Im looking for an element in $\ell^p$ which is not in $\ell^q$ for all $1\leq q&lt;p$. Does anybody know such a series?
For every $p\in [1,\infty)$, consider the set $\ell^p$ of real sequences $x=(x_n)=(x_1,x_2,\ldots)$ such that $\sum_{n=1}^\infty |x_n|^p&lt;\infty$. This is a Banach space with the norm $\Vert x\Vert_p=(\sum|x_n|^p)^{1/p}$.  I have the following problem:     Is it true that $\ell^p=\bigcup_{1\leq q&lt;p}\ell^q$ for $p&gt;1$?   I don't think it is true, but I'm having problems to show this. Clearly, the statement is true for some $p&gt;1$ iff it is true for every $p&gt;1$.  I'm trying to find a sequence $(a_n)$ such that $\sum |a_n|^s$ converges iff $s\geq 1$. I've tried many sequences (such as $(n^{1+1/n})$, involving $\log$, etc...), but none of them worked.
for (i=0 ; i&lt;=10; i++) {     ..     .. }  i=0; while(i&lt;=10) {     ..     ..     i++; }   In for and while loop, which one is better, performance wise? Hydrogen radial wave function infinity at $r=0$ When trying to solve the Schrödinger equation for hydrogen, one usually splits up the wave function into two parts: $$\psi(r,\phi,\theta)= R(r)Y_{\ell,m}(\phi,\theta).$$ I understand that the radial part usually has a singularity for the $1s$ state at $r=0$ and this is why you remove it by writing: $$R(r) = \frac{u(r)}{r}$$ But what is the physical meaning of $$R(r=0) = \infty~?$$ Wouldn't this mean that the electron cloud is only at the centre of the atomic nucleus? The Megahertz Myth became a promotional tactic due to differences between the PC's INTEL 8086 processor and Apple's Rockwell 6502 processor.  The 8086 ran at 4.77MHz while the 6502 ran at 1MHz.  However, instructions on the 6502 needed fewer cycles; so many fewer, in fact, that it ran faster than the 8086.  Why do some instructions need fewer cycles?  And why can't the instructions of the 6502, needing fewer cycles, be combined with a fast cycling processor of the 8086?  Wikipedia's article for instructions per cycle (IPC) says     Factors governing IPC   A given level of instructions per second can be achieved with a high IPC and a low clock speed...or from a low IPC and high clock speed.   Why can't you have both high instructions per cycle and high clock speed?  Maybe this has to do with what a clock cycle is?  Wikipedia mentions synchronization of circuits?  Not sure what that means.  Or maybe this has to do with how a pipeline works?  I'm not sure why instructions in a short pipeline are different from instructions in a long pipeline.  Any insight would be great!  Just trying to understand the architecture behind the myth.  Thanks!  References: Is there a black hole in the centre of the Milky Way? Reflection of a line in a plane The line $l_1$ has the equation $r=(6i+2-2k)+\lambda(4i+5j-k)$ and the plane $\pi_1$: $2x-y+4z=4$, the line $l_2$ is the reflection of $l_1$ in the plane $\pi_1$. Find the exact vector equation of line $l_2$   So the line intersects the plane when $\lambda=-2$, giving the point $(-2,-8,0)$ which will be common on $l_1$ and $l_2$. But I am unsure on how to find the direction vector for $l_2$. Any help would be appreciated. What does "$$" mean in the shell? What does the output of the command below mean? (result is positive integer)  echo $$ Are Tolkien Elves taller or shorter than Men?  In general fantasy, elves are frequently smaller than humans. But I seem to recall that in Tolkien they aren't.  Based on How do I specify unique constraint for multiple columns in MySQL? Why does my material panel look different? Complete beginner here.  I'm trying to follow  tutorial on Anisotropic shaders, but my property panel looks nothing like the property panel in the tutorial.  For example, when creating a material, this is what his property panel looks like:    But this is what mine looks like, if I follow the same path:    I'm sure I'm missing something obvious.  How can I see the property panel shown in the tutorial? Inequality $\left(a-1+\frac{1}b\right)\left(b-1+\frac{1}c\right)\left(c-1+\frac{1}a\right)\leq1$ Why are some faces in my mesh darker? While rendering a mesh with smooth shading on, I get a dark region/shadow on certain faces. The effect is more evident when smooth shading is applied to a mesh containing ngons, with the ngons turning black when rendered.   The blackening effect is visible in the viewport too.  How can I solve this problem? I try to resolve a problem, how to implement a xor gate with nand's gate.  A xor B = A'B + AB'  So, this becomes :  A xor B = A'B + AB' + AA' + BB' = A(A' + B') + B(A' + B') = (A + B)(A' + B') =   (with De Morgan apllied on the second term)  = (A + B)(AB)' = ..... ?  In this point i am blocked. If someone can help me, please. Thank you. Maxing out gear with maxed out smithing, enchanting and alchemy I have level 100 smithing, enchanting and alchemy. I also have some nice enchanted gear, fortify smithing and alchemy.   How high should I be able to get my fortified equipment?  Currently I'm around 29% for alchemy and smithing.  How high can I get fortify smithing and fortify enchanting potions without using the console (cheating)? And how? Currently I get +147% and +32% respectively.  Is something like this possible without using the console(or photoshop)? Counting integer partitions of n into exactly k distinct parts size at most M How can I find the number of partitions of $n$ into exactly $k$ distinct parts, where each part is at most $M$?  The number of partitions $p_k(\leq M,n)$ of $n$ into at most $k$ parts, each of size at most $M$, is given by the generating function: $$  \binom{M+k}{k}_{x} = \prod_{j=1}^{k}\frac{1-x^{M+k-j+1}}{1-x^j}= \sum_{n=0}^{kM} p_{k}(\leq M,n) x^n $$  For the number of the partitions $p_k(\mathcal{D},n)$ of $n$ into at most $k$ parts there is the recurrence relationship: $$   p_{k}(\mathcal{D},n) = p_{k}(\mathcal{D},n-k) + p_{k-1}(\mathcal{D},n) $$  But what, if I want to count only the partitions with distinct parts and restricted number of parts and restricted part size?  Update: Now I know the generating function for the number of distinct restricted partitions $p_k(\leq M, \mathcal{D},n)$ of $n$ into exactly $k$ distinct parts, all at most $M$ is $$    \prod_{j=1}^{M} (1+xq^{j}) = \sum_{k,n=0}^{\infty}p_k(\leq M, \mathcal{D},n)x^{k}q^{n} $$ and there is also a recurrence relation $$    p_k(\leq M, \mathcal{D},n) =         p_{k-1}(\leq M-1, \mathcal{D},n-k) +         p_k(\leq M-1, \mathcal{D},n-k) $$ How can I prove this? Could you recommend a book, where I could read about this? How to make Blender use Python2.x? OS X El Capitan installation error: AMD-Action:authenticate:SP When doing a clean re-installation of El Capitan (highest supported version) from a bootable USB on an older Mac (MBP, 15-inch, Late 2008), I get this error:  An unexpected error occured while signing in. AMD-Action:authenticate:SP  The error shows up, while logging in with the Apple ID into the App Store in the installer.  I tried with different Apple IDs,  Also tried running these commands in the Recovery Terminal and restarting: csrutil clear csrutil enable Shorten title on each beamer slide In a beamer presentation the title is long but ok for the first slide. The problem is that when the title appears below on each slide it is very long and I would like to use here some sort of running or shorter title, how can this be done? Are "if" and "iff" interchangeable in definitions? In some books the word "if" is used in definitions and it is not clear if they actually mean "iff" (i.e "if and only if"). I'd like to know if in mathematical literature in general "if" in definitions means "iff".  For example I am reading "Essential topology" and the following definition is written:     In a topological space $T$, a collection $B$ of open subsets of $T$ is said   to form a basis for the topology on $T$ if every open subset of $T$ can be written as a union of sets in $B$.   Should I assume the converse in such a case? Should I assume that given a basis $B$ for a topological space, every open set can be written as a union of sets in $B$?  This is just an example, I am not asking specifically about this sentence. Sound of a next scene starts one moment earlier than video How can I force my browser to search Google in English? I'm too bored of seeing sites like Google and such show up in my native language, I would rather like them to be in English. Yet, I have to explicitly change the URL to .com and en and that kind of parameters in order for them to show up in English. Can I somehow force this?  So, how is Google configured?  However, it is set to English on the site itself so it has to be my browser:    Then, how does my browser land up on non-english pages, like Google?  It usually shows up in non-English when I'm performing a search, which uses:  {google:baseURL}search?{google:RLZ}{google:acceptedSuggestion}{google:originalQueryForSuggestion}{google:searchFieldtrialParameter}{google:instantFieldTrialGroupParameter}sourceid=chrome&amp;ie={inputEncoding}&amp;q=%s   When performing a search, it fills these variables in with non-english values.  How can I tell my browser to fill these in with the English values?  My Google Chrome options give preference to English: