Show that $\|A\|_\infty \leq \sqrt{n}\|A\|_2$ for any $m \times n$ matrices.  I know $\|A\|_\infty$ is the largest row sum of the matrix, but I'm not sure how to compare with $\|A\|_2$ especially since matrix 2-norm is induced by some other vector. All hints will be appreciated!
I am trying to prove the matrix norm equivalence for norms 1, 2, $\infty$ and Frobenius.  I have managed to solve find the constants for $||.||_{1}$ and  $||.||_{2}$ but I cannot see how to continue if I want to prove the following:  $||A||_{\infty} \leq \sqrt{n}||A||_{2}$   and the ones for Frobenius:  $||A||_{F} \leq \sqrt{n}||A||_{1}$  $||A||_{F} \leq \sqrt{n} ||A||_{2}$  Any help is appreciated. Thank you for your time
Electrostatic charge leakage What are the ways electrostatic charged objects leak charge in humid conditions?  Can airborne particles pick up charge by contact, then be repelled hence removing charge? If so would it be a significant factor? Showing that a function is a tensor I am trying to solve question 4 in Munkres Analysis on Manifolds section 26.  The question is determine if the following is a tensor on $\mathbb{R}^4$ and express those that are in terms of the elementary tensors on $\mathbb{R}^4$.   $f(x,y) = 3 x_1 y_2 + 5 x_2 x_3$   I am trying to follow the definition of a tensor (which I don't think I fully understand) to show this. I know I must find a function $T: \mathbb{R}^4 \rightarrow \mathbb{R}$ that is linear on the $i^{th}$ variable. Since there are only 2, then I believe I have to somehow write $f$ as a projection. I'm just not sure where to  go from here.  Any hints would be great, thanks! Automating the appearance of email addresses as images Publishing pdf manuscripts on the web with emails embedded in the file results in spam.  More generally, I am looking for a way to automate the conversion of a string to an image.  It's quite easy to solve this old problem using  (more ). The trouble is that you need a distinct file and typesetting run, and you end up with unnecessary image files that you cannot delete carelessly lest you also delete non-generated images.  Can you think of a way to do this on the fly? It's understood that people will have to type the email addresses if they need to message you, but that's the point.  More briefly/concretely...  With what can one replace #1  \documentclass{article} \newcommand\showasimage[1]{#1} \begin{document} \showasimage{joe@foo.bar} \end{document}   such that #1 appears as an image?  Edit  It would be really nice if the pdf file remains vectorial. Why do prisms work (why is refraction frequency dependent)? Subsample of a random sample: random sample? Own table of contents? Let $f: \Bbb C \to \Bbb C$ be an entire (analytic on the whole plane) function such that exists $\omega_1,\omega_2 \in \mathbb{S}^1$, linearly independent over $\Bbb R$ such that: $$f(z+\omega_1)=f(z)=f(z+\omega_2), \quad \forall\,z\in \Bbb C.$$Prove that $f$ is constant.   The intuition seems clear to me, we have the three vertices of a triangle given by $0$, $\omega_1$ and $\omega_2$. All points in the plane are one of the vertices of that triangle under a suitable parallel translation. The constant value will be $f(0)$, fine. Throwing values for $z$ there, I have found that $$f(n\omega_1) = f(\omega_1) = f(0)=f(\omega_2) = f(n\omega_2), \quad \forall\, n \in \Bbb Z.$$  I don't know how to improve the above for, say, rationals (at least). Some another ideas would be:   Checking that $f' \equiv 0$. I don't have a clue of how to do that. Write $w = a\omega_1+b\omega_2$, with $a,b \in \Bbb R$, do stuff and conclude that $f(w) = f(0)$. This approach doesn't seem good, because I only have a weak result with integers above. Finding that $f$ coincides with $f(0)$ on a set with an accumulation point. This seems also bad: the set on with $f$ coincides with $f(0)$ by which I found above is discrete.    Nothing works and this is getting annoying... And I don't see how analyticity comes in there. I'll be very thankful if someone can give me an idea.  (On a side note.. I know that this title is not informative at all. Feel free to edit if you come up with something better.) Clarify an idiom in a political text Please carefully read the text below:     On 24 November, 1993, a meeting of Leftist intellectuals occurred in   London under the auspices of the Institute for Public Policy   Research (IPPR), which is a Labour-leaning think-tank.  A short document   was circulated in advance of the said meeting to clarify its   purpose.        Among other things, the document declared that the task of the   IPPR was: "To do what the Right did in the seventies, namely to break   through the prevailing parameters of debate and offer a new   perspective on contemporary British politics."    The explanatory document also said, "Our concern is not to engage in a   philosophical debate about foundations of socialism."      If this meant that those foundations were not the appropriate thing to   talk about at the 24 November meeting, then that might have been   right; not every thing has to be discussed at every meeting.  But if   what was meant was that discussion of philosophical foundations is not   what the Left now needs, then I disagree, and if that indeed is what   was meant, then it is curious that the breakthrough by the Right should   have been invoked as an achievement for the left to emulate, for if   there is a lesson for the Left in the Right's breakthrough, it is that   the Left must repossess itself of its traditional foundations on   pain of continuing along its present, politically feeble, reactive   course.  If the Left turns its back on its foundations, it will be   unable to make statements that are truly its own.   Since I've heard only one definition of the part identified in bold, I am at a dead end.  As you know, and according to Oxford's dictionary, the verb "repossess" means: To take back property or goods from SB who has arranged to buy them and can not pay.  Here, the object of verb is a thing not a person, but in the text, first,itself comes before of and its traditional foundations. As a result, I thought that this phrase might be an idiom and so has a different meaning with the original verb.  Please explain the meaning of this phrase. Hosting multiple distinct folders for distinct domains I'd like to build a version of Ubuntu that has absolutely no ability to recognize any USB device inserted. Is this possible?  There's the issue that mouses and keyboards nowadays are connected via usb, so I wonder if it's possible to recognize these but not everything else.  In the end it's a security measure, so different solutions to achieve this goal are also welcome. Can I legally downgrade from Win 7 64 bits to Win 7 32 bits? Possible Duplicate:          Tired of the unstable Win 7 64bit I got when I purchased my laptop (it was impossible to find a powerful laptop with Win 7 32 bits).  Can I legally downgrade from Win 7 64 bits to Win 7 32 bits? The key/activation will work?    Details: Everybody comments about the stability of Win 64.  Well, I made a mistake. Is not that Win 64 is unstable (it never crashed). It just works very poor. It feels choppy. Compiling a large program (on an old AMD desktop PC) takes 1.5 seconds on Win 32. The same program compiles on Win 64 in 4-5 seconds in this new and "powerful" laptop. All programs are less responsive. How does Wild Shape work with special ways to calculate your AC? Among the effects of Wild Shapes, these two:     Your game Statistics are replaced by the Statistics of the beast, but   you retain your alignment, personality, and Intelligence, Wisdom, and   Charisma scores. You also retain all of your skill and saving throw   proficiencies, in addition to gaining those of the creature. If the   creature has the same proficiency as you and the bonus in its stat   block is higher than yours, use the creature's bonus instead of yours.   If the creature has any legendary or lair actions, you can't use them.      You retain the benefit of any features from your class, race, or other   source and can use them if the new form is physically capable of doing   so. However, you can't use any of your Special senses, such as   Darkvision, unless your new form also has that sense (PHB p.67, "Wild Shape")   Seem to come into conflict with each other when things like Draconic Resilience:     When you aren't wearing armor, your AC equals 13 + your Dexterity   modifier. (PHB p.102, "Draconic Bloodline")   and Barbarians:     While you are not wearing any armor, your Armor Class equals 10 + your Dexterity modifier + your Constitution modifier. You can use a Shield and still gain this benefit.   and Monks:     Beginning at 1st level, while you are wearing no armor and not wielding a Shield, your AC equals 10 + your Dexterity modifier + your Wisdom modifier.   and Stone Sorcerers:     As an action, you can gain a base AC of 13 + your Constitution   modifier if you aren’t wearing armor, and your skin assumes a stony   appearance. This effect lasts until you end it as a bonus action, you   are incapacitated, or you don armor other than a shield.   How would these resolve? How to prove that we are living in a 3+1D world? Can a pixie fly higher than their altitude of one? In the pixie description on the compendium:     Altitude Limit: You fall at the end of your turn if you are using your racial fly speed and are more than 1 square above the ground (see the rules for flying and falling in the Rules Compendium).   So can a pixie choose to go two or more squares above the ground?   Here are the relevant rules from the compendium:  Fly speed:     A creature that has a fly speed can fly a number of squares up to that speed as a move action. If it is stunned or knocked prone while flying, it falls. See also “Flying” and "Hover".   Flying:     Flight      Some creatures have the innate ability to fly, whereas others gain the ability through powers, magic items, and the like. The rules for flight in the DUNGEONS &amp; DRAGONS game stress abstraction and simplicity over simulation. In real life, a flying creature’s ability to turn, the speed it must maintain to stay aloft, and other factors put a strict limit on flight. In the game, flying creatures face far fewer limitations.      FLIGHT      Flight follows the basic movement rules, with the following clarifications.         Fly Speed: To fly, a creature takes the walk, run, or charge action but uses its fly speed in place of its walking speed. A creature that has a fly speed can also shift and take other move actions, as appropriate, while flying.   Moving Up and Down: While flying, a creature can move straight up, straight down, or diagonally up or down. There is no additional cost for moving up or down.   Falling Prone: If a creature falls prone while it is flying, it falls. This means a flying creature falls when it becomes unconscious or suffers any other effect that knocks it prone. The creature isn’t actually prone until it lands and takes falling damage.   Remaining in the Air: A flying creature does not need to take any particular action to remain aloft; the creature is assumed to be flying as it fights, moves, and takes other actions. However, a flying creature falls the instant it is stunned, unless it can hover.   Landing: If a creature flies to a surface it can hold onto or rest on, the creature can land safely.   Terrain: Terrain on the ground does not affect a flying creature if the terrain isn’t tall enough to reach it. Because of this rule, flying creatures can easily bypass typical difficult terrain, such as a patch of ice on the ground. Aerial terrain can affect flying creatures.      Crashing        Falling while Flying: If a creature falls while it is flying, it descends the full distance of the fall but is likely to take less damage than a creature that can’t fly. Subtract the creature’s fly speed (in feet) from the distance of the fall, then figure out falling damage. If the difference is 0 or less, the creature lands without taking damage from the fall. For example, if a red dragon falls when it is 40 feet in the air, subtract its fly speed of 8 (8 squares = 40 feet) from its altitude. The difference is 0, so the dragon lands safely and is not prone.         If a creature is flying when it starts a high-altitude fall, it has one chance to halt the fall by making a DC 30 Athletics check as an immediate reaction, with a bonus to the check equal to the creature’s fly speed. On a success, the creature falls 100 feet and then stops falling. On a failure, the creature falls as normal.         High-Altitude Falls: Some encounters take place very high above the ground. In such an encounter, it is possible for a creature to spend more than one round falling to the ground. As a rule of thumb, a creature falls up to 500 feet during its first turn of falling. If it is still falling at the start of its turn, it can take actions on that turn as normal, then falls up to 500 feet at the end of the turn. If none of those actions expressly halts a fall, the creature falls up to 500 feet at the end of the turn. This sequence continues until the creature lands. Are the tags shown in a particular order? sed fails to remove newline character What is $\int_{-\pi}^\pi \cos(nx)\cos(mx)\,dx$? I'm pretty sure that there's a theorem that says that the Fourier coefficients of a sum of $\cos(nx)$ and $\sin(nx)$ 's are the coefficients of the sum itself.   I tried to prove that in the specific case of $f(x) = \sum_{n=1}^{\infty}\frac{\cos(nx)}{n^2}$. In calculating the Fourier coefficients I got to calculating  $\cos(nx)\cdot\cos(mx)$, but I'm now stuck. Any tips would be appreciated! onClick to get the ID of the clicked button On the bash command-line, ↑ gives me the previous command. On the command-lines in ipython or matlab, when I type a few characters, ↑ gives me the previously entered command starting with those characters. How can I enable exactly this behaviour in bash? I am aware of more advanced ways of searching through the command-line history, but sometimes a simple way is more convenient. Suppose $\{a_n\}$ is a sequence such that the subsequences $\{a_{2n−1}\}$ and $\{a_{2n}\}$ converge to the same limit, say $a$. Show that $\{a_n\}$ also converges to $a$.