Let $X,Y,Z$ be Banach spaces, and let $T:X\to Y$ be linear. Let $J:Y\to Z$ be linear, bounded and injective. If $JT:X\to Z$ is bounded, then T is bounded.
Let $X,Y,Z$ Banach spaces and $A:X\rightarrow Y$ and $B:Y\rightarrow Z$ linear maps with $B$ bounded and injective and $BA$ bounded. Prove that $A$ is bounded as well.   If I knew that $B(Y)$ is closed I'd have a bounded linear map $B^{-1}:B(Y)\rightarrow Y$ by the bounded inverse theorem. Therefore $A=B^{-1}BA$ is bounded. How to prove the claim if $B(Y)$ is not closed.
Here are some examples.  "Until the Son of God appear" What I was taught: Subjunctive of indefinite future.  "It is required that the applicant be 18 years old and possess a valid driver's license." What I was taught: mandative subjunctive.  "Where be ye going," said the false knight on the road?" What I was taught: subjunctive of direct discourse.  "If I were you, which we both are thankful I am not, I wouldn't do that. What I was taught: The subjunctive mood is used for supposition contrary to fact.  "I dreamed I were a fireman in my Maidenform bra." What I was taught: The subjunctive mood can be used with words like dream.  "God bless you." I don't remember, so obviously I wasn't taught.  I can live with Pluto's not being a planet, but I must say it's irksome to find that so much of what I was told about English turns out to be nonsense. Nicomachus theorem proof - what did I do wrong? The exercise asked me to proof $1^3=1$ $2^3=3+5$ $3^3=7+9+11$ $...$ I formulate the equation as (1)$$a^3 = \sum_{i=0}^{a-1} (a-1)a+1+2i$$ Proof base case: $$1^3=\sum_{i=0}^{1-1} (1-1)(1)+1+2i=1$$ Proof (a+1) case: (2) $$(a+1)^3=a^3+3a^2+3a+1$$ (3) $$(a+1)^3=\sum_{i=0}^{(a+1)-1}((a+1)-1)(a+1)+1+2i=\sum_{i=0}^{a}(a)(a+1)+1+2i$$ factor out last summation term, so number of iterations match (1): $$\sum_{i=0}^{a-1}[(a)(a+1)+1+2i]+(a)(a+1)+1+2a=\sum_{i=0}^{a-1}[(a)(a+1)+1+2i]+a^2+3a+1$$ rearrange the summation term, so part of equation match (1): $$\sum_{i=0}^{a-1}[(a)(a-1)+1+2i]+\sum_{i=0}^{a-1}(2a)+a^2+3a+1$$ substitute summation with (1): (4) $$a^3+\sum_{i=0}^{a-1}(2a)+a^2+3a+1=a^3+(2a)(a-1)+a^2+3a+1=a^3+3a^2+a+1$$ From (2) I got $a^3+3a^2+3a+1$ but from (3) &amp; (4) I got $a^3+3a^2+a+1$ My result with induction is off by 2a. Can someone please help me to understand where I did wrong? Small snap loop devices visible in gnome-disk-utility OR what is the function of snap ubuntu-core $x^p-c$ has no root in a field $F$ if and only if $x^p-c$ is irreducible? Hungerford's book of algebra has exercise $6$ chapter $3$ section $6$ [Probably impossible with the tools at hand.]:     Let $p \in \mathbb{Z}$ be a prime; let $F$ be a field  and let $c \in  F$. Then $x^p - c$ is irreducible in $F[x]$ if and only if $x^p - c$   has no root  in $F$. [Hint: consider two cases: char$(F) = p$ and   char$(F)$ different of $p$.]   I have attempted this a lot. Anyone has an answer? When referring to flash exposure, what does "max sync speed" mean? Idempotent linear operators are projections I'm working on this problem: Suppose $V$ is a Hilbert space and $P : V \to V$ is a linear map such that $P^2= P$ and $\Vert Pf \Vert \leq \Vert f \Vert$ for every $f \in V$. Prove that there exists a closed subspace $U$ of $V$ such that $P = P_U$. I did do some searching online and I know that $P$ is an idempotent matrix and I know that $U$ is supposed to be $\text{range}P$, which I can show is closed. However, I'm having a hard time showing $P_{\text{range}P} = P$. I know that I can decompose $f = g + h$, where $g \in \text{range}P$ and $h$ is in the orthogonal complement of $\text{range}P$. I'm not sure how to proceed. Most of the online searches require that $P$ be self-adjoint, but this question doesn't have this condition. Can I get some help? Is three hours enough for an international-domestic transfer at Delhi? My International flight will be reaching New Delhi IGI at 2:45 AM and I am planning to book a domestic flight at 5:45 AM.  Is three hours sufficient for me to move from International to Domestic?  I believe there must be a shuttle service inside the airport to transfer me from International to Domestic but still I don't know if Customs time at IGI could cause any delay or not. Why is gravity such a unique force? My knowledge on this particular field of physics is very sketchy, but I frequently hear of a theoretical "graviton", the quantum of the gravitational field.  So I guess most physicists' assumption is that gravity can be described by a QFT?  But I find this weird, because gravity seems so incredibly different from the other forces (yes, I know "weirdness" isn't any sort of scientific deduction principle).  For relative strengths:   Strong force: $10^{38}$ Electromagnetic force: $10^{36}$ Weak force: $10^{25}$ Gravity: $1$   Not only does gravity have a vastly weaker magnitude, it also has a very strange interaction with everything else.  Consider the Standard Model interactions:    No particle (or field) interacts directly with all other fields.  Heck, gluons only barely interact with the rest of them.  So why is it then that anything that has energy (e.g. everything that exists) also has a gravitational interaction?  Gravity seems unique in that all particles interact through it.  Then there's the whole issue of affecting spacetime.  As far as I'm aware, properties such as charge, spin, color, etc. don't affect spacetime (only the energy related to these properties). Complex number operations: telling Mathematica variables are real What is the Python equivalent of static variables inside a function? What is the idiomatic Python equivalent of this C/C++ code?  void foo() {     static int counter = 0;     counter++;     printf("counter is %d\n", counter); }   specifically, how does one implement the static member at the function level, as opposed to the class level? And does placing the function into a class change anything? "Temporarily unreachable" when using fetch as Googlebot from Webmasters Tools The page I'm trying to fetch is public.  The server logs show no request.  Google reports that any page on my site is "Temporarily unreachable". It won't even fetch the verification file, a static HTML file that was successfully used to verify with the webmaster tool application itself.   My site is running on the .  What is causing this? Where did folder "merge" or "keep both" go in OS X 10.9 Mavericks? Before Mavericks one could hold the ALT key and move a folder "A" onto a folder "B" where already an "A" existed. In this case Mac OS X asked in Lion if I want to merge - or on Mountain Lion if I want to keep both (which was also a merge).  I can't find this option in Mavericks.  Did Apple remove this useful feature?  Any hints?  (And yes: I own ForkLift, which can do it - I just sometimes still prefer the Finder). Is the universe fundamentally deterministic? Equivalent Resistance? How do you find the equivalent resistance here? (I realized it's a Wheatstone bridge, but it's unbalanced, so that doesn't help.) I tried using Kirchhoff's laws, but there is no cell given. I'm trying to approximate the computer's write speed using dd:  dd if=/dev/urandom of=/dev/null bs=1K count=10000   which gives the following output  10000+0 records in 10000+0 records out 10240000 bytes (10 MB) copied, 0.746313 s, 13.7 MB/s   How can I get the '13.7 MB/s' into a bash variable? I've tried piping the output from dd to progs like awk, sed and grep to no avail.  Ultimately, I'm calling this via os.system(...) from within a python script. If anyone knows of a more direct way to get a similar result inside python I'd be interested in that also. I'm trying to predict how long a file copy will take, based on the file size.  Any help would be greatly appreciated. Fit mixture of distributions to your time-series data in R I have time-series data containing 1440 observations and the plot of the data is   I want to fit the Gaussian Mixture Models (GMM) to the above plot, and for the same I am using Mclust function of  package. Finally, I want a fit somewhat like this:   On using Mclust function, I do get following statistics     mclus_data &lt;- Mclust(givendataseries)    &gt; summary(mclus_data) ---------------------------------------------------- Gaussian finite mixture model fitted by EM algorithm  ----------------------------------------------------  Mclust E (univariate, equal variance) model with 8 components:   log.likelihood    n df      BIC      ICL        9525.438 1440 16 18934.52 18183.67  Clustering table:    1    2    3    4    5    6    7    8  1262    0    0    0    0   13  114   51    In the above statistic, I can not understand following:   Significance of log.likelihood, BIC and ICL. I can understand what each of them is, but what their magnitude/value refers to? It shows there are 8 clusters, but why cluster no. 2,3,4,5 has 0 values? What does this mean? From the plot it is clear that there must be two Guassians, but why Mclust function shows there are 8 Guassians?   Update: Actually, I want to do model based clustering of time series data. But currently  I want to fit the distribution to my raw data, as shown in Figure 1 on page no. 3 of  paper. For your quick reference, mentioned figure in said paper is How do I hide an element when printing a web page? I have a link on my webpage to print the webpage. However, the link is also visible in the printout itself.  Is there javascript or HTML code which would hide the link button when I click the print link?  Example:   "Good Evening"  Print (click Here To Print)   I want to hide this "Print" label when it prints the text "Good Evening". The "Print" label should not show on the printout itself. Probability of an even number of sixes We throw a fair die $n$ times, show that the probability that there are an even number of sixes is $\frac{1}{2}[1+(\frac{2}{3})^n]$. For the purpose of this question, 0 is even.   I tried doing this problem with induction, but I have problem with induction so I was wondering if my solution was correct:   The base case: For $n=0$, our formula gives us $\frac{1}{2}[1+(\frac{2}{3})^0] =1$. This is true, because if we throw the die zero times, we always get zero sixes.  Suppose it's true for $n=k$. Then the odds of an even number of sixes is $\frac{1}{2}[1+(\frac{2}{3})^n]$, and thus the odds of an odd number of sixes is $1 - \frac{1}{2}[1+(\frac{2}{3})^n]$.    For $n=k+1$, there are two ways the number of sixes are even:  a. The number of sixes for $n=k$ was even, and we do not throw a six for $n=k+1$: $ \frac{5}{6} \cdot  \frac{1}{2}[1+(\frac{2}{3})^n]$  b. The number of sixes for $n=k$ was odd, and we throw a six for $n=k+1$: $\frac{1}{6}(1 - \frac{1}{2}[1+(\frac{2}{3})^n])$  So the probability $p$ for an even number of sixes at $n=k+1$ is $ \frac{5}{6} \cdot  \frac{1}{2}[1+(\frac{2}{3})^n] + \frac{1}{6}(1 - \frac{1}{2}[1+(\frac{2}{3})^n])$  I have two questions   How do I get from $ \frac{5}{6} \cdot  \frac{1}{2}[1+(\frac{2}{3})^n] + \frac{1}{6}(1 - \frac{1}{2}[1+(\frac{2}{3})^n])$ to $\frac{1}{2}[1+(\frac{2}{3})^n]$? I seem to have done something wrong, I can't get the algebra correct, I get $p = \frac{1}{3}[1+(\frac{2}{3})^n] + \dfrac{1}{6}$ Other than that, is my use of induction correct? Is it rigorous enough to prove the formula? Study the convergence of the following series: $$\sum_{n=2}^\infty \frac{1}{n^\alpha \cdot\ln^\beta(n)} \text{ where }\alpha,\beta \geq 0 $$   Applying d'Alembert criterion I have that $$ \lim_{n\to\infty} \frac{n^\alpha  \ln^\beta(n)}{(n+1)^\alpha  \ln^\beta (n+1)} = \lim_{n\to\infty} \left(\frac{n}{n+1}\right)^\alpha\left(\frac{\ln(n)}{\ln (n+1)}\right)^\beta = 1$$ so the nature of the series is inconclusive.  If $\alpha = \beta = 0$, then the series diverges, since $\sum_{n=2}^\infty 1 = \infty$.  Should I study the rest of the cases (i.e. if $\alpha = 0, \beta &gt; 0$ the root test and the ratio test are also inconclusive). What is the best form to study the series?.  Thanks in advance What is the trajectory of a photon moving through a vacuum? Since electromagnetic energy is carried by photons and moves in forms of waves, does it mean that a single photon when propagating through space doesn't follow the straight path but instead always moves up and down, up and down like a wave. If so another question arises the speed of propagation of light in vacuum is fixed meaning that it will always take the same amount of time for it to travel from point A to point B, but if a photon always moves up and down it will also mean that it travels longer distance than the distance between A and B and so it ill travel faster than light propagates, is it even possible, could you please clarify these concepts for me?