What features should I look for in a filter system to avoid buying duplicates later
What should I do when I have lenses with different filter thread sizes?
Let $A,B \subset \mathbb{R}$. Show that $\sup(A \cup B) = \max\{\sup A, \sup B\}$ Chrome's default search engine is .  How we can force chrome to use  without https?  I need a solution that allows proxies to rewrite my URLs, but https currently makes this impossible... Is redirecting homepage to a page with keyword it's targeting helps in SEO? I have firefox already running on my system, and now in minimised state. What I want is to close the current tab running in firefox. How can I do this by simply using commands in terminal? Calculate $\tan9^{\circ}-\tan27^{\circ}-\tan63^{\circ}+\tan81^{\circ}$ Calculate $\tan9^{\circ}-\tan27^{\circ}-\tan63^{\circ}+\tan81^{\circ}$?  The correct answer should be 4. How to make document custom page dimensions? The definite article implying the generic idea of something I know that the definite article "the" can be used to mean a generic idea of something as in "the lion is the king of the animals." In this sentence, the in "the lion" is used to imply the general image of lions, so not specifying a particular lion.   However, I am confused in identifying where a "the" in a sentence is used for a general idea or a specific reference. For example, in the sentence "the price of gas is soaring", I heard that "the" in the sentence is an example of the generic use of the. Although I can easily understand the generic use of "the" in the lion example, but I do not understand how there is an general idea about gas price. Soaring gas price is something real,and gas prices in different regions( in a hypothetical situation) can be different. So, if I say "the price of gas is soaring", do I mean the price of gas that is determined by the world's oil system or can I say "the gas price" even when I am indicating the state of gas price in "a particular region"?    I know it is even difficult to understand my question, but I hope someone could find what I am confused about and give an answer. What does the Canonical Commutation Relation (CCR) tell me about the overlap between Position and Momentum bases? I'd like to modify the names used to typeset some of my document elements. For example, the caption of figure floats should change from "Figure" to "Fig.", and my \tableofcontents sectioning heading shouldn't read "Contents", but "Table of Contents". How can I do that?  \documentclass{article} % \usepackage[english]{babel} \begin{document} \tableofcontents \section{foo} \begin{figure}[h] \centering \rule{1cm}{1cm}% placeholder for graphic \caption{A figure} \end{figure} \end{document} Can I enter the UK with a Schengen visa? Well, just another idea came up into my mind and i have no idea how to solve it :D Is there infinitely many prime numbers, which are not repunits and their inverse is also prime? (For example, inverse of 31 is 13 which is also prime. i didn't have any other name to describe the function!) P.S:I now know they are called s. Suppose I have a dataframe consisting of six time series. In this dataframe, some observations are missing, meaning at some timepoints all time series contain a NA-value. In R, one possible imputation package that can be used to impute time series data is Amelia. However, this package does not work for observations that are completely missing. Are there other ways to impute my data? For what it's worth, the amount of missing observations is less than 20% of all observations. Are there any non-green, non-artifact mana ramp cards? I have looked and can't seem to find any.  I have seen many green cards and some artifacts that do this, but nothing in Red, Blue, Black or White.    Are there any?  Note: By "Mana Ramp" I mean a card that generates mana, or one that lets you play more than one land per turn.  The end result, for example, would be on turn 4, playing a white only deck, I could tap more than 4 mana.  (Normally you only get one land per turn so a max of 4 mana on turn 4.) If a variable is declared as static in a function's scope it is only initialized once and retains its value between function calls. What exactly is its lifetime? When do its constructor and destructor get called?  void foo()  {      static string plonk = "When will I die?"; } How do I install make? I have read answers to questions like  but I couldn't find a general rule for using "you" or "yourself", "him" or "himself" that could explain things I see in some examples.  For example, I think that one normally say (according to examples I see in ):   You bring something with you. You have something in you. You have something on you. You pressed her to you.   But on the other hand one would say:   You draw attention to yourself. He killed himself. You can try it out for yourself.   As a general rule, it seems that the reflexive pronoun is used when the object is the same as the subject of the verb. But for prepositional phrases, it seems more complicated. Do some prepositions (like "with", "in", "on") always come with the accusative form of the pronoun, and some other (like "for") don't? What about "to", that seems to be followed by both in different circumstances? Elastic collision in two dimensions Suppose a particle with mass $m_1$ and speed $v_{1i}$ undergoes an elastic collision with stationary particle of mass $m_2$. After the collision, particle of mass $m_1$ moves with speed $v_{1f}$ in a direction of angle $\theta$ above the line it was moving previously. Particle with mass $m_2$ moves with speed $v_{2f}$ in a direction of angle $\phi$ below the line which particle with mass $m_1$ was moving previously. Using equations for conservation of momentum and kinetic energy, how can we prove these two equations  $\frac{v_{1f}}{v_{1i}}=\frac{m_1}{m_1+m_2}[\cos \theta \pm  \sqrt{\cos^2 \theta - \frac{m_1^2-m_2^2}{m_1^2}}]$  and  $\frac{\tan(\theta +\phi)}{\tan(\phi)}=\frac{m_1+m_2}{m_1-m_2}$ ?  EDIT. Here is what I've done:  For the first one, set the $xy$ coordinate system so that the positive direction of the $x$ axis points toward the original path of the particle with mass $m_1$. So we have three equations:  $m_1v_{1i}=m_1v_{1f}\cos \theta + m_2v_{2f} \cos \phi$  $0=m_1v_{1f}\sin \theta - m_2v_{2f}\sin \phi$  $m_1v_{1i}^2=m_1v_{1f}^2+m_2v_{2f}^2$.  From the second one, we get:  $v_{2f}=\frac{m_1v_{1f}\sin \theta}{m_2 \sin \phi}$  Plotting this into third equation, we get  $v_{1i}^2=v_{1f}^2(1+\frac{m_1 \sin^2 \theta}{m_2 \sin^2 \phi})$ (1)  From the first equation, we have  $\cos \phi =\frac{m_1(v_{1i}-v_{1f}\cos \theta)}{m_2v_{2f}}$  which after applying the equation we have for $v_2f$ becomes  $\sin^2 \phi = \frac{1}{1+\frac{(v_{1i}-v_{1f}\cos \theta)^2}{\sin^2 \theta \times v_1f^2}}$  Plotting this into equation (1), gives us an equation in terms of $m_1$, $m_2$, $v_{1f}$, $v_{1i}$ and $\theta$, but it is too far from what I expected.  For the second one, assigning the $xy$ coordinate in a way that the positive direction of the $x$ axis points toward the final path of the particle $m_2$, will give us three equations (two for conservation of linear momentum and one for conservation of kinetic energy), but I don't know what to do next. Is Pauli-repulsion a "force" that is completely separate from the 4 fundamental forces? Arbitrary intersection of closed, connected subsets of a compact space connected? Let $(B_i)_{i\in I}$ be an indexed family of closed, connected sets in a compact space X. Suppose $I$ is ordered, sucht that $i &lt; j \implies B_i \supset B_j$.  Is $B = \bigcap_i B_i$ necessarily connected?  I can prove it, if I assume $X$ to be Hausdorff as well: If $B$ is not connected, then there are two disjoint, closed, nonempty sets $C$, $D$ in $B$, such that $C \cup D = B$. Now these sets are also closed in $X$, hence by normality there exist open disjoint neighborhoods $U$, $V$ of $C$ and $D$, respectively.  Then for all $i$: $B_i \cap U^c \cap V^c \ne \emptyset$, since $B$ is contained in $B_i$ and $B_i$ is connected. Thus we must also have  $$ B \cap U^c \cap V^c = \bigcap_i B_i \cap U^c \cap V^c \ne \emptyset $$  by compactness and the fact that the $B_i$ satisfy the finite intersection property. This is a contradiction to the choice of $U$ and $V$.  I can neither see a counterexample for the general case, nor a proof. Any hints would be greatly appreciated!  Thanks,  S. L. How can blackbody radition be explained by quantization? I don't understand why quantization makes a peak on the blackbody radiation curve (so there is no ) and the relationship between that peak and quantization concept.  When the blackbody is heated, it starts to glow. All atoms start to vibrate. Total heating energy must be divided and equally shared by all atoms of blackbody. Right?  But what happens next?  Do some atoms get much more energy to radiate at higher frequency and waste the energy? What happens at the atomic level, so quantization can explain it? What does classical physics claim, so it can not explain the radiation curve?