I am dealing with the following exercise     A set $\mathcal M$ in a normed linear space $\mathcal R$ is said to be convex if $\mathcal M$ contains all elements of the form $\alpha x + \beta y$, where $\alpha, \beta \geq 0$, $\alpha+\beta=1$, provided that $\mathcal M$ contains x and y. Prove that the set of all elements $x \in \mathcal R$ satisfying the inequality $\|x-x_0 \| \leq c$, where $x_0$ is a fixed element of $\mathcal R$ and $c &gt; 0$, is convex.   Assume $\|x-x_0 \|\leq c$, and $\|y-x_0 \| \leq c$, we have to proof that   $$\| \alpha x + (1-\alpha) y - x_0 \| \leq c.$$  By the triangular inequality,   $$\begin{align*}\| \alpha x + (1-\alpha) y - x_0 \| &amp;  \leq \| \alpha x \| + \|(1-\alpha) y - x_0 \| \\ &amp;  \leq \| \alpha x \| + \|(1-\alpha) y \| +\|- x_0 \| \\ &amp; \leq |\alpha| \| x \| + |(1-\alpha)| \| y \| +\| x_0 \| \\ &amp; = c.\end{align*}$$  Is it enough for proof the theorem? If not, what path should I take?
If $(X,\|\cdot\|)$ is a normed linear space, then how to show any ball $B(x,r)$ is convex? I know that if $x,y\in A\subset V$ then $[x,y]\subset A$, where $A$ is a convex subset of vector space $V$ and $[x,y]=\{(1-t)x+ty\mid 0\leq t \leq 1\}$. Please give me some hint.  I tried the following:  Claim. $[a,b] \subset B(x,r)$ Let $a,b \in B(x,r).$ Then I get $\|x-a\|&lt;r$ and $\|x-b\|&lt;r$.  Is it right?
Let $a,b,c&gt; 0$ be three real numbers such that $a+b+c=1$. I want to prove that   $$\frac{a}{a^2+b^3+c^3}+\frac{b}{b^2+a^3+c^3}+\frac{c}{c^2+a^3+b^3}\le\frac{1}{5abc}.$$  My attempt: Using AM-GM on each denominator gives (here, LHS denotes the left-hand side) $$LHS\le\frac{1}{3a^{\frac12}bc}+\frac{1}{3b^{\frac12}ac}+\frac{1}{3c^{\frac12}ab}.$$  However, I think that my attempt doesn't work because the last expression can get larger than $\frac{1}{5abc}$. In fact, if we multiply with $3abc$ then the original inequality is: $$\sqrt a+\sqrt b+\sqrt c\le\frac35$$  However, as $a,b,c&lt;1$, $$\sqrt a +\sqrt b + \sqrt c&gt; a+b+c=1&gt;\frac35.$$ So my upper bound is always larger than the given one. When creating accounts on Windows 10, you get a choice between "Your family" and "Other users".  If you choose "Your family" you are forced to use a Microsoft account to sign in. There are various reasons why one wouldn't want to do that, such as too much sharing of private information with Microsoft, etc.  If you choose "Other users" you can create regular accounts just fine. However, if you need to setup time limits for some accounts, it is impossible to do so. It is a feature exclusive to "Your family". Not all time-restricted users I want to create are my family, and not everyone is okay with sharing their info with Microsoft.  So, is there a way to setup login time limits for the "Other users" type of accounts? I have a button created by my script. I wonder how I can display what the system console displays but inside blender's console. CoDEmanX did this by running a script inside it as shown , but I want to display what happens in the system console inside blender's console.  My .blend file can be found here: Why does polynomial factorization generalize to matrices I'm reading about linear algebra and I came across with the following theorem where I have a problem convincing myself:      Theorem 2.1 $\,$ Every linear operator on a finite-dimensional complex vector space has an eigenvalue.      Proof:       To show that $T$ (our linear operator on $V$) has an   eigenvalue, fix any nonzero vector $v \in V$. The vectors $v, Tv,  T^2v,..., T^nv$ cannot be linearly independent, because $V$ has   dimension $n$ and we have $n + 1$ vectors. Thus there exist complex   numbers $a_0,...,a_n$, not all $0$, such that      $$a_0v + a_1Tv + ··· + a_nT^nv = 0.$$ Make the $a$’s the coefficients   of a polynomial, which can be written in factored form as $$a_0 + a_1z  + ··· + a_nz^n = c(z − r_1)\cdots(z − r_m),$$ where $c$ is a non-zero complex number, each $r_j$ is complex, and the equation holds for all   complex $z$. We then have      $${\color{red}{ 0=(a_0I + a_1T + ··· + a_nT^n)v= c(T − r_1I)\cdots(T −  r_mI)v}},$$ which means that $T − r_j$ I is not injective for at least   one $j$. In other words, $T$ has an eigenvalue.$\;\blacksquare$   I'm having trouble with the factorized form of the matrix polynomial (in red). I understood that the factorization holds for a polynomial by the fundamental theorem of algebra but why does this also hold for matrices?  In other words, why is the the part I highlighted true? Does such an factorization always exist?    Could I have some help to see this? Thank you =)   P.S.  is my reference (page 3).  UPDATE:   Someone else also has asked the  before it seems. Who built the fortress of Dol Goldur, and when was it built? And was it always a place for evil, or Sauron just occupied it?  Despite the fact that Mordor is much better fortified; artificially by the old fortifications,  and geographically, as it is surrounded by the mountains. Also considering the presence of mount doom that seem to enhance Sauron's power, Sauron chose to start his campaign from Dol Goldur, despite Mordor being ready for quite some time.  Why did he choose this enigmatic fortress, and not go straight for Mordor or even go earlier? I have recently purchased a hybrid bike with suspension (not sure if that is relevant to this question). It came with two pieces of plastic. Any ideas what they may be for? Dual booting Windows and Ubuntu My laptop has 500gb hard disk in which for 250gb I installed Windows xp on C drive(primary partition). I want to install Ubuntu 12.04 in the remaing hard drive with dual boot option. When I am going to install Ubuntu it is showing something like  /boot /boot/   Can anybody help me in this regard that where and how to install Ubuntu? Get the coordinates from ContourPlot and RegionPlot Using $g = \frac{Gm}{r^2}$, the force on a point mass located at 1 AU from the Sun ($m = 2 \cdot 10^{30} \text{ kg}$) is about ~0.006 N/kg.  Does that mean that, e.g., a 70 kg person is ~42g lighter during the day, and ~42g heavier at night? That seems like it could make a big difference for things like measuring gold bars or other weight-sensitive items. (Gold arbitrage: buy your gold during the day and sell it during the night! Risk-free profit!)  This makes me suspect that I'm overlooking something obvious, because a ~0.05% weight difference seems like something everyone would have noticed long ago. So what am I missing?    Edit: A few answers below indicate that there shouldn't be any weight difference, because the Earth orbits the Sun in free-fall. But if that's the reason, does that mean that a 1:1 tidally-locked Earth-Sun system wouldn't experience differential gravity from the Sun on opposite sides? That doesn't seem right. For an ecology project, my lab group added vinegar to 4 tanks containing equal volumes of pond water, 1 control with no elodea (an aquatic plant) and 3 treatments with the same amount of elodea in each. The purpose of adding the vinegar was to reduce the pH. The hypothesis was that the tanks with elodea would go back to their normal pH quicker. This was indeed the case. We measured the pH of each tank daily for about two weeks. All the tanks eventually returned to their natural pH, but the length of time that this took was much shorter for the tanks with elodea.  When we told our professor about our experimental design, he said that there exists no statistical test that can be performed on the data to compare the control to the treatment. That because there was no replicate for the control (we only used one control tank) we cannot calculate variance and so we can't compare the sample means of the control and the treatment. So my question is, is this true? I definitely understand what he means. For example, if you took the height of one man and one woman, you can't draw conclusions about their respective populations. But we did 3 treatments, and the variance was small. It seems reasonable to assume that the variance would be similar in the control?  Update:  Thank you for the excellent answer. We got more water and elodea from the wetland and decided we would run the experiment again with smaller tanks but this time with 5 controls and 5 treatments. We were going to combine this with our original data but the starting pH of the tanks was different enough that it doesn't seem valid to consider the new experiment to be sampled from the same population as the original experiment.  We considered adding different amounts of elodea and trying to correlate speed of pH remediation (measured as time elapsed until pH returned to its original value) with amount of elodea, but we decided that wasn't necessary. Our objective is only to show that the elodea makes a positive difference, not to construct some kind of predictive model for exactly how the pH responds to differing amounts of elodea. It would be interesting to determine the optimal amount of elodea, but that's probably just the maximum amount that can survive. Trying to fit a regression curve to the data wouldn't be especially illuminating because of the various complicated changes that occur to the community when adding a large amount. The elodea dies, decomposes, new organisms start to dominate, and so on. In which cases is a word, or a group of words written in italics? Is italics used in specific contexts, or it is quite normal to write words in italics? I've been using the Linksys USB6300 Adapter but when I switched to Ubuntu it stopped working since its only compatible with Windows. I've tried using ndiswrapper and although it says my windows driver is installed, my device isn't showing up. Anyone know how I could get this wireless network adapter working? Thanks.  lsusb  Bus 002 Device 001: ID 1d6b:0003 Linux Foundation 3.0 root hub Bus 001 Device 012: ID 05c6:6766 Qualcomm, Inc.  Bus 001 Device 006: ID 0bda:0153 Realtek Semiconductor Corp. Mass Storage Device Bus 001 Device 005: ID 05e3:0608 Genesys Logic, Inc. USB-2.0 4-Port HUB Bus 001 Device 004: ID 04ca:007d Lite-On Technology Corp.  Bus 001 Device 003: ID 04ca:008a Lite-On Technology Corp.  Bus 001 Device 010: ID 13b1:003f Linksys  Bus 001 Device 001: ID 1d6b:0002 Linux Foundation 2.0 root hub I just installed Ubuntu 14.04 Server and cannot get wifi configured correctly to work with WPA2 personal and could use some help.    There was a simple wizard during install where I selected my SSID from a list and entered my passphrase and that worked great.  Now that the install is done I am having trouble configuring wifi.  My Access Point is setup to WPA2 Personal TKIP or AES. Any advice would be greatly appreciated.  I have been messing around with WPA supplicant ant my /etc/network/interfaces file with no luck.  Thanks Dedicated mysql server slow response Find the condition on $a$ and $b$ so that the two tangents drawn to the parabola $y^2=4ax$ from a point are normals to the parabola $x^2=4by$ Find the condition on $a$ and $b$ so that the two tangents drawn to the parabola $y^2=4ax$ from a point are normals to the parabola $x^2=4by.$  The required condition is $a^2&gt;8b^2$.I dont know how to prove it.I tried.  Let $(h,k)$ be the point from where tangents are drawn to the parabola $y^2=4ax$.Let $m_1,m_2$ be the slopes of the tangents.Equations of tangents are $y=m_1x+\frac{a}{m_1}$ and $y=m_2x+\frac{a}{m_2}$ and equations of normal to the parabola $x^2=4by$ is $y=m_1x-2bm_1-bm^3_1$ and $y=m_2x-2bm_2-bm^3_2$.  As $y=m_1x+\frac{a}{m_1}$ and $y=m_1x-2bm_1-bm^3_1$  are the same lines.Therefore,$\frac{a}{m_1}=-2bm_1-bm^3_1$  $bm^4+2bm^2+a=0$  And I am stuck.I cannot get the desired condition.Please help me. Prove a graph Containing $2k$ odd vertices contains $k$ distinct trails Algorithm for minimization of sum of squares in regression packages Is it acceptable to begin a declarative sentence with "Am"? Homework - set theory infinite union A question from my homework I'm having trouble understanding. We are given:  $A(1) = \{\varnothing\}$, $A(n+1) = A(n)\cup (A(n)\times A(n))$  $A=A(1)\cup A(2)\cup A(3)\cup \cdots \cup A(n)\cup A(n+1) \cup \cdots$ to infinity  The questions are:  1) show that $A\times A \subseteq A$  2) Is $A \times A = A$?  Thank you for your help.  I've tried writing $A(2)$ but it gets really complicated and I'm having trouble understanding what the sets are. Let alone solve the question. Determinant of an $n\times n$ complex matrix as an $2n\times 2n$ real determinant If $A$ is an $n\times n$ complex matrix. Is it possible to write $\vert \det A\vert^2$ as a $2n\times 2n$ matrix with blocks containing the real and imaginary parts of $A$?  I remember seeing such a formula, but can not remember where. Any details, (and possibly references) for such a result would be greatly appreciated.