Let $A$, $B$ be symmetric real matrices $n \times n$-type. Let's consider the polynomial $$ f(t)=\det(A-tB). $$  If $B$ is positive definite, then it is known that $f$ has only real roots. Is the same true for nonsingular $B$ (and $A$, $B$ symmetric)?
Let $\mathbf A, \mathbf B$ be Hermitian matrices of the same size. What is the characterization of $\mathbf A, \mathbf B$ such that $p(\lambda)=\det(\mathbf A-\lambda\mathbf B)=0$ has only real roots?   If $\mathbf B$ is positive definite (I corrected this), it is easy to see $p(\lambda)$ has only real roots.
Convergence in measure and almost everywhere In a finite measure space, let $\{f_{n}\}$ be a sequence of measurable functions. Show that $f_{n}  \rightarrow f$ in measure if and only if every subsequence $\{f_{n_{k}}\}$ contains a subsequence $\{f_{n_{k_{j}}}\}$, that converges almost everywhere to f.   I can prove from converge in measure to converge almost everywhere, but I don't know how to write it down for the other direction. download.01.org certificate verification failed - CRLfile missing? So I had intel-Linux-graphics-installer on my computer  but lately when I   sudo apt-get update   I get the following error  Err https://download.01.org trusty/main amd64 Packages                          server certificate verification failed.  CAfile: /etc/ssl/certs/ca-certificates.crt CRLfile: none   Tried reinstalling intel-linux-graphics-installer but nothing happend. And various solutions found here on askubuntu.com on certificates problems but didnt end up anywhere.  Any ideas on how to get that CRLfile? Also is this problem on my side or on the servers side? Can I read and write to the same file in Linux without overwriting it? Java Socket why server can not reply client I want to find an example of a homomorphism $f:G\to H$ such that $A$ is a normal subgroup of $G,$ but $f(A)$ is not so in $H.$  I know that if $f$ was onto $f(A )$ must be normal , but otherwise i want to find an example! Is reusing old code for a new assignment considered self plagiarism? How to protect yourself if you consider it to be, and a group partner does not? Prove $\int_0^{\infty}\! \frac{\mathbb{d}x}{1+x^n}=\frac{\pi}{n \sin\frac{\pi}{n}}$ using real analysis techniques only I have found a proof using complex analysis techniques (contour integral, residue theorem, etc.) that shows $$\int_0^{\infty}\! \frac{\mathbb{d}x}{1+x^n}=\frac{\pi}{n \sin\frac{\pi}{n}}$$ for $n\in \mathbb{N}^+\setminus\{1\}$   I wonder if it is possible by using only real analysis to demonstrate this "innocent" result?  Edit A more general result showing that  $$\int\limits_{0}^{\infty} \frac{x^{a-1}}{1+x^{b}} \ \text{dx} = \frac{\pi}{b \sin(\pi{a}/b)}, \qquad 0 &lt; a &lt;b$$ can be found in Limit of $(x_n)$ with $0<x_1<1$ and $x_{n + 1} = x_n - x_n^{n + 1}$ Let $0 &lt; x_1 &lt; 1$ and $x_{n + 1} = x_n - x_n^{n + 1}$ for $n \geqslant 1$.      Prove that the limit exists and find the limit in terms of $x_1$.   I have proved the existence but cannot manage the other part.  Thanks for any help. I want to describe my business and want to refer to it in a sentence.  I am trying to say:      "In 2015 we established a business (that is) named X".   Intuitively, the "that is" part is redundant, and makes the sentence more cumbersome. But I cannot think of a reason why we should be able to drop the verb.  Is the version without the "that is" part grammatically correct? How can it be explained? Why is menstruation in wild animals not a disadvantage to organismal survival? When I watch wilderness specials for more than a few minutes, I notice a familiar pattern: predators are depicted as being alerted by the scent of blood. Wounded animals seem to make the best prey and predators seem to have a heightened awareness for the smell of blood.   It would seem that it would then be a disadvantage to evolutionary survival for female organisms to menstruate. How do females of a species not attract predators to the signal of menstrual blood?   Do female organisms in the wild actually menstruate? If so, is there a difference in the composition of menses that does not alert predators? How would this not at the very least give predators an idea of the general vicinity of the female?  If wild animals excrete their menses, do they have ways in which they are able to mask the evidence of its release or are they somehow internalizing the tissue so that they are not released?  If female organisms in the wild do menstruate, then why is this not a survival disadvantage? "Important to me" or "Important for me" I cannot easily figure out which one is more appropriate to use:     It's important to me.        It's important for me.   Are they the same? If not, what's the difference? Everybody knows :-) that in Windows plain text files lines are terminated with CR+LF, and in Unix&amp;Linux - with LF only. How can I quickly convert all my source code files from one format to another and back? Do lens hoods need to be matched to focal length? Is it true that not all generic lens hoods work on all lenses (provided the filter diameter matches, of course)? I read that the focal length plays a role in it - but I cannot find such readings on any lens hoods I see available online. Is it ever bad to use \left and \right? For mathematics, is it ever disadvantageous to use \left(  and \right)  in equations?  I can't see any way that doing this would hinder the typesetting, so why isn't () just converted into \left( \right) automatically? This would save a lot of typing.  Cheers! proving topological equivalence How would I show that $$d_E = \sqrt{\sum_{i=1}^n(x_i-y_i)^2}$$ and $$d_\infty= \sum_{i=1}^n|x_i-y_i|$$ and $$\sup\{|x_1-y_1|,|x_2-y_2|,...,|x_n-y_n|\}$$ are topologically equivalent on $\mathbb{R}^n$? I can do it for $\mathbb{R}^2$, but I'm not sure how a generalization looks like. How can I get the Cartesian product (every possible combination of values) from a group of lists?  Input:   somelists = [    [1, 2, 3],    ['a', 'b'],    [4, 5] ]   Desired output:  [(1, 'a', 4), (1, 'a', 5), (1, 'b', 4), (1, 'b', 5), (2, 'a', 4), (2, 'a', 5) ...] Halo particles don't render in Cycles Can anyone enter as a student at the Citadel ? Or must one be recommended by Lords or a powerful institution?   Aemon Targaryen was a son of a King, so the Citadel would let him access without questioning. Samwell Tarly was sent by Lord Commander Jon Snow, from the Night's Watch. , a novice of the Citadel. A common born but it is not known how he entered the Citadel. We can notice that he is not a good student, but has not been fired.    I don't remember how the other Maesters were recruited, if they are always relatives of Lords, Kings... Can any lowborn commoner just "knock the door", and enter the Citadel? I know cell phone cameras are the lowest rung, but when reading reviews of phones, the reviewers always have sample shots that are much better than what I can manage.  What are some ways to take decent shots with a cell phone camera?  (I'm thinking of the higher end Android or iPhone cameras). How to compare the strings using < (Greater than symbol) Below is my input file:  PA43410-2,1,3,/vobs/atlas-idc/src/utils/logger/IDCLogger/IDC.cpp,48,19:16:46.523.177, 2 PS Sensor Value = -5.501000 , Min = -5.583000 , Max = -5.319000 PA43410-2,1,3,/vobs/atlas-idc/src/utils/logger/IDCLogger/IDC.cpp,48,19:16:46.523.210, 3 PS Sensor Value = 15.996000 , Min = 15.814000 , Max = 16.078000 PA43410-2,1,3,/vobs/atlas-idc/src/utils/logger/IDCLogger/IDC.cpp,48,19:16:46.523.231, 4 PS Sensor Value = -16.505000 , Min = -16.587000 , Max = -16.323000 PA43410-2,1,3,/vobs/atlas-idc/src/utils/logger/IDCLogger/IDC.cpp,48,19:16:46.523.263, 5 PS Sensor Value = 6.509000 , Min = 6.327000 , Max = 6.591000 PA43410-2,1,3,/vobs/atlas-idc/src/utils/logger/IDCLogger/IDC.cpp,48,19:16:46.523.302, 6 PS Sensor Value = 4.002000 , Min = 3.820000 , Max = 4.084000 PA43410-2,1,3,/vobs/atlas-idc/src/utils/logger/IDCLogger/IDC.cpp,48,19:17:46.481.557, 1 PS Sensor Value = 6.199000 , Min = 6.017000 , Max = 6.281000 PA43410-2,1,3,/vobs/atlas-idc/src/utils/logger/IDCLogger/IDC.cpp,48,19:17:46.518.691, 2 PS Sensor Value = -5.503000 , Min = -5.585000 , Max = -5.321000 PA43410-2,1,3,/vobs/atlas-idc/src/utils/logger/IDCLogger/IDC.cpp,48,19:17:46.523.156, 3 PS Sensor Value = 15.996000 , Min = 15.814000 , Max = 16.078000 PA43410-2,1,3,/vobs/atlas-idc/src/utils/logger/IDCLogger/IDC.cpp,48,19:17:46.523.195, 4 PS Sensor Value = -16.505000 , Min = -16.587000 , Max = -16.323000 PA43410-2,1,3,/vobs/atlas-idc/src/utils/logger/IDCLogger/IDC.cpp,48,19:17:46.523.221, 5 PS Sensor Value = 6.509000 , Min = 6.327000 , Max = 6.591000 PA43410-2,1,3,/vobs/atlas-idc/src/utils/logger/IDCLogger/IDC.cpp,48,19:17:46.523.240, 6 PS Sensor Value = 4.002000 , Min = 3.820000 , Max = 4.084000 PA43410-2,1,3,/vobs/atlas-idc/src/utils/logger/IDCLogger/IDC.cpp,48,19:18:46.480.644, 1 PS Sensor Value = 6.199000 , Min = 6.017000 , Max = 6.281000 PA43410-2,1,3,/vobs/atlas-idc/src/utils/logger/IDCLogger/IDC.cpp,48,19:18:46.522.615, 2 PS Sensor Value = -5.501000 , Min = -5.583000 , Max = -5.319000 PA43410-2,1,3,/vobs/atlas-idc/src/utils/logger/IDCLogger/IDC.cpp,48,19:18:46.522.729, 3 PS Sensor Value = 15.996000 , Min = 15.814000 , Max = 16.078000 PA43410-2,1,3,/vobs/atlas-idc/src/utils/logger/IDCLogger/IDC.cpp,48,19:18:46.522.765, 4 PS Sensor Value = -16.505000 , Min = -16.587000 , Max = -16.323000 PA43410-2,1,3,/vobs/atlas-idc/src/utils/logger/IDCLogger/IDC.cpp,48,19:18:46.522.788, 5 PS Sensor Value = 6.509000 , Min = 6.327000 , Max = 6.591000 PA43410-2,1,3,/vobs/atlas-idc/src/utils/logger/IDCLogger/IDC.cpp,48,19:18:46.522.810, 6 PS Sensor Value = 4.002000 , Min = 3.820000 , Max = 4.084000   I need to compare the PS Sensor Value, Min and Max value. (Min greater than PS Sensor greater than Max)   I'm expecting the output like below and here I should have only the normal lines.  What is normal lines? :  If Min value greater than PS Sensor value &amp;&amp; PS Sensor value  is greater than Max value then the line is normal lines and that should removed.        Expected Output:                                                                                                                                    PA43410-2  PS Sensor Value = 6.509000 , Min = 6.327000 , Max = 6.591000 PA43410-2 PS Sensor Value = 6.199000 , Min = 6.017000 , Max = 6.281000  Eg:     If   Min &lt; PS sensor value &lt; Max,    // Don’t care the normal  lines.          Throw this line away.    Else           Pull this line to consolidated new file. //Only focus on abnormal lines.