Explaining why proof by induction works
Why is mathematical induction a valid proof technique?
CloudPagesURL() value not showing in HTML I am trying to pass the email value from a data extension to HTML. I am using the CloudPagesURL() in my email in attempt to grab the subscriber's email.  Here is my code syntax that I am attempting to use:  Email Code:  &lt;p&gt;&lt;a title="MyPage" href="%%=RedirectTo(CloudPagesURL(505,'Email Address',@EmailAddress))=%%" alias="ThisAlias" conversion="false"&gt;Click to unsubscribe page 505&lt;/a&gt;&lt;/p&gt;   CloudPage AMPScript Code:  %%[  VAR @emailAddress  SET @emailAddress = RequestParameter("emailaddr")  ]%%   Cloudpage HTML Code:  &lt;td width='155' valign='bottom'&gt;&lt;input type='text' name='Name1' value='%%=v(@emailAddress)=%%' id='Name1' /&gt;&lt;/td&gt;   Any reason as to why this doesn't work? My syntax might just be wrong or I might be using the CloudPagesURL() function incorrectly. What does the , operator do in C? Port error while changing chrome extension from manifest v1 to v2 So just now, I had a bug in my site that caused my custom navigation bar's buttons to be sized and spaced incorrectly. I had no idea where this bug came from, and had to search around for awhile to find the answer. It turns out that SharePoint decided it would be a good idea to inexplicably add a few lines of:  nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;  nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;   between my   in my DevHome.aspx (I'm using Angular for this particular project). So, why does Sharepoint do this, and how do I prevent it? This is not the first time SharePoint has randomly added an nbsp to my page.  Thanks in advance. I am currently in hard mode and during a blood moon, several of my NPCs including my guide and my healer died. it's been several in game "days" now and they have yet to respawn. Is there a way to check why they aren't respawning?   Is there a list of reasons for the NPCs to not respawn?   Edit in reponse to lunbok's answer:  1. They don't have appropriate housing - There is an in-game housing query function (changes your cursor into a ?) that lets you click on your house to see if it is still a suitable house and I clicked in every room in my house and they are all listed as "This housing is suitable" or "This housing is already occupied".  2. Their spawn conditions aren't met anymore. - This doesn't explain why my guide isn't respawning since they don't need a condition to respawn (other than having a suitable housing which the point above already prove isn't the problem).  3. There's a goblin invasion going on. - Do the invasions last that long? It has been about two reals days and a lot more in-game "days" now. As far as I can tell there shouldn't be an invasion going on.  4. It could be because you built your house in a part of the world that is now corrupted Meaning of syntax $N(\mathbf{y} \mid \mathbf{0}, \mathbf{K})$ (multivariate normal distribution) How to move windows that open up offscreen? I have a dual monitor setup, and I recently played around with the positioning settings, and some of my single window applications do the favour of preserving where they were last closed, and opening in the same position later. Unfortunately, that now places them out of the viewable area of my screens!  Is there some way to force a particular window into the viewable area? How to avoid perceived circularity when defining a formal language? Suppose we want to define a first-order language to do set theory (so we can formalize mathematics). One such construction can be found . What makes me uneasy about this definition is that words such as "set", "countable", "function", and "number" are used in somewhat non-trivial manners. For instance, behind the word "countable" rests an immense amount of mathematical knowledge: one needs the notion of a bijection, which requires functions and sets. One also needs the set of natural numbers (or something with equal cardinality), in order to say that countable sets have a bijection with the set of natural numbers.  Also, in set theory one uses the relation of belonging "$\in$". But relation seems to require the notion an ordered pair, which requires sets, whose properties are described using belonging...  I found the following in Kevin Klement's,  (pages 2-3).  "You have to use logic to study logic. There’s no getting away from it. However, I’m not going to bother stating all the logical rules that are valid in the metalanguage, since I’d need to do that in the metametalanguage, and that would just get me started on an infinite regress. The rule of thumb is: if it’s OK in the object language, it’s OK in the metalanguage too."  So it seems that, if one proves a fact about the object language, then one can also use it in the metalanguage. In the case of set theory, one may not start out knowing what sets really are, but after one proves some fact about them (e.g., that there are uncountable sets) then one implicitly "adds" this fact also to the metalanguage.  This seems like cheating: one is using the object language to conduct proofs regarding the metalanguage, when it should strictly be the other way round.  To give an example of avoiding circularity, consider the definition of the integers. We can define a binary relation $R\subseteq(\mathbf{N}\times\mathbf{N})\times(\mathbf{N}\times\mathbf{N})$, where for any $a,b,c,d\in\mathbf{N}$, $((a,b),(c,d))\in R$ iff $a+d=b+c$, and then defining $\mathbf{Z}:= \{[(a,b)]:a,b\in\mathbf{N}\}$, where $[a,b]=\{x\in \mathbf{N}\times\mathbf{N}: xR(a,b)\}$, as in  question or  on Wikipedia. In this definition if set theory and natural numbers are assumed, then there is no circularity because one did not depend on the notion of "subtraction" in defining the integers.  So my question is:     Question Is the definition of first-order logic circular?   If not, please explain why.   If the definitions are circular, is there an alternative definition which avoids the circularity?   Some thoughts:   Perhaps there is the distinction between what sets are (anything that obeys the axioms) and how sets are expressed (using a formal language). In other words, the notion of a set may not be circular, but to talk of sets using a formal language requires the notion of a set in a metalanguage. In foundational mathematics there also seems to be the idea of first defining something, and then coming back with better machinery to analyse that thing. For instance, one can define the natural numbers using the Peano axioms, then later come back to say that all structures satisfying the axioms are isomorphic. (I don't know any algebra, but that seems right.) Maybe sets, functions, etc., are too basic? Is it possible to avoid these terms when defining a formal language? Number of reflexive relations defined on a set A with n elements Why do different wavelength get impeded more or less when in different materials? Moving with the same speed, but a longer physical distance would imply that the fields oscillate less times in the material, but I don't know why a difference in the number of oscillations would impede the wave- I don't even know why things slow down in general. Why some electromagnetic wave would slow down just because it's entering other electromagnetic fields... It would seem to me that the only factor would be time taken to physically move some electron or something in the direction of the fields... But that seems to simple of an explanation to me. How do I resolve a recurrence relation when the characteristic equation has fewer roots than terms? I know how to solve "simple" recurrence relations. For instance, say you have:  $$c_0 = 20$$ $$c_1 = 30$$ $$c_n = 3 c_{n-1} - 2 c_{n-2}$$  We can write the characteristic equation as:  $$3x^{n-1} - 2x^{n-2} = x^n$$  Solving this with $n=2$, we get $x = 1$ or $x = 2$. This lets us write the relation $c_n = \alpha_1 1^n + \alpha_2 2^n$, and we can solve for $\alpha_1$ and $\alpha_2$ with the initial states $c_0$ and $c_1$.  However, this depends on the fact that $3x^{n-1} - 2x^{n-2} = x^n$ has two roots.  Now, I'm stuck on another problem where the characteristic equation has fewer roots than terms.  Say I have this recurrence relation instead:  $$a_0 = 0$$ $$a_1 = 2$$ $$a_2 = −1$$ $$a_n = 9a_{n-1} - 15a_{n-2} - 25a_{n-3}$$  The characteristic equation would be:  $$9x^{n-1} - 15x^{n-2} - 25x^{n-3} = x^n$$  However, solving with $n=3$, we only get two roots: $x=-1$ or $x=5$. There are not enough roots to write a relation in the form of $a_n = \alpha_1 r_1^n + \alpha_2r_2^n + \alpha_3r_3^n$. How do I proceed? I've installed Xcode and MacPorts on a brand new mid-2013 MacBook Air, but MacPorts doesn't work, saying that it cannot find anything.  mba:~ cnst$ sudo port install wget Warning: The Command Line Tools for Xcode don't appear to be installed; most ports will likely fail to build. Warning: See http://guide.macports.org/chunked/installing.xcode.html for more information. ---&gt;  Computing dependencies for wgetError: Unable to execute port: can't read "build.cmd": Failed to locate 'make' in path: '/opt/local/bin:/opt/local/sbin:/bin:/sbin:/usr/bin:/usr/sbin' or at its MacPorts configuration time location, did you move it? mba:~ cnst$   And, indeed, there is no make or clang in the path, but it is definitely installed, though:  mba:~ cnst$ find /Applications -name "clang" -or -name "make" /Applications/Xcode.app/Contents/Developer/Toolchains/XcodeDefault.xctoolchain/usr/bin/clang /Applications/Xcode.app/Contents/Developer/Toolchains/XcodeDefault.xctoolchain/usr/lib/clang /Applications/Xcode.app/Contents/Developer/usr/bin/make mba:~ cnst$   Why is there no make or clang in the path?  Why MacPorts doesn't work?  I notice that  mentions that you have to install something from ADC, Apple Command Line Developer Tools, but ADC doesn't work since last Thursday or Friday due to their security breach, so, the developer tools are nowhere to be found.  What's the proper solution here? Help! My master database is corrupt, I can't even bring the SQL instance online! What are my options to get my server back up?   I do have a backup of master, but the  asks me to start the instance in single-user mode, which I can't do!  (Note: I'm leaving this question unspecified as to SQL version, so as to be a more widely-applicable reference. There are some similar questions on DBA.SE, but none that involve the server unable to start.) What is the difference between sector and cluster? I have gone through wikipedia and other sites and here's the information I have obtained.  Sector: Smallest Addressable Unit (mostly 512 bytes) Cluster: Smallest Allocation Unit (equals to 1 or more sectors and the number of clusters depends on the file system).  Let's say in the below example:  sector =512 bytes and a cluster =8 sectors=4096 bytes. Now, I am having a hard time figuring out, when I am trying to create a file what does the file system do? Does it consider 4096 bytes as the smallest write/read unit or 512 bytes?  If the answer is 4096 bytes: When the manufacturer of the disk drives has provided 512 bytes as the smallest addressable unit, why does the file system consider cluster as the smallest unit? I mean, what are the advantages in doing so? How to pipe the result of 'date' command to 'mv' command I want to rename my file to the current timestamp.I know how to get the current timestamp("date +%s") but how can I pipe it to 'mv' command? Why 220V and 110V? Why did the industry choose these seemingly unintuitive values of 110V and 220V for the mains AC? Is there a physical reason why they aren't simply 100V and 200V? How to import a jar in Eclipse The RSA signature operation is basically the same as encrypting with the private key.  In particular, both operations use the same kind of keys.  Is it safe to use the same RSA keypair both for encryption / decryption and for signing / verification? Windows 10 stuck in install loop I was having a time getting my bootable USB 3, 32GB W10 install USB to be recognized. It recognized one time but I had to restart for something and it never came up again. I eventually removed all other boot up options to make it easier and faster to try again (no F12 needed).  Finally I gave up and tried another drive (16GB this time FWIW) and then it worked.  HOWEVER, it's stuck on a reboot loop now. I can choose to install, then to select the right disk, then it says it's loading file and then next thing I know it's saying do you want to install or upgrade. Can I be admitted into a PhD program with non-thesis MS degree? If I do a non-thesis MS program in statistics, will I be eligible for an admission in to a PhD program in Statistics later in my life? The MS program I am looking into is a non-thesis program but I still have to complete a research project requirement (which involves oral presentation as well as some form of a written report) towards the end of the program...I am still not sure if it can lead to a PhD program though.  Thanks!