I have the following problem when I want use minimum  among some numbers.   $\min_{1}$ works well but   $$\min_{1} $$ or    \begin{eqnarray*} \min_{1}  \end{eqnarray*}   are looking very odd.
121356
I am wondering how can I ensure that an argument gets placed underneath \min when I use an align environment.   When I use math mode and write   \[ min_{x_1,\dots x_T}  \]   the {x_1,\dots x_T} is placed directly underneath min.  However when I use an align environment and write  \begin{align*}   min_{x_1\dots x_T}  \end{align*}   the {x_1,\dots x_T} is placed to the bottom left of \min.
49230
I want to write this equation in this way down but I have been unsuccessful. How can I improve this codes in order to get the equation below?  \[ DeltaC{t} = \left\{ \begin{array}{ll}  -1.40- 0.588w_{t-1} - 0.5351\DeltaC_{t-1}+ 0.169\DeltaBC_{t-1} &amp; \mbox{ if $w_{t-1}&lt;- 0.62$} \\ -1.40 - 0.588w_{t-1} - 0.5351\DeltaC_{t-1} + 0.169\DeltaBC_{t-1} &amp; \mbox{$w_{t-1}&gt;0.62$} \end{array} \right. \]
12628
Is there a benefit to using one over the other? In Python 2, they both seem to return the same results:  &gt;&gt;&gt; 6/3 2 &gt;&gt;&gt; 6//3 2
67825
I want to get the following:    I tried the following  \[    a(i,k) \leftarrow     \min\Big\{0, r(k,k) + \sum_{i^{'} | i^{'}\notin \{i,k\}}\max\{0, r(i{'},k)\}\Big\} \]   and got    How can I make the part under the summation extend to left and right without making a space between plus and max as in the first figure?
97642
When $$ \min_{x \in X,y \in Y} f(x,y) = \min_{x \in X} \min_{y \in Y} f(x,y) \qquad? $$  I mean when we are minimizing a function with respect to two variables, under what conditions we are allowed to minimize over one variable and then minimize over the other one?
94030
I f $$A= \begin{pmatrix}1&amp; 0 &amp; 0 \\  1 &amp; 0 &amp; 1\\ 0 &amp; 1 &amp; 0 \end{pmatrix}$$  then $ A^{50} $  is   $$ \begin{pmatrix}1&amp; 0 &amp; 0 \\  50 &amp; 1 &amp; 0\\ 50 &amp; 0 &amp; 1 \end{pmatrix}$$ $$\begin{pmatrix}1&amp; 0 &amp; 0 \\  48 &amp; 1 &amp; 0\\ 48 &amp; 0 &amp; 1 \end{pmatrix}$$ $$\begin{pmatrix}1&amp; 0 &amp; 0 \\  25 &amp; 1 &amp; 0\\ 25 &amp; 0 &amp; 1 \end{pmatrix}$$ $$\begin{pmatrix}1&amp; 0 &amp; 0 \\  24 &amp; 1 &amp; 0\\ 24 &amp; 0 &amp; 1\end{pmatrix}$$   I am stuck on this problem. Can anyone help me please...............
107389
What is the minimum of $$f(a,b,c):=\sqrt{\dfrac{a}{b+c}}+\sqrt{\dfrac{b}{c+a}}+\sqrt{\dfrac{c}{a+b}}$$ where $a,b,c$ are positive real numbers?  When $a=b=c$, we have $f(a,b,c)=\dfrac{3}{\sqrt{2}}\approx 2.12$  When $a=1,b=c\rightarrow\infty$, we have $f(a,b,c)\rightarrow 2$. So the minimum is at most $2$.
52067
I'm looking for someone to walk me through how to solve E[ZW] if Z = max[X,Y] &amp; W = min[X,Y]. X &amp; Y are independent, uniform variables over the interval [0,1].  I solved E[Z] = 2/3 and E[W] = 1/3.   E[ZW] should equal 1/4 but I keep getting the wrong answer. I have searched the forums and used other people's answers to try and solve it but have gotten the incorrect numbers.  Thank you in advance!
15740
Ratio and root tests won't help. And I can't use the comparison test because $ |a_n| $ is not necessarily smaller than $ n^2a_n $.   Can I use limits? We know:    $\lim\limits_{n \to \infty} a_n = 0 $ $\lim\limits_{n \to \infty} |a_n| \ne 0 $   And we need to prove:    $\lim\limits_{n \to \infty} n^2a_n \ne 0 $   Any ideas/hints?
105944
Show that the condition number of an invertible matrix must be at least 1. What matrices have  condition number equal to 1.  If someone could help me with this and give an explanation that would be very helpful. I do not know where to start
121232
I have this tex: \(\mu_{A}\left(x\right)\in \[0,1  \right\]\)  The issue here is that [ and ] are not showing around 0,1, why is that?  Thanks.
99506
How to make several lines of equations under one curly brace from the left side, such that it point to a reference number.  Now I do something like this:  \begin{align} \left. \begin{array}{ll} a = b + c \\ &amp; \dots  \\ a = b + c \\ &amp; \dots  \\ a = b + c  \end{array} \right\} \label{formulas}\\ \end{align}   But equation number appears in the bottom of curly brace.
69405
If $x&gt;0$, $y&gt;0$, $z&gt;0$ and $x+y+z=6$ then find the minimum value of    $$ f(x,y,z)=\left(x+\frac{1}{y}\right)^2+\left(y+\frac{1}{z}\right)^2+\left(z+\frac{1}{x}\right)^2. $$   Thanks in advance.
49109
Suppose we have a set $S$ of real numbers. Show that   $$\sum_{s\in S}|s-x| $$   is minimal if $x$ is equal to the median.    This is a sample exam question of one of the exams that I need to take and I don't know how to proceed.
68438
I want to write this:    I tried with $x_n \to_{n \to 0} 0$ but the $n \to 0$ doesn't go under the right arrow, it goes under but on the right, and I want it under but in the center, as in the picture above.
120809
So, I'm writing a mathematics paper, and let's say I have the following:  \begin{equation*} 2+2=4 \end{equation*}   Is this the proper way to do an unnumbered equation, or is it preferred to do the following:  \[ 2+2=4 \]   My question really is: what's the difference between doing \begin{equation*} and \begin{equation*} versus doing \[ \]? Is there a setting in which one may be preferred over the other?  Thanks in advance.
79935
The question is as in the title: what's the minimum of $\int_0^1 f(x)^2 \: dx$, subject to $\int_0^1 f(x) \: dx = 0, \int_0^1 x f(x) \: dx = 1$? (Assume suitable smoothness conditions.)  A problem in the textbook for the course I am TEACHING (not taking) reduces to minimizing $w_1^2 + w_2^2 + w_3^2$ subject to $w_1 + w_2 + w_3 = 0, w_1 + 2w_2 + 3w_3 = 1$. Of course there's nothing special about the number $3$ here, and so one can ask for the minimum of $\sum_{i=1}^n w_i^2$ subject to $\sum_{i=1}^n w_i = 0, \sum_{i=1}^n iw_i = 1$. At least when $n = 3, 4, 5$, we get $w_i = c_n(i-(n+1)/2)$, for some constant $c_n$ which depends on $n$. So for fixed $n$, $w_i$ is a linear function of $i$. (This is a bit of an annoying computation, so I won't reproduce it here.)  So it seems like there should be a continuous analogue of this. If we have  $$ \int_0^1 f(x) \: dx = 0, \int_0^1 x f(x) \: dx = 1 $$ and $f(x)$ is linear, then we get $f(x) = 12(x-1/2)$, and $\int_0^1 f(x)^2 \: dx = 12$.  Is this the function satisfying these integral conditions with smallest $\int_0^1 f(x)^2 \: dx$? That is, is it the case that $$ \int_0^1 f(x)^2 \: dx \ge 12 $$ for every $f(x)$ satisfying the two conditions above and whatever smoothness conditions are necessary?  I've tagged this calculus-of-variations because that's what it looks like to me. But I don't know the calculus of variations, which is why I can't just solve the problem myself.