text:
• 0 Vote(s) - 0 Average
• 1
• 2
• 3
• 4
• 5
 An asymptotic expansion for \phi JmsNxn Ultimate Fellow Posts: 902 Threads: 111 Joined: Dec 2010 02/06/2021, 03:18 AM (This post was last modified: 02/06/2021, 05:22 AM by JmsNxn.) Hey everyone, so I've recently been trying to streamline some of my proofs in my paper. It's a bit of a clusterfuck at the moment; fairly disorganized. But I thought I'd point out an interesting asymptotic expansion I've been fiddling with. If we define, $ -h_n(x) = x - \log(x+1) + \log \log(x+2) +...+ (-1)^n\log^{\circ n}(x+n)\\$ Then, $ \lim_{x\to\infty} \frac{\phi(t+\pi i) - h_n(x)}{\log^{\circ n}(x+n)} = (-1)^{n}\\$ Now this comes from the initial statement that, Sheldon noticed which I thought to be too good to be true, $ \frac{\phi(x+\pi i)}{x} \to (-1)\\$ This has me asking if there are asymptotic expansions elsewhere. And it led me to an algorithm which may help future work on $\phi$. Suppose that, $ \psi_y(t) = e^{-iy}\phi(t+iy)\\ \lim_{t\to\infty} \frac{\psi_y(t)}{A(t)} = 1\\$ Then, $ \frac{\psi_y(t+1)}{A(t+1)} = \frac{e^{t + e^{iy}\psi_y(t)}}{A(t+1)} \to 1\\$ So we should expect, $ t + e^{iy}\psi_y(t) = \log(A(t+1)) + o(\log(A(t+1)))\\$ Which we can iterate to get successive asymptotic expansions. It may not look as nice for the case $y=\pi$ but I think this may be helpful for iterations. Where we can begin to  approximate successive iterations better. And we'll get something like a log asymptotic series.  If we continue with an assumption of superexponential, $ \log(A(t+1))/A(t) \to \lambda \in\mathbb{C}\\$ Then, $ \psi_y(t) = \lambda e^{-iy}A(t) -e^{-iy}t-o(A(t))\\$ And then, $ \frac{\psi_y(t+2)}{A(t+2)} = \frac{e^{t+1+e^{iy}e^{t+e^{iy}\psi_y(t)}}}{A(t+2)} \to 1\\$ meaning, $ t+1+e^{iy}e^{t+e^{iy}\psi_y(t)} \sim \log A(t+2)\\ t+1+e^{iy}e^{\lambda A(t) + o(A(t))} \sim \log A(t+2)\\ $ This may be a nicer way of thinking about growth. I'm not sure yet. But with super exponentials the asymptotics behave properly. It should also say that exponential asymptotic growth is impossible. It's either dead in the water growth, or super-exponential growth; which means it's in for a penny or in for a dime. JmsNxn Ultimate Fellow Posts: 902 Threads: 111 Joined: Dec 2010 02/08/2021, 12:25 AM ACK, So this result is only true if $\phi(t+\pi i) / t \to -1$. The correct statement without this is, $ \psi_m(t,x) = \Omega_{j=1}^m e^{t-j-x}\bullet x\\$ Then, $ \psi_m(t+m,h_m(t)) = t+m\\$ Where since $h_m\to\infty$ we really can say much, unless $|h_m(t-m)|< M$ is bounded fixed $t$ and $m>0$. Which, is doubtful. So damn close. « Next Oldest | Next Newest »

 Possibly Related Threads… Thread Author Replies Views Last Post Searching for an asymptotic to exp[0.5] tommy1729 203 378,066 08/07/2022, 10:42 PM Last Post: tommy1729 Tetration Asymptotic Series Catullus 18 892 07/05/2022, 01:29 AM Last Post: JmsNxn Using a family of asymptotic tetration functions... JmsNxn 15 6,547 08/06/2021, 01:47 AM Last Post: JmsNxn Reducing beta tetration to an asymptotic series, and a pull back JmsNxn 2 1,433 07/22/2021, 03:37 AM Last Post: JmsNxn A Holomorphic Function Asymptotic to Tetration JmsNxn 2 1,797 03/24/2021, 09:58 PM Last Post: JmsNxn Merged fixpoints of 2 iterates ? Asymptotic ? [2019] tommy1729 1 4,036 09/10/2019, 11:28 AM Last Post: sheldonison Another asymptotic development, similar to 2sinh method JmsNxn 0 4,332 07/05/2011, 06:34 PM Last Post: JmsNxn weird series expansion tommy1729 2 7,455 07/05/2010, 07:59 PM Last Post: tommy1729 matrix function like iteration without power series expansion bo198214 15 35,465 07/14/2008, 09:55 PM Last Post: bo198214 Woon's expansion bo198214 3 9,711 09/07/2007, 03:13 PM Last Post: bo198214

Users browsing this thread: 1 Guest(s)


text:
, 12.11.2019 04:31, lin43

# 50 points! im so confused i'll mark write a linear function f with f(−4)=2 and f(6)=−3.

### Other questions on the subject: Mathematics

Mathematics, 21.06.2019 17:40, jackieespi1461
Mathematics, 21.06.2019 18:00, ciara180
Place the following steps in order to complete the square and solve the quadratic equation,
Mathematics, 21.06.2019 18:00, bryan9892
Atrain is traveling at a constant speed of 60 miles per hour the nummber of hours the train has been traveling is t. the number of miles the train has traveled is d
Mathematics, 21.06.2019 20:00, sahaitong2552
Write each of the following numerals in base 10. for base twelve, t and e represent the face values ten and eleven, respectively. 114 base 5 89t base 12

format: plain


text:
Priority Research Centre for Computer-AssistedResearch Mathematics and its Applications
 Member Links: (login required) Members Area CARMA News Manager CARMA Web Logs CARMA Seminar Manager CARMA Wiki Subscribe to our seminar mailing list Subscribe to our events calendar (iCal format)

### CARMA-Sponsored Events: Colloquia, Seminars, Workshops, Conferences and More.

Details of upcoming events are listed on this page.

## Upcoming CARMA Events

 CARMA RHD MEETING Location: V205, Mathematics Building (Callaghan), The University of Newcastle Dates: 1:00 pm - 3:00 pm, Thu, 23rd Oct 2014 [Permanent link] CARMA SEMINAR Speaker: Nathan Clisby, The University of Melbourne Title: Monte Carlo simulation of self-avoiding walks Location: V205, Mathematics Building (Callaghan), The University of Newcastle Time and Date: 4:00 pm, Thu, 23rd Oct 2014 Abstract: Self-avoiding walks are a widely studied model of polymers, which are defined as walks on a lattice where each successive step visits a neighbouring site, provided the site has not already been visited. Despite the apparent simplicity of the model, it has been of much interest to statistical mechanicians and probabilists for over 60 years, and many important questions about it remain open. One of the most powerful methods to study self-avoiding walks is Monte Carlo simulation. I'll give an overview of the historical developments in this field, and will explain what ingredients are needed for a good Monte Carlo algorithm. I'll then describe how recent progress has allowed for the efficient simulation of truly long walks with many millions of steps. Finally, I'll discuss whether lessons we've learned from simulating self-avoiding walks may be applicable to a wide range of Markov chain Monte Carlo simulations. [Permanent link] CARMA AND AMSI CONFERENCE Number Theory Down Under Location: , Harbourview Function CentreNewcastle, NSW Dates: Fri, 24th Oct 2014 - Sat, 25th Oct 2014 For more information, please visit the conference website. [Permanent link] CARMA FINANCE MEETING Location: V205, Mathematics Building (Callaghan), The University of Newcastle Time and Date: 1:00 pm, Mon, 27th Oct 2014 [Permanent link] CARMA OANT SEMINAR Speaker: Yohei Tachiya, Graduate School of Science and Technology, Hirosaki University Title: Algebraic independence results for the generating functions of pattern sequences Location: V205, Mathematics Building (Callaghan), The University of Newcastle Access Grid Venue: CARMA [ENQUIRIES] Time and Date: 4:00 pm, Mon, 27th Oct 2014 Abstract: We first introduce the notations of pattern sequence, which is defined by the number of (possibly) overlapping occurrences of a given word in the $\langle q,r\rangle$-numeration system. After surveying several properties of pattern sequence, we will give necessary and sufficient criteria for the algebraic independence of their generating functions. As applications, we deduce the linear relations between pattern sequences. The proofs of the theorem and the corollaries are based on Mahler's method. [Permanent link] TOOLS FOR MATHEMATICAL COMPUTATION Location: V206, Mathematics Building (Callaghan), The University of Newcastle Dates: 10:00 am - 12:00 pm, Tue, 28th Oct 2014 As part of Jon Borwein's Experimental Mathematics Course, he is exploring the uses of the currently available tools for mathematical computation. Each lecture should be independently accessible, so feel free to drop in. [Permanent link] CARMA OANT SEMINAR Speaker: Dr Dzmitry Badziahin, Department of Mathematical Sciences, Durham University Title: On continued fraction expansion of potential counterexamples to mixed Littlewood conjecture Location: V205, Mathematics Building (Callaghan), The University of Newcastle Access Grid Venue: SeeVogh (non-AG) [ENQUIRIES] Time and Date: 10:00 am, Wed, 29th Oct 2014 Abstract: Mixed Littlewood conjecture proposed by de Mathan and Teulie in 2004 states that for every real number $x$ one has $\liminf q * |q|_D * ||qx|| = 0,$ where $|q|_D$ is a so called pseudo norm which generalises the standard p-adic norm. In the talk we'll consider the set mad of potential counterexamples to this conjecture. Thanks to the results of Einsiedler and Kleinbock we already know that the Haudorff dimension of mad is zero, so this set is very tiny. During the talk we'll see that the continued fraction expansion of every element in mad should satisfy some quite restrictive conditions. As one of them we'll see that for these expansions, considered as infinite words, the complexity function can neither grow too fast nor too slow. [Permanent link] CARMA RHD MEETING Location: V205, Mathematics Building (Callaghan), The University of Newcastle Dates: 1:00 pm - 3:00 pm, Thu, 30th Oct 2014 [Permanent link] TOOLS FOR MATHEMATICAL COMPUTATION Location: V206, Mathematics Building (Callaghan), The University of Newcastle Dates: 10:00 am - 12:00 pm, Tue, 4th Nov 2014 As part of Jon Borwein's Experimental Mathematics Course, he is exploring the uses of the currently available tools for mathematical computation. Each lecture should be independently accessible, so feel free to drop in. [Permanent link] CARMA RHD MEETING Location: V205, Mathematics Building (Callaghan), The University of Newcastle Dates: 1:00 pm - 3:00 pm, Thu, 6th Nov 2014 [Permanent link] CARMA RHD MEETING Location: V205, Mathematics Building (Callaghan), The University of Newcastle Dates: 1:00 pm - 3:00 pm, Thu, 13th Nov 2014 [Permanent link] CARMA RHD MEETING Location: V205, Mathematics Building (Callaghan), The University of Newcastle Dates: 1:00 pm - 3:00 pm, Thu, 20th Nov 2014 [Permanent link] AMSI EVENT EViMS 2: Effective use of Visualisation in the Mathematical Sciences Location: (Canberra), Australian National University Dates: Fri, 21st Nov 2014 - Sun, 23rd Nov 2014 The Mathematical Sciences Institute will host a three day workshop on more effective use of visualization in mathematics, physics, and statistics, from the perspectives of education, research and outreach. This is the second EViMS meeting, following the highly successful one held in Newcastle in November 2012. Our aim for the workshop is to help mathematical scientists understand the opportunities, risks and benefits of visualization, in research and education, in a world where visual content and new methods are becoming ubiquitous. Visit the conference website for more information. [Permanent link] MATHEMATICAL SCIENCES INSTITUTE CONFERENCE New Directions in Fractal Geometry Location: (Canberra), Australian National University Dates: Sun, 23rd Nov 2014 - Fri, 28th Nov 2014 This conference will cover some emerging hot topics in Fractal Geometry and will stimulate collaboration between theory and applications. Details are available on the conference website. [Permanent link] CARMA RHD MEETING Location: V205, Mathematics Building (Callaghan), The University of Newcastle Dates: 1:00 pm - 3:00 pm, Thu, 27th Nov 2014 [Permanent link] CARMA RHD MEETING Location: V205, Mathematics Building (Callaghan), The University of Newcastle Dates: 1:00 pm - 3:00 pm, Thu, 4th Dec 2014 [Permanent link] EXTERNAL EVENT 2014 ANZAMP annual meeting Location: The University of Melbourne Dates: Mon, 8th Dec 2014 - Fri, 12th Dec 2014 The annual meeting of the Australian and New Zealand Association of Mathematical Physics will be incorporated into the 8th Australia New Zealand Mathematics Convention at the University of Melbourne. For details, please visit the convention website. [Permanent link] CARMA RHD MEETING Location: V205, Mathematics Building (Callaghan), The University of Newcastle Dates: 1:00 pm - 3:00 pm, Thu, 11th Dec 2014 [Permanent link] CARMA RHD MEETING Location: V205, Mathematics Building (Callaghan), The University of Newcastle Dates: 1:00 pm - 3:00 pm, Thu, 18th Dec 2014 [Permanent link] AMSI EVENT AMSI Summer School in the Mathematical Sciences Location: Dates: Sat, 3rd Jan 2015 - Mon, 26th Jan 2015 Details will be available later. [Permanent link] AMSI EVENT 2015 AMSI Summer School Location: (Callaghan), The University of Newcastle Dates: Mon, 5th Jan 2015 - Thu, 29th Jan 2015 The AMSI Summer School is an exciting opportunity for mathematical sciences students from around Australia to come together over the summer break to develop their skills and networks. Further details are available on AMSI's website. [Permanent link] CARMA WORKSHOP HMEC: Hunter Maths Educators Conference Location: , Mathematics Building (Callaghan), The University of Newcastle Dates: 9:00 am - 5:00 pm, Fri, 6th Feb 2015 Details will be available on the conference webpage. [Permanent link] CARMA AND UNISA CONFERENCE SPCOM 2015: South Pacific Continuous Optimization Meeting Location: (Adelaide), University of South Australia Dates: Sun, 8th Feb 2015 - Thu, 12th Feb 2015 SPCOM 2015 will offer a rich scientific program consisting of Conference Talks, a Fitzpatrick Workshop, two half-day Tutorials, and a Poster Session. Further details and a list of speakers are available on the conference website. [Permanent link] EXTERNAL EVENT Spring School on Variational Analysis VI Location: Paseky nad Jizerou Dates: Sun, 19th Apr 2015 - Sat, 25th Apr 2015 Ffollowing a longstanding tradition, the Faculty of Mathematics and Physics of Charles University in Prague and the Academy of Sciences of the Czech Republic will organize the Spring School on Variational Analysis VI. The School will be held in Paseky nad Jizerou, in a chalet in the Krkonose Mountains, April 19 - 25, 2015. The program will consist of series of lectures on Variational Analysis and its Applications. The lectures will be given by Jonathan M. Borwein (The University of Newcastle, Australia) Theory and Applications of Convex and Non-convex Feasibility Problems Marian Fabian (Academy of Sciences of the Czech Republic) Separable Reductions and Rich Families in Theory of Frechet Subdifferentials Alexander Ioffe (Technion, Haifa, Israel) Variational Analysis and Optimization Theory David Russel Luke (Georg-August-Universitaet Goettingen, Germany) Variational Methods in Numerical Analysis Further informations can be found on the website http://www.karlin.mff.cuni.cz/katedry/kma/ss/apr15. [Permanent link] CARMA CONFERENCE ANZAMP 2015 Location: Dates: Wed, 9th Dec 2015 - Fri, 11th Dec 2015 More details will be added closer to the event. [Permanent link] CARMA WORKSHOP ANZAMP Satellite Meeting Location: Dates: 9:00 am - 5:00 pm, Sat, 12th Dec 2015 More details will be added closer to the event. [Permanent link]

This space intentionally left blank.


text:
# Complete measure space

1. Jan 22, 2008

### johnson123

[SOLVED] complete measure space

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

Assume that ($$\Omega$$,$$\Sigma$$,$$\mu$$) is a complete
measure space, let $$\mu_{e}$$ be the outer measure defined by $$\mu$$
. Prove that if $$\mu_{e}$$(S)=0 $$\Rightarrow$$ S$$\in$$$$\Sigma$$ .

I know that $$\mu_{e}$$ = $$\mu$$ when restricted on $$\Sigma$$

and that if $$\mu$$(A)=0, then every subset of A has measure zero from completeness.

I intuitively can see this to be true, since if S where not in $$\Sigma$$ and
$$\mu_{e}$$ = 0 , then this would contradict the fact that ($$\Omega$$,$$\Sigma$$,$$\mu$$) is complete, since the fact that
$$\mu_{e}(A_{1})$$=$$\mu_{e}(A_{2})$$...=$$\mu_{e}(A_{n})$$=0, for $$A_{1}\subseteq A_{2}\subseteq.......\subseteq A_{n}\subseteq S$$
would imply that $$\Sigma$$ is incomplete, but not sure how to rigorously prove
it.

Last edited: Jan 22, 2008


text:
# Biblatex and authoryear broken but why? [closed]

After a reinstallation of a newer version of MiKTeX the citations created by biblatex are broken. I have no idea what causes this. A similar issue could be fixed by deleting all auxillary files but that does not help here.

Here is my non-working MWE:

\documentclass{article}

\usepackage{filecontents}
\begin{filecontents*}{test.bib}
@BOOK{test,
author= {A. Author},
title = {A Title for a Book},
date = {2012-10-08}
},
@ONLINE{test2,
author = {A. Author},
title = {Some title},
url = {http://example.com},
urldate = {2013-01-08}
}
\end{filecontents*}

\usepackage[%
backend=biber,
citestyle=authoryear,
bibstyle=authoryear,
]{biblatex}

\begin{document}
Here is a first citation: \cite{test}.
\nocite{*}

\printbibliography
\end{document}

With the output:

Here is a first citation: A Title for a Book.

References

Author, A. Some title. url: http://example.com (visited on 01/08/2013).

A Title for a Book.

How can I get the normal citations back?

## closed as too localized by Alexander, barbara beeton, egreg, Claudio Fiandrino, QrrbrbirlbelJun 25 '13 at 16:42

This question is unlikely to help any future visitors; it is only relevant to a small geographic area, a specific moment in time, or an extraordinarily narrow situation that is not generally applicable to the worldwide audience of the internet. For help making this question more broadly applicable, visit the help center. If this question can be reworded to fit the rules in the help center, please edit the question.

• What do the logs say (*.blg, *.log)? – Audrey Jun 25 '13 at 13:21
• @Audrey: Oh, I found something that might be a hint: the .blg contains: [115] Utils.pm:167> WARN - Warning: Found biblatex control file version 2.4, expected version 2.3 and the .log: Package biblatex Warning: File 'no_month.bbl' is wrong format version - expected 2.2. This is all on an uptodate MiKTeX 2.9 install. – Alexander Jun 25 '13 at 13:27
• @Audrey: That turned out to be a great hint. It seems that MiKTeX is going through some restructuring at the moment. It installed biblatex 2.6 together with biber 1.5. This causes the error. Installing a newer biber 1.6 by hand fixes the issue. Thank you! – Alexander Jun 25 '13 at 13:40
• No problem. biber 1.5 is indeed not compatible with biblatex 1.6. You can find a table of compatible versions in the biblatex manual. – Audrey Jun 25 '13 at 14:15
• I already have for some time biber 2.6 in a current miktex and I don't see any update in the manager which could revert this to biber 2.5. But if you have a 64bit miktex then it doesn't come with biber and you will have to update it manually everytime biblatex is updated. – Ulrike Fischer Jun 25 '13 at 14:24


text:
1. ## Integral with limit

Express the limit as a definite integral on the given interval.
lim n
 n xi ln(1 + xi2) Δx, [0, 4] i = 1

My work: I figured out that f(x) = xln(1+x^2)

therefore we need to find the integral from 0 to 4 of xln(1+x^2)dx

I'm a little confused. Do I start by making a graph? I haven't memorized the log graphs so sorry for my naivety. I appreciate any help.

2. ## Re: Integral with limit

Originally Posted by Steelers72
Express the limit as a definite integral on the given interval.
lim n
 n xi ln(1 + xi2) Δx, [0, 4] i = 1

My work: I figured out that f(x) = xln(1+x^2)

therefore we need to find the integral from 0 to 4 of xln(1+x^2)dx

Your work is correct. What more do you need to do?
It does not ask you to evaluate it. Does it?

3. ## Re: Integral with limit

Originally Posted by Steelers72
Express the limit as a definite integral on the given interval.
lim n
 n xi ln(1 + xi2) Δx, [0, 4] i = 1

My work: I figured out that f(x) = xln(1+x^2)

therefore we need to find the integral from 0 to 4 of xln(1+x^2)dx

I'm a little confused. Do I start by making a graph? I haven't memorized the log graphs so sorry for my naivety. I appreciate any help.
If you need to evaluate the integral \displaystyle \begin{align*} \int_0^4{x\ln{\left( 1 + x^2 \right)} \, dx } = \frac{1}{2} \int_0^4{ 2x\ln{ \left( 1 + x^2 \right) } \,dx } \end{align*}, let \displaystyle \begin{align*} u = 1 + x^2 \implies du = 2x\, dx \end{align*} and note that \displaystyle \begin{align*} u(0) = 1 \end{align*} and \displaystyle \begin{align*} u(4) = 17 \end{align*}, then

\displaystyle \begin{align*} \frac{1}{2}\int_0^4{2x\ln{\left( 1 + x^2 \right) }\, dx} &= \frac{1}{2} \int_1^{17} { \ln{(u)} \, du } \end{align*}

which can now be solved by integration by parts.

4. ## Re: Integral with limit

Thank you both; the work I had done was all we needed to do haha...sorry for that!


text:
# Problem: How does the concept of pressure follow from kinetic molecular theory?

###### FREE Expert Solution

Pressure is defined as force divided by area.

$\overline{){\mathbf{P}}{\mathbf{=}}\frac{\mathbf{F}}{\mathbf{A}}}$

98% (393 ratings)
###### Problem Details

How does the concept of pressure follow from kinetic molecular theory?


text:
IMO – GRADE 6 – FRACTIONS

1. The below figure was cut and separated into three pieces.

Which of the following groups could be the three pieces of the cube ?

2. Tarun gave $\frac{1}{4}$ of his salary to charity and spent $\frac{1}{8}$ of the remainder.What fraction of his salary was left ?

3. Observe the given figure. How many more triangles should be shaded such that $\frac{3}{4}$ of the figure is shaded ?

4. The value of   $\frac{\sqrt[6]{0.001}\sqrt[^{6}]{x^{1296}}}{\sqrt{10}}$

×

Cart


text:
Courses

# Test: Physics Mock- 8

## 45 Questions MCQ Test NEET Mock Test Series | Test: Physics Mock- 8

Description
This mock test of Test: Physics Mock- 8 for NEET helps you for every NEET entrance exam. This contains 45 Multiple Choice Questions for NEET Test: Physics Mock- 8 (mcq) to study with solutions a complete question bank. The solved questions answers in this Test: Physics Mock- 8 quiz give you a good mix of easy questions and tough questions. NEET students definitely take this Test: Physics Mock- 8 exercise for a better result in the exam. You can find other Test: Physics Mock- 8 extra questions, long questions & short questions for NEET on EduRev as well by searching above.
QUESTION: 1

Solution:
QUESTION: 2

Solution:
QUESTION: 3

### Two particles of masses 1kg and 3kg have position vectors and respectively. The centre of mass has a position vector

Solution:
QUESTION: 4
A string breaks if its tension exceeds 10 newtons. A stone of mass 250 gm tied to this string of length 10 cm is rotated in a horizontal circle. The maximum angular velocity of rotation can be
Solution:
QUESTION: 5
What is increased in a step-down transformer
Solution:
QUESTION: 6

An explosion blows a rock into three parts. Two parts go off at right angles to each other. These two are, 1 kg first part moving with a velocity of 12 ms−1 and 2 kg second part moving with a velocity of 8 ms−1 . If the third part flies off with a velocity of 4 ms −1 , its mass would be:

Solution:
QUESTION: 7

Energy needed to establish an alternating current I in a coil of self inductance L is

Solution:

E = 1/2 LI2

QUESTION: 8
Three identical cells each of e.m.f. 2 V and unknown internal resistances are connected in parallel. This combination is connected to a 5 Ω resistor. If the terminal voltage across the cell is 1.5 V, then internal resistance of each cell is
Solution:
QUESTION: 9
What is the de Broglie wavelength of the α-particle accelerated through a potential difference V
Solution:
QUESTION: 10
The resistance of a 5 cm long wire is 10Ω. It is uniformly stretched so that its length becomes 20cm. The resistance of the wire is
Solution:
QUESTION: 11
Five equal capacitors connected in series have a resultant capacitance of 4 $\mu$F. The total energy stored in these, when these are connected in parallel and charged to 400V is
Solution:
QUESTION: 12

If an unchanged capacitor is charged by connecting it to a battery, then the amount of energy lost as heat is

Solution:
QUESTION: 13

300 Joule of work is done in sliding a 2 kg. block on an inclined plane to a height of 10 meters. Taking value of acceleration due to gravity 'g' to be 10 m/s2, work done against friction is

Solution:

Work done in lifting 2 kg mass upto height 10 metres is given by
W = mgh
= 2 x 10 x 10 = 200 J
Work done against friction
= 300 - 200 = 100 joule

QUESTION: 14
What will be the acceleration due to gravity on the surface of the moon, if its radius is 1/4th the radius of the earth and its mass is 1/80th the mass of the earth?
Solution:
QUESTION: 15
Average density of the earth
Solution:
QUESTION: 16
On giving equal amount of heat to equal masses of a monoatomic and a diatomic gas the rise in temprature (∆T) is more for
Solution:
QUESTION: 17

A current carrying coil when freely suspended will allign itself

Solution:
QUESTION: 18

The temperature of a body falls from 50oC to 40oC in 10 minutes. If the temperature of the surroundings is 20oC Then temperature of the body after another 10 minutes will be

Solution:
QUESTION: 19
A positively charged particle moving due east enters a region of uniform magnetic field directed vertically upwards. The particle will
Solution:
QUESTION: 20
If iron pieces loaded in a ship, are thrown into the water, then level of water will
Solution:
QUESTION: 21
The force between two short bar magnets with magnetic moments M₁ and M₂ whose centres are r meter apart is 8 N, when their axes are in same line. If the separation is increased to 2 r, the force between them is reduced to
Solution:
QUESTION: 22
Shearing stress produces a change in
Solution:
QUESTION: 23

A cord is wound round the circumference of wheel of radius r. The axis of the wheel is horizontal and moment of inertia about it is I. A weight mg is attached to the end of the cord and falls from rest. After falling through a distance h, the angular velocity of the wheel will be

Solution:
QUESTION: 24
The ratio of the numerical values of the average velocity and average speed of a body is always
Solution:
QUESTION: 25

Which of the following equations is definitely wrong?

Solution:
QUESTION: 26

A block of metal weighing 2 kg is resting on a frictionless plane. It is struck by a jet releasing water at a rate of 1kg/sec and at a speed of 5 m/sec. The initial acceleration of the block will be

Solution:
QUESTION: 27
A river is flowing from east to west at a speed of 5 m/min. A man on South bank river, capable of swimming 10 m/min in still water, wants to swim across the river in shortest time he should swim
Solution:
QUESTION: 28
The magnifying power of a telescope can be increased by increasing
Solution:
QUESTION: 29

The potential energy of a particle executing simple harmonic motion from a distance x from its equilibrium position is

Solution:
QUESTION: 30

Two projectiles are fired at different angles with the same magnitude of velocity, such that they have the same range. At what angles they might have been projected?

Solution:

For equal range, the particle should either be projected at an angle θ or (90 - θ)

QUESTION: 31
An object is placed 40 cm infront of a convex mirror of radius of curvature 20 cm. The image
Solution:
QUESTION: 32
Angular momentum is
Solution:
QUESTION: 33

If î denotes a unit vector along incident light, r̂ a unit vector along refracted ray into medium of refractive index μ and n̂ a unit vector normal to the boundary of the media, directed towards incident medium, then the law of refraction can be written as

Solution:
QUESTION: 34
A boat, which has a speed of 5 km/hr in still water, crosses a river of width 1 km along the shortest possible path in 15 minutes. The velocity of the river water in km/hr is
Solution:
QUESTION: 35
The rate of steady volume flow of water through a capillary tube of length 'l' and radius 'r' under a pressure difference of P is V . This tube is connected with another tube of same but half the radius in series. Then the rate of steady volume flow through them is (pressure across the combination is P)
Solution:
QUESTION: 36

Consider the two following statements A and B and identify the correct choice even in the answers (A) Duddells thermo-galvanometer is suitable to measure direct current only (B) Thermopile can measure temperature differences of the order of 10⁻3 oC

Solution:
QUESTION: 37

An electric kettle takes 4 A current at 220 V. How much time will it take to boil 1 kg of water from room temperature 20oC ? The temperature of boiling water is 100oC

Solution:
QUESTION: 38

Heat given to a body which rises its temperature by 1oC is

Solution:
QUESTION: 39
When the amount of work done is 300 J and change in internal energy is 100 J, then the heat energy supplied is
Solution:
QUESTION: 40

Magnetic field is measured in

Solution:
QUESTION: 41

The force F on a sphere of radius 'a' moving in a medium with velocity 'V' is given by F = 6 π η aV. The dimensions of η are

Solution:
QUESTION: 42
When a source is linear, the wavefront is
Solution:
QUESTION: 43
Quality of a musical note depends on
Solution:
QUESTION: 44
In which case does the potential energy decrease
Solution:
QUESTION: 45

A block of mass 10 kg is moving in x-direction with a constant velocity of 10 m−s⁻1. It is subjected to a retarding force F = 0.1x J-m⁻1 during its travel from x = 20 m to x = 30 m. Its final kinetic energy will be

Solution:


text:
## Publications

Export 1507 results:
B
Berti M. Soluzioni periodiche di PDEs Hamiltoniane. Bollettino dell\\\'Unione Matematica Italiana Serie 8 7-B (2004), p. 647-661 [Internet]. 2004 . Available from: http://hdl.handle.net/1963/4582
. Sobolev quasi-periodic solutions of multidimensional wave equations with a multiplicative potential. Nonlinearity. 2012 ;25:2579-2613.
. Multiplicity of periodic solutions of nonlinear wave equations. Nonlinear Anal. 56 (2004) 1011-1046 [Internet]. 2004 . Available from: http://hdl.handle.net/1963/2974
. Cantor families of periodic solutions for wave equations via a variational principle. Advances in Mathematics. 2008 ;217:1671-1727.
. Sobolev periodic solutions of nonlinear wave equations in higher spatial dimensions. Archive for Rational Mechanics and Analysis. 2010 ;195:609-642.
. Quasi-periodic solutions with Sobolev regularity of NLS on Td with a multiplicative potential. Journal of the European Mathematical Society. 2013 ;15:229-286.
. Cantor families of periodic solutions of wave equations with C k nonlinearities. Nonlinear Differential Equations and Applications. 2008 ;15:247-276.
. Optimal stability and instability results for a class of nearly integrable Hamiltonian systems. Atti.Accad.Naz.Lincei Cl.Sci.Fis.Mat.Natur.Rend.Lincei (9) Mat.Appl.13(2002),no.2,77-84 [Internet]. 2002 . Available from: http://hdl.handle.net/1963/1596
. Fast Arnold diffusion in systems with three time scales. Discrete Contin. Dyn. Syst. 8 (2002) 795-811 [Internet]. 2002 . Available from: http://hdl.handle.net/1963/3058
. An abstract Nash-Moser theorem with parameters and applications to PDEs. Annales de l'Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis. 2010 ;27:377-399.
. Effective inverse spectral problem for rational Lax matrices and applications. Int. Math. Res. Not. IMRN. 2007 :Art. ID rnm103, 39.
. The partition function of the extended $r$-reduced Kadomtsev-Petviashvili hierarchy. J. Phys. A [Internet]. 2015 ;48:195205, 20. Available from: http://dx.doi.org/10.1088/1751-8113/48/19/195205
. Mixed correlation functions of the two-matrix model. J. Phys. A. 2003 ;36:7733–7750.
Bertola M. The Malgrange form and Fredholm determinants. SIGMA Symmetry Integrability Geom. Methods Appl. [Internet]. 2017 ;13:Paper No. 046, 12. Available from: http://dx.doi.org/10.3842/SIGMA.2017.046
. Universality for the focusing nonlinear Schrödinger equation at the gradient catastrophe point: rational breathers and poles of the \it Tritronquée solution to Painlevé I. Comm. Pure Appl. Math. [Internet]. 2013 ;66:678–752. Available from: http://dx.doi.org/10.1002/cpa.21445
. A general construction of conformal field theories from scalar anti-de Sitter quantum field theories. Nuclear Phys. B. 2000 ;587:619–644.
Bertola M. The dependence on the monodromy data of the isomonodromic tau function. Comm. Math. Phys. [Internet]. 2010 ;294:539–579. Available from: http://0-dx.doi.org.mercury.concordia.ca/10.1007/s00220-009-0961-7
. Universality Conjecture and Results for a Model of Several Coupled Positive-Definite Matrices. Commun. Math. Phys. [Internet]. 2015 ;337:1077–1141. Available from: http://link.springer.com/article/10.1007/s00220-015-2327-7
. The partition function of the two-matrix model as an isomonodromic τ function. J. Math. Phys. [Internet]. 2009 ;50:013529, 17. Available from: http://0-dx.doi.org.mercury.concordia.ca/10.1063/1.3054865
Bertola M. Two-matrix model with semiclassical potentials and extended Whitham hierarchy. J. Phys. A. 2006 ;39:8823–8855.
. Noncommutative Painlevé Equations and Systems of Calogero Type. Comm. Math. Phys. 2018 .
. Zeros of Large Degree Vorob'ev-Yablonski Polynomials via a Hankel Determinant Identity. International Mathematics Research Notices. 2014 ;rnu239.
. Spectra of random Hermitian matrices with a small-rank external source: the critical and near-critical regimes. J. Stat. Phys. [Internet]. 2012 ;146:475–518. Available from: http://0-dx.doi.org.mercury.concordia.ca/10.1007/s10955-011-0409-2
. Correspondence between Minkowski and de Sitter quantum field theory. Phys. Lett. B. 1999 ;462:249–253.


text:
RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Forthcoming papers Archive Impact factor Subscription Guidelines for authors License agreement Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Mat. Sb.: Year: Volume: Issue: Page: Find

 Mat. Sb. (N.S.), 1971, Volume 85(127), Number 4(8), Pages 586–609 (Mi msb3280)

On positive solutions of elliptic equations

T. G. Pletneva, S. D. Èidel'man, V. A. Kondrat'ev

Abstract: In this paper the authors study weak solutions of elliptic equations of the form
$$Pu\equiv\sum_{|k|\leqslant m}(-1)^kD_x^k(a_k(x)u(x))=f(x)$$
in a bounded domain $\Omega$. It is assumed known about these solutions either that they are positive, or that estimates in certain norms hold for their negative parts. It is assumed moreover that an estimate on the $L_1$-norm of the solution holds on some subdomain $\Omega'\subset\Omega$. Summability of such solutions with a weight function that vanishes at the boundary is established, and with the use of the results of Ya. A. Roitberg integral representations are given in terms of the Green's function for the Dirichlet problem.
Bibliography: 8 titles.

Full text: PDF file (2185 kB)
References: PDF file   HTML file

English version:
Mathematics of the USSR-Sbornik, 1971, 14:4, 587–613

Bibliographic databases:

UDC: 517.946
MSC: Primary 35J30; Secondary 35C15, 35D10

Citation: T. G. Pletneva, S. D. Èidel'man, V. A. Kondrat'ev, “On positive solutions of elliptic equations”, Mat. Sb. (N.S.), 85(127):4(8) (1971), 586–609; Math. USSR-Sb., 14:4 (1971), 587–613

Citation in format AMSBIB
\Bibitem{PleEidKon71} \by T.~G.~Pletneva, S.~D.~\Eidel'man, V.~A.~Kondrat'ev \paper On positive solutions of elliptic equations \jour Mat. Sb. (N.S.) \yr 1971 \vol 85(127) \issue 4(8) \pages 586--609 \mathnet{http://mi.mathnet.ru/msb3280} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=283372} \zmath{https://zbmath.org/?q=an:0222.35019} \transl \jour Math. USSR-Sb. \yr 1971 \vol 14 \issue 4 \pages 587--613 \crossref{https://doi.org/10.1070/SM1971v014n04ABEH002823} `

• http://mi.mathnet.ru/eng/msb3280
• http://mi.mathnet.ru/eng/msb/v127/i4/p586

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. V. A. Kondrat'ev, T. G. Pletneva, S. D. Èidel'man, “Positive solutions of linear quasielliptic evolution equations”, Math. USSR-Sb., 18:1 (1972), 15–44
2. V. A. Kondrat'ev, S. D. Èidel'man, “On the summability of positive solutions of partial differential equations of arbitrary order in a neighborhood of a characteristic manifold”, Math. USSR-Sb., 28:4 (1976), 521–531
3. Khung N., “The Absence of Positive Solutions of Second-Order Nonlinear Elliptic Equations in Conical Domains”, Differ. Equ., 34:4 (1998), 532–539
•  Number of views: This page: 286 Full text: 104 References: 55


text:
TheInfoList

In
set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, i ...
, the union (denoted by ∪) of a collection of sets is the set of all
element Element may refer to: Science * Chemical element Image:Simple Periodic Table Chart-blocks.svg, 400px, Periodic table, The periodic table of the chemical elements In chemistry, an element is a pure substance consisting only of atoms that all ...
s in the collection. It is one of the fundamental operations through which sets can be combined and related to each other. A refers to a union of sets and it is by definition equal to the
empty set #REDIRECT Empty set #REDIRECT Empty set#REDIRECT Empty set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry ...

. For explanation of the symbols used in this article, refer to the
table of mathematical symbols A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object, an action on mathematical objects, a relation between mathematical objects, or for structuring the other symbols that occur in a formula. ...
.

Union of two sets

The union of two sets ''A'' and ''B'' is the set of elements which are in ''A'', in ''B'', or in both ''A'' and ''B''. In symbols, :$A \cup B = \$. For example, if ''A'' = and ''B'' = then ''A'' ∪ ''B'' = . A more elaborate example (involving two infinite sets) is: : ''A'' = : ''B'' = : $A \cup B = \$ As another example, the number 9 is ''not'' contained in the union of the set of
prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
s and the set of
even number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
s , because 9 is neither prime nor even. Sets cannot have duplicate elements, so the union of the sets and is . Multiple occurrences of identical elements have no effect on the
cardinality In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
of a set or its contents.

Algebraic properties

Binary union is an
associative In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
operation; that is, for any sets $A, B, \text C,$ $A \cup (B \cup C) = (A \cup B) \cup C.$ Thus the parentheses may be omitted without ambiguity: either of the above can be written as $A \cup B \cup C.$ Also, union is
commutative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
, so the sets can be written in any order. The
empty set #REDIRECT Empty set #REDIRECT Empty set#REDIRECT Empty set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry ...

is an
identity element In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
for the operation of union. That is, $A \cup \varnothing = A,$ for any set $A.$ Also, the union operation is idempotent: $A \cup A = A.$ All these properties follow from analogous facts about
logical disjunction In logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents st ...
. Intersection distributes over union $A \cap (B \cup C) = (A \cap B)\cup(A \cap C)$ and union distributes over intersection $A \cup (B \cap C) = (A \cup B) \cap (A \cup C).$ The
power set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
of a set $U,$ together with the operations given by union,
intersection In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...
, and Complement (set theory), complementation, is a Boolean algebra (structure), Boolean algebra. In this Boolean algebra, union can be expressed in terms of intersection and complementation by the formula $A \cup B = \left(A^\text \cap B^\text \right)^\text,$ where the superscript $^\text$ denotes the complement in the Universe (mathematics), universal set $U.$

Finite unions

One can take the union of several sets simultaneously. For example, the union of three sets ''A'', ''B'', and ''C'' contains all elements of ''A'', all elements of ''B'', and all elements of ''C'', and nothing else. Thus, ''x'' is an element of ''A'' ∪ ''B'' ∪ ''C'' if and only if ''x'' is in at least one of ''A'', ''B'', and ''C''. A finite union is the union of a finite number of sets; the phrase does not imply that the union set is a finite set.

Arbitrary unions

The most general notion is the union of an arbitrary collection of sets, sometimes called an ''infinitary union''. If M is a set or Class (set theory), class whose elements are sets, then ''x'' is an element of the union of M if and only if there is existential quantification, at least one element ''A'' of M such that ''x'' is an element of ''A''. In symbols: : $x \in \bigcup \mathbf \iff \exists A \in \mathbf,\ x \in A.$ This idea subsumes the preceding sections—for example, ''A'' ∪ ''B'' ∪ ''C'' is the union of the collection . Also, if M is the empty collection, then the union of M is the empty set.

Notations

The notation for the general concept can vary considerably. For a finite union of sets $S_1, S_2, S_3, \dots , S_n$ one often writes $S_1 \cup S_2 \cup S_3 \cup \dots \cup S_n$ or $\bigcup_^n S_i$. Various common notations for arbitrary unions include $\bigcup \mathbf$, $\bigcup_ A$, and $\bigcup_ A_$. The last of these notations refers to the union of the collection $\left\$, where ''I'' is an index set and $A_i$ is a set for every $i \in I$. In the case that the index set ''I'' is the set of natural numbers, one uses the notation $\bigcup_^ A_$, which is analogous to that of the infinite sums in series. When the symbol "∪" is placed before other symbols (instead of between them), it is usually rendered as a larger size.

Notation encoding

In Unicode, union is represented by the character . In TeX, $\cup$ is rendered from \cup.


text:
# Another question on probabilities with algebra

I'm here again, and now with more doubts on algebra on probability functions;

$Let$ $\Omega{{a1, a2, a3}},$find $P(a1)$whether: $P(a1) = 2*P(a2)$ and $P(a3) = \dfrac{1}{4}$

I've got now after notice that $P(a1) + P(a2) + P(a3) = 1$ that $P(a1) + 2*P(a2) = \dfrac{3}{4}$ but I've got stuck on this;

what I do now ? Thanks in advance.

-
What you do now is learn how to accept answers, and then do it! – Gerry Myerson Apr 12 '12 at 0:26

You did not substitute the information $P(a_1)=2P(a_2)$ correctly.
You should start by writing $$\color{maroon}{P(a_1)}+P(a_2)+\color{darkgreen}{P(a_3)}=1\iff \color{maroon}{2P(a_2)} +P(a_2)+\color{darkgreen}{{1\over 4}}=1.$$ Of course, finish by solving the equation on the right hand side of the above for $P(a_2)$. Then you can find $P(a_1)$.
Alternatively you could start with $${P(a_1)}+\color{maroon}{P(a_2)}+\color{darkgreen}{P(a_3)}=1\iff {P(a_1)} +\color{maroon}{{1\over 2}\cdot P(a_1)}+\color{darkgreen}{{1\over 4}}=1.$$


text:
# Evaluating a “binomial-like” sum

I suspect there is a way to do the following sum by hand, but I'm having some trouble: $$\sum_{x=0}^{n} x^{2} {n \choose x} p^{x}(1-p)^{n-x}$$

There are a couple questions like this, but for general $x^{k}$, I wanted to know for k = 2.

• $=E(X^2)=(E(X))^2+\sigma^2_X=(np)^2+np(1-p)$ for $X\sim B(n,p)$. For general $k$, you could probably use $k$th derivative of moment generating function for a binomial. – A.S. Mar 19 '16 at 18:22
• @A.S. ya this is how i ended up doing it when i answered orginially – yoshi Mar 19 '16 at 19:05
• I'm assuming $k$ and $n$ are the same thing? And $x$ does not generally represent an integer variable, but that's more of a convention. – Teepeemm Mar 20 '16 at 0:21

\begin{align*} \sum_kk^2\binom{n}kp^k(1-p)^{n-k}&=n\sum_kk\binom{n-1}{k-1}p^k(1-p)^{n-k}\\ &=n\sum_k(k+1)\binom{n-1}kp^{k+1}(1-p)^{n-k-1}\\ &=n\sum_kk\binom{n-1}kp^{k+1}(1-p)^{n-k-1}+n\sum_k\binom{n-1}kp^{k+1}(1-p)^{n-k-1}\\ &=n(n-1)\sum_k\binom{n-2}{k-1}p^{k+1}(1-p)^{n-k-1}+np\sum_k\binom{n-1}kp^k(1-p)^{(n-1)-k}\\ &=n(n-1)\sum_k\binom{n-2}kp^{k+2}(1-p)^{n-2-k}+np\\ &=n(n-1)p^2+np\\ \end{align*}
Take $q$ to be a constant in the following, and use the identity $$\sum_{x=0}^n \left(n\atop x\right)p^x q ^{n - x} = (p +q )^n.$$ Apply the operator $D = p \,{d\over dp}$ to both sides $k$ times (twice, here), and then set $q = 1-p$, and you'll get the same answer (for $k=2$) as is on this page... i.e., $p^2n (n-1) +pn = (pn)^2 + np(1-p).$


text:
# Find a set of vectors {u, v} in $R^4$ that spans the solution set of the equations

Find a set of vectors {u, v} in $\mathbb R^4$ that spans the solution set of the equations

$x – y + 2z +3w = 0$

$4x + 2y – z + 3w = 0$

$u =\begin{bmatrix}\\\\\\\end{bmatrix}, v =\begin{bmatrix}\\\\\\\end{bmatrix}$

How exactly can I do that, what do they mean by “that spans

#### Solutions Collecting From Web of "Find a set of vectors {u, v} in $R^4$ that spans the solution set of the equations"

First off, to solve this you do exactly what you’ve been doing in all of your other problems. Since you weren’t sure of your answer, I went ahead and worked it out for you.

The augmented matrix equation that you’re trying to solve is this one:

$$\left[\begin{array}{cccc|c} 1 & -1 & 2 & 3 & 0 \\ 4 & 2 & -1 & 3 & 0\end{array}\right]$$

Now I’ll do Gaussian elimination to solve it. Note that I’m going to leave off the final column of $0$’s because no elementary row operation will actually change them. So just imagine that column still being there:

\begin{align}\begin{bmatrix} 1 & -1 & 2 & 3 \\ 4 & 2 & -1 & 3\end{bmatrix} &\sim_{R_2 \to R_2-4R_1} \begin{bmatrix} 1 & -1 & 2 & 3 \\ 0 & 6 & -9 & -9\end{bmatrix} \\ &\sim_{R_2\to \frac 16R_2} \begin{bmatrix} 1 & -1 & 2 & 3 \\ 0 & 1 & -\frac 32 & -\frac 32\end{bmatrix} \\ &\sim_{R_1\to R_1+R_2} \begin{bmatrix} 1 & 0 & \frac 12 & \frac 32 \\ 0 & 1 & -\frac 32 & -\frac 32\end{bmatrix}\end{align}

This is the RREF of your matrix. Now we see that columns $3$ and $4$ don’t have pivots so we set $z=s$, $s\in \Bbb R$ and $w=t$, $t\in \Bbb R$. Then we write down the equations that the above matrix represents and substitute these new variables:

$$\begin{cases} x +\frac 12z+\frac 32w=0 \\ y-\frac 32z-\frac 32w=0\end{cases} \iff \begin{cases} x= -\frac 12s-\frac 32t \\ y=\frac 32s+\frac 32 t\end{cases}$$

So each element of the solution set is of the form $$\begin{bmatrix} x \\ y \\ z \\ w\end{bmatrix} = \begin{bmatrix} -\frac 12s-\frac 32t \\ \frac 32s+\frac 32 t \\ s \\ t\end{bmatrix} = s\begin{bmatrix} -\frac 12 \\ \frac 32 \\ 1 \\ 0\end{bmatrix} + t\begin{bmatrix} -\frac 32 \\ \frac 32 \\ 0 \\ 1\end{bmatrix},\quad s,t\in\Bbb R$$

Thus the set $\{(-\frac 12, \frac 32, 1, 0),(-\frac 32,\frac 32, 0, 1)\}$ spans the space. But fractions are a little annoying. So what we can do is multiply each vector by $2$. Really what we’re doing here is defining new variables $s’=\frac 12s$ and $t’=\frac 12t$. Then plugging these into the above and moving the constant inside the vectors we get $$\bbox[5px,border:2px solid red]{\{(-1,3,2,0),(-3,3,0,2)\}}$$

Note that these vectors are not unique. There are an infinite number of other pairs of vectors which will span this space. For instance, Vineet’s solution is another perfectly valid pair.

Now that we see this exercise just asked to you solve it the same way as all the others you’ve done so far, I’ll tackle your question about what the word span means. Let $S=\{\mathbf u_1, \dots, \mathbf u_k\}\subset \Bbb R^n$.

Definition: A linear combination of $S$ is a vector $\mathbf w$ such that $$\mathbf w = w_1\mathbf u_1 + \cdots + w_k\mathbf u_k$$ for some $w_1, \dots, w_k \in \Bbb R$.

Example: The vector $(-7,-2,3)$ is a linear combination of $\{(1,2,3),(3,2,1)\}$ because $$(-7,-2,3) = 2(1,2,3)-3(3,2,1)$$

However $(1,0,0)$ is not a linear combination of $\{(1,2,3),(3,2,1)\}$ because there are no numbers $a,b$ such that $$(1,0,0) = a(1,2,3)+b(3,2,1)$$

Definition: The span of $S$ is the set of all linear combinations of $S$. I.e. $$\operatorname{span}(S) = \{\mathbf w \in \Bbb R^n \mid\mathbf w = w_1\mathbf u_1 + \cdots + w_k\mathbf u_k\}$$

Examples:
\begin{align}(1)\quad &\operatorname{span}\{(1,0,0),(0,1,0)\} = \operatorname{span}\{(10,9,0),(e,-\pi,0)\} = \text{the xy-plane} \\ (2)\quad &\operatorname{span}\{(1,2,3)\} = \text{the line containing O parallel to (1,2,3)}\end{align}

Lemma: The span of any set of vectors in $\Bbb R^n$ is a subspace of $\Bbb R^n$. (Note: I’m not going to prove this lemma here, it’s just important to know.)

So this question is asking you to find a set of two vectors whose span is the solution set to your system of linear equations.

Let $x$ and $y$ be the free variables. To get the $u$, put $x=0,y=1$ and for $v$, put $x=1,y=0$ and solve two equations in $z$ and $w$. Finally you will get

$u=[0,3,3,-1]^T$ & $v=[1,0,1,-1]^T$

Please note the magnitude of these vectors are irrelevant. So you can multiply it with any scaler you want. In case of $u$, I multiplied vectore with $3$.

Hope this will be helpful !

what means: the set of solutions $(x, y, z, w)$ in $\Bbb{R}^4$, of an
equation of type $ax + by + cz + dw = 0$ , $a,b,c,d\in \Bbb{R}$, is the sub space kernel in $\Bbb{R}^4$
of the linear form $(x, y, z, w)\rightarrow a x + by + cz + dw$ ,
so this kernel is by rank theorem (case not a, b, c, d, all zero)
a hyperplane of $\Bbb{R}^4$, so a sub space of dimension $3$.

The solution set $(x, y, z, w)$ of $n$ equations of the same type
as above, is the subspace obtained by the intersection of the
solutions set of each equation.

in your case $n=2$,
the question suppose that this intersection of solution can it be
generated by two vectors $u$ and $v$, that is all solutions are in
the form $a u+b v$ where $a,b\in\Bbb{R}$


text:
# Message Log class

This topic is 4852 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.

## Recommended Posts

Okay, here's what I've got so far:
#ifndef _CLOG_H_
#define _CLOG_H_

#include <string>
#include <iostream>
#include <fstream>
using namespace std;
#include <time.h>

#define DEFAULT_STREAM_FILE "default_stream.txt"
#define TIME_BUFFER_SIZE 100

/**
* This is a singleton class that will allow objects to output debug and other messages
*/
class CLog
{
private:
static ofstream streams[10];	///< Output file stream

public:

/// Enumeration representing the different message (to different streams)
enum StreamType{
STREAM_DEBUG=0,	///< debug stream
STREAM_2,
STREAM_3,
STREAM_4,
STREAM_5,
STREAM_6,
STREAM_7,
STREAM_8,
STREAM_9,
STREAM_10,

NUM_OF_STREAMS	///< Implicit number of streams
};

/// Enumeration representing the different write modes
enum WriteMode{

WRITE=0,		///< truncates file and starts writing
APPEND,			///< append to existing file

NUM_OF_MODES	///< Implicit number of write modes
};

/**
* Function that opens the specified file for the specified stream in the specified
* write mode.
* \param file The file name of the file into which will be written the messages logged for this stream.
* \param type The stream you want to associate with this file.
* \param mode The write mode you wish to open this file in.
*/
static void openStream(string file = DEFAULT_STREAM_FILE, StreamType type = STREAM_DEBUG, WriteMode mode = WRITE )
{
// Check if it is a valid write mode
if( mode < WRITE || mode >= NUM_OF_MODES ){
cout << "CLog::openStream: Error: Invalid WriteMode: not opening file." << endl;
return;
}

if( type < STREAM_DEBUG || type >= NUM_OF_STREAMS ){
cout << "CLog::openStream: Error: Invalid StreamType: not opening file." << endl;
return;
}

// Open file in mode specified
if( mode == APPEND )
// Open file stream in append mode
streams[(int)type].open(file.c_str(), ios::app);
else if( mode == WRITE )
// Open file stream in write mode
streams[(int)type].open(file.c_str() );

}

/**
* Flush and close the selected file stream.
* \param type The stream to close.
*/
static void closeStream(StreamType type)
{
streams[(int)type].flush();
streams[(int)type].close();
}

/**
* Function that will write a message to the open specified stream.
* \param type A StreamType variable indicating which stream you want to write the message to.
* \param s A string object containing the message you want to log.
*/
static void write(StreamType type, string s)
{
// If file is open then write the string to file
if( streams[(int)type].is_open() ){
writeTime(streams[(int)type]);
streams[(int)type] << s;
}
else
// Else file is not open -> object has not been created
cout << "CLog::write: Warning: attempted writing to stream " << (int)type <<
" when stream is not open." << endl;
}

/**
* Function that will write the time and date in the form of
* day/month/year hour:minute:second into the stream passed.
* \param s A reference to the stream to write the date and time into.
*/
static void writeTime(ofstream& s)
{
// get time and print it to file before message
time_t t = time(NULL);
struct tm* now = localtime( &t );
char buf[TIME_BUFFER_SIZE];
strftime(buf, TIME_BUFFER_SIZE, "%d/%m/%y %I:%M:%S\t ", now);

s << buf;
}

};

/** Out of class scope initialization for the streams */
ofstream CLog::streams[10];

#endif

I just re-wrote it from some wierd singleton mash that seemed to make sense about half-way through last year, fairly close to the start of my C++ experience. I haven't tested it yet, because even for something this small, it would take some time, and well, I feel a little more like sleeping. But I wanted to ask some advice about this class. Nothing specific, anything you felt like giving that might be useful, but, since a question is usually asked, which do you think is better for a message logging class? An actual object? A singleton? Or just a bunch of related functions?

##### Share on other sites
I personally think a singleton is fine, but others may disagree.

Generally, you should check these things before using a singleton:

* Will every portion of the program use the class exactly the same way?

* Will every program using the class use it exactly the same way?

* Will every program never need more than one instance of the class?

* Can clients use the class without knowledge of what application they're in?

If you answer yes to all those, you pretty much OK to use a singleton.

EDIT: Whether you go with a singleton or not, one thing is sure: don't just make a bunch of global functions with 'Log_' prefixed to them. If you're going to do that, the least you can do is put them into a 'Log' namespace or something.

[Edited by - nilkn on July 4, 2005 10:55:12 AM]

##### Share on other sites
Personally, I would [and do] use an interface. That is have a pure virtual class which has virtual members for all the stuff you require a logger class to have. [for me it's merely write(const char *)] Then you can make various loggers to see what works best [or change what works best depending on the scenario] and just have your classes write to the base interface. Let inheritance do the rest.

##### Share on other sites
Instead of having many anonymous streams, I have found it very useful and easier just to give a couple pre-built names streams. This way you can control what the output is according to the type of message, etc.

It's not the most flexible, but I don't need it to be any more. I could change how it logs each type of message if I want (right now it logs all of them the same way).

Here it is if you want to take a look.

.h
/* --------------------------------------------------------------	File:			AXLogger.h	Description:	This file defines a class that logs messages.  Depending on the type of message,					it logs it to the appropriate output.					For now, it logs everything to both the console and the file.  I think it's important					to separate the types of messages, however, so later I can come back and change the type of					outputs for each type of message.					You specify the type of message by passing in MESSAGE_USER, MESSAGE_APP, or MESSAGE_SYSTEM					when you write the message.  A line with the type of message, calling function, and message					will appear in the appropriate output.  You are allowed printf style formatting.	Date:			June 29, 2005	Author:			James Long-------------------------------------------------------------- */#if !defined(AXLOGGER_INCLUDE)#define AXLOGGER_INCLUDE// The different log typesenum LogType {	LOG_TYPE_USER = 1,	LOG_TYPE_APPLICATION = 2,	LOG_TYPE_SYSTEM = 4,	LOG_TYPE_DEBUG = 8};// Define some macros to make logging much easier#define MESSAGE_USER LOG_TYPE_USER, __FUNCTION__#define MESSAGE_APP LOG_TYPE_APPLICATION, __FUNCTION__#define MESSAGE_SYS LOG_TYPE_SYSTEM, __FUNCTION__#define MESSAGE_DBG LOG_TYPE_DEBUG, __FUNCTION__class AXLogger : AXSingleton<AXLogger> {public:	AXLogger();	virtual ~AXLogger();	AXResult Initialize();	void Shutdown();	void Write(LogType target, const char* func, const char* msg, ...);private:	char _Temp_Buf[5012];	ofstream _FileLog;};#endif

.cpp
/* --------------------------------------------------------------	File:			AXLogger.cpp	Description:	Implementation for the AXLogger class.  See AXLogger.h for details.	Date:			June 29, 2005	Author:			James Long-------------------------------------------------------------- */#include "..\\AXCore.h"AXLogger::AXLogger() : AXSingleton<AXLogger>() {}AXLogger::~AXLogger() {}AXResult AXLogger::Initialize() {	// Open the log file	_FileLog.open("log.txt");	return AXSUCCESS;}void AXLogger::Shutdown() {	// Close the log file	_FileLog.close();}void AXLogger::Write(LogType target, const char *FunctionName, const char *Msg, ...) {	va_list args;	va_start(args, Msg);	vsprintf(_Temp_Buf, Msg, args);	va_end(args);	// Detect the log type and log accordingly (for now they basically do the same thing)	if(target & LOG_TYPE_USER) {		printf("\nAX User> *** %s *** \n\n", _Temp_Buf);		_FileLog << "\nAX User> *** " << _Temp_Buf << " *** \n\n";	}	if(target & LOG_TYPE_APPLICATION) {		printf("AX Application (%s)> %s\n", FunctionName, _Temp_Buf);		_FileLog << "AX Application (" << FunctionName << ")> " << _Temp_Buf << "\n";	}	if(target & LOG_TYPE_SYSTEM) {		printf("AX System (%s)> %s\n", FunctionName, _Temp_Buf);		_FileLog << "AX System (" << FunctionName << ")> " << _Temp_Buf << "\n";	}	if(target & LOG_TYPE_DEBUG) {#ifdef _DEBUG		printf("*** AX Debug (%s)> %s\n", FunctionName, _Temp_Buf);#endif	}}

1. 1
2. 2
3. 3
Rutin
16
4. 4
5. 5

• 14
• 9
• 9
• 9
• 10
• ### Forum Statistics

• Total Topics
632912
• Total Posts
3009186
• ### Who's Online (See full list)

There are no registered users currently online

×


text:
# The complex of partial bases for Fn and finite generation of the Torelli subgroup of Aut (Fn)

```@article{Day2010TheCO,
title={The complex of partial bases for Fn and finite generation of the Torelli subgroup of Aut (Fn)},
author={Matthew B. Day and Andrew Putman},
journal={Geometriae Dedicata},
year={2010},
volume={164},
pages={139-153}
}```
• Published 9 December 2010
• Mathematics
• Geometriae Dedicata
We study the complex of partial bases of a free group, which is an analogue for Aut(Fn) of the curve complex for the mapping class group. We prove that it is connected and simply connected, and we also prove that its quotient by the Torelli subgroup of Aut(Fn) is highly connected. Using these results, we give a new, topological proof of a theorem of Magnus that asserts that the Torelli subgroup of Aut(Fn) is finitely generated.
On the second homology group of the Torelli subgroup of Aut(Fn)
• Mathematics
• 2017
Let IA_n be the Torelli subgroup of Aut(F_n). We give an explicit finite set of generators for H_2(IA_n) as a GL_n(Z)-module. Corollaries include a version of surjective representation stability for
Cutting and Pasting in the Torelli subgroup of Out(\$F_n\$)
Using ideas from 3-manifolds, Hatcher–Wahl defined a notion of automorphism groups of free groups with boundary. We study their Torelli subgroups, adapting ideas introduced by Putman for surface
On finite generation of the Johnson filtrations
• Mathematics
Journal of the European Mathematical Society
• 2021
We prove that every term of the lower central series and Johnson filtrations of the Torelli subgroups of the mapping class group and the automorphism group of a free group is finitely generated in a
The complex of partial bases of a free group
We prove that the simplicial complex whose simplices are the nonempty partial bases of \$\mathbb{F}_n\$ is homotopy equivalent to a wedge of \$(n-1)\$-spheres. Moreover, we show that it is Cohen-Macaulay.
A small normal generating set for the handlebody subgroup of the Torelli group
We prove that the handlebody subgroup of the Torelli group of an orientable surface is generated by genus one BP-maps . As an application, we give a normal generating set for the handlebody subgroup
Generators for the hyperelliptic Torelli group and the kernel of the Burau representation at \$\$t=-1\$\$t=-1
• Mathematics
• 2015
We prove that the hyperelliptic Torelli group is generated by Dehn twists about separating curves that are preserved by the hyperelliptic involution. This verifies a conjecture of Hain. The
N ov 2 01 7 On finite generation of the Johnson filtrations
• Mathematics
• 2017
We prove that every term of the lower central series and Johnson filtrations of the Torelli subgroups of the mapping class group and the automorphism group of a free group is finitely generated in a
The complex of partial bases of a free group
We prove that the simplicial complex whose simplices are the nonempty partial bases of Fn is homotopy equivalent to a wedge of (n−1) ‐spheres. Moreover, we show that it is Cohen–Macaulay.
Generating the Johnson filtration
• Mathematics
• 2013
For k >= 1, let Torelli_g^1(k) be the k-th term in the Johnson filtration of the mapping class group of a genus g surface with one boundary component. We prove that for all k, there exists some G_k

## References

SHOWING 1-10 OF 46 REFERENCES
Cerf Theory for Graphs
• Mathematics
• 1998
We develop a deformation theory for k‐parameter families of pointed marked graphs with fixed fundamental group Fn. Applications include a simple geometric proof of stability of the rational homology
An Infinite Presentation of the Torelli Group
In this paper, we construct an infinite presentation of the Torelli subgroup of the mapping class group of a surface whose generators consist of the set of all “separating twists”, all “bounding pair
Automorphisms of free groups with boundaries
• Mathematics
• 2004
The automorphisms of free groups with boundaries form a fam- ily of groups An,k closely related to mapping class groups, with the stan- dard automorphisms of free groups as An,0 and (essentially) the
The structure of the Torelli group I : A finite set of generators for J
This is the first of three papers concerning the so-called Torelli group. Let M = Mg be a compact orientable surface of genus g having n boundary components and let 9 = Xg . be its mapping class
COHOMOLOGICAL ASPECTS OF MAGNUS EXPANSIONS.
We generalize the notion of a Magnus expansion of a free group in order to extend each of the Johnson homomorphisms defined on a decreasing filtration of the Torelli group for a surface with one
The Johnson homomorphism and its kernel
We give a new proof of a celebrated theorem of Dennis Johnson that asserts that the kernel of the Johnson homomorphism on the Torelli subgroup of the mapping class group is generated by separating
Two theorems on the mapping class group of a surface
The mapping class group of a closed surface of genus > 3 is perfect. An infinite set of generators is given for the subgroup of maps that induce the identity on homology. Let Tg be the boundary of a


text:
# Will magnesium undergo a redox reaction with zinc(II) or copper(II)?

Magnesium can do a redox reaction with $$\ce{Cu^2+}$$ as well as with $$\ce{Zn^2+}$$, since Mg has the lowest reduction potential $$E_0$$. (I am aware that a reaction may or may not be feasible as redox, and this is according the reduction potential of the two half-equations)

$$\begin{array}{l|l|r} \ce{Cu^2+ & Cu} &\pu{0.34 V} \\ \ce{Zn^2+ & Zn} &\pu{-0.76 V} \\ \ce{Mg^2+ & Mg} &\pu{-2.37V} \\ \end{array}$$

I'm supposing we have in one beaker, a solution of $$\ce{Cu^2+}$$ ions and $$\ce{Zn^2+}$$ ions and a piece of magnesium. So the possible reactants are: $$\ce{Cu^2+}$$, $$\ce{Zn^2+}$$, and $$\ce{Mg}$$.

The potential difference between $$\ce{Mg}$$ and $$\ce{Cu^2+}$$ is greater than with $$\ce{Zn^2+}$$. So what's the answer?

My guess is "with $$\ce{Zn^2+}$$", since by comparing with another example, ethanol ($$\ce{CH3CH2OH}$$) reduces $$\ce{MnO4-}$$ to give $$\ce{CH3CHO}$$ instead of $$\ce{CH3COOH}$$, if the quantity of $$\ce{MnO4-}$$ was limited. $$\begin{array}{l|l|r} \ce{MnO4- & Mn^2+} & \pu{1.51 V} \\ \ce{CH3CHO & CH3CH2OH} & \pu{0.19 V} \\ \ce{CH3COOH & CH3CH2OH}& \pu{0.05 V} \end{array}$$

So $$\ce{CH3CH2OH}$$ reacted with $$\ce{MnO4-}$$ to give $$\ce{CH3CHO}$$, that leads me to deduce that in the case of multiple possibilities (i.e. feasible redox reactions), the reaction with the least potential difference takes place (since $$1.51 − 0.19 < 1.51 − 0.05$$).

From another hand, I guess it's familiar that the other option of having $$\ce{Mn^2+}$$ reacting with $$\ce{Mg}$$ is true as being held between the most oxidant which is $$\ce{Mn^2+}$$ and the most reductant which is solely $$\ce{Mg}$$ in our case.

• I wonder if an answer could be in antioxidant in the body. – YoussefDir Dec 24 '15 at 10:36

Let’s assume for a second that magnesium reacts only with the zinc. The reaction would occur according the following equation:

$$\ce{Mg + Zn^2+ -> Mg^2+ + Zn}$$

You have already identified, if I read your question correctly that the redox potentials allow this and that the reaction is spontaneous etc. Assume that you have equimolar amounts of magnesium, zinc and copper for the rest of this thought experiment.

Now our new solution would contain magnesium(II), copper(II) and zinc(0) (okay, the zinc would deposit itself somewhere, but you get the picture). Notice anything? Yes, zinc and copper can react according to the well-known and often-displayed electrochemical reaction (depositing copper on a blank piece of zinc metal). The equation for that one is:

$$\ce{Zn + Cu^2+ -> Zn^2+ + Cu}$$

Again, check out the standard potentials to see that this reaction is possible and spontaneous.

If we add the two reactions together, we get an effective reaction:

$$\ce{Mg + Zn^2+ + Zn + Cu^2+ -> Mg^2+ + Zn + Zn^2+ + Cu}$$

$$\ce{Mg + Cu^2+ -> Mg^2+ + Cu}$$

You notice that the zinc is little more than a spectator ion.

Now how does this differ from the ethanol case? Well, permanganate can (and will) react with ethanol to form ethanal, and it can (and will) react with ethanal to form ethanoic acid — more rapidly. However, if permanganate (and therefore: electrons) is the limiting reagent, we end up with a mixture of ethanol, maybe ethanal and ethanoic acid. By a reaction analogous to the reverse Cannizarro reaction, these can comproportionate:

$$\ce{CH3CH2OH + CH3COOH <=>> 2 CH3CHO}$$

Therefore, the overall reaction seems to be one where ethanol only reacted with permanganate to only form ethanal, while in fact ethanoic acid was formed, but in the end the equilibrium favoured ethanal.

$$\ce{Mg^2+}$$ will undergo a redox reaction with both the species.

Have a look at the electrode potential values at this link: http://www.physchem.co.za/data/electrode_potentials.htm via the Internet Archive

What you'll find is that the half reaction of $$\ce{Mg^2+}$$ has a highly negative electrode potential $$-2.37$$ while the $$\ce{Zn^2+}$$ and $$\ce{Cu^2+}$$ half reactions have less negative $$E_0$$ values [$$-0.76$$ and $$0.34$$ respectively]. A highly negative electrode potential indicates a strong tendency to reduce other species, while it gets oxidized itself. The $$E_0$$ values for the net reactions in both cases is positive. [Try it out: $$E(\text{cathode})-E(\text{anode})$$].

Consider the reverse reaction, i.e.
\begin{align} \ce{Cu + Mg^2+ &-> Mg + Cu^2+}\\ \tag{anode} \ce{Cu &-> Cu^2+ + 2e-} & (E_0 &= +0.34)\\ \tag{cathode} \ce{Mg^2+ + 2e- &-> Mg} &(E_0 &= -2.37) \end{align}
The $$E_0$$ of the net reaction is $$E(\text{cathode})-E(\text{anode})$$ which turns out to be negative. The change in Gibbs free energy- given by the expression $$-nFE_0$$, is thus positive for this reaction, implying that it is not feasible - i.e. $$\ce{Mg^2+}$$ will not be reduced by $$\ce{Cu^2+}$$. If you try this out with $$\ce{Zn^2+}$$, you'll end up with the same conclusion.

So, in response to your question - $$\ce{Mg^2+}$$ will participate in a redox reaction with both the species in which $$\ce{Mg}$$ gets oxidized while simultaneously reducing the other species.

• Yash, I'm aware that E(cathode)−E(anode) must be positive so the reaction is feasible. I have edited my question to clarify it. What I actually meant is that in one container we have Mg, and both the $\ce{Zn^2+}$ and $\ce{Cu^2+}$. – YoussefDir Dec 10 '15 at 21:02


text:
# Tag Info

I will just answer your first question as I do not know the details of SSVI. Total variance is more intrinsic than volatility. The BS formula can be rewritten in terms of 3 parameters: the log-strike (log-moneyness would be more accurate) $k$, the total variance $w$ and the discount factor. Volatility never appears without a $\sqrt{T}$. It is just there ...


text:
1. domwass

I would rather recommend a documentclass like memoir or scrreprt (from KOMA-Script), since they are much more flexible than report.

• I agree, my experience with them is limited though. Thanks for the addendum.
Here is the documentation: memoir, scrreprt (KOMA script)

Best, Tom.

2. Nice post Tom. I’m actually writing a two-part (or three) on Writing the PhD thesis: the tools. Feel free to comment, I hope to update it as I write my thesis, so any suggestions are welcome.

• Thanks for the link. I just saw your post and thought I should really check out git sometimes :-). Best, Tom.

• Yes, git is awesome. It can be a bit overwhelming with all the options and commands, but if you’re just working alone, and probably on several machines, then you can do everything effortlessly with few commands.

• That’s what has kept me so far. But I’ll definitely give it a try. Thanks!

3. What a great overview. Thank you, this will come handy… when I finally get myself to start writing that thesis 🙂

• Thanks and good luck with your thesis! Tom.

4. Hi, I can recommend two important packages: lineno.sty to insert linenumbers (really helpful in the debugging phase) and todonotes (allows you to insert todo-notes for things you still have to do.)

• Hey, thanks for the tip. Too bad they don’t say anything in the documentation apart from the fact that the packages are not compatible.

Best, Tom.

5. nick

good thesis template can be also found here (free):

including living headers, empty pages, two-sided with front and main matter as well as a complete structure

• Hi Nick,

Thanks for the link to the thesis template!

Best, Tom

6. Hi Tom, I’m writing a report on spanish in LaTex, using emacs, auctex, aspell (~170pags. ~70 files included by now) and this blog is my savior every time because I’m quite new with all these.

The question: Is there anyway (other than \- in every occurrence) to define the correct hyphenation for accented words (non english characters like é)?
I have three o four accented words, about the subject of my report, that occur near 100 times each, across several files, and the \hyphenation{} command can’t handle these.

• Hi!

I was wondering what packages you load in your preamble. For a better hyphenation (and easier typing), you should use these packages:

\usepackage[T1]{fontenc}
\usepackage[latin1]{inputenc}
\usepackage[spanish]{babel}

See here for more details.

If this doesn’t help, please provide a minimal working example to illustrate the problem.

Thanks, Tom.


text:
# Is this space countably compact

Let $X$ be a Tychonoff countably compact space and $A$ is a subapce of $X$ such that for any countable $B \subset A$ we have $\overline{B} \subset A$. My question is this:

Is this subspace $A$ countably compact?

Thanks very muvh for any help.

-
What if we let $A=X=\mathbb R$? –  Alex Becker Feb 25 '13 at 1:21
Also on MO: mathoverflow.net/questions/122847/… –  Asaf Karagila Feb 25 '13 at 1:36

Hint: Given an infinite $B \subseteq A \subseteq X$, every accumulation points of $B$ (as a subset of $X$) which belongs to $A$ is an accumulation point of $B$ viewed as a subset of $A$.