Are owe and due are the opposites?
Do "owed" and "due" mean the opposite?
When should you use a class vs a struct in C++? In what scenarios is it better to use a struct vs a class in C++? I checked this  but didn't fully understand it.   I know that the volume of a right circular cone is $V = \frac{1}{3}\pi x^2h$  I know that I must take the first derivative and set it equal to zero, which will find the maximum.   My problem is how to deal with the variable $h$, the height?   How can I rewrite that in terms of $x$ or $r$.      I have tried to implicitly differentiate this equation, but that wasn't helpful for me. Why is log-of-sum-of-exponentials $f(x)=\log\left(\sum_{i=1}^n e^ {x_i}\right)$ a convex function for $x \in\mathbb R^n$? How to prove $f(x)=\log\left(\displaystyle\sum_{i=1}^n e^{x_i}\right)$ is a convex function?  EDIT1: for above function $f(x)$, following inequalities hold:  $$\max\{x_1,x_2,\ldots,x_n\}\leqslant f(x)\leqslant\max\{x_1,x_2,\ldots,x_n\}+\log n$$  and I have tried proving its convexity via definition of a convex function with above inequalities, but that didn't work.  EDIT2: I have posted my answers below. Which rational numbers can be written as the sum of two rational squares?  That is, for which rational numbers $a$, are there rational numbers $x$ and $y$ such that $a = x^2 + y^2$.  It is a famous theorem that if an integer can be written as the sum of two rational squares then it can be written as the sum of two integral squares, and then the solution is the famous one by Fermat, but I didn't find anything about the general case. How to find sum of power series $\sum_{n=0}^\infty\frac{1}{n!(n+2)}$ by differentiation and integration? Let $$ S = \sum_{n=0}^\infty \frac{1}{n!(n+2)} $$  Integrate the Taylor Series of $xe^x$ to show that S = 1  Also, Differentiate the Taylor series of $\frac{e^x - 1}{x}$ to show that S = 1  For the integration one, I got $\int xe^x dx$ = $x^2\sum_{n=0}^\infty \frac{1}{n!(n+2)} x^n$ and pretty much stuck. I can see the S term but i have no idea how to move on from there To prove in a Group Left identity and left inverse implies right identity and right inverse Let $G$ be a nonempty set closed under an associative product, which in addition satisfies :      A. There exists an $e$ in $G$ such that $a \cdot e=a$ for all $a \in G$.      B. Given $a \in G$, there exists an element $y(a) \in G$ such that $a \cdot y(a) =e$.      Prove that $G$ must be a group under this product.   Attempt -Since Associativity is given and Closure also, also the right identity and right inverse is given .So i just have to prove left identity and left inverse.  Now as $ae=a$ post multiplying by a, $aea=aa$. Now pre multiply by a^{-1} I get hence $ea=a$. And doing same process for inverse Is this Right? I have a database which contains 5 tables. Each table contains 24 rows and each row contains 4 columns.  I want to display these records in Excel sheet. The heading of each table is the name of the table, but I am unable to merge the columns for heading.  Please help me. How to prove that a continuous function, homotopic to the constant map $f:S^1\to S^1$  (a) has a fixed point and that (b) has a point $x$, such that $f$ maps $x$ to its antipodal point $-x$? Override a method at instance level Calculating p-values for two tail test for population variance A sporting goods manufacturer claims that the variance of string tensions for any decent tennis racquet should be about 9 pounds. The string tensions of 18 randomly selected tennis racquets produced a variance of 8.13 pounds. Find the p-value to test the manufacturer's claim (assume the population is normally distributed).  I tried attempting the question as follows: ${H_o}: {\sigma ^2} = 9$ ${H_1}: {\sigma ^2} \ne 9$ We are also given the following: $n=18$ $s^2 = 8.13$ Test Statistics: $\chi^2 = {{(n-1)S^2}\over{\sigma^2}}$ Calculating Test Statistics under $H_o$ gives us: $\chi^2 = {{(18-1)8.13}\over 9} = 15.35667$  For calculating the p-value I proceeded as follows: $p-value = 2\min\{P[\chi^2&gt;15.35667],\ P[\chi^2&lt;15.35667]\}\ =2\times 0.430192 = 0.860384$  I am not certain if my method for calculating the p-value is correct or not. Can someone please tell if the above process is valid?  Thanks in advance! I want to display from cache for a long time and I want a slightly different behavior on page render vs loading the page from cache.  Is there an easy way I can determine this with JavaScript? Concatenating numbers and adding in leading zeros? I have three text fields populated as follows: 1, 9, 1 that I need to concatenate into a field that looks like 01.09.01.  Getting to 1.9.1 is easy but can anyone help out with how to formulate it so that each value has two places (i.e. add in those zeroes).  I used to do this in Excel but can't figure out how to work in ArcMap 10. How to create text file and insert data to that file on Android Send authors an inbox message if their question is closed Algorithm to match one input file with given numbers of file Iterating over array of objects javascript - odd behaviour? Error: mysqli_stmt::bind_param(): Number of elements in type definition string doesn't match number of bind variables A number of times I have encountered in text-books and articles that neutrinos might contribute only a small fraction to dark matter. The reason has to do with the fact that if all of the dark matter consisted of neutrinos, then small-scale structures in the Universe could not have formed yet, because, as they say, neutrinos "wash out" small fluctuations. However, none of these texts provided a reference to any specific sources explaining in detail what is meant by "washing out". After all, neutrinos are notorious in their weak interaction with baryonic matter, so if there is a small-scale fluctuation of baryons, then how background neutrinos can prevent it from growing further if they practically do not interact with baryons? I guess the question boils down to calculating cross-sections of interactions at specific temperatures. I would appreciate comments and references to sources addressing this particular issue. Fixed page header overlaps in-page anchors How to load up CSS files using Javascript? Is it possible to import css stylesheets into a html page using Javascript? If so, how can it be done?  P.S the javascript will be hosted on my site, but I want users to be able to put in the &lt;head&gt; tag of their website, and it should be able to import a css file hosted on my server into the current web page. (both the css file and the javascript file will be hosted on my server).