Replacing bathroom light fixture, no junction box
Bathroom vanity light with no junction box
What information does the trace of a matrix give? Crookshanks is Hermione's familiar in PoA, GoF, and OOTP. I cannot recall mention of Crookshanks following OOTP, though. What happened to Crookshanks? Does canon refer to Crookshanks in either HBP or DH? Every Artinian ring is isomorphic to a direct product of Artinian local rings Proposition. Let $R$ be commutative ring with $1_R$. We assume that $R$ is an Artinian ring and $M_1,\dots,M_n$ its maximal ideals. Then         $R/\mathrm{Jac}(R)\cong (R/M_1)\times \dotsb \times (R/M_n)$.   The ring $R$ is isomorphic to the direct product of a finite number of Artinian local rings.      Proof. 1. It's obvious that $M_i+M_j=R,\ \forall 1\leq i \neq j \leq n$. So, from Chinese Remainder Theorem, we have  $$R/\bigcap_{i=1}^{n}M_i = R/\mathrm{Jac}(R) \cong (R/M_1)\times \dotsb \times (R/M_n),$$ as we wanted.    Since $R$ is an Artinian ring, we have $\mathrm{Jac}(R)^m=\{0_R\}$, for some $m\in \Bbb Z^+$. But,  $$\{0_R\}\subseteq M_1^m\dotsb M_n^m=(M_1\dotsb M_n)^m\subseteq (\bigcap_{i=1}^{n}M_i=\mathrm{Jac}(R))^m=\{0_R\}.$$ So, if we apply CRT we will take  $$R\cong R/\{0_R\}\cong R/M_1^m\dotsb M_n^m\cong (R/M_1^m)\times \dots \times (R/M_n^m).$$   Questions.   1) Are these thoughts complete and correct?  2) Why $R/M_i^m$ are artinian local rings?  3) Could you please elaborate some examples as an application? Scoring matrices (BLOSUM & PAM) in BLAST and other sequence-comparison programs The  states that:      The scores are created by comparing the word in the list in step 2* with all the 3-letter words. By using the scoring matrix (substitution matrix) to score the comparison of each residue pair, there are 203 possible match scores for a 3-letter word. For example, the score obtained by comparing PQG with PEG and PQA is 15 and 12, respectively with the BLOSUM62 weighting scheme.   I do not understand how they got a score of 15 for PQG v. PEG. What is a scoring matrix, how is it computed, and how is it used?      Step 2 on that page is Make a k-letter word list of the query sequence: “Take k=3 for example, we list the words of length 3 in the query protein sequence (k is usually 11 for a DNA sequence) sequentially, until the last letter of the query sequence is included”. triggered my curiosity for this question. Why most of manga are black and white? Is this because of more cost if they add colors to it? Is  && statement() the same as if() statement()? Summer in Saint Helena The Winter Bash site is located at http://winterba.sh/.  Only one problem:   . Saint Helena is in the Southern Hemisphere. It's Summer in Saint Helena.   So shouldn't it be the Summer Bash? Don't forget our Aussie/Kiwi/South African/Southern Brazilian/Argentine/Uruguayan/Zimbabwean/Swazi/Tokelauan/Tuvaluan/Pitcairn Islander/ friends! Assume that $A$ is an $n\times n$  real matrix, i.e. $$A^T=-A.$$  Since $\det(A-\lambda I)=\det(A^T-\lambda I)$, $A$ and $A^T$ have the same eigenvalues. On the other hand, $A^T$ and $-A$ also have the same eigenvalues. Thus if $\lambda$ is an eigenvalue of $A$, so is $-\lambda$. If $n$ is odd, $\lambda = 0 $ is an eigenvalue.  A curious search in Google returns  the nonzero eigenvalues of $A$ are all pure imaginary and thus are of the form $iλ_1, −iλ_1, iλ_2, −iλ_2,$ … where each of the $λ_k$ are real.   Here is my question:     How can I prove the fact that "the nonzero eigenvalues of $A$ are all pure imaginary"? The category Set has the property that any morphism f can be decomposed into an epimorphism $e$ and a monomorphism $m$ with $f$ = $m\circ e$. (The intermediary set object is the image of $f$.)  Is there a name for this construction?  Is there a name for this property of categories?  In Set, the intermediary object is unique up to isomorphism. Is this the case for all categories with the property? System installation is stalled on Compaq Presario R3000 Leave newlines intact in bounty messages Conversion Error setting value for 'null Converter' - Why do I need a Converter in JSF? Can regular expressions be used to match nested patterns? How to read such figures with % as "up (by) 120%" and "increased (by) 120%". Do they mean the same? Sometimes when I'm making a sauce or soup that needs to be a bit spicy, I have no troubles spicing how I want to.  But if the dinner party is a bit late, or I made the sauce in advance to really draw out all the flavors of the spices, vegetables and meat, sometimes I end up with a way too spicy/hot dish.  You can always pour a bit water and try to dilute it a bit, but that also takes away all of the good flavor. So is there some trick that I'm missing I can use, when this happens to me (other than being more careful or using the spices later)? Is there a difference between foo(void) and foo() in C++ or C? Consider these two function definitions:  void foo() { }  void foo(void) { }   Is there any difference between these two? If not, why is the void argument there? Aesthetic reasons? Book a return flight a year in advance Is it possible to buy an airline ticket that has me returning to my country after a year being abroad? I can't seem to book a flight that far in advance online. Fixed point of a Function (monotonically increasing) I am going to Santa Monica for 4 weeks to polish my English in July and would like to volunteer at the Planetary Society located in Pasadena.  I am a Swiss citizen and have a biometric passport. However, I don't have any special visa, as there is no visa needed for the school I am attending and using the Visa Waiver Program, I am allowed to stay up to 90 days in the US.  Am I allowed to do volunteer work during my stay? How to determine the amount of experience needed for leveling up Related questions under this topic might include:   How should I base how much experience points it takes to get to the next level? What factors are involved in this? How do I keep the player not entertained long enough till he levels up? How can I vary the way the player gets experience points?