Hard edges during smoothing
Custom hard edges in Blender
Can somebody help me understanding the fouries series coefficients?  I know that if we have:  $$f(t) = \sum_{n=1}^N A_n \sin(2\pi nt + Ph_n) \tag{where $Ph_n$ = phase}$$  And because of the $\sin(a+b)$ formula: $$ f(t) = \sin(2\pi nt + Ph) = \sin(2\pi nt )\cos(Ph_n) + \cos(2\pi nt )\sin(Ph_n)$$ Then:  $$\sum_{n=1}^N A_n \sin(2\pi nt + Ph) = \sum_{n=1}^N (A_n\sin(2\pi nt )\cos(Ph_n) + A_n\cos(2\pi nt )\sin(Ph_n))$$ And by definition: $$a_n = A_n\cos(Ph_n)$$ $$b_n = A_n\sin(Ph_n)$$ Then:  $$f(t) = \sum_{n=1}^N A_n \sin(2\pi nt + Ph) = \sum_{n=1}^N (a_n\sin(2\pi nt ) + b_n\cos(2\pi nt ))$$ Where does the $A_n$ in the first formula come from? Could somebody explain me where does the $\frac{a_0}{2}$ in the formula below come from? $$f(x) = \frac{a_0}{2}+\sum_{n=1}^\infty (a_n \cos(nx) + b_n \sin(nx))$$  And the most important:  How can I proof the formulas for the $a_n$ and $b_n$ coefficients?  PLEASE, I'm searchinf for these answers for DAYS. Prove that $r \in R$ such that $0 \lt r \lt 1$ then $\frac{1}{r (1 - r)} \ge 4$ How can one predict whether an individual infected with the SARS-CoV-2 virus will be asymptomatic? According to  , half of the   virus carriers show no symptoms.   What are the biological factors that determine whether an infected individual will be asymptomatic or not? 16.04 Upgrade: Failed to fetch Empty files can't be valid archives Is using only HTTPS bad in a SEO perspective? I would like to set up a new business site. I plan to direct www.example.com to example.com and only use that domain. My plan is also to only use HTTPS and not HTTP.   Does this affect the SEO for the site? E.g. do Google put lower or higher rank for sites that is only available via HTTPS? I was recently told (  ):     You cannot delete a class in production directly. You will need delete   the class from your sandbox and then deploy the deletions to your   production org.   When you deploy from sandbox to production, the   missing classes will come up in red and you can choose to deploy these   deletions to Production.   However, I'd like some clarification because finding information on how to do this is like pulling teeth, unfortunately.  Am I supposed to:   Delete the Trigger in Sandbox Make a deployment package in Sandbox Add the Trigger which is going to be listed in RED now Deploy (and that deletes it from Production)?   I am not seeing deleted Triggers in Red in Sandbox.  Are they NOT listed under "Triggers" for deployment, but elsewhere?   Frustrated = true.   Does this ONLY apply to Classes??  UPDATE:  I just tried again, with a CLASS, and it still doesn't show up at all as a Deployable package (red or not) after I delete it in the Sandbox.  UPDATE 2:  I was just able to delete a class, that I know was written via IDE from a third party, in both Sandbox and Production.  So now I'm wondering if it has to do with where it was originally deployed from (if from IDE, IDE can delete it - it from Sandbox, you have to jump through hoops).  :-( Can Objective C determine the caller object reference at runtime (reflection)? What's that ship in the opening credits of 'Star Trek: Enterprise'? Why is gradient the direction of steepest ascent? $$f(x_1,x_2,...x_n):\mathbb{R}^n \rightarrow \mathbb{R}$$ The definition of the gradient is  $$ \frac{\partial f}{\partial x_1}\hat{e}_1 +\  ... +\frac{\partial f}{\partial x_n}\hat{e}_n$$  which is a vector.  Reading this definition makes me consider that each component of the gradient corresponds to the rate of change with respect to my objective function if I go along with the direction $\hat{e}_i$.  But I can't see why this vector (defined by the definition of the gradient) has anything to do with the steepest descent.  Why do I get maximal value again if I move along with the direction of gradient? How do I install fonts? Question says it all. I need them to be available in gimp. Blackboard bold variants for Greek letters It is common (or it used to be) to use a blackboard-bold variant of \mu to denote roots of unity in some mathematical papers.  However, despite searching (several times, periodically over several years) I have never been able to locate any package or technique that makes this possible, other than using Metafont or possibly hunting through the font definitions to see if it is something I can create a variant of (along the lines of the suggestions in Section 8 of the Comprehensive LaTeX symbol list).  Thus far, I haven't been sufficiently motivated to learn how to do any of this (although it probably would have resulted in a net time savings), and I feel like I cannot be the only person who would have tried.  Is anyone aware of an existing way to produce this character? In my book about Logic, which is called 'Language, Proof and Logic', by the way, there is explained that the conditional $ P \rightarrow Q $ is equivalent with $\neg P \lor Q$.   that gives as answer: 'just compare the truth tables and you can see that they are equivalent'. However, I would like to know how to prove this using formal propositional logic. This should be possible, right? How can I disown a running process and associate it to a new screen shell? I have a running program on a SSH shell. I want to pause it and be able to unpause its execution when I come back.  One way I thought of doing that was to transfer its ownership to a screen shell, thus keeping it running in there.  Is there a different way to proceed? Let $X$ be an infinite cofinite space and let $A \subset X$ be infinite. Then $\overline A = X$  I am having a tough time in trying to understand this. Why is every infinite set in an infinite cofinite space dense ? Show that $\lim_{n\to\infty}b_n=\frac{\sqrt{b^2-a^2}}{\arccos\frac{a}{b}}$ If $a$ and $b$ are positive real numbers such that $a&lt;b$ and if $$a_1=\frac{a+b}{2}, b_1=\sqrt{(a_1b)},..., a_n=\frac{a_{n-1}+b_{n-1}}{2},b_n=\sqrt{a_nb_{n-1}},$$ then show that   $$\lim_{n\to\infty}b_n=\frac{\sqrt{b^2-a^2}}{\arccos\frac{a}{b}}.$$ I tried to calculate explicitly the first few terms $a_2,b_2$ etc but the terms got too complicated really quickly and I couldn't spot any pattern. How to show that $\int_{0}^{\pi}{\sin^5 x\over 1+\cos^3 x}\mathrm dx=\ln 3?$ Given the integral $(1)$     $$\int_{0}^{\pi}{\sin^5 x\over 1+\cos^3 x}\mathrm dx=\ln 3\tag1$$   This integral seems to be quite tough, I can't even know where to start. I have try substitution $u=1+\cos^3 x$ and $u=\cos^3 x$, can seem to remove the x variables.  We have the following identities   $$\sin^5a={5\over 8}\sin a-{5\over 16}\sin 3a+{1\over 16}\sin 5a\tag2$$  $$\cos^3 a={3\over 4}\cos a+{1\over 4}\cos 3a\tag3$$  We may write $(1)$ as  $$\int_{0}^{\pi}{\sin x(1-\cos^2 x)^2\over 1+\cos x(1-\sin ^2 x)}\mathrm dx\tag4$$  $u=1-\sin^2 x\implies \mathrm du=-2\sin x\cos x\mathrm dx$ then $(4)$ becomes  $${1\over 2}\int_{0}^{1}{(1-u)^2\over \sqrt{u}+u^2}\mathrm du\tag5$$  $u=v^2\implies du=2vdv$ then $(5)$ becomes  $$\int_{0}^{1}{(1-v^2)^2\over v+v^4}\cdot{v\mathrm dv}\tag6$$ Simplify to  $$\int_{0}^{1}{(1-v)(1-v^2)\over 1-v+v^2}\mathrm dv\tag7$$  I guess using partial decomposition of fraction and integrate term by term, seem lengthy process  How else can we prove $(1)$? Setting "checked" for a checkbox with jQuery The American convention in quotations is (typically) to place punctuation inside quoted text. But I always run into situations where the punctuation of the quote interferes with the punctuation of the sentence. How would you punctuate this (American, non technical)?     When my friends ask, "What do you want for your birthday?", I never know how to respond.   It seems odd to place the last comma outside the quote simply because of the question mark. Is that the preferred (i.e. most often accepted) standard? How do I use F3 to get my coordinates when my laptop makes F3 a special key? Converting Older United States Dollar Bills I received from my Dad 150 American Dollars about 12 years ago. I never have thought to change this money but a few days I tried change it in a currency exchange in Sydney and the guy didn't change it because the series are old.     What do I have to do to change this money?