Prove that $f$ is convex if and only if $f\left(\frac{a+b}{2}\right)\leq \frac{1}{b-a}\int_a^b f(x) dx$
Characterization convex function.
Proving there are infinitely many pairs of square-full consecutive integers So the question goes like this :  A positive integer $n$ is called square-full if for every prime $p$, $p \, | \, n$ implies $p^2 \, | \, n$, i.e. every prime power factor of $n$ occurs at least at the second power. This is equivalent to saying that there is no prime $p$ such that $p \, \| \, n$.  Show that there are infinitely many consecutive pairs of square-full numbers, i.e. infinitely many $n$ such that $n$ and $n+1$ are square-full.  The context of the question highly suggests that this can be proved by induction and it is the solution that I am looking for. It is not a homework, just something that caught my attention because I couldn't find any natural answer to it... I hope I'm not missing something trivial!  I've tried assuming $n_1, n_2, \dots, n_k$ were the first $n_k$ integers such that $n_i$ and $n_i+1$ are square-full, and then generate some polynomial in those $2k$ integers to get a new couple $n_{k+1}$, $n_{k+1} + 1$ but after a few natural tries I got nothing good out of it. Any ideas? Pre-declare all private/local variables? How to custom (split in two)? Ibidem Is it correct to say "12:00am"? I've read in various places the first minute of a day described as 12:00am.  Now, whilst I personally prefer to use 24h clock notation and therefore don't have this problem as I can simply describe this minute as 00:00, how is one supposed to express this time using am/pm notation?  Is 00:00am common (strange because there is no 00:00pm), or is the least-bad way to say 12:00am (strange IMHO because it seems to imply that the previous hour was 11:00am when it was actually 11:00pm)? My HP Pavilion laptop has Ubuntu 18 version and I once tried to disable InBuilt webcam just for security reasons. But now it wasn't working and I reinstalled newer version of Ubuntu 20. But even after installing newer version the camera is still not working. When I open cheese it shows this screen. I assumed I might have deleted the drivers as I was doing package installation and removel that time. Please suggest me what should I do to get this working. A bold plus symbol in math mode Which cyan colored line is produced in the Thomson e/m apparatus? Related:   After reading Lisa Lee’s OP on an electron deflection tube, although she had some misunderstandings on its operation, I still believe that her question is still relevant. If one looks at the Thomson e/m tube filled with helium gas (made by ), the tracer light is created by de-excited electrons, not by an electron beam interacting with a phosphor as in Lisa Lee's OP. That is, the accelerated electrons scatters off of a low pressure helium gas and emit a tracer line to follow the electrons path, as shown below.     Now if you look at the bright line spectrum of helium,     One can clearly see that there are two “cyan” color lines around 500 nm. Following in Lisa Lee’s footsteps, there is a series of question one can ask: (1) which line is emitted by the Helium gas atoms? Or are both lines emitted? (2) How can a variable accelerating voltage for the electrons always produce the same tracer line color? In theory, I assume that if I decreased the energy of the electron beam, I could produce red (around 670 nm) or yellow (around 590 nm) tracer lines, but that doesn’t happen. Somehow, it appears to me that the e/m tube is “tuned” just right so that the emitted light is always this “cyan” color. Why? Given an alphabet of size $n$, how many strings of length $c$ contain every single letter of the alphabet at least once?  I first attempted to use a recurrence relation to work it out:  $$ T(c) = \left\{ \begin{array}{cr} 0 &amp;\mbox{ if $c&lt;n$} \\ n! &amp;\mbox{ if $c = n$} \\ T(c-1) \cdot n \cdot c &amp;\mbox{ if $c &gt; n$} \end{array} \right. $$  As there's no strings that contain every letter if c &lt; n, and if c = n then it's just all permutations. When c > n you can take any string of size (c-1) that contains all letters (of which there are $T(c-1)$ to choose from), you choose which letter to add (of which there are $n$ choices) and there are $c$ different positions to put it. However, this gives out results that are larger than $n^c$ (the total number of strings), so it can't be right, and I realised it was because you could count some strings multiple times, as you can make them taking different inserting steps.   Then I thought about being simpler: you choose n positions in the string, put each letter of the alphabet in one of those positions, then let the rest of the string be anything:  $$ {c\choose{n}} \cdot n! \cdot n^{c-n} $$  But again this counts strings multiple times.  I've also considered using multinomial coefficients, but as we don't know how many times each letter appears in the string it seems unlikely they would be much help. I've also tried several other methods, some complicated and some simple, but none of them seem to work.  How would you go about working out a formula for this? I'm sure there's something simple that I'm missing. Let $q \in \mathbb C$, $|q|=1$ and $q^n \neq 1, \forall n \in \mathbb N$. Show that $\{q^n:  n \in \mathbb N\}$ is dense in $S^1$.  My attempt: As $(q^n)$ is limited, there is subsequence $(q^{n_j})$ convergent. So given $\epsilon &gt; 0, \exists j_0 \in \mathbb N, \quad |q^{n_j} - q^{n_k}| &lt; \epsilon, \quad \forall j,k \geq j_0$. Supose that $n_j &gt; n_k$, then  $|q^{n_j} - q^{n_k}| = |q^{n_k}||q^{n_j - n_k} - 1| = |q^{n_j - n_k} - 1| &lt; \epsilon, \quad \forall j, k \geq j_0$. So, there is a subsequence of $(q^n)$ that converges to $1$.  Using this fact, how can I ensure that for all $w \in S^1$ there is a subsequence of $(q^n)$ that converges to  $w$? Prove that:$\int_{a}^{b} f(x) dx = b \cdot f(b) - a \cdot f(a) - \int_{f(a)}^{f(b)} f^{-1}(x) dx$ Calculate $ \int_{-2}^{0} \frac{x}{\sqrt{e^x+(x+2)^2}} dx $ $ \int_{-2}^{0} \frac{x}{\sqrt{e^x+(x+2)^2}} dx $ = -$  2.887270... $  The function has no antiderivate and there's no symmetry here to help, if you solve this I would be thankful if you would for a highschooler ^^ Why is it dangerous use a coiled extension cord Isomorphism of factor groups I came across an intuitive statement on factor groups that I realized I did not know how to prove.   Let $G$ be a group, and let $H \unlhd G, K \unlhd G$ such that $H \cong K$. I want to show that $G/H \cong G/K$.   I'm not sure how exactly to construct such an isomorphism from an isomorphism $\varphi: H \to K$. Is that the correct approach? By extending such an isomorphism to an automorphism on $\varphi^{\uparrow}:G \to G$, then we could apply the First Isomorphism Theorem to the composition of the natural homomorphism $\mu: G \to G/K$ and that automorphism (for $\ker \mu\varphi^{\uparrow} = H$, if such a $\varphi^\uparrow$ exists).  Edit: Hamou points out this is false in the infinite case: can it still be salvaged in the finite case? Are there restrictions we can put on $G, H,$ and $K$ such that this is true? Combining php arrays I have the following arrays in PHP (okay they are a bit bigger but the idea is what counts).  $array1 = array(1 =&gt; 'a', 2 =&gt; 'b'); $array2 = array(3 =&gt; 'c', 4 =&gt; 'd');   Essentially I want to combine the two arrays as if it were something like this  $array3 = array(1 =&gt; 'a', 2 =&gt; 'b', 3 =&gt; 'c', 4 =&gt; 'd');   Thanks How to get a function name as a string? In Python, how do I get a function name as a string, without calling the function?  def my_function():     pass  print get_function_name_as_string(my_function) # my_function is not in quotes   should output "my_function".  Is such function available in Python? If not, any ideas on how to implement get_function_name_as_string, in Python? How to save user-entered line breaks from a TextArea to a database? How do I save user-entered line breaks from a &lt;textarea&gt; HTML element to a database?  It always removes the line breaks. What happens to half of the energy in a circuit with a capacitor? For a simple circuit with a battery supplying a voltage $V$ to a capacitor, let us assume that the charge on the capacitor is $Q$. Now, the work done by the battery or the energy supplied is given by the relation:  $$W=QV$$  But the energy stored in the capacitor is given by:  $$U = \tfrac12 QV$$  The value of $Q$ as well as that of $V$ should be the same in both the equations.  Now my question is, where is the other half of the energy that the battery supplied? Reshape three column data frame to matrix ("long" to "wide" format) How to completely remove Notepad++ in Ubuntu 18.04 terminal?  I can't find it in Synaptic package manager, and don't know where is on system, but I can see an icon in Activity. It is probably not completely installed, because I can't open it. How do we know that certain quantum effects are random?