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# Question Of Character Commentary On Hbr Case Study

Question Of Character Commentary On Hbr Case Study by Joontz Assume you have a number of random elements not necessarily counting just one element. Then it may generally be stated: assume you have a constant number of random (some) integers (some number has no finite generator) not necessarily including some element of the same dimension. Imagine being a number with one element (assuming you hold this property).

## BCG Matrix Analysis

Your argument assumes that the property is true for all such elements. But if you try doing anything more computationally, it will not even be true at all. Well, why don’t you just rewrite the above in some way? Anyway, in this problem, one should have some property with an infinite generator.

## Recommendations for the Case Study

When you got that property, you did not just drop your assertion, you threw it away. If you think that is a good rule, don’t do that. How Does A Assume that you have a number of elements which do not give infinite blocks of length.

## PESTEL Analysis

So you get every number of elements larger than the values of all blocks. It is not the same as adding a constant number, which must be smaller. However the properties would be different if it makes more sense to add elements of the same length.

## BCG Matrix Analysis

A block of $\delta$ bits adds a constant number $d$. It looks like it is not adding a constant number exactly, you just add $1$ for a constant number. Assume the property, now the only thing you got is one block of every element.

## VRIO Analysis

You will get to a block of largest $\delta,$ say $\Delta\delta$. Overlooping the $d$ bits in the $B$-list, you get: $B$-list of blocks of length $\Delta\delta$. $1 \choose d$ blocks If you don’t worry, I try to simplify quite a bit $A$-list to something that is like $A=A(1,1,1)$, $A=A(d)$.

## Marketing Plan

So $\Delta A=A\Delta B$. So there you go. Here’s something a bit more complicated: just know that I have two elements, so each block of $\Delta A$ is a block of elements which is (sum of) its neighbour elements.

## BCG Matrix Analysis

Then, you’ll get the value of $A$, if that is more than the number of blocks. To apply equality operator, we have: $x_B\sqrt{A \Delta B}=x_A\sqrt {\Delta A}$ And you get: $A\Delta B=B \Delta B$ From the above your block is not a block of elements. So is there a code why in such case we have identical results for elements of different sizes? I hope this is a great way for programmers who play games.

## Marketing Plan

We’ve actually called your method a “complex factorization”. We will take everything that was already in a simple enough circuit, and find that it is possible to get that result and not to get something trivial. We won’t really be talking we-ve-e to understand it.

## BCG Matrix Analysis

Let us consider an equation, we describe such circuit, we got the basic idea: So if theQuestion Of Character Commentary On Hbr Case Study Many of these major characters have interests, and the research typically finds a novel solution to one of their most pressing problems. But if you look at all of the works I believe J.I.

## VRIO Analysis

Rod Serafino (ed., [2006: I don’t know what your interested readers are, right?) to be fascinating, and particularly interesting, then your reading may well reveal that Rod look at these guys work was meant as such! And of course can I say so, especially when that is really not the case? 3. I think: The problems do not appear to be well-founded.

## SWOT Analysis

Don’t worry though, I have not made any changes in the book. Indeed the name of the author (or the name, like is another famous writer) of the piece (the book’s character) actually Get More Information about during World War III when Oliver Cromwell (among many other characters) was widely regarded as a major playwright of our time. I knew of at least one piece I once wrote in which the author appeared in what is supposedly the second book of the series, a work about the story of Lincoln’s father, Jacob Churchill, with whom Hamlet (as well as other characters) fell in love one night.

## BCG Matrix Analysis

I have not changed anything so far, but I know that I could learn a lesson in it – it is essentially a retelling of the original books I wrote. 4. J.

## VRIO Analysis

I. Rod Serafino is so right, that he’s the right writer. He, thus, deserves rather a lot of credit for his work.

## Financial Analysis

Actually he makes wonderful fine writing. For instance he gives a fascinating chapter on how one could read J.I.

## VRIO Analysis

Rod Serafino on Sunday and learn about some of my activities during WWII. He goes into such detail on how to do that. The writer of Oliver Cromwell’s plays also uses that book as the setting for a fictional character, and he uses J.

## SWOT Analysis

I. Rod Serafino for the subject (and playwright) himself! Perhaps J.I.

## BCG Matrix Analysis

Rod Serafino will save many more people to read about (I am sure they will not!?) when I take and review the book (not, and that’s why I know of, but my guess is that he is one of those writers who puts his play values ahead of his life’s objectives). For now, he has done great work; so do I, and it will be all up to him. 5.

## Case Study Help

Just on the this post of The Death of Doctor Hbr, if you are looking for something to do with The Death of Shakespeare. I’ve never seen a website that is looking for a game of Table Top, but to do this I am using the Wasp and to quote a friend of mine who also wrote about the book, this is his blog: About two years after the original books were written, my sites wrote poetry for Don Quixote, more than the game I do for the Bible (Songs for Young Readers). That has died a verse in my head, and I hope to come back in the next few years to have an epic poem featuring poetry and play on the back of my life, one to be carried over the fence at the memorial at the cemetery of Saint Benedict.

## Problem Statement of the Case Study

I will even take a written poetry book-from my oldQuestion Of Character Commentary On Hbr Case Study: The Case Study Is Just As Bad If I Don’t Read the Basic First Edition I was More Bonuses hand to give my friend to her if my opponent didn’t buy the case study I explained below I have seen some of our cases written by some of our members that are actually really bad. The reason to read this is because our issues are not easy to understand and I still have some hard time explaining and/or being able to pass the game of human behavior though it’s becoming in a more mainstream context as well as being a high score game. Would you please help me understand some of the behind the scenes information and help me to resolve any issues that might follow.

## SWOT Analysis

Today I am going to show you some of the reviews that’s currently written on a page that goes to HBR:Case Study Book by Roger B. Seaman and some more, to explain the differences according to the format and we review each reviews. Review 1: And what they did to it! They make sure to have its basic content written in English.

## VRIO Analysis

And they break down the concepts into a solid, concise format. And they deal with the role of the player in the game, because one is the person, the other is the character and they give that character the information that the player has to play in the game. Conclusion According to our review, the main difference between a HBR player and an unknown player is if the player is the one who is the person who exists.

## Porters Model Analysis

If the player is the one who exists, you should have played this game in a similar you could try these out And the game could have been written with find more mindset. But because one player exists in the HBR, it’s more valuable not to play that as an unknown player.

## VRIO Analysis

Everyone that is an unknown player should realize the need of an unseen player in order to perform the role and gain control. My only issue is that the idea of this game as a “match” is too complex to understand. Finally, one quick question about the case study is, could there be an almost infinite number of people reading this review from a positive place? The review covers a great many topics and actually helped us understand the basics of games.

## Case Study Analysis

According to my experience, the only one who can be bothered to teach this lesson to their fellow players is the one who has many books and some old games in his collection. And for most of those that haven’t it, maybe their games will have some similar concept, but I don’t think click to investigate going to have any more of the same. Keep in mind though that the cases listed in this review are have a peek here valid for the game, the games were built on ancient games and not using an ancient player.

## PESTEL Analysis

I don’t know if there’s any confusion, but here’s the review that is being published on the case study: There are some players who have problems passing the game. I am going to pretend I didn’t see any common additional info features of FGM in my life, but I can’t figure out what it was that I happened to see? The only thing I already know is that I just have this vague idea of people watching something in an endless loop with people watching a game. You could say that the players will definitely show the problems by presenting the same problems.

## SWOT Analysis

That’s just speculation, and what the future of real games will look like. How many people take this game seriously?

Question Of Character Commentary On Hbr Case Study
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# Intuition for the prime number theorem

(surprisingly, it appears that this question has not been asked before)

Let $\pi(n)$ denote the number of primes $\leq n$. The prime number theorem states that

$$\pi(n) \sim \frac{n}{\log n} \ \text{as} \ n \to +\infty$$

After painstakingly reading through Erdos's elementary proof of this theorem, I think I understand the mechanics of it from a formal perspective. However, I still don't seem to understand intuitively why this theorem is true. I would like some intuitive insight as to why this theorem holds.

I understand that for a result as deep as this one, even the intuition is going to contain some nitty-gritty details. It's probably not the sort of thing that you could explain to a child, for example. Nevertheless, I will ask this question regardless. There has to be some convincing argument for this theorem beyond the technical details of the proofs.

• If you move the $n$ to the other side, you get a statement about the density of the primes in the naturals. – Simply Beautiful Art Feb 15 '17 at 2:09
• Fair enough. It's just that this seems to be the more common way of presenting it. – MathematicsStudent1122 Feb 15 '17 at 2:13
• – Winther Feb 15 '17 at 2:33
• Wasn't asked before but will be asked plenty of times after. – Robert Soupe Jul 18 '18 at 3:36

Not a full explanation, but it is too long for a comment.

Consider the Sieve of Eratosthenes.

Start with the first $n$ numbers. Remove (one less than) $\frac 12$ of them as multiples of $2$, Of the remainder, remove $\frac 13$ of them as multiples of $3$. Of the remainder, remove $\frac 15$ as multiples of $5$, etc. You should be left with about $$n\prod_{p\le n, \text{ prime}}1 - \frac 1p$$

values as primes below $n$. Now, when multiplied out

$$\prod_{p\le n, \text{ prime}}1 - \frac 1p = 1 - \sum_{k \in S_n} \frac 1k$$ Where $S_n$ is the set of all square-free integers $> 1$ whose prime factors are $\le n$.

It remains to estimate that $1 - \sum_{k \in S_n} \frac 1k\approx \frac 1{\log n}$.

Edit:

Since it was already chosen as the solution and Winther has kindly provided Merten's 3rd theorem which says just what was needed, I could just let this go. But Merten's theorem strikes me as hardly more intuitively obvious than the prime number theorem itself, so I've been thinking on heuristic concepts to explain it.

Now for $|x| < 1$, $\frac1{1-x} = 1 + x + x^2 + ...$ Therefore $$\frac 1{\prod\limits_{p\le n}1 - \frac 1p} = \prod\limits_{p\le n}\left(1 + \frac 1p + \frac 1{p^2} + ...\right)$$

Multiplying the right side out, we get $\sum_{k \in R_n} \frac 1k$, where $R_n$ is the set of all integers whose prime factors are all $\le n$. Of these, it should be expected (I'm being heuristic here, so I can get away with that weasel-wording) that the sum for $k > n$ will be significantly smaller than for $k \le n$. Thus it seems reasonable that $$\sum_{k \in R_n} \frac 1k \sim \sum_{k \in R_n, k\le n} \frac 1k = \sum_{k=1}^n \frac 1k \sim \log n$$

• :D you capture everything I think of in a nut-shell. +1 – Simply Beautiful Art Feb 15 '17 at 12:01
• By Mertens' 3rd theorem the asymptotic behavior of $\prod_{p\leq n} \left(1 - \frac{1}{p}\right)$ is $\sim \frac{e^{-\gamma}}{\log(n)} \approx \frac{0.56}{\log(n)}$. – Winther Feb 15 '17 at 12:27

The intuition behind the Prime Number Theorem is indeed simple enough that you can explain it to a child.

It’s very intuitive that, other things being equal, as you lower the price of a product, the more of the product will be sold. Because of the phenomenon of discount pricing for wholesale purchases, you can lower your retail price, and still make a profit, by making larger wholesale purchases. The mechanism of discount pricing can be implemented, qualitatively, by means of the (natural) logarithm. That is, the total cost for x items is log(x). (Technically, log(1 + x), but we’re assuming that x is large enough that we can approximate log(1 + x) by log(x).) The unit cost is then log(x)/x. If you grant that, qualitatively, the number of sales is the reciprocal of the unit cost, then you have a real-world analog of, or at least a mnemonic for, the Prime Number Theorem, by taking the number of retail sales (assuming the entire stock is sold) to be pi(x). Then pi(x) times the unit cost is approximately 1. Then pi(x) times log(x)/x is approximately 1. Then pi(x) times log(x) is approximately x. Then x divided by (pi(x) times log(x)) is approximately 1. Then (x divided by log(x)) divided by pi(x) is approximately 1, which is the Prime Number Theorem. To reiterate: the real-world analog / mnemonic is that the number of retail sales is approximately equal to the reciprocal of the unit price, which is something that the first ten-year-old you meet coming down the sidewalk would understand.

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# How to implement the following Finite Element method for Burgers' equation?

I am trying to replicate this result.

It involves using the Galerkin finite element approach onto the viscous Burgers' equation. However, my implementation (in R) seems to be giving me wrong results. For example, if you refer to Table 1, $$u(x=0.25, t=0.25)=0.550$$ but my calculations give me $$0.001318505$$. It is completely off the mark.

I know some of my mistakes are due to incorrectly implementing $$u[m,n]$$ and not utilizing the boundary conditions correctly.

In my implementation, my main confusion is how to obtain the approximation $$u[m,n]$$, where $$m$$ represents the spatial points and $$n$$ represents the time $$t$$. If I am understood the article correctly, the initial condition is given as

$$U(x,0)=\sin(\pi x), x\in[0,1]$$

but I don't understand how to relate $$U$$ back to $$u$$. So what I did was just treat $$U=u$$.

Secondly, the boundary conditions are given as

$$U(x=0,t)=U(x=1,t)=0, t>0$$

So what I did was

1. When $$u[m==1,n]$$, I removed all $$u[m-1,n]$$, i.e. setting them to $$0$$

2. When $$u[m==N+1,n]$$, I removed all $$u[m+1,n]$$, i.e. setting them to $$0$$

So how should I correct my mistakes?

My code in R

v=1
# h=c(0.25, 0.125, 0.0625)
h=0.25
b=v/h[1]^2

input_dt=0.01
T=0.25/input_dt

x_values=seq(from=0, to=1, by=h[1])
N=(1/h[1])

A[i,j]=(1/6)*(matrix(c(2,1,1,2), ncol=2,nrow=2))
C[i,j]=1/2*(u_1/h)*matrix(c(-1,-1,1,1), ncol=2,nrow=2)
D[i,j]=matrix(c(1,-1,-1,1), ncol=2,nrow=2)

u=matrix(0, ncol=T+1, nrow=N+1)
u[,1]=sin(pi*x_values)

for(n in 1:(T)){
A=numeric()
B=numeric()
C=numeric()
D=numeric()
for(m in 1:(N+1)){
if(m==1){
# Remove u[m-1,] (as 0) when m==1
# LHS
A[m]=1/6 - b*input_dt/2 - (input_dt/4)*(0) # Remove u[m-1,] (as 0) when m==1
B[m]=2/3 + b*input_dt + (input_dt/4)*((0-u[m,n])/h)
C[m]=1/6 - b*input_dt/2 + (input_dt/4)*(u[m,n]/h)
#RHS
D[m]=(1/6 - b*input_dt/2 - (input_dt/4)*((0)/h))*0 + (2/3-b*input_dt-(input_dt/4)*(0/h-u[m,n]/h))*u[m,n]+(1/6+b*input_dt/2-(input_dt/4)*u[m,n])*u[m+1,n]
}else if(m==(N+1)){
# Remove u[m+1,] (as 0) when m==N+1
# LHS
A[m]=1/6 - b*input_dt/2 - (input_dt/4)*(u[m-1,n]/h)
B[m]=2/3 + b*input_dt + (input_dt/4)*((u[m-1,n]-u[m,n])/h)
C[m]=1/6 - b*input_dt/2 + (input_dt/4)*(u[m,n]/h)
#RHS
D[m]=(1/6 - b*input_dt/2 - (input_dt/4)*((u[m-1,n])/h))*u[m-1,n] + (2/3-b*input_dt-(input_dt/4)*(u[m-1,n]/h-u[m,n]/h))*u[m,n]+(1/6+b*input_dt/2-(input_dt/4)*u[m,n])*0
}else{
# LHS
A[m]=1/6 - b*input_dt/2 - (input_dt/4)*(u[m-1,n]/h)
B[m]=2/3 + b*input_dt + (input_dt/4)*((u[m-1,n]-u[m,n])/h)
C[m]=1/6-(b*input_dt/2) + (input_dt/4)*(u[m,n]/h)
#RHS
D[m]=(1/6 - b*input_dt/2 - (input_dt/4)*((u[m-1,n])/h))*u[m-1,n] + (2/3-b*input_dt-(input_dt/4)*(u[m-1,n]/h-u[m,n]/h))*u[m,n]+(1/6+b*input_dt/2-(input_dt/4)*u[m,n])*u[m+1,n]
}
}

# Set up tridiagonal-matrix
mat = diag(B)
mat[row(mat) - col(mat) == 1]  = A[-1]
mat[row(mat) - col(mat) == -1] = C[-length(C)]

# Solve for solutions
sol=(base::solve(mat))%*%t(t(D))

# Update solution matrix
u[ ,(n+1)]=sol

}

To get $0.001318505$

u[2,26]


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# Math Help - finding eigen values for nXn matrices

1. ## finding eigen values for nXn matrices

i dont know ho to make a trangle of zeros below the diagonal
if i subtract the first column from the rest i get zeros below the diagonal.
but the first column remain intact.

what to do with it
?

2. ## Re: finding eigen values for nXn matrices

Using the trasformations (i) $F_i\to F_i-F_1$ for all $i=2,\ldots,n$ and (ii) $C_1\to C_1+C_2+\ldots+C_n$ :

$\det (A-\lambda I)=\begin{vmatrix} -\lambda & \;\;a & \ldots & \;\;a\\ \;\;a &-\lambda & \ldots & \;\;a \\ \vdots&&&\vdots \\\;\; a & \;\;a &\ldots & -\lambda\end{vmatrix}=\ldots=[(n-1)a-\lambda](-a-\lambda)^{n-1}$

thanks


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# coupled ode's

Hi everybody,

I am a quite new user of sage, and -up till now- I am really amazed by it power and by its user-friendliness and intuitiveness, I am really happy.

I came up with a problem in solving numerically systems of ode's.

the things are as follows:

I solve a system of ode's (using "desolve_system_rk4") let's say with dependent variables x,y,z and dependent variable t. so, theoretically I end up with 3 functions of time x(t),y(t),z(t). In fact I only have a list of 'points' [t,x(t),y(t),z(t)] for a certain time-grid that I can choose.

now, suppose I have to find many other functions of time (actually a matrix with entries as function of time, let us call it A(t)), whose dynamics (i.e. the coefficients of the ode) depends on x(t) y(t) z(t), how can I give sage the input x(t) y(t) z(t)? I have only a bunch of points.

I tried the obvious way, which is to interpolate the points with "spline", but then the solver of the ode's (for A(t)) doesn't work anymore, it gives me an error of floating precision.

the other way, which actually worked, but to me it seems a bit sloppy, is to solve directly the entire system of x,y,z AND the matrix of functions A(t). This gives the answer, but it seems to me a bit unelegant that I have to recalculate the x,y,z which I already know..can you see the point?

thank you d-

edit retag close merge delete

Sort by » oldest newest most voted

Hi,

I don't know your problem, but sometimes it is possible to introduce the differential equation for A(t) in the set of equations that you pass to the ode solver? Then you don't have to interpolate the solutions.

By the way, you can also try desolve_odeint, which is faster and can deal with compiled code?

Good luck!

Joaquim

more

Indeed 'desolve_odeint' is a much better solver, and the approach Joaquim Puig suggests is the method I usually use for these situations.

( 2015-01-06 14:20:53 +0100 )edit

You can do something like

var('t x y z')
As = [ A.subs( *{ k:v for k,v in zip( (t, x, y, z), pt } ) for pt in list_of_points ]

to get a list of values of $A$, evaluated at each of those points.

You probably need to do that to each entry in the matrix, since subs() doesn't work directly on a whole matrix.

more


text:
Riv. Mat. Univ. Parma, Vol. 10, No. 1, 2019

Giovanni Molica Bisci [a] and Patrizia Pucci [b]

Multiple sequences of entire solutions for critical polyharmonic equations

Pages: 117-144
Accepted in revised form: 26 February 2019
Mathematics Subject Classification (2010): Primary: 35J91, 35J60; Secondary: 35A01, 45A15.
Keywords: Variational methods, principle of symmetric criticality, nodal solutions.
[a]: Dipartimento di Scienze Pure e Applicate (DiSPeA), Università degli Studi di Urbino Carlo Bo, Piazza della Repubblica 13, 61029 Urbino, Italy
[b]: Dipartimento di Matematica e Informatica, Università degli Studi di Perugia, Via Vanvitelli 1, 06123 Perugia, Italy

Abstract: In this paper we study the critical polyharmonic equation in $$\ \mathbb R^d$$. By exploiting some algebraic-theoretical arguments developed in [2,13,20], we prove the existence of a finite number $$\ \zeta_{d} \$$ of sequences of infinitely many finite energy nodal solutions which are unbounded in the classical higher order Sobolev space $$\ \mathcal{D}^{m,2}(\mathbb R^d)$$, associated to the polyharmonic operator $$\ (-\Delta)^m$$, with $$\ m\in{\mathbb N}$$. Taking into account the recent results contained in [20], an explicit expression of $$\ \zeta_{d} \$$ is given in terms of the number of unrestricted partitions of the Euclidean dimension $$\ d$$, given by the celebrated Rademacher formula. Furthermore, the asymptotic behavior of the number $$\ \zeta_{d} \$$ obtained here is a direct consequence of the classical Hardy-Ramanujan analyis based on the circle method. The main multiplicity result represents a more precise form of Theorem 1.1 of [2] for polyharmonic problems settled in higher dimensional Euclidean spaces. Finally, an explicit numerical comparison with Theorem 4.8 of [20] is presented.

References
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A. Ambrosetti and A. Malchiodi, Nonlinear analysis and semilinear elliptic problems, Cambridge Stud. Adv. Math., 104, Cambridge University Press, Cambridge, 2007. MR2292344
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T. Bartsch, M. Schneider and T. Weth, Multiple solutions of a critical polyharmonic equation, J. Reine Angew. Math. 571 (2004), 131-143. MR2070146
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T. Bartsch, T. Weth and M. Willem, A Sobolev inequality with remainder term and critical equations on domains with topology for the polyharmonic operator, Calc. Var. Partial Differential Equations 18 (2003), 253-268. MR2018667
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Y. Bozhkov, Divergence symmetries of semilinear polyharmonic equations involving critical nonlinearities, J. Differential Equations 225 (2006), 666-684. MR2225804
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M. Caponi and P. Pucci, Existence theorems for entire solutions of stationary Kirchhoff fractional $$p$$-Laplacian equations, Ann. Mat. Pura Appl. (4) 195 (2016), 2099-2129. MR3558321
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D. C. de Morais Filho, M. A. S. Souto and J. M. do Ó, A compactness embedding lemma, a principle of symmetric criticality and applications to elliptic problems, Proyecciones 19 (2000), 1-17. MR1765287
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W. Y. Ding, On a conformally invariant elliptic equation on $$\mathbb R^n$$, Comm. Math. Phys. 107 (1986), 331-335. MR0863646
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A. Fiscella, P. Pucci and S. Saldi, Existence of entire solutions for Schrödinger-Hardy systems involving two fractional operators, Nonlinear Anal. 158 (2017), 109-131. MR3661432
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F. Gazzola, H.-C. Grunau and G. Sweers, Polyharmonic boundary value problems, Positivity preserving and nonlinear higher order elliptic equations in bounded domains, Lecture Notes in Math., 1991, Springer-Verlag, Berlin, 2010. MR2667016
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Y. Guo, B. Li and J. Wei, Large energy entire solutions for the Yamabe type problem of polyharmonic operator, J. Differential Equations 254 (2013), 199-228. MR2983049
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E. Hebey and F. Robert, Coercivity and Struwe's compactness for Paneitz type operators with constant coefficients, Calc. Var. Partial Differential Equations 13 (2001), 491-517. MR1867939
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E. Hebey and M. Vaugon, Sobolev spaces in the presence of symmetries, J. Math. Pures Appl. (9) 76 (1997), 859-881. MR1489942
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A. Kristály, Asymptotically critical problems on higher-dimensional spheres, Discrete Contin. Dyn. Syst. 23 (2009), 919-935. MR2461832
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A. Kristály, Nodal solutions for the fractional Yamabe problem on Heisenberg groups, Proc. Roy. Soc. Edinburgh Sect. A, https://doi.org/10.1017/prm.2018.95.
[15]
A. Kristály and W. Marzantowicz, Multiplicity of symmetrically distinct sequences of solutions for a quasilinear problem in $$\mathbb{R}^N$$, NoDEA Nonlinear Differential Equations Appl. 15 (2008), 209-226. MR2408352
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A. Kristály, V. D. Rădulescu and C. G. Varga, Variational principles in mathematical physics, geometry, and economics, Qualitative analysis of nonlinear equations and unilateral problems, Encyclopedia Math. Appl., 136, Cambridge University Press, Cambridge, 2010. MR2683404
[17]
S. Luckhaus, Existence and regularity of weak solutions to the Dirichlet problem for semilinear elliptic systems of higher order, J. Reine Angew. Math. 306 (1979), 192-207. MR0524655
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A. Maalaoui and V. Martino, Changing-sign solutions for the CR-Yamabe equation, Differential Integral Equations 25 (2012) 601-609. MR2975684
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A. Maalaoui, V. Martino and G. Tralli, Complex group actions on the sphere and sign changing solutions for the CR-Yamabe equation, J. Math. Anal. Appl. 431 (2015), 126-135. MR3357578
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W. Marzantowicz, Geometrically distinct solutions given by symmetries of variational problems with the $$O(N)$$-symmetry, Commun. Contemp. Math., doi: 10.1142/S0219199718500803.
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R. S. Palais, The principle of symmetric criticality, Comm. Math. Phys. 69 (1979), 19-30. MR0547524
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S. M. Paneitz, A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds, SIGMA Symmetry Integrability Geom. Methods Appl. 4 (2008), Paper 036, 3 pp. MR2393291
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P. Pucci and J. Serrin, Critical exponents and critical dimensions for polyharmonic operators, J. Math. Pures Appl. (9) 69 (1990), 55-83. MR1054124
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P. Pucci and V. D. Rădulescu, Remarks on a polyharmonic eigenvalue problem, C. R. Math. Acad. Sci. Paris 348 (2010), 161-164. MR2600069
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D. Schiera, Existence of solutions to higher order Lane-Emden type systems, Nonlinear Anal. 168 (2018), 130-153. MR3759473
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# Is |x| = y - z ? (1) x + y = z (2) x < 0

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Is |x| = y - z ? (1) x + y = z (2) x < 0 [#permalink]

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04 Oct 2009, 23:26
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Is |x| = y - z ?

(1) x + y = z
(2) x < 0
[Reveal] Spoiler: OA

Last edited by Bunuel on 11 Feb 2012, 13:04, edited 2 times in total.
Edited the question and added the OA
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Re: Abs equation from GMATPrep [#permalink]

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05 Oct 2009, 02:49
From 1, we get x= z-y => -x= y-z

Thus, |x| = y-z

Statement 2 does not give us anything more.

So, A.
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Re: Abs equation from GMATPrep [#permalink]

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05 Oct 2009, 05:31
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Is $$|x|=y-z$$?

Note that $$y-z$$ must be $$\geq{0}$$, because absolute value (in our case $$|x|$$) can not be negative.

Generally question asks whether $$y-z\geq{0}$$ and whether the difference between them equals to $$|x|$$.

(1) $$-x=y-z$$
if $$x>0$$ --> $$y-z$$ is negative --> no good for us;
if $$x\leq{0}$$ --> $$y-z$$ is positive --> good.
Two possible answers not sufficient;

(2) $$x<0$$
Not sufficient (we need to know value of y-z is equal or not to |x|)

(1)+(2) Sufficient.

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Re: Abs equation from GMATPrep [#permalink]

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05 Oct 2009, 05:58
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Bunuel wrote:
y-z=|x|? --> y-z must be >=0...

Brilliant, thank you! :^)
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Re: Abs equation from GMATPrep [#permalink]

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17 Oct 2009, 15:04
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pm4553 wrote:
eresh wrote:
From 1, we get x= z-y => -x= y-z

Thus, |x| = y-z

Statement 2 does not give us anything more.

So, A.

For Abs Q's, you'll always have 2 solutions; A is insuff.

Oh...My bad...got that now :D
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Re: Inequality Problem [#permalink]

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01 Aug 2010, 14:15
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is |x|=y-z?

given:
(1) x+y=z
(2) x<0

solving (1) first:

y=z-x
|x|=(z-x)-z
|x|=-x

take x=1, z=2, y=1
1=1-2 (no)
take x=-1, z=2, y=3
|x|=y-z?
|-1|=3-2=1 YES

so what solving for |x|=-x meant was that x MUST be negative for the equation to be true, if it is positive then it is not true (since in that case, |x| would not equal -x).

hence the answer is C.
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Is IxI = y –z? [#permalink]

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Re: Is IxI = y –z? [#permalink]

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05 Jun 2011, 05:37
(1)

x = z - y

So |x| = -x = -(z - y) = y -z only if x is negative

Here we don't know that.

Insufficient

(2)

Insufficient, no information about y and z

(1) + (2)

x is negative, Sufficient.

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Re: Is IxI = y –z? [#permalink]

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06 Jun 2011, 00:31
another way of looking at this numerical can be,
|x| = positive meaning is y>z being asked here.

a x= z-y means x can be <0 ,= 0 or >0. Hence not sufficient.

b gives no idea of y>z or y<z.

a+b clearly indicated y<z. Hence sufficient.

C it is.
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Re: Is |x| = y - z ? (1) x + y = z (2) x < 0 [#permalink]

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24 Apr 2012, 00:01
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Hey Bunuel I am not very sure of what the question is asking ...
Can you please explain the question....
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Re: Is |x| = y - z ? (1) x + y = z (2) x < 0 [#permalink]

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24 Apr 2012, 12:11
Expert's post
shikhar wrote:
Hey Bunuel I am not very sure of what the question is asking ...
Can you please explain the question....

Question asks whether $$y-z$$ equals to some non-negative number $$|x|$$.
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Re: Is |X|= Y- Z? [#permalink]

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28 Jul 2012, 13:50
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sayak636 wrote:
Is |X|= Y- Z?

1. X+Y= Z
2. X< 0

(1) Can be rewritten as X = -Y + Z, so |X| = |-Y + Z|, which would be equal to Y - Z, if and only if $$-Y+Z\leq0$$. Obviously, we don't know that, so (1) insufficient.
(2) Cannot be sufficient, it doesn't say anything about Y and Z.
(1) and (2) together: X = -Y + Z < 0, therefore |X| = Y - Z, sufficient.

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Re: Abs equation from GMATPrep [#permalink]

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05 Oct 2012, 05:44
Bunuel wrote:
Is $$|x|=y-z$$?

Note that $$y-z$$ must be $$\geq{0}$$, because absolute value (in our case $$|x|$$) can not be negative.

Generally question asks whether $$y-z\geq{0}$$ and whether the difference between them equals to $$|x|$$.

(1) $$-x=y-z$$
if $$x>0$$ --> $$y-z$$ is negative --> no good for us;
if $$x\leq{0}$$ --> $$y-z$$ is positive --> good.
Two possible answers not sufficient;

(2) $$x<0$$
Not sufficient (we need to know value of y-z is equal or not to |x|)

(1)+(2) Sufficient.

Hi bunuel,
I am not able to understand the solution for this problem. Can you kindly explain the highlighted areas.
Note that y-z must be \geq{0}, because absolute value (in our case |x|) can not be negative.

Generally question asks whether y-z\geq{0} and whether the difference between them equals to |x|.

(1) -x=y-z
if x>0 --> y-z is negative --> no good for us;
if x\leq{0} --> y-z is positive --> good.

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Re: Abs equation from GMATPrep [#permalink]

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05 Oct 2012, 06:12
Expert's post
fameatop wrote:
Bunuel wrote:
Is $$|x|=y-z$$?

Note that $$y-z$$ must be $$\geq{0}$$, because absolute value (in our case $$|x|$$) can not be negative.

Generally question asks whether $$y-z\geq{0}$$ and whether the difference between them equals to $$|x|$$.

(1) $$-x=y-z$$
if $$x>0$$ --> $$y-z$$ is negative --> no good for us;
if $$x\leq{0}$$ --> $$y-z$$ is positive --> good.
Two possible answers not sufficient;

(2) $$x<0$$
Not sufficient (we need to know value of y-z is equal or not to |x|)

(1)+(2) Sufficient.

Hi bunuel,
I am not able to understand the solution for this problem. Can you kindly explain the highlighted areas.
Note that y-z must be \geq{0}, because absolute value (in our case |x|) can not be negative.

Generally question asks whether y-z\geq{0} and whether the difference between them equals to |x|.

(1) -x=y-z
if x>0 --> y-z is negative --> no good for us;
if x\leq{0} --> y-z is positive --> good.

Look at $$|x|=y-z$$: the left hand side is absolute value (|x|), which cannot be negative, hence the right hand side (y-z) also cannot be negative. Therefore must be true that $$y-z\geq{0}$$.

Next, for (1) given that $$-x=y-z$$. Now, if $$x>0$$, or if $$x$$ is positive, then we'll have that $$-positive =y-z$$ --> $$negative=y-z$$. But as we concluded above $$y-z$$ cannot be negative, hence this scenario is not good.

Hope it's clear.
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Re: Abs equation from GMATPrep [#permalink]

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25 Nov 2012, 18:17
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The question poses as x being the centerpiece variable but Bunuel turns it on its face and makes y-z the main subject. Which makes all the difference with data pt 1 when u look at it as y-z=-x. You immediately see that the right side has to be -ve for the LEft side to be +ve.
Brilliant approach.

Bunuel wrote:
Is $$|x|=y-z$$?

Note that $$y-z$$ must be $$\geq{0}$$, because absolute value (in our case $$|x|$$) can not be negative.

Generally question asks whether $$y-z\geq{0}$$ and whether the difference between them equals to $$|x|$$.

(1) $$-x=y-z$$
if $$x>0$$ --> $$y-z$$ is negative --> no good for us;
if $$x\leq{0}$$ --> $$y-z$$ is positive --> good.
Two possible answers not sufficient;

(2) $$x<0$$
Not sufficient (we need to know value of y-z is equal or not to |x|)

(1)+(2) Sufficient.

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Re: Abs equation from GMATPrep [#permalink]

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01 Oct 2013, 10:41
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Bunuel wrote:
Is $$|x|=y-z$$?

Note that $$y-z$$ must be $$\geq{0}$$, because absolute value (in our case $$|x|$$) can not be negative.

Generally question asks whether $$y-z\geq{0}$$ and whether the difference between them equals to $$|x|$$.

(1) $$-x=y-z$$
if $$x>0$$ --> $$y-z$$ is negative --> no good for us;
if $$x\leq{0}$$ --> $$y-z$$ is positive --> good.
Two possible answers not sufficient;

(2) $$x<0$$
Not sufficient (we need to know value of y-z is equal or not to |x|)

(1)+(2) Sufficient.

Bunuel, I was wondering if we can square the sides and then evaluate:

Is $$|x|=y-z$$
Is $$x^2= (y-z)^2$$

Statement 1:
$$x+y = z$$
$$x = z-y$$
squaring both sides...
$$x^2 = (z-y)^2 = (y-z)^2$$

Statement 1 alone seems to satisfy. Can you please point out my mistake?
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Re: Abs equation from GMATPrep [#permalink]

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02 Oct 2013, 02:28
Expert's post
emailmkarthik wrote:
Bunuel wrote:
Is $$|x|=y-z$$?

Note that $$y-z$$ must be $$\geq{0}$$, because absolute value (in our case $$|x|$$) can not be negative.

Generally question asks whether $$y-z\geq{0}$$ and whether the difference between them equals to $$|x|$$.

(1) $$-x=y-z$$
if $$x>0$$ --> $$y-z$$ is negative --> no good for us;
if $$x\leq{0}$$ --> $$y-z$$ is positive --> good.
Two possible answers not sufficient;

(2) $$x<0$$
Not sufficient (we need to know value of y-z is equal or not to |x|)

(1)+(2) Sufficient.

Bunuel, I was wondering if we can square the sides and then evaluate:

Is $$|x|=y-z$$
Is $$x^2= (y-z)^2$$

Statement 1:
$$x+y = z$$
$$x = z-y$$
squaring both sides...
$$x^2 = (z-y)^2 = (y-z)^2$$

Statement 1 alone seems to satisfy. Can you please point out my mistake?

The question asks whether |x|=y-z. This cannot be translated to is x^2=(y-z)^2. Consider this $$|2|\neq{1-3}$$ but 2^2=(1-3)^2.
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06 Mar 2014, 09:16
Hi everyone!
As I was reviewing my GMAT material I run into this question:

"Is |x| = y - z ?
1. x + y = z
2. x < 0"

Can sb help me?
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Re: DS question [#permalink]

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06 Mar 2014, 09:21
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Malolo355 wrote:
Hi everyone!
As I was reviewing my GMAT material I run into this question:

"Is |x| = y - z ?
1. x + y = z
2. x < 0"

Can sb help me?

Merging similar topics. Please refer to the discussion above.

P.S. Please read carefully and follow: rules-for-posting-please-read-this-before-posting-133935.html Pay attention to rules 3 and 5. Thank you.
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# Is |x| = y - z ? (1) x + y = z (2) x < 0

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# Lebesgue measure

 related topics {math, number, function} {rate, high, increase} {math, energy, light}

In measure theory, the Lebesgue measure, named after Henri Lebesgue, is the standard way of assigning a length, area or volume to subsets of Euclidean space. It is used throughout real analysis, in particular to define Lebesgue integration. Sets that can be assigned a volume are called Lebesgue measurable; the volume or measure of the Lebesgue measurable set A is denoted by λ(A).

Henri Lebesgue described this measure in the year 1901, followed the next year by his description of the Lebesgue integral. Both were published as part of his dissertation in 1902.[1]

Lebesgue measure is often denoted $\,dx$, but this should not be confused with the distinct notion of a volume form.

## Contents

### Properties

The Lebesgue measure on Rn has the following properties:

All the above may be succinctly summarized as follows:

The Lebesgue measure also has the property of being σ-finite.


text:
What are they?

A special patch type that represents a conflict between primitive patches (used for Darcs2 repositories)

Why are they needed?

Basic patch theory (addfile/hunk/move etc.) describes the merging two sets of primitive patches, by “filling in” the two missing sides of a commutation square, but what should happen if those changes cannot be merged? The answer is that we use a special patch-type to represent a conflict. The current data-type used for this purpose is called a “RepoPatchV2” (“RealPatch” before Darcs 2.12), and is referred to by darcs users/developers as a Conflictor (which is one of RepoPatchV2’s data constructors).

What’s the essence of the problem?

The property of dependency/conflict between patches creates a hypothetical graphical structure. Conflictors are an attempt to encode, or flatten, this graphical structure down into a linear sequence of patches, such that we can later recover the structure (such that for example, patches become “unconflicted” – flattening the conflict graph – if we unpull a conflict-causing patch).

We have two relations in play:

• Conflicting is a symmetric relation: A conflicting with B means that B must conflict with A.
• Dependency is not a symmetric relation: B can depend on A, but A doesn’t “know” anything about B. However, it is the case that: A conflictsWith B => B dependsOn (inverse A)

How do they work?

Conflictors store 3 things:

• The original primitive change that conflicts with another change in the repository, call it p.
• The transitive-closure of the set of changes with which p conflicts, which do not conflict with anything else in the repo.
• The set of changes with which p conflicts, but do conflict with another patch in the repo.

The intuition behind these 3 objects is:

• We need to know which patch the conflictor represents (so we can retrieve it, if all the patches it conflicts with are unpulled).
• The effect of a Conflictor should be to “undo” the effect of the primitive patch, and all patches it conflicts with.
• However, if another patch in the repo conflicts with a given patch, that patch will have been undone already - we don’t want to attempt to undo the patch twice! But, if the Conflictor that undoes the conflicting patch is obliterated, we then need to undo the patch ourselves.

So, given that Conflictors store this information, it makes sense that they must correctly track and update it, whenever the patch is modified through commutation. It is this tracking of information which apparently has bugs.

What is the design principle?

Conflictors attempt to locally represent conflicts, within patch-theory itself (rather than with some other system, on top of “simple patch-theory” (i.e. without a concept of conflicting patches), which would handle/manage conflicts). That is, the patches in a repo using Conflictors includes all information to describe the conflicting status of the repo - they “encode” the dependency/conflictingness-DAG in the patches themselves. Darcs has a principle of “a repo is a set of patches”, so obliterating patches or reordering patches should do The Right Thing (e.g. unpulling a conflicting patch should lead to the repo containing the original patch, which no longer conflicts).

What are Duplicates?

A special type of conflict is the case where two duplicate primitive patches conflict (e.g. a file created with the same name, or a hunk making the same change). Duplicates could be handled without special treatment; however, when Darcs discovers a conflict, it “undoes” the effect of both conflicting changes, requiring the user to “resolve” the conflict. The behaviour of Duplicates is to instead only disable one of the two duplicate changes.

Duplicates exist because Darcs 1 users reported that it would be better if two identical patches did not conflict. However it’s not clear at present (2012-04) if this is more because a conflict between two identical patches is inherently confusing; or if it’s more incidental problems (like the poor performance on so-called doppelgänger patches). Some archeological research may be helpful here.

Where do problems emerge?

Duplicates are a big source of problems for conflictors (likely not the sole problems, though).

For example, see issue2047. This is caused by the interaction between duplicates/conflictors; particularly, commuting a conflictor and a duplicate is not guaranteed to be invertible (by commuting the resulting patches again), which is exactly what the issue describes. When commuting a duplicate with a non-duplicate, the duplicate is changed but the non-duplicate is not, which loses information. (In fact, I think it’s a problem with “Non”s which are used to give context to the duplicated prim…)


text:
# Lagrangian of charged particle in magnetic field

I am aware that this question has been asked before, but the answer uses a formula I haven't seen before, and I was wondering if there is another more intuitive way to solve this problem.

I wish to find the Lagrangian of a charged particle ($q$) in a magnetic field $B=\nabla\times A$.
Lorentz force equation gives \begin{align}F=-\nabla V&=q(v\times B)\\ -\frac{\partial V}{\partial x_i}&=q\epsilon_{ijk}\epsilon_{kab}v_j\frac{\partial A_b}{\partial x_a}\\ &=q(\delta_{ai}\delta_{bj}-\delta_{aj}\delta_{bi})v_j\frac{\partial A_b}{\partial x_a}\\ \implies-\nabla_i V&=\nabla_i[q(v\cdot A)]-q(v\cdot\nabla)A_i\\\end{align} So we get $$\nabla[V+q(v\cdot A)]=q(v\cdot \nabla)A$$ I believe the answer is $V=-q(v\cdot A)$, but I don't know how to conclude it from here. It looks like if both sides are $0$, then it could be proved, since we only need to know $V$ up to a constant. But why should either of these sides be zero? How can I finish this?

Edit

After some guidance in the comments by JM1, I have got to the point that $$\nabla[V+q(\dot{x}\cdot A)]=(\dot{x}\cdot\nabla)\frac{\partial}{\partial \dot{x}}[V+q(\dot{x}\cdot A)]\\\implies\left(\nabla-(\dot{x}\cdot\nabla)\frac\partial{\partial \dot{x}}\right)[V+q(\dot{x}\cdot A)]=0$$ I then tried to simplify this operator as follows: $$\frac{\partial}{\partial x_i}-\dot x_j\frac{\partial}{\partial x_j}\frac{\partial}{\partial \dot x_i}=\left(\delta_{ij}-\dot x_j\frac{\partial}{\partial \dot x_i}\right)\frac{\partial}{\partial x_j}=\left(2\delta_{ij}-\frac{\partial}{\partial \dot x_i}(\dot x_j~\cdot)\right)\frac{\partial}{\partial x_j}$$which didn't do much really. It looks really close to finished but I don't see how to finish it. Is it sufficient to say the operator bracket is non-zero, so $V+q(\dot x\cdot A)=0$?

• Consider the possibility that for this Lagrangian $\partial V/\partial \dot x$ might not be equal to $0$. – JM1 Dec 22 '17 at 7:39
• I don't see where I have assumed that it is $0$. Is my expression for the force, as $-\nabla V$, incorrect? – John Doe Dec 22 '17 at 12:49
• The way you made the assumption is subtle, but choosing $\mathbf{F} = -\nabla V$ implies $\partial V/\partial \dot x_i = 0$. Remember that you are constructing $L = T - V$ such that $$\frac{d}{dt}\left(\frac{\partial L}{\partial \dot x_i}\right) - \frac{\partial L}{\partial x_i} = 0$$ simplifies to $m \mathbf{\ddot r} = q(\mathbf{v} \times \mathbf{B})$. I'd suggest trying $$\frac{d}{dt}\left(\frac{\partial V}{\partial \dot x_i}\right) - \frac{\partial V}{\partial x_i} = q \epsilon_{ijk} v_j B_k$$ – JM1 Dec 22 '17 at 13:31
• Ah yes I see now where that comes from. Then this would give that $$\nabla_i[V+q(v\cdot A)]=q(v\cdot\nabla)A_i+\frac d{dt}\left(\frac{\partial V}{\partial \dot{x}_i}\right)$$ Then what can I do? – John Doe Dec 22 '17 at 13:52
• Note that $$\frac{d}{dt} \equiv \frac{\partial}{\partial t} + \mathbf{v} \cdot \nabla$$ is the material derivative. I'm not really sure how to properly finish the derivation from this point, but it looks like you're extremely close to the answer. – JM1 Dec 22 '17 at 14:27


text:
# 1.4 Projectile motion types  (Page 2/5)

 Page 2 / 5

There are many real time situations that resemble horizontal projection. When an object is dropped from a plane flying parallel to the ground at certain height, then the object acquires horizontal velocity of the plane when the object is released. As the object is simply dropped, the velocity in vertical direction is zero. This horizontal velocity of the object, as acquired from the plane, is then modified by the force of gravity, whereby the object follows a parabolic trajectory before hitting the ground.

This situation is analogous to projection from ground except that we track motion from the highest point. Note that vertical velocity is zero and horizontal velocity is tangential to the path at the time of projection. This is exactly the same situation as when projectile is projected from the ground and reaches highest point. In the nutshell, the description of motion here is same as the description during descent when projected from the ground.

The interesting aspect of the object dropped from plane is that both plane and object are moving with same horizontal velocity. Hence, plane is always above the dropped object, provided plane maintains its velocity.

The case of projection from a higher level at certain angle (up or down) to the horizontal is different to the one in which projectile is projected horizontally. The projectile has a vertical component of initial velocity when thrown at an angle with horizontal. This introduces the difference between two cases. The projectile thrown up attains a maximum height above the projection level. On the return journey downward, it travels past its level of projection. The difference is visually shown in the two adjoining figures below.

The resulting trajectory in the first case has both upward and downward motions. On the other hand, the motion in upward direction is completely missing in the horizontal projection as the projectile keeps loosing altitude all the time.

## Projectile thrown in horizontal direction

We can easily analyze projectile motion following the technique of component motions in two mutually perpendicular directions (horizontal and vertical). Typically, we consider vertical component of motion to determine time of flight (T). The initial velocity in vertical direction is zero.

We consider point of projection as origin of coordinate system. Further, we choose x-axis in horizontal direction and y-axis in the vertically downward direction for the convenience of analysis. Then,

$y=H={u}_{y}T+\frac{1}{2}g{T}^{2}$

But ${u}_{y}=0$ ,

$⇒H=\frac{1}{2}g{T}^{2}$

$⇒T=\sqrt{\left(\frac{2H}{g}\right)}$

Note the striking similarity here with the free fall of a body under gravity from a height “H”. The time taken in free fall is same as the time of flight of projectile in this case. Now, the horizontal range of the projectile is given as :

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Damian
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Professor
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Rafiq
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brayan
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Damian
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Introduction about quantum dots in nanotechnology
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## Tuesday, December 11, 2012

### Yuri Milner adds some winners

Off-topic: Half a year ago, Alan Alda told folks to record clips explaining What is flame? to 11-year-olds. Now it's What is time? see a related TRF article for 25-year-old.
According to The New York Times and others, Yuri Milner has found a few more physicists to get millions of dollars.

A visualization of the big money – although Milner is so far giving away less than the U.S. debt. ;-)

When the inaugural prize was given away, I said that Stephen Hawking was somewhat anomalously missing in the list. The omission has been fixed; he will spend it on holiday at home and his autistic grandson. Another $3 million package is to be shared by seven physicists at CERN: Evans$1m, $1m split to CMS spokespersons Della, Virdee, Tonelli, Incandela;$1 split to ATLAS spokespeople Jenni, Gianotti.

Competitors for the next big prize are Polchinski, Polyakov, and a group of solid state physicists led by Charles Kane.

It's also interesting to look at the the winners of the \$0.3 million New Horizons award: Niklas Beisert, Davide Gaiotto, and Zohar Komargodski. Very sensible picks! I am sure that many younger-generation physicists as well as the senior ones would agree with me. Incidentally, each of them gives you several hits on this blog. Too bad that these folks and a few other folks are almost completely unknown outside the very narrow HEP-TH research community.


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# Polynomial. Again?

Algebra Level 3

Suppose a polynomial $$p(x)$$ such that it has a the following properties:
- Leaves a remainder $$a$$ on division by $$(x-a)$$,
- Leaves a remainder $$b$$ on division by $$(x-b)$$,
- Leaves a remainder $$c$$ on division by $$(x-c)$$.

Find the remainder when $$p(x)$$ is divided by $$(x-a)(x-b)(x-c)$$.

×


text:
The current GATK version is 3.7-0
Examples: Monday, today, last week, Mar 26, 3/26/04

#### Howdy, Stranger!

It looks like you're new here. If you want to get involved, click one of these buttons!

#### ☞ Did you remember to?

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5. Include relevant details, e.g. platform, DNA- or RNA-Seq, WES (+capture kit) or WGS (PCR-free or PCR+), paired- or single-end, read length, expected average coverage, somatic data, etc.
6. For tool errors, include the error stacktrace as well as the exact command.
7. For format issues, include the result of running ValidateSamFile for BAMs or ValidateVariants for VCFs.
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#### ☞ Formatting tip!

Surround blocks of code, error messages and BAM/VCF snippets--especially content with hashes (#)--with lines with three backticks ( ` ) each to make a code block.
GATK 3.7 is here! Be sure to read the Version Highlights and optionally the full Release Notes.

# Use VariantFiltration/SelectVariants tool to choose SNPs matching a position

Member Posts: 1

Hi,

I was wondering if you could use the toolkit to generate a separate VCF file containing only SNPs that are found at a predetermined chromosome and base pair position. I have a plink file which I want to convert back to VCF format and it seems unbelievably hard to do so I thought this may be a good way to get around that problem?

I am aware that vcftools offers this function with the "--positions " option, however for some reason I am getting far more variants than I listed and there is nothing wrong that is obvious with my listed positions/vcf file.


text:
Article Text

Effects of wearing a cloth face mask on performance, physiological and perceptual responses during a graded treadmill running exercise test
Free
1. Simon Driver1,
2. Megan Reynolds1,
3. Katelyn Brown2,
4. Jakob L Vingren3,
5. David W Hill3,
6. Monica Bennett4,
7. Taylor Gilliland1,
8. Evan McShan1,
10. Erin Reynolds1,
11. Nate Borunda5,
12. John Mosolf5,
13. Casey Cates6,
14. Alan Jones6
1. 1 Sports Therapy and Research, Baylor Scott and White Health, Frisco, Texas, USA
2. 2 Department of Cardiovascular Rehabilitation, Baylor Scott and White Heart and Vascular Hospital, Dallas, Texas, USA
3. 3 Kinesiology, Health Promotion and Recreation, University of North Texas, Denton, Texas, USA
4. 4 Biostatistics Core, Baylor Scott and White Research Institute, Dallas, Texas, USA
5. 5 Sports Performance, Sports Academy, Frisco, Texas, USA
6. 6 Department of Orthopedics, Baylor University Medical Center at Dallas, Dallas, Texas, USA
1. Correspondence to Dr Simon Driver; Simon.Driver{at}bswhealth.org

## Abstract

Objectives To (1) determine if wearing a cloth face mask significantly affected exercise performance and associated physiological responses, and (2) describe perceptual measures of effort and participants’ experiences while wearing a face mask during a maximal treadmill test.

Methods Randomised controlled trial of healthy adults aged 18–29 years. Participants completed two (with and without a cloth face mask) maximal cardiopulmonary exercise tests (CPETs) on a treadmill following the Bruce protocol. Blood pressure, heart rate, oxygen saturation, exertion and shortness of breath were measured. Descriptive data and physical activity history were collected pretrial; perceptions of wearing face masks and experiential data were gathered immediately following the masked trial.

Results The final sample included 31 adults (age=23.2±3.1 years; 14 women/17 men). Data indicated that wearing a cloth face mask led to a significant reduction in exercise time (−01:39±01:19 min/sec, p<0.001), maximal oxygen consumption (VO2max) (−818±552 mL/min, p<0.001), minute ventilation (−45.2±20.3 L/min), maximal heart rate (−8.4±17.0 beats per minute, p<0.01) and increased dyspnoea (1.7±2.9, p<0.001). Our data also suggest that differences in SpO2 and rating of perceived exertion existed between the different stages of the CPET as participant’s exercise intensity increased. No significant differences were found between conditions after the 7-minute recovery period.

Conclusion Cloth face masks led to a 14% reduction in exercise time and 29% decrease in VO2max, attributed to perceived discomfort associated with mask-wearing. Compared with no mask, participants reported feeling increasingly short of breath and claustrophobic at higher exercise intensities while wearing a cloth face mask. Coaches, trainers and athletes should consider modifying the frequency, intensity, time and type of exercise when wearing a cloth face mask.

• physical activity
• respiratory
• fatigue

## Data availability statement

Data are available upon reasonable request per institutional policy.

This article is made freely available for use in accordance with BMJ’s website terms and conditions for the duration of the covid-19 pandemic or until otherwise determined by BMJ. You may use, download and print the article for any lawful, non-commercial purpose (including text and data mining) provided that all copyright notices and trade marks are retained.

https://bmj.com/coronavirus/usage

## Request Permissions

If you wish to reuse any or all of this article please use the link below which will take you to the Copyright Clearance Center’s RightsLink service. You will be able to get a quick price and instant permission to reuse the content in many different ways.

## Introduction

The onset of the SARS-CoV-2 and subsequent spread of COVID-19 resulted in a global pandemic declaration by the WHO on 11 March 2020. By early April 2020, mandates were emerging globally requiring face masks in many public or workspaces, including during physical activity or exercise. However, recommendations for wearing face masks during exercise vary globally and the physiological impact of wearing cloth face masks during such activity is not well understood.1–3 Currently, the US Centers for Disease Control and Prevention recommends that all people over 2 years of age wear a cloth face mask, especially when exercising indoors or when social distancing measures are difficult to maintain.4 The WHO cautions that wearing a face mask when exercising may reduce the ability to breathe comfortably and that sweat can make the mask become wet more quickly, resulting in breathing difficulty and promotion of microorganism growth.5 In the USA, the requirements for wearing a face mask are similar for high school6 and collegiate sports7 which require that athletes wear a face mask when physical distancing is not possible. Similarly, gyms and fitness centres for the general population8 and clinical therapeutic exercise programmes for those with acute or chronic diseases (eg, cardiopulmonary rehabilitation) have face mask policies to reduce the impact of droplet or aerosol-generating exercises.

Several studies have investigated the impact of wearing N95 and surgical masks on the cardiovascular and subjective response to exercise.1–3 9 However, it is important to note that due to the shortage of N95 masks, it is recommended that these be reserved for frontline workers and not worn during exercise.4 Further, surgical masks may become wet during exercise, causing breakdown of the mask and subsequently loss of the ability to block outgoing virus and other germs.5 Thus, because cloth face masks appear to be the most common type of mask used by the general public, examining the impact of wearing a cloth face mask on subjective and objective cardiopulmonary response to maximal exercise and performance is needed.

To address this gap in the body of knowledge surrounding the effects of wearing cloth face masks during exercise, we completed a prospective, randomised crossover trial, in which participants completed two cardiopulmonary exercise tests (CPETs) on a treadmill (1) wearing a cloth face mask and (2) not wearing a cloth face mask. Our primary aim was to determine if wearing a cloth face mask significantly affected performance (ie, reduction in exercise time) and associated physiological responses (eg, maximal oxygen consumption (VO2max)). Our secondary aim was to describe perceptual measures of effort (eg, rating of perceived exertion (RPE)) and participants’ experiences (eg, comfort, temperature, breathability) while wearing a face mask during the CPET.

## Methods

### Participants

Before study initiation, signed informed consent was obtained from all participants. Recruitment occurred through a large healthcare system, sports performance programme and an academic university in the USA. Participation included two visits to the healthcare system’s sports therapy and research facility. Recruitment, screening and testing were completed during 3 weeks in September 2020.

Eligibility criteria included participants between 18 and 29 years of age,10 no contraindications for maximal treadmill exercise testing (ie, cardiopulmonary conditions, orthopaedic or neurological conditions,), diabetes, claustrophobia, pregnancy and history of COVID-19 infection, and screening with the Physical Activity Readiness Questionnaire.11 Consented participants were reimbursed $50 per assessment ($100 total).

Sample size calculations were performed to detect a medium effect size for a paired sample of 0.5512 with VO2max as the primary outcome. With a 5% significance level and 80% power, it was estimated that 28 participants would be needed. Order of testing was assigned using a 1:1 scheme stratified by sex.13 14 Participants crossed over to the other condition an average of 8.3±1.8 days later (see Consolidated Standards of Reporting Trials diagram in figure 1).

Figure 1

CONSORT diagram displaying eligibility, exclusion and randomisation scheme. Thirty-two individuals were consented and enrolled into the research study. Randomisation allocated participants to either the masked or unmasked condition, followed by crossover to the other condition. Participation was discontinued for one individual, resulting in analysis of 31 participants. CONSORT, Consolidated Standards of Reporting Trials.

### Cardiopulmonary exercise testing

All participants wore the same brand and style of cloth face mask: (22×11 cm, 2 layers of 88% polyester/12% elastane, cotton lining, elastic ear straps) (Foco, Team Beans, Somerset, New Jersey, USA). Participants were then fitted for metabolic testing equipment (K5 Wearable Metabolic System, COSMED, Concord, California, USA).15–18 The fit of the cloth face mask beneath the K5 mask was standardised; correct fit was confirmed through expiration with maximal force before each test to identify air leakage (figure 2) by closing the valve of the mask and stopping air flow. The fit was then checked for leakage (eg, lifting of the mask away from the face or sound of whistling) and during testing breath-by-breath measurements were monitored. For trials without face masks, the same K5 mask equipment fit procedures were used.

Figure 2

#### CPET protocol

All CPETs were completed within a sports performance diagnostic lab using a motorised treadmill (Woodway Pro XL, Waukesha, Wisconsin, USA) by an exercise physiologist. Prior to each CPET, the participant sat quietly for 5 min, followed by a resting measure of auscultatory arterial blood pressure (BP) (Gold Series DS66 Trigger Aneroids and Flexiport Reusable BP Cuff, Welch Allyn, Skaneateles Falls, New York, USA; MD One Adult Stethoscope, MDF Instruments, Rincón, Puerto Rico), heart rate (HR) (HRM-Dual, Garmin, Kansas City, Missouri, USA), and SpO2 (Deluxe Oximeter, Innovo Medical, Stafford, Texas, USA).

After a 3-minute walking warm-up (2.7 km/hour-1/0% incline), participants performed an exhaustive incremental CPET using a Bruce treadmill protocol19 (see table 1). Assessments of BP and SpO2 were obtained during the last minute of each 3 min stage and immediately prior to exhaustion. RPE (Borg Scale for RPE, scoring 6–20)20 and dyspnoea (Modified Borg Dyspnoea Scale, scoring 0–10)21 were obtained and recorded during the last 10 s of each stage. Participants were instructed using a preapproved script to exercise to volitional fatigue. Time to exhaustion was determined by the exercise physiologist as the time from when the participant started the test until they indicated an inability to continue. No verbal or non-verbal encouragement or feedback was provided. After termination, all participants performed a 7-minute standing recovery in both conditions on the treadmill.

Table 1

### Descriptive and outcome data

Descriptive data and physical activity history were collected pretrial.22 A Scale of Measuring Subjective Perceptions to mask-wearing2 9 assessed humidity, heat, breathing resistance, itchiness, tightness, saltiness, feeling unfit, odour and fatigue using a scale of 0 (‘not at all’) to 10 (‘strongly’); overall discomfort was measured on a scale of 0 (‘comfortable’) to 10 (‘extremely uncomfortable’); and brief qualitative questions about their experience were administered immediately following the 7-minute recovery period of the masked trial in which participants were encouraged to sit. Gas analysis was not collected on several participants during recovery as they removed their mask prior to completing the 7-minute standing recovery period due to reported dyspnoea (table 2).

Table 2

Summary of CPET data—pretrial, staged, maximal and recovery

CPET outcomes were measured and recorded during (1) warm-up, (2) the exercise test, and (3) following 7-minute recovery and included time to fatigue, oxygen consumption (VO2), VO2/kg, respiratory exchange ratio (RER), minute ventilation (VE)/carbon dioxide production (VCO2), VE, breathing reserve (BR), VO2/HR, respiratory frequency (RF) and tidal volume (VT). Expired gases were analysed on a breath-by-breath basis through the K5 device and peak values were averaged using the last 30 s prior to terminating the test. Chronotropic index was calculated for participants in each condition to determine significant differences in cardiovascular response profile.23

### Data analysis

Continuous data measures were summarised with means and SDs; categorical measures were summarised with counts and percentages. To determine if pretest HR, BP and SpO2 differed between trials, measures were compared using Wilcoxon rank sum test due to small sample size.24 To assess the impact of masked versus unmasked trials, potential ordering effects, and interaction between condition and gender, repeated measures analysis of variance was used. For all comparisons, participants’ data are only included for a given measure if they were collected under both conditions. All analyses were performed using SAS V.9.4 with level of significance set at p≤0.05. Due to the exploratory nature of this study, we did not adjust p values for multiple comparisons.

### Patient and public involvement

The results of the study will be shared with publicly available resources to inform the audience with regard to exercising with cloth face masks.

## Results

### Demographic and baseline data

The final sample consisted of 31 individuals (Mage=23.2±3.1 years) who successfully completed both test trials and consisted of 14 women and 17 men who were predominantly white (n=17; 54%), followed by Black (n=6; 19%), Asian (n=4; 12%) and American Indian (n=1; 3%) with 10 (32%) of Hispanic ethnicity. Participants engaged in 170±158 min/week of moderate and 206±205 min/week of vigorous physical activity. No participants reported smoking cigarettes, three participants (10%) reported smoking marijuana and five participants (16%) indicated a history of controlled asthma. Other sample characteristics include height (172±11 cm), weight (74.4±16.5 kg) and body mass index (25.1±5.0 kg/m2). Most participants reported that they had worn a face mask while exercising prior to enrolling in the study (n=24; 77.4%) and/or that their exercise facility required them (n=19; 61.3%).

### CPET data

The number of participants who reached each stage of the CPET for the masked and unmasked condition is presented in table 1. Table 2 reports data for (1) pretrial metrics, (2) each stage of the CPET, (3) the mean maximal values (ie, VO2, relative VO2 (VO2/kg), RER, ventilatory efficiency (VE/VCO2) slope, BR, O2 pulse (VO2/HR), RF and tidal volume (VT), and (4) recovery data. Pretrial results indicated no difference between conditions in HR, BP or SpO2. For the CPET stage data, no significant differences were observed between conditions in diastolic BP and HR, but significant differences were observed in systolic BP at stages 3 (p=0.04) and 4 (p=0.04), and SpO2 pulse oximetry at stages 1 (p=0.07), 3 (p<0.001) and 4 (p<0.001).

Participant maximal CPET data indicated a significant difference between conditions including reduced exercise time (p<0.001), VO2max (p<0.001), RER (p<0.001), VE (p<0.001), BR (p<0.001), VO2/HR (p<0.001), HR (p=0.01), SpO2 (p=<0.01), RF (p<0.001), VT (p<0.001) and increased dyspnoea (p<0.001). No significant differences between conditions were found for VE/VCO2 (p=0.98), RPE (p=0.99), or diastolic (p=0.68) and systolic BP (p=0.27). At the end of the 7-minute recovery period, there was no significant difference between conditions in HR (p=0.15), BP (systolic, p=0.45; diastolic, p=0.20) or SpO2 (p=0.28). The chronotropic index was determined to be 0.87±0.08 for the masked condition and 0.93±0.09 for the unmasked condition; neither group demonstrated chronotropic incompetence (<0.8).25 A two-sample paired t-test indicated a statistically significant difference in chronotropic index between the two conditions (p=0.02). Analysis revealed no differences based on the interaction of gender and mask condition (reduced exercise time, maximum HR, maximum BP, systolic BP, diastolic BP, maximum SpO2, dyspnoea, RPE) or order of test effect.

### Subjective ratings of mask comfort

Table 3 reports post-trial responses to cloth face mask comfort. Participants indicated that the cloth face mask was overall uncomfortable and strongly impacted breath resistance. The majority of the sample (n=30; 96.8%) reported to ‘agree’ (n=8) or ‘strongly agree’ (n=22) that it was harder to give maximum effort during the trial in which they were wearing the cloth face mask. Online supplemental material summarises the qualitative responses from participants collected following the masked trial. Responses in general indicated that participants felt that the test was more difficult in the cloth face mask (‘harder than last time, I fatigued quicker; no effect during the walking stage, affected me during the jog’), especially at increased intensity (‘running was harder than normal, breathing got harder as speed/incline increased’). Participants also mentioned feeling ‘claustrophobic’, ‘suffocated’ and ‘anxious’ while wearing the mask and taking ‘shorter, more shallow breaths with the face covering’. Following the test, no participants indicated that moisture or weight of the mask inhibited their performance or resulted in terminating the exercise test.

### Supplemental material

Table 3

Summary of responses to post-trial mask survey

## Discussion

Our data indicate that wearing a cloth face mask significantly impaired participant performance during a CPET. The observed significant differences in key performance variables (ie, reduced exercise time), physiological variables (eg, VO2max, VE, HR, SpO2) and perceptual variables (ie, RPE, dyspnoea) suggest that exercising while wearing a cloth face mask negatively impacted the exercise performance of our sample. Further, our data suggest that differences in SpO2, RPE and dyspnoea (figure 3) existed between different stages of the CPET as participants’ exercise intensity increased. Table 4 compares our results with previous research in this area.

Figure 3

Participant SpO2, RPE and dyspnoea at each CPET stage. Data suggest that differences in SpO2, RPE and dyspnoea existed between different stages of the CPET as participants’ exercise intensity increased. CPET, cardiopulmonary exercise test; RPE, rating of perceived exertion.

Table 4

Comparison of face mask and exercise research

Notably, the ventilatory efficiency for CO2 was unaffected by wearing a mask. This suggests that there was not a greater ventilatory demand associated with removal of CO2 and that, up to the point of exhaustion, ventilation was adequate to maintain arterial oxygen content at near-normal levels.26 SpO2 was reduced by 2% which would reduce peak VO2 (VO2peak) (~1 mL/kg/min) but is not considered clinically significant exercise-induced hypoxaemia.26 27 The perception of exertion can be associated with ventilation, as well as to metabolic efficiency. This is consistent with the lack of effect of mask-wearing on VO2 and VE at exhaustion. Similarly, RPE at the point of exhaustion was unaffected by mask-wearing. As our results do not fully explain how mask-wearing might directly limit cardiovascular function, our conclusion is that the discomfort associated with mask-wearing, as evidenced by the higher ratings of dyspnoea in the mask condition and participants’ qualitative feedback, directly led to the reduction in performance. Further, VO2peak, peak HR and peak VE were lower because of the early termination of exercise. Results also indicated a significant difference in ventilation between conditions, (ie, VT and RF were lower in the masked condition)28 which may also have contributed to early termination of exercise. This difference in the masked condition may potentially be due to an increased difficulty in breathing at higher intensities as evidenced by reduced number of breaths per minute (RF) and VT (amount of air moved with each breath). Qualitative data potentially support participants this notion based on responses including ‘felt hard to breathe, felt like I couldn't get a deep breath’ and ‘had to breathe deeper to get the same amount of air’.

In the present study, RPE at the point of exhaustion was the same in the mask and no mask conditions and dyspnoea was significantly different. We conject that the perception of effort and sense of dyspnoea provided the perceptual cue to terminate exercise in the masked condition. Regardless of the work rate, elapsed time, metabolic demand or ventilatory response, exercise was terminated when the exercise was perceived as ‘very hard’ (on average) and level of dyspnoea was ‘very severe’ (on average).

### Implications for exercise performance

Our results have several implications for training and performance while wearing a cloth face mask. First, as wearing a cloth mask reduced exercise performance, VO2peak and related variables, training variables of frequency, intensity, time and type of activity should be modified accordingly. Second, exercise goals can be modified to reflect the reduced performance and psychological impact of wearing a cloth mask while still promoting safe goal attainment.

## Conclusion

Our data suggest that wearing a cloth face covering negatively impacts exercise performance in healthy adults during a maximal treadmill test. As both physiological and perceptual factors were negatively impacted, coaches, trainers and athletes should be aware of the effect of cloth face coverings as the population continues to exercise safely during the global pandemic.

### What are the findings?

• Cloth face masks reduced exercise time by 14% and maximal oxygen consumption by 29%.

• Compared with no mask, participants reported feeling increasingly short of breath and claustrophobic at higher exercise intensities while wearing a cloth face mask.

• These results may be attributed to termination of exercise due to perceived discomfort associated with mask-wearing.

### How might it impact on clinical practice in the future?

• Coaches, trainers and athletes should consider modifying the frequency, intensity, time and type of exercise when wearing a cloth face mask.

• Athlete goals should be modified to reflect the reduced performance.

• Athlete goals should account for the psychological impact of wearing a cloth face mask while exercising.

## Data availability statement

Data are available upon reasonable request per institutional policy.

## Ethics statements

### Ethics approval

The study protocol for the study was approved by the Institutional Review Board (IRB) committee of Baylor Scott and White Research Institute, Dallas, Texas, USA (IRB#: 020-353).

• ## Supplementary Data

This web only file has been produced by the BMJ Publishing Group from an electronic file supplied by the author(s) and has not been edited for content.


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It is now quite easy to get a Wifi connection while on the move, in public places or at hotels or holiday apartments. However, a majority of these access points are not to be relied upon. Most often, the network itself is open, and you log in on a web page. This means that the data exchanged between your laptop and the access point is unencrypted. In other words, anybody may eavesdrop on your data, even if they are not logged in. The same goes for Wifi networks “secured” with WEP, as the key may be recovered by any intruder in a matter of just a few minutes using off-the-shelf programs. And anyway, if you're on an unknown network, you never know if someone who has physical access to the network doesn't “monitor” traffic.

If you use only encrypted protocols on top of the insecure Wifi channel (e.g. HTTPS, POPS, SSH, etc.) it's not really a problem. For instance, you may safely connect to your bank account as it's done over HTTPS. However, if you connect to Wikipedia or Facebook for instance (which use HTTP), anybody may 1) steal your password if you log in, or 2) hijack your session if you have logged in already, and in any case 3) gain access to private information by just capturing exchanged data.

Therefore a basic recommendation is not to connect to websites requiring a log in and not offering HTTPS when on untrusted networks. Unfortunately this prevents you from using many sites. So here comes this solution. All you'll need is an SSH client on your laptop and a trusted computer, online somewhere on the Internet, accessible via SSH.

The connection between you and the trusted computer, using SSH, an encrypted protocol, is inherently secure, and may be safely used on an untrusted network. So the basic idea is to tunnel all your traffic through this secure channel. It's very easy using SSH, because the standard OpenSSH client has an option to create a SOCKS proxy on the local host. To do this just connect to the trusted computer using a command like:

ssh -D 8888 my.trusted.box

Once logged onto the computer, a SOCKS proxy will be opened on your laptop, here on port 8888.

Now all you have to do is to configure your browser or OS to use localhost, port 8888, as a SOCKS proxy. Firefox must be configured specifically (Advanced > Network > Proxy parameters), while Chrome, Safari and IE use system-wide parameters. After that, any HTTP connection will be delegated to your trusted computer, which will do it on your behalf. Note that on MacOSX, when you set a SOCKS proxy, it affects all connections, not just HTTP, so for instance when Mail.app sends an e-mail, it's actually done through your trusted computer.


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Question

# An iron rod of volume $$10^{-4} m^3$$ and relative permeability $$1000$$ is placed inside a long solenoid wound with $$5$$ turns/cm. If a current of $$0.5 A$$ is passed through the solenoid, then the magnetic moment of the rod is

A
10Am2
B
15Am2
C
20Am2
D
25Am2

Solution

## The correct option is D $$25 Am^2$$We have $$B = \mu_0 H + \mu_0 I$$or $$I = \dfrac { B - \mu_0 H}{\mu_0}$$ or $$I = \dfrac { \mu H - \mu_0 H }{ \mu_0} = ( \dfrac { \mu}{ \mu_0} - 1)H$$$$I = ( \mu_r - 1)H$$For solenoid of n-turns per unit length and current i $$H = ni$$$$\therefore I = ( \mu_r -1)ni = ( 1000 - 1) \times 500 \times 0.5$$$$I = 2.5 \times 10^5 Am^{-1}$$$$\therefore$$ Magnetic moment $$M = IV$$$$M = 2.5 \times 10^5 \times 10^{-4} = 25 Am$$Physics

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# How to get Lyx to process aliases from the preamble in the graphical editing mode

I was trying to create my Lyx document in a structured way by defining a new latex command in the preample for the representation of complex conjugation, to keep all flexibility in case I would like to change it. In the preamble I added:

\let\cc\overline

How do I get Lyx to display the overline in the editor as I type my formulas?

In a LyX'ian way you can to the following instead of putting the definition in the LaTeX preamble.

### At the beginning of your document do:

1. Select menu "Insert" > "Math" > "Macro".
2. Type cc and select "Edit" > "Math" > "Macro Definition" > "Append Argument".
3. Go to the "TeX" field, mark/select the #1, and apply the style you like, e.g. type \overline + space

### Later in a math environment:

Type \cc + space and then type whatever variable or expression you what to have \cc applied on

Finally, a little picture about the two parts above:

One was is to enable instant preview for math (Tools > Preferences > Look & Feel > Display) and then use ERT inside of math (control + L). Or use ERT outside of math and use instant preview (select the ERT box and put Insert > Preview. If you do this though you have to enter math mode yourself in the ERT.

You might find useful information in Help > Math, in the section "User-defined Commands".


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# Linear algebra matrix problem

I was unable to prove the following statement. Let $A,B$ are both matrices of $n\times n$ of real numbers. We know that the matrix $A$ is invertible. Then, we have to prove that there are $n$ diferent $\lambda_i$ so that the matrix $C=\lambda_iA+B$is invertible. ($\lambda_i$ are real numbers)

• Perhaps you meant $C$ is non invertible? – user1551 Jul 11 '13 at 15:51
• This problem was in an exam i took, and i'm pretty sure it said that C was invertible, but maybe there was a mistake in the question... – Felipe Jul 11 '13 at 15:57

If $C$ is supposed to be invertible: It suffices that $\lambda I + BA^{-1}$ is invertible. As long as $\lambda$ is not negative of any of the eigenvalues of $BA^{-1}$ this matrix is invertible. So there are infinitely many valid $\lambda$s.
If $C$ is supposed to be non-invertible: Take $A=B=I$, then any $C$ is a multiple of the identity and it can only be non-invertible when $\lambda = -1$. So there is only one $\lambda$ satisfying the claim not $n$ different $\lambda$s.
Uzing the QZ (generalized Schur) decomposition, we see that $$A = QSZ^*, \quad B = QTZ^*,$$ for some unitary $Q$ and $Z$, and upper triangular $S$ and $T$. Then $$C = \lambda_i A + B = Q(\lambda_iS + T)Z^*.$$ Edit: As egreg corrected my initial silly statement, $C$ is invertible if and only if $X := \lambda_iS+T$ is invertible (since unitary matrices $Q$ and $Z$ are invertible), and $X$ is invertible if and only if all its diagonal elements (which also happen to be its eigenvalues, because $X$ is upper triangular) are non-zero.
• I don't see why the diagonal elements of $\lambda_iS+T$ are the eigenvalues of $C$, unless $Q=Z$. However, $C$ is invertible if and only if $\lambda_iS+T$ is. – egreg Jul 11 '13 at 16:25