X is r.v. on ($\Omega$,F,P) where $E[\left\lvert X \right\rvert^a] &lt; \infty$. Prove $E[\left\lvert X \right\rvert^b] &lt; \infty$ for all $0&lt;b\le a$  How would I go about proving this using something in the realm of Lebesgue Integration or Domination thm. If I am off base with this please let me know. Having trouble even visualizing how to prove, would appreciate any help or hints.  edit: added abs value. sorry for that oversight!
Suppose $\mu(X)&lt;\infty$. Show that if $\ f\in\ L^q(X,\mu)$ for some $1\le q&lt;\infty$ then $\ f\in\ L^r(X,\mu)$ for $1\le r&lt;q$   In the solution the assistant wrote this:  If $1\le r&lt;q$ then $q/r\ge 1$. Let p be the dual exponent to $q/r$ (i.e. $\frac{1}{q/r}+\frac{1}{p}=1)$  then by Hölder Inequality  $\int_{X}|f|^rd\mu\le\underbrace{(\int_{X}|f|^qd\mu)^{r/q}}_{??????}(\int_X1^pd\mu)^{1/p}&lt;\infty$  for me the middle part doesn't make sense (it is not the hölder inequality) can somebody show it ?  instead of this can't we say (only in this case) that   $\int_{X}|f|^rd\mu\le (\int_{X}|f|d\mu)^r$ then  $(\int_{X}|f|d\mu)^r\le ((\int_{X}|f|^qd\mu)^{1/q}(\int_X1^pd\mu)^{1/p})^r&lt;\infty$  ?
Pattern matching with LIKE, SIMILAR TO or regular expressions in PostgreSQL I had to write a simple query where I go looking for people's name that start with a B or a D :   SELECT s.name  FROM spelers s  WHERE s.name LIKE 'B%' OR s.name LIKE 'D%' ORDER BY 1   I was wondering if there is a way to rewrite this to become more performant. So I can avoid or and / or like? Let $B$ be a commutative domain and $A \subseteq B$ a subring. Suppose that the extension $A \to B$ is finite, i.e. $B$ is finitely generated as an $A$-module. If $I \cap A = \lbrace 0 \rbrace$ for some ideal $I$ of $B$, can we conclude that $I = \lbrace 0 \rbrace$?  I know that if $I$ contains a nonzero prime ideal, the statement holds since $A \to B$ is an integral extension, whereby no two comparable prime ideals of $B$ can have the same intersection with $A$. But does the statement still hold if $I$ does not contain a nonzero prime ideal? And what if $A \to B$ is only assumed to be an integral extension? In my book, it proves that an infinite subset of a coutnable set is countable. But not all the details are filled in, and I've tried to fill in all the details below. Could someone tell me if what I wrote below is valid?  Let $S$ be an infinite subset of a countable set $T$. Since $T$ is countable, there exists a bijection $f: \mathbb{N} \rightarrow T$. And $S \subseteq T = \{ f(n) \ | \ n \in \mathbb{N} \}$. Let $n_{1}$ be the smallest positive integer such that $f(n_{1}) \in S$. And continue where $n_{k}$ is the smallest positive integer greater than $n_{k-1}$ such that $f(n_{k}) \in S$. And because $S$ is infinite, we continue forever. Now consider the function $\beta : \mathbb{N} \rightarrow S$ which sends $k \rightarrow f(n_{k})$.   So in order for $S$ to be countable, $\beta$ would have to be a bijection. $\beta$ is injective since if $f(n_{k}) = f(n_{j})$, then $n_{k} = n_{j}$ because $f$ is injective. We also can conclude $k = j$ because the various $n_{i}$ chosen were strictly increasing.   Edit: $\beta$ also needs to be surjective. So for every $r \in S$ there exists a $q \in \mathbb{N}$ such that $\beta(q) = f(n_{q}) = r$. We know that $f^{-1}(r) \in \{n_{1}, n_{2}, ..., n_{k} ... \}$, so we can let $f^{-1}(r) = n_{d}$. Thus $\beta(d) = f(n_{d}) = r$. Open source computer algebra systems for general relativity I would like to use an open source computer algebra system (CAS) for research in general relativity. Kindly suggest a good choice between Ipython and Sage. I am more interested in the theoretical aspect rather than numerical computations. Is there any continuity between James Bond movies that have different actors portraying the title character? There was a foolish (in my opinion) question asked on Scifi.SE       James bond keeps returning with a different face, albeit usually a similar one, and often operates with high-tech gadgetry that seem to be on par with the Doctor's sonic screw driver.   In my opinion each new actor represents a reboot of the franchise using the new face, not a continuation of the story.   Is there any continuity between versions of James Bond which change actors? I'm not talking about characters like M or Q, I'm talking about events or past advisories being referenced in a different Bond movie where the main actor was someone else. Relationship between SVD and PCA. How to use SVD to perform PCA? Principal component analysis (PCA) is usually explained via an eigen-decomposition of the covariance matrix. However, it can also be performed via singular value decomposition (SVD) of the data matrix $\mathbf X$. How does it work? What is the connection between these two approaches? What is the relationship between SVD and PCA?  Or in other words, how to use SVD of the data matrix to perform dimensionality reduction? Are there any rules relevant to involuntary movement and effects other than Opportunity Attacks? PHB p. 195 states that involuntary movement does not provoke :     You also don't provoke an opportunity attack when you teleport or when someone or something moves you without using your movement, action, or reaction.   Is there a general rule somewhere about involuntary movement and how it interacts with other effects?  The question arose at our table about what happens if a creature is dragged via the Thorn Whip spell across an area of Spike Growth.  Does the involuntarily moving creature take the damage from the spikes as if he'd voluntarily moved through them?    Logically, yes, if you're being dragged through spikes, you take damage from them as if you'd walked through them. But logically, if I drag you past my ally, my ally would be able to make an Opportunity Attack, likely more easily than if you were moving under your own power.  Presumably that is not allowed for balance reasons rather than versimilitude reasons, so I wonder if the same should apply to other effects? Spheres with csv file Understanding the "Birthday Problem" "It was me" or "It was I" It was ... salted the earth around your flower deck.    I wrote the second form "It was me". I have chosen the most "appropriate" answer to this question as somebody who is learning English.  This decision was made on a phonetic basis so it's for sure I made a mistake.   Surely you have a rational explanation to this exercise, maybe a grammar rule, or is it an English expression? I'm trying to prove that if there are to numbers $n,m$ (natural numbers), and their smallest common multiple is $k$, so that $k = n·i$ and $k = m·j$ for some $i,j$ natural numbers, any common multiple $q$ of $n,m$ is a multiple of $k$, so that there exists a $g$ so that $q=k·g$.  Logically it seems correct, but I can't seem to prove it... Any help? Bash Script if condition is always true SPSite and SPWeb object dispose inside foreach loop Preview on the right please, optionally? How do some Ws on transcripts affect PhD admissions in mathematics? Can there be only one extension to the factorial? Usually, when someone says something like $\left(\frac12\right)!$, they are probably referring to the Gamma function, which extends the factorial to any value of $x$.  The usual definition of the factorial is $x!=1\times2\times3\times\dots x$, but for $x\notin\mathbb{N}$, the Gamma function results in $x!=\int_0^\infty t^xe^{-t}dt$.  However, back a while ago, someone mentioned that there may be more than one way to define the factorial for non-integer arguments, and so, I wished to disprove that statement with some assumptions about the factorial function.    the factorial   is a  for $x\in\mathbb{C}$ except at $\mathbb{Z}_{&lt;0}$ because of singularities, which we will see later. is a monotone increasing function that is concave up for $x&gt;1$. satisfies the relation $x!=x(x-1)!$ and lastly $1!=1$     From $3$ and $4$, one can define $x!$ for $x\in\mathbb{N}$, and we can see that for negative integer arguments, the factorial is undefined.  We can also see that $0!=1$.  Since we assumed $2$, we should be able to sketch the factorial for $x&gt;1$, using our points found from $3,4$ as guidelines.  At the same time, when sketching the graph, we remember $1$, so there can be no jumps or gaps from one value of $x$ to the next.  Then we reapply $3$, correcting values for $x\in\mathbb{R}$, since all values of $x$ must satisfy this relationship.  Again, because of $1$, we must re-correct our graph, since having $3$ makes the derivative of $x!$ for $x\in\mathbb N$ undefined.  So, because of $1$ and $3$, I realized that there can only be one way to define the factorial for $x\in\mathbb R$.  Is my reasoning correct?  And can there be only one extension to the factorial?    Oh, and here is a  to how I almost differentiated the factorial only with a few assumptions, like that it is even possible to differentiate.  Putting that in mind, it could be possible to define the factorial with Taylor's theorem? I've heard many times that it's bad to use display: none for SEO reasons, as it could be an attempt to push in irrelevant popular keywords. A few questions:   Is that still received wisdom? Does it make a difference if you're only hiding a single word, or perhaps a single character? If you should avoid any use of it, what are the preferred techniques for hiding (in situations where you need it to become visible again on certain conditions)?   Some references I've found so far:       If you're straight-out using CSS to hide text, don't be surprised if that is called spam. I'm not saying that mouseovers or DHTML text or have-a-logo-but-also-have-text is spam; I answered that last one at a conference when I said "imagine how it would look to a visitor, a competitor, or someone checking out a spam report. If you show your company's name and it's Expo Markers instead of an Expo Markers logo, you should be fine. If the text you decide to show is 'Expo Markers cheap online discount buy online Expo Markers sale ...' then I would be more cautious, because that can look bad."   And  on the same article     We can flag text that appears to be hidden using CSS at Google. To date we have not algorithmically removed sites for doing that. We try hard to avoid throwing babies out with bathwater.   Eric Enge      The legitimate use of this technique is so prevalent that I would rarely expect search engines to penalize a site for using the display: none attribute. It’s just very difficult to implement an algorithm that could truly ferret out whether the particular use of display: none is meant to deceive the search engines or not. Combination with repetitions. How does Annihilation work? If $n\geq m$ then $(x^m+y^m)^{1/m} \ge (x^n+y^n)^{1/n}$