XX) 1 General topology (54-XX)


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# Math Help - Basic rearranging

1. ## Basic rearranging

I am currently solving questions preparing for my mechanics exam and I am struggling with the final part of the question I have the following equation

y-yo=Vo*sin(beta)*x^2/Vo*cos(beta)-1/2*g*x^2/Vo^2cos^2(beta)

I need to solve this for Vo any help would be much appreciated

2. I'm sorry but this is ambigious. I'll just put what I think the equation is into proper LaTeX form, and please tell us if this is the right one.

$y - y_0=V_0 \times \sin{(\beta)} \times \frac{x^2}{V_0} \times \cos{(\beta)} - \frac{1}{2} \times g \times \frac{x^2}{V_0^2} \times \cos^2{(\beta)}$

Is this the one ?

3. Sorry your right that is the correct equation could you please point me in the right direction on how to solve for Vo

4. Assuming that is correct the $V_0$ and $\frac{1}{V_0}$ in the first term will cancel leaving
$y- y_0= sin(\beta)cos(\beta)-\frac{1}{2}g\frac{x^2}{V_0^2}cos^2(\beta)$

Subtract $sin(\beta)cos(\beta)$ from both sides:
$y- y_0- sin(\beta)cos(\beta)= -\frac{1}{2}g\frac{x^2}{V_0^2}cos^2(\beta)$

Multiply both sides by $V_0^2$:
$V_0^2(y- y_0- sin(\beta)cos(/beta))= -\frac{1}{2}gx^2cos^2(\beta)$

Divide both sides by $y- y_0- sin(\beta)cos(/beta)$:
$V_0^2= -\frac{gx^2cos^2(\beta)}{2(y- y_0- sin(\beta)cos(\beta)}$

And, finally, take the square root of both sides:
$V_0= \sqrt{-\frac{gx^2cos^2(\beta)}{2(y- y_0- sin(\beta)cos(\beta))}}$

$V_0= \sqrt{\frac{gx^2 cos^2(\beta)}{2(sin(\beta)cos(\beta)- y+ y_0)}}$

5. thankyou this has been very helpful

format: plain


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## anonymous one year ago A square piece of fabric has an area of 595 cm2. To the nearest tenth, what is the length of 1 side of the fabric? A. 21.3 cm B. 23.2 cm C. 24.4 cm D. 25.6 cm

1. campbell_st

just take the square root of the area. That will give the side length remember is a square side length l $Area = l^2$ so if you know the area $595 = l^2$ hope it helps

2. anonymous

okay so do i divide 595 by the sq root?

3. anonymous

@campbell_st ?

4. campbell_st

no you take the square root... as an example $\sqrt{16} = 4$

5. anonymous

oh okay thats what i ment THANKS

6. anonymous

c?

7. campbell_st

I think so... 25^2 = 625... so its between 24 and 25

8. anonymous

wait if there are four sides would i have to separate 595 into four

9. campbell_st

no... in a square every side is the same length... so area is Area = length x length or A = l^2 so take the square root of the area to get the side length you answer is correct

10. anonymous

thanks for helping me understand XD


text:
# Q:Is it possible to do a coordinate transfomation in momentum space?

1. Sep 9, 2014

### FinalCatch

Q: How does one do a coordinate transformation in momentum space while insuring conservation of momentum?

I have a several particles with momentum components $P_x , P_y , P_z$.
I would like to rotate the x, y, and z axis. By angle θ in the x/y and angle Θ in the y/z .
So giving new momentum $P_x' , P_y' , P_z'$.

Is it possible to do this an conserve momentum while remaining in the lab frame? (The particles are relativistic but I don't believe this matters). What are the coordinate transformations?

2. Sep 9, 2014

### HomogenousCow

The conserved quantity is a vector, you can rotate the basis vectors however you want and it wouldn't change the magnitude of the vector.

3. Sep 9, 2014

### FinalCatch

just to double check $P^2=P_x^2 +P_y^2 +P_z^2=P_x'^2 +P_y'^2 +P_z'^2$ correct?

Last edited by a moderator: Sep 9, 2014
4. Sep 9, 2014

### Staff: Mentor

If you've done the transformation to the primed coordinates correctly, yes.

In fact, HomogeneousCow has understated how much is conserved; the direction of the vector is also conserved. Of course it's a bit tricky talking about the "direction" of a vector when you don't have coordinate axes to make angles with - (1,0) in coordinates in which the x-axis points to the northeast is the same vector in the same direction as $\sqrt{2}/2(1,1)$ in coordinates in which the x-axis points east, but it's not obvious at all from the coordinates that that is so.

However, the dot-product of two vectors is invariant under these coordinate transformations, and as the dot-product depends on the angle between the vectors, that gives us a coordinate-independent way of claiiming that direction is also invariant.

Last edited: Sep 9, 2014
5. Sep 9, 2014


text:
# DC Motor Losses MCQ || Losses in DC motor Questions and Answers

1. Overheating of DC motors is mainly due to:

1. Rough commutator
3. Loose parts
4. Absence of brushes

Explanation:

The reason for DC motor overheating are

2. Poor power condition
3. High effective service factor
4. Frequent stops and starts
5. Environmental reasons
• Over-Heating in motors can cause a number of performance problems.
• Overheating causes the motor winding insulation to deteriorate quickly.
• For every ten centigrade rise in temperature, the insulation life is cut in half. It has been concluded that more than 55% of the insulating failures are caused by over-heating.
• Every electric motor has a design temperature. If a motor is started off at a bad current value, it starts operating in a much warmer condition than the design temperature.
• It is very important that the motors should be matched with their ideal current values.
• Overheating also occurs when an electric motor is forced to operate in a high-temperature environment. This causes the rate at which heat can be conducted to reduce at an alarming rate.
• The area where electric motors are operating must have a proper cooling system and a ventilation system should be there in case the cooling system stops working.

2. Which of the following is one of the categories of core loss in a DC machine?

1. Friction loss
2. Loss due to series field winding
3. Windage loss
4. Hysteresis loss

Explanation:

Core Losses or Iron Losses in DC Machine or Magnetic Losses:

• As the iron core of the armature is rotating in the magnetic field, some losses occur in the core which is called core losses.
• Normally, machines are operated with constant speed, so these losses are almost constant. These losses are categorized in two forms; Hysteresis loss and Eddy’s current loss.

Hysteresis Loss in DC Machine

• Hysteresis losses occur in the armature winding due to the reversal of magnetization of the core. When the core of the armature is exposed to the magnetic field, it undergoes one complete rotation of magnetic reversal.
• The portion of the armature is under S-pole, after completing half the electrical revolution, the same piece will be under the N-pole, and the magnetic lines are reversed in order to overturn the magnetism within the core.
• The constant process of magnetic reversal in the armature, consume some amount of energy which is called hysteresis loss. The percentage of loss depends upon the quality and volume of the iron.

3. Iron losses in a D. C. machine are independent of variations in

1. Speed
3. Voltage
4. Speed and voltage

Explanation:

Iron Loss or Core Loss in DC machine:

• These losses occur in the armature of a d.c. machine and are due to the rotation of armature in the magnetic field of the poles.
• Mechanical losses depend on speed, iron losses depend on field flux and speed, and copper losses depend on armature current. The field copper losses, iron, and mechanical losses are the constant types of losses for a d.c. shunt machine.
• It depends on the frequency (Speed) and voltage and does not depend on load or load current

4. The loss that takes place due to winding resistance is called ______.

1. Iron loss
2. Hysteresis loss
3. Eddy current loss
4. Copper loss

Explanation:

• The winding losses taking place during the current flowing through the winding are known as Copper losses.
• Copper losses are directly proportional to the square of the current.
• These are variable losses dependent on load current.
• Windage losses are the losses sustained by a machine due to the resistance offered by air to the rotation of the shaft.
• These are mechanical losses and independent of load.
• As the armature core of the electrical machine is made of iron and rotates in a magnetic field, a small current gets induced in the core.
• Due to this current, eddy current loss and hysteresis loss occur in the armature iron core.
• Iron losses are also called core losses or magnetic losses.
• These are constant losses and independent of load.

5. Torque developed by a motor while running at 1000 rpm is 106 N-m and the shaft torque available is 100 N-m. then the iron and mechanical losses in watts are:

1. 300 π
2. 50 π
3. 200 π
4. 100 π

Explanation:

Torque developed by a motor shaft is given by

Tsh = Psh

Where

Tsh = shaft Torque or output torque

Psh = Shaft power or output power

ω = angular frequency = 2πN/60

N = shaft speed

Tloss = T– Tsh

Where,

Tloss = Torque due to losses

T= Developed Torque

Tsh = shaft or output Torque

Calculation:

Given:

Tsh = 100 N-m, Td = 106 N-m, N = 1000 rpm

Tloss = T– Tsh  = 106 – 100 = 6 N-m

Therefore

Ploss = Tloss × ω

= 6 × 2π × 1000/60

Ploss = 200π W

6. In electric motor _________ is loss due to fans and the bearing friction.

1. Iron loss
2. Core loss
3. Stray loss
4. Friction and windage loss

Explanation:

Friction and Windage Losses in DC machine:

• Friction and windage losses result from bearing friction, windage, and circulating air through the motor and account for 8 – 12 % of total losses. These losses are independent of load.
• The reduction in heat generated by stator and rotor losses permits the use of a smaller fan.
• The windage losses also reduce with the diameter of the fan leading to a reduction in windage losses.

7. _______ combined together are called stray losses.

1. Frictional loss and copper loss
2. Field copper loss and iron loss
3. Iron and mechanical losses
4. Copper and iron losses

Explanation:

Stray load loss:- If the machine is tested under load conditions, the measured total loss is consistently greater than the sum of the above losses, even using the most accurate computational and experimental methods available to determine these losses individually. To account for this additional loss, a fifth loss (called the stray load loss) is defined.

The combination of friction losses due to moving of bearing and shaft and windage losses due to air gap is called stray loss.

8. Which of the following is considered as mechanical loss in induction motor losses?

1. Stator copper loss
2. Iron loss
3. Windage and friction losses
4. Rotor copper loss

Explanation:

No-load test or open circuit test:

• A no-load test or open circuit test is used to find the shunt branch parameters of the equivalent circuit and the constant losses.
• We can separate these constant losses into iron losses and mechanical losses.

Constant losses = Iron loss + Mechanical losses

Windage and frictional losses come under mechanical losses.

Blocked rotor test or short circuit test:

• It is used to find out series branch parameters (R01 and X01) of equivalent circuit parameters when referred to the primary side (stator) and the full load copper losses (I2R).
• By applying rated voltage under a blocked rotor test, the motor will take a very high short circuit current which is equivalent to 5 to 8 times the full load current.
• We need to apply reduced voltage initially and increase the voltage up to the current drawn by the motor to reach the rated current.

Variable losses = Copper loss + Stray load Loss

9. In a D.C. motor the windage losses are proportional to

1. Supply voltage
2. Square of supply voltage
3. Square of flux density
4. Square of the armature speed

Answer:4. Square of the armature speed

Explanation:

Friction and windage loss: Since parts of the machine are in motion, frictional forces must occur. The corresponding losses are localized at the bearing surfaces. In the wound rotor machine, additional friction loss is caused by the brushes sliding on the slip rings.

“Windage” is basically “wind friction” and occurs since machine motion occurs in an atmosphere. Machines with fans mounted on the shaft for forced ventilation cooling have greatly increased windage losses.

Additional windage loss is created by the rotor fan blades which circulate air within the motor, and, of course, there is some friction loss in the bearings.

Energy-efficient motors have less losses to be dissipated than a standard motor and, therefore, smaller external fans can frequently be used as less airflow is required. The smaller fans have a favorable effect on friction and windage loss and, therefore, contribute to increased motor efficiency.

• If N is the speed, then the windage loss in the dc motor is proportional to the square of armature speed.
• Windage losses occur due to the air inside the rotating coil of the machine.
• Eddy’s current losses are proportional to the square of speed.

10. Which of the following is NOT one of the categories of copper loss in a DC machine?

1. Field winding loss
2. Brush contact resistance loss
3. Armature loss
4. Eddy current loss

Explanation:

The primary I2r is the ohmic loss caused by current passing through the stator winding.

Increasing the cross-sectional area of copper to reduce the primary resistance is an effective means of reducing the primary I2r loss and this method is used in nearly all energy-efficient motor designs.

Type of Copper Loss in DC machine

Armature copper losses:

• Armature copper losses = I2aRa
• These losses are about 30% of the total full load losses.
• Armature copper losses in a DC generator vary significantly with the load current.

Field copper losses:

• Field copper losses = I2shRsh
• These losses are about 25% theoretically, but practically it is constant.

Brush Contact Loss

Brush contact loss is attributed to the resistance between the surface of the brush and the commutator. It is not a loss that could be calculated separately as it is a part of variable losses. Generally, it contributes to both types of copper losses.

Note:- Eddy current Losses is a type of core or Rotational loss

11. For a dc motor operating under the condition of maximum transfer of power, the efficiency of the motor is

1. 100 %
3. 75%
4. less than 50%

Explanation:

Maximum power condition:

If the value of $\frac{dP_m}{dI_a}=0,$ the motor will deliver maximum power

$\frac{dP_m}{dI_a}= V – 2I_aR_a = 0$

Ia Ra = V / 2 ….(2)

From equation (1)&(2)

Eb = V / 2

The efficiency of the motor is = Output Power(Pm) / Input power(P)

$\eta= \frac{E_b I_a }{VI_a }\times 100=\frac{E_b }{V}\times 100=50$%

For a dc motor operating under the condition of maximum transfer of power, the efficiency of the motor is less than or equal to 50%

12. Which of the following losses occur due to the induction of current in a DC Machine?

1. Copper Loss
2. Iron Loss
3. Mechanical Loss
4. Windage Loss

Explanation:

Core Losses or Iron Losses in DC Machine or Magnetic Losses:

• As the iron core of the armature is rotating in the magnetic field, some losses occur in the core which is called core losses.
• Iron Loss occurs due to the induction of current in a DC Machine.
• Normally, machines are operated with constant speed, so these losses are almost constant. These losses are categorized in two forms; Hysteresis loss and Eddy’s current loss.

Hysteresis Loss in DC Machine

• Hysteresis losses occur in the armature winding due to the reversal of magnetization of the core. When the core of the armature is exposed to the magnetic field, it undergoes one complete rotation of magnetic reversal.
• The portion of the armature is under S-pole, after completing half the electrical revolution, the same piece will be under the N-pole, and the magnetic lines are reversed in order to overturn the magnetism within the core.
• The constant process of magnetic reversal in the armature, consume some amount of energy which is called hysteresis loss. The percentage of loss depends upon the quality and volume of the iron.

Eddy Current Loss in DC Machine

• According to Faraday’s law of electromagnetic induction, when an iron core rotates in the magnetic field, an emf is also induced in the core.
• Similarly, when armature rotates in the magnetic field, a small amount of emf is induced in the core which allows the flow of charge in the body due to the conductivity of the core.
• This current is useless for the machine. This loss of current is called eddy current. This loss is almost constant for the DC machines. It could be minimized by selecting the laminated core.

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Title: Drinfeld double of deformed quantum algebras
Abstract: We provide a deformation, f_{\alpha, \beta} , of Lusztig algebra f. Various quantum algebras in literature, including half parts of two-parameter quantum algebras, quantum superalgebras, and multi-parameter quantum algebras/superalgebras, are all specializations of f_{\alpha, \beta} . Moreover, f_{\alpha, \beta} is isomorphic to Lusztig algebra f up to a twist. As a consequence, half parts of those quantum algebras are isomorphic to Lusztig algebra f over a big enough ground field up to certain twists. We further construct the entire algebra U _{\beta, \xi} by Drinfeld double construction. As special cases, above quantum algebras all admit a Drinfeld double construction under certain assumptions. This is a joint work with Junjing Xing.

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and representation algebra, and then report some new trends in representation

algebras of some classes of Hopf algebras and applications.

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# TheilU

0th

Percentile

##### Theil's U Index of Inequality

Calculate Theil's U index of inequality.

Keywords
multivar
##### Usage
TheilU(a, p, type = c(2, 1), na.rm = FALSE)
##### Arguments
a
a numeric vector with the actual observed values.

p
a numeric vector containing the predictions.

type
defining the type of Theil's two U measures, see Details. Default is 2.

na.rm
logical, indicating whether NA values should be stripped before the computation proceeds. If set to TRUE complete cases of cbind(x, y) will be used. Defaults to FALSE.
##### Details

Theil proposed two error measures, but at different times and under the same symbol U, which has caused some confusion. U type = 1 is taken from Theil (1958, pp. 31-42). The argument a represents the actual observations and p the corresponding predictions. He left it open whether a and p should be used as absolute values or as observed and predicted changes. Theil (1966, chapter 2) proposed U type = 2 as a measure of forecast quality: "...where $A_i$ and $P_i$ stand for a pair of predicted and observed changes. ..." As $U_1$ has some serious disadvantages (see Bliemel 1973) it is recommended to use $U_2$.

##### References

Theil, H. (1958): Economic Forecasts and Policy. Amsterdam: North Holland.

Thiel, H. (1966): Applied Economic Forecasting. Chicago: Rand McNally.

Bliemel, F. (1973): Theil's Forecast Accuracy Coefficient: A Clarification, Journal of Marketing Research Vol. 10, No. 4 (Nov., 1973), pp. 444-446

Gini

• TheilU
##### Examples
TheilU(1:10, 2:11, type=1)
TheilU(1:10, 2:11, type=2)

Documentation reproduced from package DescTools, version 0.99.19, License: GPL (>= 2)

### Community examples

navid.nobani@gmail.com at Oct 27, 2017 DescTools v0.99.19

library(forecast) a <- c(1,2,3,5,6,8,9) #actual p <- c(1,3,3,4,6,7,9) #forecasted TheilU(a,p)  [1] 0.1167748


text:
# Mass density problem

1. Nov 5, 2009

### jdw_

Hi, I am wondering if somebody could explain how to answer these questions, it is from "Foundations of astrophysics" ch.15 q15.10

Suppose the mass density of a star as a function of radius is

p(r)=p0[1-(r/R)^2]

where R is the radius of the star

a) Find the mass M of the star in terms of p0 and R
b) Find the mean density of the star in terms of p0
c) Show that the central pressure of the star is:

Pc=15/16pi x GM^2/R^4

I have a good idea on how to do b) and c) but i need help with a)

useful equations :
P(r)= -GM(r)p(r)/r^2

M(r) = 4pi r^2 p(r)

2. Nov 5, 2009

### cepheid

Staff Emeritus
jdw,

It looks like you need to integrate the density over the volume in order to get the total mass. The problem is made simpler by the fact that there is symmetry (rho depends only on r, so that the density is constant everywhere on a sphere of radius r). Therefore, the integrals over the angular coordinates will just turn into 4pi, and you'll be integrating over successive spherical shells of surface area 4pi r^2, and thickness dr. To get this result, you need to know what a volume element is in terms of spherical coordinates.

3. Nov 5, 2009

### qraal

Spherical coordinates weren't specified so why use them?

Mass is straight forward. First the mass element is...

dM = 4.pi.ρ(r).r2 dr

...sub in ρ(r) = ρo(1-(r/R)2) and expand out the brackets to make it easier to integrate.

From there it's straight forward. I think you might've been confused by what you've quoted as the mass equation:

M(r) = 4pi r^2 p(r) [sic]

...except that's the equation for dM(r) NOT M(r). Was that the problem?

Also 'p(r)' should've been 'ρ(r)' else you'd be thinking "pressure as a function of radius" instead of "density as a function of radius".

Last edited: Nov 5, 2009
4. Nov 6, 2009

### cepheid

Staff Emeritus
I just meant that you implicity already *have* used them if you do the integration over "shells" having surface area 4pi r^2. It's just that the integral over the angular coordinates has already been done. Of course the spherical symmetry means that it reduces to a 1D integral -- I just thought it would be good if the OP could see how that comes about by starting with the full volume integral.


text:
# Vector potential of current flowing to a point from all directions

Homework Statement:
This is from Griffith , Introduction to Electrodynamics 4th Edition question 10.7

A time-dependent point charge q(t) at the origin, ##\rho(r, t) = q(t)\delta^3(r)##, is fed by a current ##J(r, t) = −\frac{1}{4π} \frac{\dot q}{r^2}) \hat r##
(a) Check that charge is conserved, by confirming that the continuity equation is obeyed.
(b) Find the scalar and vector potentials in the Coulomb gauge. If you get stuck, try working on (c) first.
(c) Find the fields, and check that they satisfy all of Maxwell’s equations.
Relevant Equations:
$$A = \frac{\mu_0}{4\pi}\int \frac{ J}{|r-r'|}d^3r'$$
$$J = −\frac{1}{4π} \frac{\dot q}{r^2}) \hat r$$
I am having problem with part (b) finding the vector potential. More specifically when writing out the volume integral,
$$A = \frac{\mu_0}{4\pi r}\frac{dq}{dt}\int_{0}^{2\pi}\int_{0}^{\pi}\int_{0}^{?}\frac{1}{4\pi r'^2} r'^2sin\theta dr'd\theta d\phi$$
How do I integrate ##r'##?

The solution says because ##B## can only have ##r## component and only depend on ##r## and ##t## due to symmetry, and ##\nabla \cdot B = 0##, ##B = \nabla \times A = 0## also by using coulomb gauge we set ## \nabla\cdot A = 0##, ##A=0## or it could be any constant.
The argument makes sense to me, but I am not sure what is wrong with the integral, and how does it end up with 0.

Delta2

jasonRF
Gold Member
When you setup the integral,
$$\mathbf{A }(\mathbf{r})= \frac{\mu_0}{4\pi}\int \frac{ \mathbf{ J}(\mathbf{r'})}{|\mathbf{r}-\mathbf{r'}|}d^3r'$$
you left out the factor ##\frac{1}{|\mathbf{r}-\mathbf{r'}|}## in the integrand. The integral is messy.

Last edited:
Delta2
When you setup the integral,
$$\mathbf{A }(\mathbf{r})= \frac{\mu_0}{4\pi}\int \frac{ \mathbf{ J}(\mathbf{r'})}{|\mathbf{r}-\mathbf{r'}|}d^3r'$$
you left out the factor ##\frac{1}{|\mathbf{r}-\mathbf{r'}|}## in the integrand. The integral is messy.
Yes that's right. I'd fix it but I can't seem to edit it anymore. The correct integral should have an extra ##\frac{1}{r'}## factor in it, but I am still not sure what is the upper bound of integral so that it come out to be a constant.

TSny
Homework Helper
Gold Member
Homework Equations: $$A = \frac{\mu_0}{4\pi}\int \frac{ J}{|r-r'|}d^3r'$$
This equation is not valid in general when you have time-dependent fields and currents. In the Lorentz gauge (also called Lorenz gauge) there is a similar formula for A except the integrand is evaluated at "retarded times". But, you want A in the Coulomb gauge. In this gauge, it can be shown that $$\mathbf{A}(\mathbf r) = \frac{\mu_0}{4\pi}\int \frac{\mathbf{J_t}(\mathbf{r}', t_r)}{|\mathbf{r}-\mathbf{r'}|}d^3r'$$where again you use retarded times ##t_r##, but the integrand involves only the so-called "transverse component" ##\mathbf{J_t}## of ##\mathbf{J}##. If you want to read more about this, see section 5.2 here. For your problem ##\mathbf{J_t}=0##, which is consistent with ##\mathbf{A} = 0## in the Coulomb gauge for this particular problem.

vanhees71, RedDeer44 and Delta2


text:
# Extent of reaction vs driving force

For an endothermic reaction with a negative change in entropy, based on dG = dH - TdS increasing the temperature should make the process "less spontaneous". However, if the reaction is endothermic then classic Le Chatelier principle says that the reaction should shift right with increasing temperature. How can we reconcile this?

• Are you familiar with the Van't Hopf Equation and with its derivation? – Chet Miller Mar 7 '18 at 20:56
• Hi Chester, yes I am. How does this help? – Andrew U Mar 8 '18 at 7:38
• The derivation explains the correct way to take into account the effect of temperature on the equilibrium constant. The predictions from Van't Hopf are consistent with Le Chatelier. Let's see your analysis using Van't Hopf. – Chet Miller Mar 8 '18 at 12:27
• dGo = dHo – TdSo = -RT lnK Rearranging the second half of the equation gives (–dHo/RT) + (dSo/R) = lnK So we have a linear relationship for lnK vs 1/T, with slope –dHo/R – Andrew U Mar 8 '18 at 22:33
• So, from this equation, what is the sign of the derivative of lnK with respect to temperature if the reaction is exothermic? – Chet Miller Mar 9 '18 at 1:36

The change in free energy $\Delta G^0$ in these equations represents the change from molar quantities of reactants (in separate containers) at temperature T and pressure 1 bar, and corresponding molar quantities of products (in separate containers) at temperature T and pressure 1 bar. So writing the equation in terms of differentials makes no sense. It should more properly represented by $$\Delta G^0=H^0_{P}-H^0_R-T(S^0_P-S^0_R)$$where the subscript P represents products and the subscript R represents reactants. So, $$\frac{\Delta G^0}{T}=\frac{H^0_{P}-H^0_R}{T}-(S^0_P-S^0_R)$$Now, for $H^0_P$ and $H^0_R$, since the pressure is constant between the two states of the products and reactants, respectively, at different temperatures, we have $$dH^0_P=(C_p)_PdT=TdS^0_P$$and $$dH^0_R=(C_p)_RdT=TdS^0_R$$So, algebraically, by the product rule for differentiation, we have$$\frac{d(\Delta G^0/T)}{dT}=-\frac{(H_P^0-H_R^0)}{T^2}+\frac{T(dS_P^0-dS_R^0)}{TdT}-\frac{(dS^0_P-dS^0_R)}{dT}=-\frac{(H_P^0-H_R^0)}{T^2}$$This is what you get when you do the math correctly.


text:
# diophantus

Hello, this is beta version of diophantus. If you want to report about a mistake, please, write to hello@diophantus.org

#### Confirmation of Cylindrical Perfect Invisibility Cloak Using Fourier-Bessel Analysis

09 Apr 2007 physics.optics arxiv.org/abs/0704.1183
Abstract. A cylindrical wave expansion method is developed to obtain the scattering field for an ideal two-dimensional cylindrical invisibility cloak. A near-ideal model of the invisibility cloak is set up to solve the boundary problem at the inner boundary of the cloak shell. We confirm that a cloak with the ideal material parameters is a perfect invisibility cloak by systematically studying the change of the scattering coefficients from the near-ideal case to the ideal one. However, due to the slow convergence of the zero$^{th}$ order scattering coefficients, a tiny perturbation on the cloak would induce a noticeable field scattering and penetration.

# Reviews

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text:
This document describes and comments Org syntax as it is currently read by its parser (Org Elements) and, therefore, by the export framework. It also includes a few comments on that syntax.

A core concept in this syntax is that only headlines, sections, planning lines and property drawers are context-free12. Every other syntactical part only exists within specific environments.

Three categories are used to classify these environments: "Greater elements", "elements", and "objects", from the broadest scope to the narrowest. The word "element" is used for both Greater and non-Greater elements, the context should make that clear.

The paragraph is the unit of measurement. An element defines syntactical parts that are at the same level as a paragraph, i.e. which cannot contain or be included in a paragraph. An object is a part that could be included in an element. Greater elements are all parts that can contain an element.

Empty lines belong to the largest element ending before them. For example, in a list, empty lines between items belong are part of the item before them, but empty lines at the end of a list belong to the plain list element.

Unless specified otherwise, case is not significant.

#+BEGIN_EXAMPLE STARS KEYWORD PRIORITY TITLE TAGS #+END_EXAMPLE

STARS is a string starting at column 0, containing at least one asterisk (and up to org-inlinetask-min-level if =org-inlinetask= library is loaded) and ended by a space character. The number of asterisks is used to define the level of the headline. It's the sole compulsory part of a headline.

KEYWORD is a TODO keyword, which has to belong to the list defined in org-todo-keywords-1. Case is significant.

PRIORITY is a priority cookie, i.e. a single letter preceded by a hash sign # and enclosed within square brackets.

TITLE can be made of any character but a new line. Though, it will match after every other part have been matched.

TAGS is made of words containing any alpha-numeric character, underscore, at sign, hash sign or percent sign, and separated with colons.

#+BEGIN_EXAMPLE ,*

,** DONE

,*** Some e-mail

,**** TODO [#A] COMMENT Title :tag:a2%: #+END_EXAMPLE

If the first word appearing in the title is "COMMENT", the headline will be considered as "commented". Case is significant.

If its title is org-footnote-section, it will be considered as a "footnote section". Case is significant.

If "ARCHIVE" is one of its tags, it will be considered as "archived". Case is significant.

A headline contains directly one section (optionally), followed by any number of deeper level headlines.

A section contains directly any greater element or element. Only a headline can contain a section. As an exception, text before the first headline in the document also belongs to a section.

As an example, consider the following document:

#+BEGIN_SRC org An introduction.

Some text.

,** Sub-Topic 1

,** Sub-Topic 2

Its internal structure could be summarized as:

Affiliated Keywords

:PROPERTIES: :CUSTOM_ID: Affiliated_keywords :END:

With the exception of inlinetasks, items, planning, clocks, node properties and table rows, every other element type can be assigned attributes.

This is done by adding specific keywords, named "affiliated keywords", just above the element considered, no blank line allowed.

Affiliated keywords are built upon one of the following patterns: "#+KEY: VALUE", "#+KEY[OPTIONAL]: VALUE" or "#+ATTR_BACKEND: VALUE".

KEY is either "CAPTION", "HEADER", "NAME", "PLOT" or "RESULTS" string.

BACKEND is a string constituted of alpha-numeric characters, hyphens or underscores.

OPTIONAL and VALUE can contain any character but a new line. Only "CAPTION" and "RESULTS" keywords can have an optional value.

An affiliated keyword can appear more than once if KEY is either "CAPTION" or "HEADER" or if its pattern is "#+ATTR_BACKEND: VALUE".

"CAPTION", "AUTHOR", "DATE" and "TITLE" keywords can contain objects in their value and their optional value, if applicable.

Greater Elements

:PROPERTIES: :CUSTOM_ID: Greater_Elements :END:

Unless specified otherwise, greater elements can contain directly any other element or greater element excepted:

Greater Blocks

:PROPERTIES: :CUSTOM_ID: Greater_Blocks :END:

Greater blocks consist in the following pattern:

#+BEGIN_EXAMPLE ,#+BEGIN_NAME PARAMETERS CONTENTS ,#+END_NAME #+END_EXAMPLE

NAME can contain any non-whitespace character.

PARAMETERS can contain any character other than new line, and can be omitted.

If NAME is "CENTER", it will be a "center block". If it is "QUOTE", it will be a "quote block".

If the block is neither a center block, a quote block or a block element, it will be a "special block".

CONTENTS can contain any element, except : a line #+END_NAME on its own. Also lines beginning with STARS must be quoted by a comma.

Drawers and Property Drawers

:PROPERTIES: :CUSTOM_ID: Drawers :END:

Pattern for drawers is:

#+BEGIN_EXAMPLE :NAME: CONTENTS :END: #+END_EXAMPLE

NAME can contain word-constituent characters, hyphens and underscores.

CONTENTS can contain any element but another drawer.

Dynamic Blocks

:PROPERTIES: :CUSTOM_ID: Dynamic_Blocks :END:

Pattern for dynamic blocks is:

#+BEGIN_EXAMPLE ,#+BEGIN: NAME PARAMETERS CONTENTS ,#+END: #+END_EXAMPLE

NAME cannot contain any whitespace character.

PARAMETERS can contain any character and can be omitted.

Footnote Definitions

:PROPERTIES: :CUSTOM_ID: Footnote_Definitions :END:

Pattern for footnote definitions is:

#+BEGIN_EXAMPLE 3 CONTENTS #+END_EXAMPLE

It must start at column 0.

LABEL is either a number or follows the pattern "fn:WORD", where word can contain any word-constituent character, hyphens and underscore characters.

CONTENTS can contain any element excepted another footnote definition. It ends at the next footnote definition, the next headline, two consecutive empty lines or the end of buffer.

Inlinetasks are defined by org-inlinetask-min-level contiguous asterisk characters starting at column 0, followed by a whitespace character.

Optionally, inlinetasks can be ended with a string constituted of org-inlinetask-min-level contiguous asterisk characters starting at column 0, followed by a space and the "END" string.

Inlinetasks are recognized only after org-inlinetask library is loaded.

Plain Lists and Items

:PROPERTIES: :CUSTOM_ID: Plain_Lists_and_Items :END:

Items are defined by a line starting with the following pattern: "BULLET COUNTER-SET CHECK-BOX TAG", in which only BULLET is mandatory.

BULLET is either an asterisk, a hyphen, a plus sign character or follows either the pattern "COUNTER." or "COUNTER)". In any case, BULLET is follwed by a whitespace character or line ending.

COUNTER can be a number or a single letter.

COUNTER-SET follows the pattern [@COUNTER].

CHECK-BOX is either a single whitespace character, a "X" character or a hyphen, enclosed within square brackets.

TAG follows "TAG-TEXT ::" pattern, where TAG-TEXT can contain any character but a new line.

An item ends before the next item, the first line less or equally indented than its starting line, or two consecutive empty lines. Indentation of lines within other greater elements do not count, neither do inlinetasks boundaries.

A plain list is a set of consecutive items of the same indentation. It can only directly contain items.

If first item in a plain list has a counter in its bullet, the plain list will be an "ordered plain-list". If it contains a tag, it will be a "descriptive list". Otherwise, it will be an "unordered list". List types are mutually exclusive.

For example, consider the following excerpt of an Org document:

#+BEGIN_EXAMPLE
1. item 1
2. X] item 2
3. some tag
item 2.1
#+END_EXAMPLE

Its internal structure is as follows:

#+BEGIN_EXAMPLE (ordered-plain-list (item) (item (descriptive-plain-list (item)))) #+END_EXAMPLE

Property Drawers

:PROPERTIES: :CUSTOM_ID: Property_Drawers :END:

Property drawers are a special type of drawer containing properties attached to a headline. They are located right after a headline and its planning information.

PROPERTYDRAWER follows the pattern

#+BEGIN_EXAMPLE :PROPERTIES: CONTENTS :END: #+END_EXAMPLE

where CONTENTS consists of zero or more node properties.

Tables

:PROPERTIES: :CUSTOM_ID: Tables :END:

Tables start at lines beginning with either a vertical bar or the "+-" string followed by plus or minus signs only, assuming they are not preceded with lines of the same type. These lines can be indented.

A table starting with a vertical bar has "org" type. Otherwise it has "table.el" type.

Org tables end at the first line not starting with a vertical bar. Table.el tables end at the first line not starting with either a vertical line or a plus sign. Such lines can be indented.

An org table can only contain table rows. A table.el table does not contain anything.

One or more "#+TBLFM: FORMULAS" lines, where "FORMULAS" can contain any character, can follow an org table.

Elements

:PROPERTIES: :CUSTOM_ID: Elements :END:

Elements cannot contain any other element.

Only keywords whose name belongs to org-element-document-properties, verse blocks , paragraphs and table rows can contain objects.

Babel Call

:PROPERTIES: :CUSTOM_ID: Babel_Call :END:

Pattern for babel calls is:

#+BEGIN_EXAMPLE ,#+CALL: VALUE #+END_EXAMPLE

VALUE is optional. It can contain any character but a new line.

Blocks

:PROPERTIES: :CUSTOM_ID: Blocks :END:

Like greater blocks, pattern for blocks is:

#+BEGIN_EXAMPLE ,#+BEGIN_NAME DATA CONTENTS ,#+END_NAME #+END_EXAMPLE

NAME cannot contain any whitespace character.

If NAME is "COMMENT", it will be a "comment block". If it is "EXAMPLE", it will be an "example block". If it is "EXPORT", it will be an "export block". If it is "SRC", it will be a "source block". If it is "VERSE", it will be a "verse block".

DATA can contain any character but a new line. It can be ommitted, unless the block is either a "source block" or an "export block".

In the latter case, it should be constituted of a single word.

In the former case, it must follow the pattern "LANGUAGE SWITCHES ARGUMENTS", where SWITCHES and ARGUMENTS are optional.

LANGUAGE cannot contain any whitespace character.

SWITCHES is made of any number of "SWITCH" patterns, separated by blank lines.

A SWITCH pattern is either "-l "FORMAT"", where FORMAT can contain any character but a double quote and a new line, "-S" or "+S", where S stands for a single letter.

ARGUMENTS can contain any character but a new line.

CONTENTS can contain any character, including new lines. Though it will only contain Org objects if the block is a verse block. Otherwise, CONTENTS will not be parsed.

Clock, Diary Sexp and Planning

:PROPERTIES: :CUSTOM_ID: Clock,_Diary_Sexp_and_Planning :END:

A clock follows either of the patterns below:

#+BEGIN_EXAMPLE CLOCK: INACTIVE-TIMESTAMP CLOCK: INACTIVE-TIMESTAMP-RANGE DURATION #+END_EXAMPLE

INACTIVE-TIMESTAMP, resp. INACTIVE-TIMESTAMP-RANGE, is an inactive, resp. inactive range, timestamp object.

DURATION follows the pattern:

#+BEGIN_EXAMPLE => HH:MM #+END_EXAMPLE

HH is a number containing any number of digits. MM is a two digit numbers.

A diary sexp is a line starting at column 0 with "%%(" string. It can then contain any character besides a new line.

A planning is an element with the following pattern:

where HEADLINE is a headline element and PLANNING is a line filled with INFO parts, where each of them follows the pattern:

#+BEGIN_EXAMPLE KEYWORD: TIMESTAMP #+END_EXAMPLE

KEYWORD is either "DEADLINE", "SCHEDULED" or "CLOSED". TIMESTAMP is a timestamp object.

In particular, no blank line is allowed between PLANNING and HEADLINE.

A "comment line" starts with a hash signe and a whitespace character or an end of line.

Comments can contain any number of consecutive comment lines.

Fixed Width Areas

:PROPERTIES: :CUSTOM_ID: Fixed_Width_Areas :END:

A "fixed-width line" start with a colon character and a whitespace or an end of line.

Fixed width areas can contain any number of consecutive fixed-width lines.

Horizontal Rules

:PROPERTIES: :CUSTOM_ID: Horizontal_Rules :END:

A horizontal rule is a line made of at least 5 consecutive hyphens. It can be indented.

Keywords

:PROPERTIES: :CUSTOM_ID: Keywords :END:

#+BEGIN_EXAMPLE ,#+KEY: VALUE #+END_EXAMPLE

KEY can contain any non-whitespace character, but it cannot be equal to "CALL" or any affiliated keyword.

VALUE can contain any character excepted a new line.

If KEY belongs to org-element-document-properties, VALUE can contain objects.

LaTeX Environments

:PROPERTIES: :CUSTOM_ID: LaTeX_Environments :END:

Pattern for LaTeX environments is:

#+BEGIN_EXAMPLE \begin{NAME} CONTENTS \end{NAME} #+END_EXAMPLE

NAME is constituted of alpha-numeric or asterisk characters.

CONTENTS can contain anything but the "\end{NAME}" string.

Node Properties

:PROPERTIES: :CUSTOM_ID: Node_Properties :END:

Node properties can only exist in property drawers. Their pattern is any of the following

#+BEGIN_EXAMPLE :NAME: VALUE

:NAME+: VALUE

:NAME:

:NAME+: #+END_EXAMPLE

NAME can contain any non-whitespace character but cannot end with a plus sign. It cannot be the empty string.

VALUE can contain anything but a newline character.

Paragraphs

:PROPERTIES: :CUSTOM_ID: Paragraphs :END:

Paragraphs are the default element, which means that any unrecognized context is a paragraph.

Empty lines and other elements end paragraphs.

Paragraphs can contain every type of object.

Table Rows

:PROPERTIES: :CUSTOM_ID: Table_Rows :END:

A table rows is either constituted of a vertical bar and any number of table cells or a vertical bar followed by a hyphen.

In the first case the table row has the "standard" type. In the second case, it has the "rule" type.

Table rows can only exist in tables.

Objects

:PROPERTIES: :CUSTOM_ID: Objects :END:

Objects can only be found in the following locations:

Most objects cannot contain objects. Those which can will be specified.

Entities and LaTeX Fragments

:PROPERTIES: :CUSTOM_ID: Entities_and_LaTeX_Fragments :END:

An entity follows the pattern:

#+BEGIN_EXAMPLE \NAME POST #+END_EXAMPLE

where NAME has a valid association in either org-entities or org-entities-user.

POST is the end of line, "{}" string, or a non-alphabetical character. It isn't separated from NAME by a whitespace character.

A LaTeX fragment can follow multiple patterns:

#+BEGIN_EXAMPLE \NAME BRACKETS $$CONTENTS$$ $CONTENTS$ $$CONTENTS$$ PRE$CHAR$POST PRE$BORDER1 BODY BORDER2$POST #+END_EXAMPLE

NAME contains alphabetical characters only and must not have an association in either org-entities or org-entities-user.

BRACKETS is optional, and is not separated from NAME with white spaces. It may contain any number of the following patterns:

#+BEGIN_EXAMPLE [CONTENTS1] {CONTENTS2} #+END_EXAMPLE

where CONTENTS1 can contain any characters excepted "{" "}", "[" "]" and newline and CONTENTS2 can contain any character excepted "{", "}" and newline.

CONTENTS can contain any character but cannot contain "\)" in the second template or "\]" in the third one.

PRE is either the beginning of line or a character different from $. CHAR is a non-whitespace character different from ~.~, ~,~, ~?~, ;, ' or a double quote. POST is any punctuation (including parentheses and quotes) or space character, or the end of line. BORDER1 is a non-whitespace character different from ~.~, ~,~, ;~ and ~$.

BODY can contain any character excepted $, and may not span over more than 3 lines. BORDER2 is any non-whitespace character different from ~,~, ~.~ and $.

#+ATTR_ASCII: :width 5

#+BEGIN_QUOTE It would introduce incompatibilities with previous Org versions, but support for $...$ (and for symmetry, $$...$$) constructs ought to be removed.

They are slow to parse, fragile, redundant and imply false positives. --- ngz #+END_QUOTE

Export Snippets

:PROPERTIES: :CUSTOM_ID: Export_Snippets :END:

Patter for export snippets is:

#+BEGIN_EXAMPLE @@NAME:VALUE@@ #+END_EXAMPLE

NAME can contain any alpha-numeric character and hyphens.

VALUE can contain anything but "@@" string.

Footnote References

:PROPERTIES: :CUSTOM_ID: Footnote_References :END:

There are four patterns for footnote references:

#+BEGIN_EXAMPLE 4 5 6 #+END_EXAMPLE

LABEL can contain any word constituent character, hyphens and underscores.

DEFINITION can contain any character. Though opening and closing square brackets must be balanced in it. It can contain any object encountered in a paragraph, even other footnote references.

If the reference follows the second pattern, it is called an "inline footnote". If it follows the third one, i.e. if LABEL is omitted, it is an "anonymous footnote".

Inline Babel Calls and Source Blocks

:PROPERTIES: :CUSTOM_ID: Inline_Babel_Calls_and_Source_Blocks :END:

Inline Babel calls follow any of the following patterns:

NAME can contain any character besides (, ~)~ and "\n".

HEADER can contain any character besides ~]~ and "\n".

ARGUMENTS can contain any character besides ~)~ and "\n".

Inline source blocks follow any of the following patterns:

#+BEGIN_EXAMPLE src_LANG{BODY} src_LANG[OPTIONS]{BODY} #+END_EXAMPLE

LANG can contain any non-whitespace character.

OPTIONS and BODY can contain any character but "\n".

Line Breaks

:PROPERTIES: :CUSTOM_ID: Line_Breaks :END:

A line break consists in "\\SPACE" pattern at the end of an otherwise non-empty line.

SPACE can contain any number of tabs and spaces, including 0.

There are 4 major types of links:

PRE1 and POST1, when they exist, are non alphanumeric characters.

RADIO is a string matched by some radio target. It may contain entities, latex fragments, subscript and superscript.

PROTOCOL is a string among org-link-types.

PATH can contain any character but ~]~, <, > and \n.

PRE2 and POST2, when they exist, are non word constituent characters.

PATH2 can contain any non-whitespace character excepted (, ~)~, < and >. It must end with a word-constituent character, or any non-whitespace non-punctuation character followed by /.

DESCRIPTION must be enclosed within square brackets. It can contain any character but square brackets. It can contain any object found in a paragraph excepted a footnote reference, a radio target and a line break. It cannot contain another link either, unless it is a plain or angular link.

DESCRIPTION is optional.

PATH3 is built according to the following patterns:

#+BEGIN_EXAMPLE FILENAME ("file" type) PROTOCOL:PATH4 ("PROTOCOL" type) PROTOCOL://PATH4 ("PROTOCOL" type) id:ID ("id" type) #CUSTOM-ID ("custom-id" type) (CODEREF) ("coderef" type) FUZZY ("fuzzy" type) #+END_EXAMPLE

FILENAME is a file name, either absolute or relative.

PATH4 can contain any character besides square brackets.

ID is constituted of hexadecimal numbers separated with hyphens.

PATH4, CUSTOM-ID, CODEREF and FUZZY can contain any character besides square brackets.

Macros

:PROPERTIES: :CUSTOM_ID: Macros :END:

#+BEGIN_EXAMPLE {{{NAME(ARGUMENTS)}}} #+END_EXAMPLE

NAME must start with a letter and can be followed by any number of alpha-numeric characters, hyphens and underscores.

ARGUMENTS can contain anything but "}}}" string. Values within ARGUMENTS are separated by commas. Non-separating commas have to be escaped with a backslash character.

#+BEGIN_EXAMPLE <<>> #+END_EXAMPLE

CONTENTS can be any character besides <, > and "\n". It cannot start or end with a whitespace character. As far as objects go, it can contain text markup, entities, latex fragments, subscript and superscript only.

#+BEGIN_EXAMPLE <> #+END_EXAMPLE

TARGET can contain any character besides <, > and "\n". It cannot start or end with a whitespace character. It cannot contain any object.

#+BEGIN_EXAMPLE [PERCENT%] [NUM1/NUM2] #+END_EXAMPLE

PERCENT, NUM1 and NUM2 are numbers or the empty string.

Subscript and Superscript

:PROPERTIES: :CUSTOM_ID: Subscript_and_Superscript :END:

Pattern for subscript is:

#+BEGIN_EXAMPLE CHAR_SCRIPT #+END_EXAMPLE

Pattern for superscript is:

#+BEGIN_EXAMPLE CHAR^SCRIPT #+END_EXAMPLE

CHAR is any non-whitespace character.

SCRIPT can be * or an expression enclosed in parenthesis (respectively curly brackets), possibly containing balanced parenthesis (respectively curly brackets).

SCRIPT can also follow the pattern:

#+BEGIN_EXAMPLE SIGN CHARS FINAL #+END_EXAMPLE

SIGN is either a plus sign, a minus sign, or an empty string.

CHARS is any number of alpha-numeric characters, commas, backslashes and dots, or an empty string.

FINAL is an alpha-numeric character.

There is no white space between SIGN, CHARS and FINAL.

Table Cells

:PROPERTIES: :CUSTOM_ID: Table_Cells :END:

#+BEGIN_EXAMPLE CONTENTS SPACES| #+END_EXAMPLE

CONTENTS can contain any character excepted a vertical bar.

SPACES contains any number of space characters, including zero. It can be used to align properly the table.

The final bar may be replaced with a newline character for the last cell in row.

Timestamps

:PROPERTIES: :CUSTOM_ID: Timestamp :END:

There are seven possible patterns for timestamps:

#+BEGIN_EXAMPLE <%%(SEXP)> (diary) (active) [DATE TIME REPEATER-OR-DELAY] (inactive) -- (active range) (active range) [DATE TIME REPEATER-OR-DELAY]--[DATE TIME REPEATER-OR-DELAY] (inactive range) [DATE TIME-TIME REPEATER-OR-DELAY] (inactive range) #+END_EXAMPLE

SEXP can contain any character excepted > and \n.

DATE follows the pattern:

#+BEGIN_EXAMPLE YYYY-MM-DD DAYNAME #+END_EXAMPLE

Y, M and D are digits. DAYNAME can contain any non whitespace-character besides +, -, ~]~, >, a digit or \n.

TIME follows the pattern H:MM. H can be one or two digit long and can start with 0.

REPEATER-OR-DELAY follows the pattern:

#+BEGIN_EXAMPLE MARK VALUE UNIT #+END_EXAMPLE

MARK is + (cumulate type), ++ (catch-up type) or ~.+~ (restart type) for a repeater, and - (all type) or -- (first type) for warning delays.

VALUE is a number.

UNIT is a character among h (hour), d (day), w (week), m~ (month), ~y (year).

MARK, VALUE and UNIT are not separated by whitespace characters.

There can be two REPEATER-OR-DELAY in the timestamp: one as a repeater and one as a warning delay.

Text Markup

:PROPERTIES: :CUSTOM_ID: Emphasis_Markers :END:

Text markup follows the pattern:

#+BEGIN_EXAMPLE PRE MARKER CONTENTS MARKER POST #+END_EXAMPLE

PRE is a whitespace character, (, { ' or a double quote. It can also be a beginning of line.

MARKER is a character among * (bold), = (verbatim), /~ (italic), ~+ (strike-through), _ (underline), ~ (code).

CONTENTS is a string following the pattern:

#+BEGIN_EXAMPLE BORDER BODY BORDER #+END_EXAMPLE

BORDER can be any non-whitespace character excepted ~,~, ' or a double quote.

BODY can contain contain any character but may not span over more than 3 lines.

BORDER and BODY are not separated by whitespaces.

CONTENTS can contain any object encountered in a paragraph when markup is "bold", "italic", "strike-through" or "underline".

POST is a whitespace character, -, ~.~, ~,~, :, ~!~, ~?~, ', ~)~, ~}~ or a double quote. It can also be an end of line.

PRE, MARKER, CONTENTS, MARKER and POST are not separated by whitespace characters.

#+ATTR_ASCII: :width 5

#+BEGIN_QUOTE All of this is wrong if org-emphasis-regexp-components or org-emphasis-alist are modified.

This should really be simplified.

Also, CONTENTS should be anything within code and verbatim emphasis, by definition. --- ngz #+END_QUOTE

Footnotes

quoted by a comma when they do not define a headline.

recognized just by looking at the beginning of the line. Planning lines and property drawers can be recognized by looking at one or two lines above.

As a consequence, using org-element-at-point or ~org-element-context~ will move up to the parent headline, and parse top-down from there until context around original location is found.

1. In particular, the parser requires stars at column 0 to be
2. It also means that only headlines and sections can be


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# Frequency table in A-Level Statistic book S1

I am confused with the answer given in the A-Level book S1. Please see the following question and please tell me why answer is 38 and not 24. Is it misprinting or am I missing out something?

Question: The nurse at a health centre records the heights, h cm, to the nearest cm, of a group of boys in the same school age group. The frequency table shows the result

\begin{array}{|c|c|c|} \hline h& Frequency & Cumulative freqency \\ \hline 165& 8& 8&\\ \hline 166& 7& 15&\\ \hline 167& 9& 24&\\ \hline 168& 14& 38&\\ \hline 169& 18& 56&\\ \hline 170& 16& 72&\\ \hline \end{array}.

State the number of boys who are less than 168 cm tall.

-
Usually, "less than" implies "less than but not equal to". Otherwise, the explicit "less than or equal to" is used. –  Alfonso Fernandez Dec 29 '12 at 0:02
so answer is 8 + 7 + 9 = 24. but book gives answer 38. So I am right to say that 24 boys have height less than 168cm. –  user18096 Dec 29 '12 at 0:03
Oh, I read it wrong... If the book included 168, then either the book is wrong, or the conventions in the A-levels (they're British, right?) are different. Are you sure the original wasn't "at most 168cm" or something like that? –  Alfonso Fernandez Dec 29 '12 at 0:14
No I have copied exactly what is written in A-level (yes, they're British) book. I am pretty sure it is misprinting as it doesn't make sense to me. I just wanted to double-check. –  user18096 Dec 29 '12 at 0:18
I would still check other places in the book or ask a teacher to make sure this is not used consistently, but you're most probably fine. Good luck with you A-levels! –  Alfonso Fernandez Dec 29 '12 at 0:22

Less than $168$ presumably includes numbers like $167.92$. Since $168$ (exactly) is unlikely, the answer $38$ is substantially more reasonable than $24$. Less than $168$ does not mean $167$ or below.


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## Understanding the Meaning of Absolute Value

One important concept in algebra that we learn is the distance between two points on the number line. In particular, we study the distance of a point that corresponds to a number to the point that corresponds to 0. In the following figure, the point (or circle) on the left represents 0, while the point on the right represents 5. To simplify our language, we will use coordinates to refer to its corresponding point on the number line. For instance, we will use -8 to refer to the point that corresponds to -8.

Looking at the number line, it is easy to see that 5 is 5 units away from 0 and that -8 is 8 units away from 0. We can also see that 0 is 0 units away from 0. The distance of a number from 0 on the number line is called its absolute value. Hence, the absolute value of 5 is equal to 5, the absolute value of -8 is equal to 8, and the absolute value of 0 is equal to 0. » Read more

## How to Add Integers Using the Number Line

There are several ways that we can visualize addition of integers. One way is to use the number line. If you can see the pattern in this method, it is easier to see the sign of the sum of two integers. To be able to use this method to add integers, first we should think of integers as movements on the number line. We can think of the positive integers as movement to the right and the negative integers as movement to the left.  » Read more

## Making Connection Between Distance and Absolute Value

We have learned about the distance formula and we also have discussed how to get the distance between two points given their coordinates.  In this post, we are going to explore a simpler concept: getting the distance between two points on the number line.

Suppose we want to get the distance between two p0ints on the number line whose coordinates are integers, we can just do it by counting. In the figure below, the distance from $0$ to $5$ is $5$, and the distance from $5$ to $9$ is $4$.

image via nctm illuminations

It is also obvious that we can get the distance by subtraction: $5-0=5$ and $9-5=4$.  It is clear that if we let $a$ be the larger integer, and $b$ be the smaller, then the distance can between the two integers is $a-b$. It is also clear that this formula applies to non-integral coordinates.  » Read more

1 2


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## Smooth functions invariant under the action of a compact Lie group.(English)Zbl 0297.57015

### MSC:

 57S15 Compact Lie groups of differentiable transformations 58C25 Differentiable maps on manifolds
Full Text:


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# How to calculate per capita income of a country?

## Question:

How to calculate per capita income of a country?

## National Income Statistics:

National income statistics are statistics that are computed by the country and they include the gross domestic product, per capita value among others. These statistics are used to compare the living standards of individuals in different countries and to show the level of living standards in a country.

Per capita income of a country refers to that average level of income per person in a country. There per capita income is computed using the following formula:

• {eq}\text{Per capita income}=\dfrac{\text{Total income}}{\text{Total population}} {/eq}

For example, taxing the total income or the GDP of a country to be $4,550,000,000 and the country's population is 3000000 people the per capita income will be: • {eq}\text{Per capita income}=\dfrac{$4,550,000,000}{3000000 } {/eq}
• {eq}\text{Per capita income}=\$1516.667 {/eq}


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# Describe how you would separate a mixture of water, sand and iron metal filings. Which separation techniques would you use?

Mar 25, 2016

#### Answer:

Water may be simply filtered off. The iron filings may be dissolved in acid, and this solution filtered from the sand.

#### Explanation:

$F e \left(s\right) + 2 H C l \left(a q\right) \rightarrow F e C {l}_{2} \left(a q\right) + {H}_{2} \left(g\right) \uparrow$

Sand, $S i {O}_{2}$ to a first approximation, is very inert, and only specialty acids (for instance $H F$) would touch it.


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# Hyperelliptic curve

Fig. 1. A hyperelliptic curve

In algebraic geometry, a hyperelliptic curve of genus g > 1 is an algebraic curve given by an equation of the form

${\displaystyle y^{2}+h(x)y=f(x)}$

where f(x) is a polynomial of degree n = 2g + 1 > 4 or with n = 2g + 2 > 4 distinct roots and h(x) is a polynomial of degree < g + 2 (if the characteristic of the ground field is not 2, one can take h(x) = 0). A hyperelliptic function is an element of the function field of such a curve or possibly of the Jacobian variety on the curve, these two concepts being the same in the elliptic function case, but different in the present case. Fig. 1 is the graph of ${\displaystyle C:y^{2}=f(x)}$ where

${\displaystyle f(x)=x^{5}-2x^{4}-7x^{3}+8x^{2}+12x=x(x+1)(x-3)(x+2)(x-2).}$

## Genus of the curve

The degree of the polynomial determines the genus of the curve: a polynomial of degree 2g + 1 or 2g + 2 gives a curve of genus g. When the degree is equal to 2g + 1, the curve is called an imaginary hyperelliptic curve. Meanwhile, a curve of degree 2g + 2 is termed a real hyperelliptic curve. This statement about genus remains true for g = 0 or 1, but those curves are not called "hyperelliptic". Rather, the case g = 1 (if we choose a distinguished point) is an elliptic curve. Hence the terminology.

## Formulation and choice of model

While this model is the simplest way to describe hyperelliptic curves, such an equation will have a singular point at infinity in the projective plane. This feature is specific to the case n > 4. Therefore, in giving such an equation to specify a non-singular curve, it is almost always assumed that a non-singular model (also called a smooth completion), equivalent in the sense of birational geometry, is meant.

To be more precise, the equation defines a quadratic extension of C(x), and it is that function field that is meant. The singular point at infinity can be removed (since this is a curve) by the normalization (integral closure) process. It turns out that after doing this, there is an open cover of the curve by two affine charts: the one already given by

${\displaystyle y^{2}=f(x)\,}$

and another one given by

${\displaystyle w^{2}=v^{2g+2}f(1/v)\,}$ .

The glueing maps between the two charts are given by

${\displaystyle (x,y)\mapsto (1/x,y/x^{g+1})}$

and

${\displaystyle (v,w)\mapsto (1/v,w/v^{g+1}),}$

wherever they are defined.

In fact geometric shorthand is assumed, with the curve C being defined as a ramified double cover of the projective line, the ramification occurring at the roots of f, and also for odd n at the point at infinity. In this way the cases n = 2g + 1 and 2g + 2 can be unified, since we might as well use an automorphism of the projective line to move any ramification point away from infinity.

## Using Riemann–Hurwitz formula

Using the Riemann–Hurwitz formula, the hyperelliptic curve with genus g is defined by an equation with degree n = 2g + 2. Suppose the bijective morphism f : X → P1 with ramification degree 2, where X is a curve with genus g and P1 is the Riemann sphere. Let g1 = g and g0 be the genus of P1 ( = 0 ), then the Riemann-Hurwitz formula turns out to be

${\displaystyle 2-2g_{1}=2(2-2g_{0})-\sum _{s\in X}(e_{s}-1)}$

where s is over all ramified points on X. The number of ramified points is n, so n = 2g + 2.

## Occurrence and applications

All curves of genus 2 are hyperelliptic, but for genus ≥ 3 the generic curve is not hyperelliptic. This is seen heuristically by a moduli space dimension check. Counting constants, with n = 2g + 2, the collection of n points subject to the action of the automorphisms of the projective line has (2g + 2) − 3 degrees of freedom, which is less than 3g − 3, the number of moduli of a curve of genus g, unless g is 2. Much more is known about the hyperelliptic locus in the moduli space of curves or abelian varieties,[clarification needed] though it is harder to exhibit general non-hyperelliptic curves with simple models.[1] One geometric characterization of hyperelliptic curves is via Weierstrass points. More detailed geometry of non-hyperelliptic curves is read from the theory of canonical curves, the canonical mapping being 2-to-1 on hyperelliptic curves but 1-to-1 otherwise for g > 2. Trigonal curves are those that correspond to taking a cube root, rather than a square root, of a polynomial.

The definition by quadratic extensions of the rational function field works for fields in general except in characteristic 2; in all cases the geometric definition as a ramified double cover of the projective line is available, if it[clarification needed] is assumed to be separable.

Hyperelliptic curves can be used in hyperelliptic curve cryptography for cryptosystems based on the discrete logarithm problem.

Hyperelliptic curves also appear composing entire connected components of certain strata of the moduli space of Abelian differentials.[2]

Hyperellipticity of genus-2 curves was used to prove Gromov's filling area conjecture in the case of fillings of genus =1.

### Classification

Hyperelliptic curves of given genus g have a moduli space, closely related to the ring of invariants of a binary form of degree 2g+2.[specify]

## History

Hyperelliptic functions were first published[citation needed] by Adolph Göpel (1812-1847) in his last paper Abelsche Transcendenten erster Ordnung (Abelian transcendents of first order) (in Journal für reine und angewandte Mathematik, vol. 35, 1847). Independently Johann G. Rosenhain worked on that matter and published Umkehrungen ultraelliptischer Integrale erster Gattung (in Mémoires des sa vanta etc., vol. 11, 1851).


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Oort Cloud Parabolic Comet

A comet in the Oort cloud falls towards the Sun. As it picks up speed from the Sun's gravity, it accidentally picks up a very tiny extra velocity boost from a nearby object in the solar system, so that its trajectory is now a true parabola. It eventually whips around the Sun and begins to travel back out into the far reaches of space.

Let $$\theta$$ be the angle $$\angle POA,$$ where point $$O$$ is the center of the Sun, point $$P$$ is where the comet has come closest to the Sun, and point $$A$$ is where it is on the trajectory outward.

The radial velocity of the comet relative to the Sun is the speed in which the comet is moving away from the Sun in the direction of the line $$OA$$. For what angle $$\theta$$ is this radial velocity the greatest?

Express your answer to the nearest degree.

Note: Kepler orbits are commonly described by the polar equation $r(\theta)=\dfrac{a(1-e^2)}{1+e \cos(\theta)},$ where $$a$$ is the semi-major axis, and $$e$$ is the eccentricity. For a circle, $$e=0;$$ for an ellipse, $$0<e<1;$$ for a parabola, $$e=1;$$ for a hyperbola, $$e>1$$. Also, you may want to look up Kepler's laws and vis-viva equation.

×


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# What is the most effective way to learn, understand, and utilize an algorithm?

In order to keep my interviewing skills sharp, I solve one algorithm/data-structure problem every day, from sites like Leetcode or HackerRank.

If I am not able to solve it after 45 minutes or so, I usually refer to the solution. I review it, understand it, and move on.

However, I have found that I do not really learn anything at all by looking at the solution. Even though I understand the solution at that point in time, it doesn't really stick with me. I am not able to learn the approach and trick of that solution, for purposes of retaining and utilizing it later. And, if I ran into a similar question in the future, I would not remember how to utilize the approach that I have definitely seen before.

What is your technique or suggestion in learning such approaches/techniques in solving algorithm/data-structure problems?

If this question is not in the right Stack Exchange please let me know.

• How long have you been trying your approach? It might not be bad as you may think it is. I'll suggest in order to strengthen your comprehension of the material: try to make an overview per algorithm technique you have studied. Also, keep a list of a few, key, examples. You might want to check some cheatsheet to guide you, but you must try to do it yourself. At the same time, keep working solving problems. As with any other matter, mastery becomes with practice. Feb 14 '20 at 10:38
• I have usually done it for like up to 1 month at a time. This month, I have been doing problems every day. However, I do not feel that I am getting on the next gear. A cheatsheet would be nice, but that worries me because it might make me take a memorization approach. But I suppose if I was able to come up with some clear way to specify how a specific solution works. That could be beneficial in the future. Do you have any approaches or samples of you doing this? Feb 14 '20 at 23:37

To be an algorithm designer, you have to do some steps.

First, learn the basics of algorithmic thinking. You can learn it by using a simple programming language book. Choosing the language is not important. You can take a look at this page for finding a book. After finishing this step, you have to be able to present pseudocode (or code written in a specific language) for simple problems, for example:

• Searching in a list: there is a list of numbers, and we want to see a specific number is in the list or not.
• Sorting a list
• Finding the greatest common divisor of two numbers

Second, learn the important data structures— link-list, stack, queue, and tree— and their algorithms. For doing this step, you can read the books of data structure written for bachelor students of computer science. For example, Fundamentals of Data Structures by Ellis Horowitz and Sartaj Sahni is a good book. At the end of this step, you have to know/do the following things:

• What is a data structure? How we can iterate over the elements of a data structure? How we can add a new item(/remove an item) from the data structure?
• What are the main features of each data structure?

Third, learn conventional algorithm design techniques— divide and conquer, dynamic programming, greedy, back-track, branch and bound. For doing this step, you can read the algorithm design books written for bachelor students of computer science. For example:

• Foundations Of Algorithms, Richard Neapolitan
• Introduction to Algorithms, Thomas H. Cormen, et al. (Chapters 1-17) After doing this step, you can solve a large range of algorithmic problems, like the problems that can be found on this website: onlinejudge.org.

Forth, learn specialized algorithm design techniques, like randomized algorithmic techniques, parallel algorithmic techniques, distributed algorithmic techniques, approximation algorithmic techniques, etc. For each case, there are some books written for graduate students of computer science.

Besides, there are some other steps, like learning the basics of the theory of computing, complexity classes (NP-complete and P-complete theories), and a lot of other things. It depends on your goal and the level that you want to be reached.

• To round out your skills, learn the language you use most well, but check out some other languages with different objectives for perspective (i.e., learn a bit of Python -- it is a very flexible language, that is used quite a bit "in earnest", search for Downey's "Think Python", 2nd edition for a gentle introduction --, take a peek at some functional language and see how you can do magic with recursion) Feb 16 '20 at 1:45


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# Integral solutions of linear equation

How does one find the number of integral solutions of equations like,

$2x + 5y + 4z = 20$

I know how to do this counting if the coefficient of every variable on the LHS is $1$. But I don't know how to do this counting with arbitrary coefficients.

-
Do you mean positive integral solutions? (Or nonnegative?) There are infinitely many integral solutions. –  Jonas Meyer Mar 25 '11 at 15:31
I don't know if this helps, but what you have there is a linear Diophantine equation in which Bézout's identity applies (en.wikipedia.org/wiki/B%C3%A9zout%27s_identity). It doesn't give any guidance with respect to counting though. –  Gilead Mar 25 '11 at 16:24

The linear equation $2x + 5y + 4z = 20$ has a two-parameter family of infinite solutions in integers. Note that $y$ has to be even. Further any even $y$ will give a one parameter family of infinite solutions for the equation $2x + 4z = 20 - 5y$.

Let $y = 2y_1$. Then we need to solve $2x + 4z = 20 - 10y_1 \implies x + 2z = 10 -5y_1$.

Hence, the set of all integer solutions is the two parameter family $$(10-5y_1-2z_1,2y_1,z_1)$$ where $y_1,z_1 \in \mathbb{Z}$

If you are interested in non-negative integral solutions for $(x,y,z)$, then we need $y_1 \geq 0$,$z_1 \geq 0$ and $5y_1 + 2z_1 \leq 10$.

This implies $y_1 \in \{0,1,2\}$.

If $y_1 = 0$, then $z_1 = \{0,1,2,3,4,5\}$.

If $y_1 = 1$, then $z_1 = \{0,1,2\}$.

If $y_1 = 2$, then $z_1 = \{0\}$.

Hence, the set of all non-negative solutions are $$\{ (10,0,0),(8,0,1),(6,0,2),(4,0,3),(2,0,4),(0,0,5),(5,2,0),(3,2,1),(1,2,2),(0,4,0) \}$$

The set of all positive solutions are $$\{(3,2,1),(1,2,2)\}$$

-

here is a paper on the problem i just happened to see somewhere else

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What is the connection? I can't see how that helps solves my question! –  Anirbit Mar 31 '11 at 8:32
While this link may answer the question, it is better to include the essential parts of the answer here and provide the link for reference. Link-only answers can become invalid if the linked page changes. –  Arkamis Aug 31 '12 at 21:11

Let $x' = 2x$, $y' = 5y$ and $z' = 4z$. Then $2 \leq x' \leq 18$, $5 \leq y' \leq 15$ and $4 \leq z' \leq 16$ for $x',y',z' \in \mathbb{Z}^{+}$. So you can apply the same techniques for counting the number of integral solutions if all the coefficients on the LHS are $1$.

So the number of solutions would be the coefficient of $x^{20}$ in $$\left[(x^2)^{a_1}+(x^2)^{a_1+1}+ \cdots+ (x^a)^{a_2} \right] \cdot \left[(x^5)^{b_1}+(x^5)^{b_1+1}+ \cdots+ (x^5)^{b_2}\right] \cdot \left[(x^4)^{c_1}+(x^4)^{c_1+1}+ \cdots+ (x^4)^{c_2} \right]$$

where $a_1 \leq x \leq a_2$, $b_1 \leq y \leq b_2$ and $c_1 \leq z \leq c_2$.

-
So you're assuming that each is positive. But then your bounds are quite weak relative to what is given. E.g. $y'>4$ and $z'>3$ implies $x'<12$. How do you know you can apply the same techniques? I don't know what techniques the OP uses when the coefficients are 1, but note that modified unknowns aren't allowed to take on all integer values, only multiples of 2,5,4, respectively, so it isn't exactly reduced to the previous problem. –  Jonas Meyer Mar 25 '11 at 15:36
I didn't understand your answer. What is $a_1$, $a_2$, $b_1$, $b_2$, $c_1$ and $c_2$ ? Can you give some background, references and some more explanations? –  Anirbit Mar 31 '11 at 8:25


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Approximating $\sqrt2$ using Taylor polynomials

I have to approximate $\sqrt2$ using Taylor expansion with an error $<10^{-2}$.

I noticed that I can do MacLaurin expansion of $\sqrt{x+1}$ then put $x=1$

So: $$\sqrt{x+1}=1 + \dfrac{x}{2} - \dfrac{x^2}{8} + \dfrac{x^3}{16} + {{\frac1{2}}\choose{n+1}}x^{n+1}(1+\xi)^{-\frac1{2}-n}$$

I have to find the order of the polynomial at which

$\sqrt2 -($the value I found with the polynomial$)<10^{-2}$

I can check the value of the polynomial in $x=1$ order by order until I find that

$\sqrt2 -($the value I found with Taylor$)<10^{-2}$

Or is there a faster way to find the desired order?

The series$\sum_{k=0}^\infty\binom{1/2}{k}$ is alternating (which is the reason why it converges, by the way). For such series this estimate holds $$\left\lvert\sum_{k=0}^\infty a_k-\sum_{h=0}^n a_h\right\rvert\le \rvert a_{n+1}\rvert$$
The Taylor series of the function $f(x):=(1+x)^{1/2}$ just barely converges for $x:=1$. Evaluate the Taylor expansion of $g(x):=(1+x)^{-1/2}$ at $x:=-{1\over2}$ instead.


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# Find the number of binary strings of length N with at least 3 consecutive 1s in C++

C++Server Side ProgrammingProgramming

Suppose, we have an integer N, We have to find the number of all possible distinct binary strings of the length N, which have at least three consecutive 1s. So if n = 4, then the numbers will be 0111, 1110, 1111, so output will be 3.

To solve this, we can use the Dynamic programming approach. So DP(i, x) indicates number of strings of length i with x consecutive 1s in position i + 1 to i + x. Then the recurrence relation will be like −

DP(i, x) = DP(i – 1, 0) + DP(i – 1, x + 1).

The recurrence is based on the fact, that either the string can have 0 or 1 present at position i.

• If it has 0, at ith index then for (i – 1)th position value of x = 0
• If it has 1, at ith index then for (i – 1)th position value of x = 1 + value of x at position i

## Example

Live Demo

#include<iostream>
using namespace std;
int n;
int numberCount(int i, int x, int table[][4]) {
if (i < 0)
return x == 3;
if (table[i][x] != -1)
return table[i][x];
table[i][x] = numberCount(i - 1, 0, table);
table[i][x] += numberCount(i - 1, x + 1, table);
return table[i][x];
}
int main() {
n = 4;
int table[n][4];
for (int i = 0; i < n; i++)
for (int j = 0; j < 4; j++)
table[i][j] = -1;
for (int i = 0; i < n; i++) {
table[i][3] = (1 << (i + 1));
}
cout << "The number of binary strings with at least 3 consecutive 1s: " << numberCount(n - 1, 0, table);
}

## Output

The number of binary strings with at least 3 consecutive 1s: 3
Published on 19-Dec-2019 11:52:32


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# Turning a T1 text-mode character into a maths-mode character

I would like to use the T1 font character Ð, \DH, as a mathematical symbol, as so in a quick and dirty way I defined

\newcommand{\dH}{\text{\DH}}

However, in my theorem environments, where the default is italic text, the Ð is italicised, which is not the case for my equation symbols which are standard roman characters. For instance,

\begin{lemma}\label{depth_bounds}
The depth function $\dH_{Y'}$ satisfies the inequalities
$\dH_{Y'}(x,y) \le lcm(\dH_{G/K}(x),\dH_Y(y))$
\end{lemma}

The disparity with \dH_Y outside the lemma environment, for example, is blatantly obvious: ÐY vs ÐY, modulo HTML spacing issues.

-
I should clarify that I want the upright character throughout, i.e. the one outside the lemma environment. –  David Roberts Nov 2 '12 at 2:43
Adding \normalfont in front of \DH should solve your problem -- \newcommand{\dH}{\text{\normalfont\DH}} –  PattaFeuFeu Nov 2 '12 at 3:16

So for the actually intended way you want to use it, here’s your solution:

\newcommand{\dH}{\text{\normalfont\DH}}

This should print the \DH in “upright character” while the rest of the lemma is italic, as it’s supposed to be.

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Worked a treat, thanks. –  David Roberts Nov 2 '12 at 3:39
Also \newcommand{\dH}{\textnormal{\DH}} should work. –  egreg Nov 2 '12 at 9:26

I’m not 100% sure whether I got your question right, to be honest, but here is my attempt.

This:

\documentclass[paper=a4, fontsize=11pt]{article}

\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}

\usepackage{amsmath, amsthm, amssymb}

\newcommand{\dH}{\text{\DH}}

\begin{document}

\newtheorem{lem}{Lemma}
Lemma environment:
\begin{lem}\label{depth_bounds}
The depth function $\dH_{Y'}$ satisfies the inequalities
$\dH_{Y'}(x,y) \le lcm(\dH_{G/K}(x),\dH_Y(y))$
\end{lem}

Equation environment:
$\dH_{Y'}(x,y) \le lcm(\dH_{G/K}(x),\dH_Y(y))$
\end{document}

results in

While a small change – namely: \newcommand{\dH}{\text{\itshape\DH}}– results in:

-
Ah, I should have said. I want the upright version as default, not the italic version. I tried using \textrm but that gave the same result at \text –  David Roberts Nov 2 '12 at 2:41


text:
Editing for missing dollar signs

So, I'm a new user and trying to figure out how things work around here. I saw a post in which someone forgot to put one of their math equations between dollar signs, so it came out in raw LateX rather than in a readable form.

I saw an 'edit' button, and went "I'll just be nice and edit this in". Saw that it'd then enter a queue to have the edit be confirmed, which is fine...but then when I went to try to submit it, I got an error message saying edits had to be at least 6 characters long. I can see why one might not want people going in to edit minor spelling mistakes, but it seems like this is a pretty common edit to want to make....should I just put a comment in for the original poster to fix, or is there some other way around this, or...?

My suggestions, in decreasing order of preference.

1. Try to improve the post in other ways. Rare is a question that cannot be improved. Even rarer is an impeccable question by a user who forgets dollar signs.

2. Leave the post as is. Some 2k user will notice the missing markup and fix it.

3. Work around the restriction in some way: insert whitespace into a formula, or invisible HTML entities in text  , etc. Adding an empty formula is probably the worst option in terms of rendering speed.

• There are some posts I've seen that were well-written, but the user thought that MathJax would render \begin{align}\end{align} like LaTeX does (and thus they didn't enter dollar signs). I thought 2k users were still subject to character edit limits... but maybe not. (Aside: do HTML comments work for the character limit? I don't want to test it and bump a post by accident.) – apnorton Oct 1 '14 at 4:16
• huh. I guess MathJax made the dollar signs on align environments optional now. Nevermind about that. – apnorton Oct 1 '14 at 4:17
• #3 is terrible, in my opinion. There's no point in going through the whole edit review process just for adding two dollar signs. – Najib Idrissi Oct 1 '14 at 7:38
• 1. is often not possible. I almost got banned from editing because of some 2/3 edits that added $s and changed the grammar. – ghosts_in_the_code Nov 8 '14 at 4:48 The workaround, which should only be used when absolutely necessary, is to write ${}{}{}\$ at the bottom. It renders invisibly.

• Thanks, working on learning the ropes :) – Alan Oct 1 '14 at 1:37
• Please don't. At least use  . Including junk mathjax slows the page down for everyone. – Willie Wong Oct 1 '14 at 8:51
• Fair enough--I'll leave this answer here so people can see that it's not recommended. – apnorton Oct 1 '14 at 20:29


text:
4:24 AM
@ReubenStern Thanks Reuben!

5 hours later…
9:20 AM
@DenisNardin I am in the process of elaborating in a TeX file.

2 hours later…
11:19 AM
Does anyone know a reference (not a proof) for the following fact?
A small ∞-category is stable iff it has all finite limits and finite colimits and admits a conservative exact functor to the ∞-category of spectra.

11:49 AM
(I ask because I just invoked this fact in a MO answer: mathoverflow.net/a/374704/57405)

@AlexanderCampbell I don't know a reference but it's literally half a line using HA.1.4.2.27, so I would use that as a reference

@DenisNardin I don't see how to prove the 'only if' direction (which is the interesting direction) from that reference. (I give a proof in the linked answer.)

Ahh I missed the only if. Sorry for the noise

Not at all :)
@ScottBalchin I see you're giving a lecture series on model categories next month! sarah-whitehouse.staff.shef.ac.uk/ttt/…
2

@AlexanderCampbell Yes, the LMS decided it would be a good idea to run some introductory style things for incoming PhD students :) Of course it gives me a good chance to talk about some of the more exotic examples from the book

12:03 PM
I look forward to watching it. It'll be late at night in my time zone, so I'm glad they'll be recorded.

4 hours later…
4:03 PM
@DanielBruegmann i'd be interested to see this, too! that is a cool fact.
@ScottBalchin congrats, that looks like fun! what book is @AlexanderCampbell referring to?
@TimCampion in re your question, i believe i've seen a characterization of stable ∞-categories $C$ as follows: they admit finite limits and colimits, and moreover for any functor $I \times J \to C$ where $I$ and $J$ are finite ∞-categories, the operations of (fiberwise) colimit over I and (fiberwise) limit over J commute. that is, LKan followed by RKan in the composite $I \times J \to J \to pt$ is equivalent to RKan followed by LKan in the composite $I \times J \to I \to pt$.
i think i learned this from moritz groth
maybe it's due to him, as well
here's a question i'm scared to know the answer to: does anybody know whether NSF grant funding this year will be the same as in past years? (or will it be reduced due to covid for some reason or another?) the program officers weren't willing to say anything definitive, but i figure there could still be people out there who have a better guess than i do (based on an undertanding of how the system works), or even inside info.

4:29 PM
@DanielBruegmann I'd love to see it too (if that wasn't clear :) )

4:40 PM
@AaronMazel-Gee Moritz Rahn* (these days) proves at least in derivators: a derivator (admitting finite limits and colimits) is stable if and only if all finite limits commute with finite colimits. (Co)limits of quasicategories are the same in their associated derivator

5:22 PM
Hi friends, I have another most likely dumb question about a line in HA (also do let me know if these aren't appropriate for this channel). On page 1142 of HA (latest version), above remark 14.1.1.5, he talks about the homotopy fiber of a map Map(A\otimes B, k) -> Map(A, k) \times Map (B, k). My question is, since the tensor product is the coproduct in Alg_k, isn't that map just an equivalence and hence the fiber is contractible? What am I missing?
Ah wait I'm dumb, Alg_k are E_1 not E_infty algebras, ignore the above

Ok I can't even find a version of HA with a chapter 14 so I'm glad you figured it out

Ah I meant SAG, sorry, thx

1 hour later…
6:43 PM
If I want to have a relative category, or category with w.e.'s, but it's simplicially enriched, are there requirements that need to be checked for these two things to be "compatible?"

@JonathanBeardsley In order to do what? A simplicially enriched category presents an oo-category. The week equivalences then give you some maps (just need homotopy classes thereof even), and you can ask for the universal category inverting those maps.

Maybe you should make sure that degenerations of w.e. are still w.e.?

I suppose my initial thought was just about the difference between the simplicial localization of the category with w.e.'s and the simplicially enriched category. Is there any reason to believe these things coincide, for instance?
(I think this is the case for a simplicially enriched model category, right?)

It is clear that you need some assumptions for that (consider the case where only isomorphisms are weak equivalences), but I don't know what they are.

I agree.
Here's another question: if I have a Kan complex X and I just restrict it to the full subcategory of Δ^{op} spanned by [0] and [1], can I call the resulting thing the fundamental groupoid of X? Is there any obvious flaw in this?

6:59 PM
I don't think you can recover pi_1 of your space from this, can you?

@JonathanBeardsley I'm not sure what you mean, but whatever you mean should probably include the value at [2] to know anything about composition

Right, yeah, the composition is going to get lost.

I think if you then right Kan extend back up then this would be the (nerve of the) fundamental groupoid

7:38 PM
If you have a Kan complex, you can recover its fundamental groupoid from the restriction to Δ≤2
iirc
this is because for quasicategories, the homotopy category can be computed via a coskeleton
for Kan complexes, there's also a general statement that the n+1-coskeleton computes the nth truncation
I might be indexing off-by-1

7:53 PM
Okay right, yeah, great! You extend back to sSet and you get the fundamental groupoid. I guess I was sort of implicitly thinking that but not, you know, saying it, which is a problem I have.

8:21 PM
It's true of any simplicial set that its fundamental groupoid can be recovered from its 2-skeleton.

8:42 PM
@AlexanderCampbell you mean the fundamental groupoid of the realization?

If you like, but I really mean the left adjoint to the nerve functor from groupoids to simplicial sets.

9:15 PM
@IanColey ah, right! thanks for that reminder. (including the reminder that moritz changed his last name.)
@JonathanBeardsley yes, this is true for simplicial model categories. this is a rather nontrivial theorem of dwyer--kan. one sort of compatibility you can ask for is that the weak equivalences become equivalences in the simplicial localization. equivalently, for any weak equivalence $X \to Y$, the map $hom(-,X) \to hom(-,Y)$ is a componentwise weak equivalence of simplicial sets.
(but of course, that's still very far from saying that the localization at the weak equivalences coincides with the given simplicial enrichment.)

2 hours later…
11:35 PM
i've got a relative $\infty$-category $(C,W)$, and i know that the localization $C[W^{-1}]$ is in fact an ordinary category. i'd like to conclude that $C$ itself is an ordinary category. is this true? it feel to me like it's probably false as stated.
in my actual situation of interest, $C$ just has one equivalence class of object. so i'm talking about an $\infty$-monoid $M$ whose $\infty$-group completion $G$ is in fact a discrete group. and a special feature of my situation is that the map $M \to G$ is injective on $\pi_0$. (i believe this is not automatic?) but so, i'm asking whether i can conclude that $M$ is in fact a discrete monoid.


text:
Ceci est une ancienne révision du document !

# Referring to things by their name

LaTeX's labelling mechanism is designed for the impersonal world of the academic publication, in which everything has a number: an extension is necessary if we are to record the name of things we've labelled. The two packages available extend the LaTeX sectioning commands to provide reference by the name of the section.

The titleref package is a simple extension which provides the command `\titleref`; it is a stand-alone package — don't use it in a document in which you also need to use hyperref.

The byname package is part of the smartref bundle and works well with smartref, and works (to an extent) with hyperref, but the links it defines are not hyperlinks.

The memoir class incorporates the functionality of titleref, but doesn't work with byname (though a search of `comp.text.tex` on [groups.google.com] will find a patch to byname to remedy the problem).

The hyperref bundle includes a package nameref, which will work standing alone (i.e., without hyperref: of course, in this mode its references are not hyperlinked). If you load hyperref itself, nameref is automatically loaded. Memoir requires the memhfixc when running with hyperref; following the sequence:

```\documentclass[...]{memoir}
...
\usepackage[...]{hyperref}
\usepackage{memhfixc}```

nameref commands may be used in a memoir document.

Zref defines a proposed replacement for all of the LaTeX reference mechanisms, and among other things provides name-referencing mechanisms:

```\usepackage[user,titleref]{zref}
...
\section{hello}\zlabel{sec:one}
The section name is: \ztitleref{sec:one}.```

(One might hope that something of this sort would be the “way of the future”, but things move slowly in the LaTeX world: don't hold your breath.)

Each of titleref, byname and nameref defines a reference command with the same name as the package: `\titleref`, `\byname` and `\nameref`. The nameref package also defines a command `\byshortnameref`, which uses the optional “short” title argument to the chapter and section commands. (Although it comes from the same author, zref doesn't define a short-name variant.)

3_composition/texte/renvois/se_referer_aux_sections_par_leur_titre.1543881440.txt.gz · Dernière modification: 2018/12/04 00:57 par jejust