Does influence on gravity increase with mass when velocities approach the speed of light? In other words, when a particle's mass increases significantly as it approaches the speed of light (c), does its influence on gravity increase, and if so, does it increase proportionally?
If a mass moves close to the speed of light, does it turn into a black hole? I'm a big fan of the podcast Astronomy Cast and a while back I was listening to a Q&amp;A episode they did. A listener sent in a question that I found fascinating and have been wondering about ever since.   From the show :      Arunus Gidgowdusk from Lithuania asks: "If you took a one kilogram mass and accelerated it close to the speed of light would it form into a black hole? Would it stay a black hole if you then decreased the speed?"   Dr. Gay, an astrophysicist and one of the hosts, explained that she'd asked a number of her colleagues and that none of them could provide a satisfactory answer. I asked her more recently on Facebook if anyone had come forward with one and she said they had not.  So I thought maybe this would be a good place to ask.
Most common method for deciding when to stop training a neural net on a batch I have created my own neural net which is using batch gradient descent. In other words, it trains on batches of examples all at once.  My issue is trying to figure out when to stop the training of the batch. I'll try to make things as understandable as possible since there are so many options, but please comment if you need clarification.  Each batch allows me to compute a list of output errors that correspond to the error the computer has on each example. Each error is a vector with elements equal to the number of output neurons.  Let each error vector be notated by e1, e2, e3, ... so the list of output errors is [e1, e2, ...]. Let an error vector in general be notated by e.  Now there are 2 things we can do to a vector or a list:   Average: average all of the elements in a vector or list. This will be notated by the function a(). For example to find the average error of an example we would do a(e) Check: make sure that each element in the vector or list is below a certain value. If any of the elements are too high, we continue to train on the entire batch. (This function can also be applied to a single value.) This will be notated by the function c(). For example, if we wanted to make sure the average error in e was low enough, we would do c(a(e)) Sum: add up all of the elements in the vector or list. This will be notated by the function s(). For example, s(e) would be totaling the error found in error vector e.   Here are the methods I have determined for deciding when to stop gradient descent:   Iterative: just run the training a certain number of times Average of averages: Find the average error of each example, then find the average of those and make sure it's below a certain amount. This is given by the formula c(a([a(e1), a(e2), ...])) Specific error: Check to make sure the error in each output neuron is below a certain amount. This is given by the formula c(e1), c(e2), ... Specific averages: Check to make sure the average error in each example is below a certain amount. This is given by the formula c([a(e1), a(e2), ...]) Sum of sums: Total up all the error in the entire batch and then make sure it's low enough. This is given by the formula c(s([s(e1), s(e2), ...]) Sum of averages: Find the average error of each example, then add up all those and make sure it's below a certain amount. This is given by the formula c(s([a(e1), a(e2), ...])) Average of sums: Compute the average total error in each example, and then make sure it's below a certain amount. This is given by the formula c(a([s(e1), s(e2), ...])) Specific sums: Check to make sure the total of the error in each example is below a certain amount. This is given by the formula c([s(e1), s(e2), ...])   So my question is this:  Which one of these options is most commonly used when training neural nets? Why? (Is there another option that I've missed?)  (Edit: I have found a very general version of this question , where the answers seem to say there is no specifically good way. If that is still true, please say so in your answer. Note that this is not a duplicate, as I explain the specific methods I would like to use.)  Edit: as A.D pointed out, summing is pretty useless because it depends on how many values are in each list/array/batch. And if you know how many examples there are, you can just use averaging. So, the methods involving "summing" have been cut. Stack Exchange & Area 51 sign up/ log in update I understand that multivariate regression models contain more than 1 dependent variable, but what is the difference between running a multivariate regression with dependent a and b to a set of independent variables, rather than running two regression models, one for a and one for b against the same set of independent variables? How to add line break to caption without using caption package I wish to add a line break to a caption, but I would wish to avoid adding of caption package. Is it possible?  \documentclass{article} \usepackage[format=plain,  justification=raggedright,singlelinecheck=false]{caption} \begin{document} \begin{figure} \caption{This is the caption\\ This is the second line} \end{figure} \end{document} How to access an object from another class? $f$ holomorphic on $D\setminus \{0\}$ and takes no values in $(-\infty,0],$ then $0$ removable If $f$ is holomorphic on $D\setminus \{0\}$ and takes no values in $(-\infty,0]$ then $0$ is a removable singularity.   I thought to prove this by elimination, but I can't really tell anything about the behavior of $f$ around $0$. How can one translate the information about the definition of $f$ in semi-open interval $(-\infty,0]$. I asked a question that has been marked as a duplicate. I read a lot of similar questions and explanations in advance but I couldn't find a solution. In the question that should answer my question the asker used a different (not working) approach. There is even an answer that would work for my case however I'd rather consider it as a workaround than a solution since I have seen examples using my approach.  So I would like your opinion in that case. Would it be ok to repost my question while explaining why other answers didn't help me? I edited my question but I think it's not getting any attention anymore. Naturally I will remove the old question. Is there a criterion to differentiate "thriller" from "horror"? The genres, in my opinion, have a very slim boundary separating one from another. I see a lot of movies being categorized as thriller or horror almost interchangeably, which might seem like lack of thought or criteria. However, is there a true criterion used to differentiate between the two film genres? Conditional Formatting: 3 color scale showing text relevance in Excel for Mac Is there a way to use conditional formatting in the following manner in Excel? (For those who are curious, I am using Excel 2010 for Mac)  I want Cell B2 to be green when A2 says "John likes green apples" and B2 also says "John likes green apples", yellow if A2 says "John likes green apples" and B2 says "Bill likes blue apples", and red if A2 says "John likes green apples" and B2 says "Bill hates red oranges".  I am just wondering if there is a way to something to that effect.  Thanks The binomial sum   $$s_n=\binom{n+1}{0}+\binom{n}{1}+\binom{n-1}{2}+\cdots$$   satisfies the Fibonacci relation.   I failed to prove that $\binom{n-k+1}{k}=\binom{n-k}{k}+\binom{n-k-1}{k}$... Any hints or suggestions? Suppose I have a node in the shape of a circle. How can I make its border thicker?   \node[circle, draw=blue!80, inner sep=0pt, minimum size=12pt] (1) at (0,0) {1}; Thats my . I loaded this to PostgreSQL DB using PostGIS shape file loader. I ran pgr_anlayzeGraph() on it and found it had a lot of isolated segments.  Next I ran pgr_nodeNetwork() to create a new table 'ways_ways'. I made the topology  for it and ran the following query in Qgis sql window.   SELECT * FROM ways_ways JOIN (SELECT seq, id1 AS node, id2 AS edge, cost FROM  pgr_astar('SELECT id, source, target, cost, x1, y1, x2, y2 FROM ways_ways',  1, 108, false, false) )AS route ON ways_ways.id = route.node;   Given below are the screenshots: Original floorplan and the prescribed route  query ouput   Clearly , I don't have a continuous output. I ran pgr_analyseGraph on ways_ways to find that it has no more isolated segments . It has 51 deadends, which is acceptable.   Did I miss any step in pgrouting? Is there anyway to see which node has id 1 and id 108 in qgis?  i ran the same query in another map(which i downloaded) and got output for all possible source and target ids. So how can i create a way(a route) that is completely noded? I've got a question about how to drive the car in Europe on Highways. In the US, I can overtake another car on either side, and I am fine.  In Europe I was told that I have to drive on the right side (Germany), and to overtake on the left side of the other driver.  Now I had following situation, and I did not know how to handle this properly: I was driving along on the right lane, the Autobahn was 3 lanes wide on my side, and was going around 110 km/h. Someone overtook me, but he stayed on the middle lane, and suddenly got slower. He wasn't really in front of me, so I could not change to the left lane, but I was told I may not overtake on the right side.  Would I have to brake in such a situation, or should I just ignore it, and overtake him again on the right side?  (I overtook him, and he was on the phone, so I guess that was why he got distracted and slowed down that much). How can I prove that if $G$ is an Abelian group with elements $a$ and $b$ with orders $m$ and $n$, respectively, then $G$ contains an element whose order is the least common multiple of $m$ and $n$?  It's an exercise from Hungerford's book, but it's not homework. I could not solve it when I take the course on groups and I think it should be easy. There is a hint that says to first prove when $m$ and $n$ are coprimes. I did this part. But I have no idea how to solve the general case.  Thanks. Number of surjections from $\{1,...,m\}$ to $\{1,...,n\}$ Continuous v. per-chapter/section numbering of figures, tables, and other document elements Some document elements (e.g., figures in the book class) are numbered per chapter (figure 1.1, 1.2, 2.1, ...). How can I achieve continuous numbering (figure 1, 2, 3, ...)?  And vice versa: Some document elements (e.g., figures in the article class) are numbered continuously. How can I achieve per-section numbering?  \documentclass{book}% for "vice versa" variant, replace `book` with `article`  \begin{document}  \chapter{foo}% for "vice versa" variant, replace `\chapter` with `\section`  \begin{figure} \centering \rule{1cm}{1cm}% placeholder for graphic \caption{A figure} \end{figure}  \end{document}   Bonus question: Is it possible to adjust the numbering of the sectioning headings themselves? Can I, e.g., switch from per-chapter to continuous numbering of sections in the book class? I am a UI designer and developer and have been designing websites for aound 10 years. I want to get into user experience design. I know how to do a design, but I am lagging the terminologies and the technical things in the user experience field.   So how do I go about getting the skills and mindset necessary for being a User Experience Designer? How much to tip in the United States? Proving $\gcd \left(\frac{a}{\gcd (a,b)},\frac{b}{\gcd (a,b)}\right)=1$ How would you go about proving that $$\gcd \left(\frac{a}{\gcd (a,b)},\frac{b}{\gcd (a,b)}\right)=1$$  for any two integers $a$ and $b$?  Intuitively it is true because when you divide $a$ and $b$ by $\gcd(a,b)$ you cancel out any common factors between them resulting in them becoming coprime. However, how would you prove this rigorously and mathematically? Proving $\gcd((a+b)^m,(a-b)^m)\le2^m$ for coprime $a,b$ and $m \in \Bbb N$