Counting solutions to a five-variable equation
How to use stars and bars for solving number of integral solutions to an equality?
Find a circle's radius with three known tangent lines I need to find the equation for a circle (mainly its radius) which is tangent to the following three lines:  $y = 0$  $y = \tan(70)x$  $y = -1.428148x + 0.790201$  For the last tangent line equation, I know that it is tangent at the point $(0.371272, 0.259968)$ However, for the other two I do not know the exact point of tangency, only that the circle is tangent to them. Is the integral $\int_0^\infty \frac{\mathrm{d} x}{(1+x^2)(1+x^a)}$ equal for all $a \neq 0$? Let $a$ be a non-zero real number. Is it true that the value of $$\int\limits_0^\infty \frac{\mathrm{d} x}{(1+x^2)(1+x^a)}$$  is independent on $a$? What regex pattern would need I to pass to  java.lang.String.split()  to split a String into an Array of substrings using all whitespace characters (' ', '\t', '\n', etc.) as delimiters? How to extract and join files xxx.zip, xxx.z01 and xxx.z02 I have three files 1.zip, 1.z01 1.z02. I would like to extract and join them into a file, by right clicking 1.zip and selecting "Extract Here". But then there comes an error:     7-Zip 9.04 beta  Copyright (c)   1999-2009 Igor Pavlov  2009-05-30   p7zip Version 9.04   (locale=en_US.utf8,Utf16=on,HugeFiles=on,2   CPUs)      Error: /home/tim/Desktop/1.zip:   Can not open file as archive      Errors: 1   I was wondering what the problem it is and how I can solve it?  Thanks and regards!    ADDED: Is my usage of "unzip -FF" wrong? It seems not work:  $ unzip -FF 1.zip  Archive:  1.zip warning [1.zip]:  zipfile claims to be last disk of a multi-part archive;   attempting to process anyway, assuming all parts have been concatenated   together in order.  Expect "errors" and warnings...true multi-part support   doesn't exist yet (coming soon). file #1:  bad zipfile offset (local header sig):  0   Thanks again!  2nd ADDED: Still ain't right.  $ cat 1.z01 1.z02 1.zip &gt; combined.zip &amp;&amp; unzip -FF combined.zip Archive:  combined.zip warning [combined.zip]:  zipfile claims to be last disk of a multi-part archive;   attempting to process anyway, assuming all parts have been concatenated   together in order.  Expect "errors" and warnings...true multi-part support   doesn't exist yet (coming soon). warning [combined.zip]:  209829313 extra bytes at beginning or within zipfile   (attempting to process anyway) file #1:  bad zipfile offset (local header sig):  209829313   (attempting to re-compensate)  extracting: 1.wmv   bad CRC ee181eef  (should be f3c61875)   It generated 1.wmv, which can play well only until  midway in Gnome Mplayer.  3rd ADDED: Am I using 7z wrongly:  $ 7z e 1.z01  7-Zip 9.04 beta  Copyright (c) 1999-2009 Igor Pavlov  2009-05-30 p7zip Version 9.04 (locale=en_US.utf8,Utf16=on,HugeFiles=on,2 CPUs)  Processing archive: 1.z01  Error: Can not open file as archive  $ 7z e 1.zip  7-Zip 9.04 beta  Copyright (c) 1999-2009 Igor Pavlov  2009-05-30 p7zip Version 9.04 (locale=en_US.utf8,Utf16=on,HugeFiles=on,2 CPUs)  Processing archive: 1.zip  Error: Can not open file as archive What is the difference when you say "get off of something" and "get off something"? Connectivity, Path Connectivity and Differentiability I have two questions which pertain to differentiability, connectivity and path connectivity. Ocasionally, I will encounter an author who defines connectivity in the following way:   An open subset $U$ of $\mathbb{R}^n$ is said to be connected if and only if given two points $a$ and $b$ of $U$ there exists a differentiable mapping $\phi: \mathbb{R} \rightarrow U$ such that $\phi(0) = a$ and $\phi(1) = b$.   This particular example is from Edward's Advanced Calculus of Several Variables p 84. Now, this is obviously not the standard definition we learn from topology which has nothing to do with differentiability but rather whether there exists two nonempty open subsets that comprise a separation. It also seems to me that what the author is really defining what it means for a space to be "smoothly path connected", which of course implies connectivity and, it seems to me, considerably more.  My first question is: Is "smoothly path connected", as defined above actually equivalent to "connected", in the topological sense, in $\mathbb{R}^n$?  Next, in Vector Calculus by Baxandall and Liebeck on p 150 the authors state the existence of a continuous path $\alpha$ from the closed interval $[0,1]$ to an open subset $D$ of $\mathbb{R}^n$ where $\alpha (0) = a \in D$ and $\alpha(1) = b \in D$ guarantees the existence of a differentiable path with the same properties. This claim is stated without proof.   My second question is: Can someone provide a reference to a proof of the above claim or explain why it is so? Power of a generator generates the group iff this power is coprime to the group order. Let $\langle g\rangle$ be a cyclic group of order $n$. Suppose $1\leq q \leq n-1$, I want to show that $g^q$ generates $\langle g\rangle$ if and only if $\gcd(n,q)=1$.  Suppose $g^q$ generates $\langle g\rangle$, then   $$ 1,g^q,g^{2q},...,g^{(n-1)q} $$  are all distinct and $g^{nq}=1$, since $n$ is the order of the group. Note that this is the lowest multiple of $q$ for which this is the case. Therefore $\operatorname{lcm}(q,n)=qn$ and it follows that $q$ and $n$ are coprime.  On the other hand, if $\gcd(q,n)=1$, then $\operatorname{lcm}(q,n)=qn$, we have $g^{nq}=1$ and hence all of  $$ 1,g^q,g^{2q},...,g^{(n-1)q} $$  are distinct again and $g^q$ is a generator.  Is my proof correct at all and is there maybe a more elegant argument? According to Hubble's Law, how can the expansion of the Universe be accelerating? Scientists today think the expansion of the universe is accelerating.   According to Hubble's law, objects further away are moving faster than objects closer to us. The further away an object is, the further back in time we are seeing, so in the past objects moved faster (is this sentence correct?).  So because objects moved faster before than they do now, surely that's deceleration not acceleration?  my textbook says "if Scientists today think the rate of expansion of the Universe were decreasing then distant objects should appear different to Hubble's Law predictions: universe is accelerating. More distant objects would seem to be receding faster (since expansion was faster in the past)" Live CD asks for a username and password I have two custom taxonomies applied to two custom post types. the terms list on the sidebar just fine and will list all posts associated with it. However, if you search one of the terms in specific, it doesn't bring up a post with that term.  Example:  Search for term "PQRI"  I get nothing. Any ideas? I've tried using various search plugins but they either break my custom search parameters or just don't work. How to accomplish SQL Server 2012+2014 high availability Maybe I'm searching for the wrong thing, but I can't see how I can do this.  I have two servers:   Server A (has SQL Server 2012 installed) Server B (has SQL Server 2014 installed)   Server A provides an API via IIS. The code both inserts and gets data from the database depending on the API call.  Server B does exactly the same for its local database.  I would like to have it so that the two databases are consistent copies of each other. I do believe this is what HA accomplishes.   So a load balancer, for example, might direct API calls to A sometimes, and B sometimes, but the data needs to be consistent either way.  I found some documentation on HA for SQL Server 2012, but not on how to do this for a mixed environment of 2012 and 2014.  Any help or pointers to documentation would be great! Thanks. Form of the Classical EM Lagrangian So I know that for an electromagnetic field in a vacuum the Lagrangian is $$\mathcal L=-\frac 1 4 F^{\mu\nu} F_{\mu\nu},$$ the standard model tells me this. What I want to know is if there is an elementary argument (based on symmetry perhaps) as to why it has this form. I have done some searching/reading on this, but have only ever found authors jumping straight to the expression and sometimes saying something to the effect that it is the "simplest possible". How do I install pdfedit? How do I disable fog on specific camera/s $a, b, c$ are positive real numbers.   Prove that $$8abc\le (a+b)(b+c)(c+a)$$  I haven't really gotten anywhere, just tried to open up $(a+b)(b+c)(c+a)$ and got null. "Same as I do" or "same as I have" She has the same book as I have.  Isn't it supposed to be She has the same book as I do?  But I often hear the former.  Why? Solving of a seperable differential equation Given $a_1, a_2, a_3, \ldots , a_n$ not all zero, show $\gcd(a_1, a_2, a_3, \ldots , a_n)$ is the least positive integer that can be expressed in the form $a_1x_1+a_2x_2+ \ldots +a_nx_n$. Also deduce $a_1x_1+a_2x_2+ \ldots +a_nx_n = b$ has solutions $\iff \gcd(a_1, a_2, a_3, \ldots , a_n)|b$.   I know that $ax+by=n$ has integer solutions $\iff \gcd(a,b)|n$. I'll need to use induction to show $a_1x_1+a_2x_2+ \ldots +a_nx_n = \gcd(a_1, a_2, a_3, \ldots , a_n)$ has solutions, and then I will be able to say that as $\gcd(a_1, a_2, a_3, \ldots , a_n) | a_1x_1+a_2x_2+ \ldots +a_nx_n $ for any $x_1,x_2, x_3 \ldots x_n$ then $c \leq \gcd(a_1, a_2, a_3, \ldots , a_n)$ and $c \geq \gcd(a_1, a_2, a_3, \ldots , a_n)$, and thus $c=\gcd(a_1, a_2, a_3, \ldots , a_n)$. How can I assign custom icons to folders? Of Half-elven and mortal life