Counter example to $b_1^c : b_2^c \subset (b_1 : b_2)^c$
Contraction of quotient ideal is quotient of contractions?
On a series involving harmonic numbers Finding $$\sum^{\infty}_{n=1}\bigg[\bigg(\frac{(-1)^{n+1}}{n+1}\bigg)\bigg(\sum^{n}_{r=1}\frac{1}{r}\bigg)\bigg]$$   Try: Let $\displaystyle H_{n} =\sum^{n}_{r=1}\frac{1}{r},$ then series $\displaystyle \sum^{\infty}_{n=1}\frac{(-1)^{n+1}}{n+1}\cdot H_{n}=\frac{H_{1}}{2}-\frac{H_{2}}{3}+\frac{H_{3}}{4}\cdots\cdots$  could some help me to solve it, thanks How to understand Cauchy's proof of AM-GM inequality(the last step) I am using a plugin to help "build" a customized product. There are different tabs/steps the user has to go through with back and continue buttons. At the last step I would like to add an "agree to terms" box with a checkbox "Agree". I have this done. But I would like to hide the add to cart button in Woocommerce until the user selects the checkbox agreeing to the terms.  This will be beneficial to me to make sure they reviewed the return policy as well as avoid confusion with the add to cart button appearing during all the steps, assuring they go through all the steps.  Here is the checkbox coding i pulled from console:  &lt;ul data-rules="{&amp;quot;Agree_0&amp;quot;:[&amp;quot;&amp;quot;]}" data-original-rules="{&amp;quot;Agree_0&amp;quot;:[&amp;quot;&amp;quot;]}" data-rulestype="{&amp;quot;Agree_0&amp;quot;:[&amp;quot;&amp;quot;]}" data-tm-validation="{&amp;quot;required&amp;quot;:true}" class="tmcp-ul-wrap tmcp-elements tm-extra-product-options-checkbox tm-element-ul-checkbox element_16 termsbox-ul"&gt;     &lt;li class="tmcp-field-wrap tmhexcolorimage-li-nowh"&gt;         &lt;label for="tmcp_choice_16_0_35"&gt;    &lt;input class="tmcp-field termsbox tmhexcolor_16_0_35 tm-epo-field tmcp-checkbox" name="tmcp_checkbox_16_0" data-limit="" data-exactlimit="" data-minimumlimit="" data-image="" data-imagec="" data-imagep="" data-imagel="" data-image-variations="[]" data-price="" data-rules="[&amp;quot;&amp;quot;]" data-original-rules="[&amp;quot;&amp;quot;]" data-rulestype="[&amp;quot;&amp;quot;]" value="Agree_0" id="tmcp_choice_16_0_35" tabindex="35" type="checkbox"&gt;     &lt;span for="tmcp_choice_16_0_35"&gt;&lt;/span&gt;&lt;span class="tc-label tm-label"&gt;Agree&lt;/span&gt;&lt;/label&gt; &lt;span class="price tc-price  hidden"&gt;     &lt;span class="amount"&gt;$0.00&lt;/span&gt; &lt;/span&gt;&lt;i data-tm-tooltip-html="This is were the description will be to agree" class="tm-tooltip tc-tooltip tcfa tcfa-question-circle"&gt;&lt;/i&gt;     &lt;/li&gt;&lt;/ul&gt;   Here is the add to cart section with the quantity  &lt;div class="quantity buttons_added"&gt;     &lt;input value="-" class="minus button is-form" type="button"&gt;    &lt;input class="input-text qty text" step="1" min="1" max="9999" name="quantity" value="1" title="Qty" size="4" pattern="[0-9]*" inputmode="numeric" type="number"&gt;     &lt;input value="+" class="plus button is-form" type="button"&gt;  &lt;/div&gt; &lt;button type="submit" name="add-to-cart" value="267" class="single_add_to_cart_button button alt"&gt;Add to cart&lt;/button&gt;   I am self taught with html and css, but javascript is to foreign for me  I tried the following in CSS but it does nothing.Tried is on codepen and works (i know needs to show instead of hide when checked)....  input[type=checkbox]:checked + .single_add_to_cart_button {     display:none !important;  } Why cannot a set be its own element? Which characters are valid in CSS class names/selectors? In windows we have an option to view recently opened media files as my recent documents, I was  wondering does there exists something like that in Ubuntu?  via terminal or via GUI? How do I set a background image while I am editing? How do I put a backdrop behind the model that I am creating so that I can compare the picture to the model while editing? Proving $v(s,p)=2^{p-1}(2s-1)$ is a bijection of natural numbers and $f(s)=2s-1$ is a bijection between natural numbers and odd numbers. How do I prove this function is bijective? $$ v(s,p)=2^{p-1}(2s-1). $$ The domain is natural numbers and the codomain is also the natural numbers.  And this one: $$ f(s)=2s-1. $$ The domain is the natural numbers, and the codomain is the odd numbers in the natural numbers.  With this one I would do this to show it's injective: $$ \begin{align} v(s)&amp;=v(s_1)\\ \implies 2s-1&amp;=2s_1-1\\ \implies (2s_1)/2&amp;=(2s_2)/2\\ \implies s=s_1 \end{align} $$ So it's injective since if $v(s)=v(s_1)$ then $s=s_1$.  And to show it's surjective $f(s)=y$: $$ \begin{align} y&amp;=2s-1\\ \implies s&amp;=(y+1)/2 \end{align} $$ Then the function must be surjective since every $y$ is the same as the codomain for $f$.  Am I correct? What happens if you use a Clone Effect on a Double Faced Card that's Transformed? Say someone has , which is a 0/3 Defender, and after some mechanics, transforms into a 13/13 Lizard Horror with Trample.  Say this someone managed to transform said card into its 13/13 Trample form, and then cast  on it (or any other clone type card if it would matter between different ones). Does the Metamorph/Clone/etc. come into play transformed, or as the 0/3 Defender? Allow rejected editors to be @replied to Checking for valid floating point number Recently, I was wondering how division of aleph numbers would work. First, I thought about how finite cardinality division would work. What I came up with was that the result of $A/B$ where $A$ and $B$ are both cardinalities, is the number of times that each element of $B$ had to be mapped to an element of $A$ in order to ensure that all elements of $A$ were mapped to.  Extending this to aleph numbers, specifically, I thought that $\aleph_1/\aleph_0$ would be $\aleph_1$. The reasoning behind this is that there are an infinite ($\aleph_1$) number of real numbers between any two natural numbers. As such, that mapping would need to be applied.  Can anyone validate this idea? Is this how division of aleph numbers actually works, or am I totally off base?  Thanks. I'm working from the URL I found here:    My ssh client is Ubuntu 64 bit 11.10 desktop and my server is Centos 6.2 64 bit. I have followed the directions. I still get a password prompt on ssh.  I'm not sure what to do next. Bound for the correlation of three random variables There are three random variables, $x,y,z$. The three correlations between the three variables are the same. That is,  $$\rho=\textrm{cor}(x,y)=\textrm{cor}(x,z)=\textrm{cor}(y,z)$$  What is the tightest bound you can give for $\rho$? How is evolution possible in contemporary humans? I would like to know if evolution is continuing to happen in modern humans, assuming things like existence of the nuclear family structure, fidelity to one partner, etc. It seems to me the answer would be NO because evolution depends on differential reproductive rates, but in the modern world, all male humans have roughly 2.5 (or whatever the number) kids. Add in the process of culturally modified selection pressure, and it seems to me that even an "unfit" male would end up having a couple of offspring. The fittest male (or female) is no better off than his or her contemporaries because of this "leveling" effect.   However, the impression I get from the popular science media is that scientists think evolution is continuing to happen. I would like to know what the actual scientific consensus is, and why. Thanks. Splitting pieces of a mesh into a new object I've just made a set of armor by using a male body mesh model I downloaded and selecting certain faces from the model and duplicating them to create viable armor pieces.   Since the pieces are just duplicated from the same mesh male model, then it's still part of the same object.   How do I separate the armor piece from the model to make the pieces separate? Convergence test of the series $\sum\sin100n$ I need to prove that $$\sum_{n=1}^{\infty} {\sin{100n}} \; \text{diverges}$$ I think the best way to do it is to show that $\lim_{n\to \infty}{\sin{100n}}\not=0$. But how do I prove it? Proof that the real numbers are countable: Help with why this is wrong I was just thinking about this recently, and I thought of a possible bijection between the natural numbers and the real numbers.  First, take the numbers between zero and one, exclusive.  The following sequence of real numbers is suggested so that we have bijection.    0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 0.01, 0.02, ... , 0.09, 0.10, 0.11, ... , 0.99, 0.001, 0.002, ... , 0.999, 0.0001, etc.  Obviously, this includes repeats, but this set is countable.  Therefore, the set of all numbers between zero and one is a subset of the above countable set, and is thus countable.  Then we simply extend this to all real numbers and all the whole numbers themselves, and since the real numbers, as demonstrated above, between any two whole numbers is countable, the real numbers are the union of countably many countable sets, and thus the real numbers are countable.    Please help me with this.  I understand the diagonalization argument by Cantor, but I am curious specifically about this proof which I thought of and its strengths and flaws.  Thanks. Find a file when you know its checksum? Parentheses vs. double commas vs. dashes to provide additional detail When do you use commas and when do you use parentheses to provide more detail about something?  For example:   The suspect, Tom Wilson, is now being charged with murder. The suspect (Tom Wilson) is now being charged with murder. John Smith, a member of the jury, agreed with the verdict. John Smith (a member of the jury) agreed with the verdict. The first example, the one about the fox, is the more valid one. The first example (the one about the fox) is the more valid one.   I realize that the first example, the one that uses parentheses, is probably more valid in this case (if anyone has a better example, please include it), but when, if ever, would you favor parentheses?  Update: I realize that dashes are sometimes used for similar situations, so I'd appreciate information on these as well.  For example:   The children—who are wearing shoes—will go on the trip The children, who are wearing shoes, will go on the trip The children (who are wearing shoes) will go on the trip