I am looking for a general rule for all possible integers $k\in(a,b)$ such that $k$ is can be expressed as a linear combination of $a$ and $b$.   I understand that k must be a multiple of $\gcd(a,b)$, since any linear combination of $a$ and $b$ will be a multiple of $\gcd(a,b)$. Is there a general formula for $k$ in terms of $a,b$?  Thanks in advance!
I want to find a set of integer solutions of Diophantine equation: $ax + by = c$, and apparently $\gcd(a,b)|c$. Then by what formula can I use to find $x$ and $y$ ?   I tried to play around with it: $x = (c - by)/a$, hence $a|(c - by)$.     $a$, $c$ and $b$ are known. So to obtain integer solution for $a$, then $c - by = ak$, and I lost from here, because $y = (c - ak)/b$. I kept repeating this routine and could not find a way to get rid of it? Any hint?  Thanks, Chan
What is the reason that the value "47" is so popular in electrical engineering? We often see component values of 4.7K Ohm, 470uF, or 0.47uH.  For example, digikey has millions of 4.7uF ceramic capacitors, and not a single 4.8uF or 4.6uF and only 1 listed for 4.5uF (specialty product).  What's so special about the value 4.7 that sets so far apart from say 4.6 or 4.8 or even 4.4 since in the 3.. series we usually 3.3,33, etc.  How did these numbers come to be so entrenched?  Perhaps a historical reason? Find the ratio of $\frac{\int_{0}^{1} \left(1-x^{50}\right)^{100} dx}{\int_{0}^{1} \left(1-x^{50}\right)^{101} dx}$ Short example of regular expression converted to a state machine? laptop fails to suspend or hibernate (wakes up after 2-5 seconds without user interaction) Shortcut to open Terminal and run some commands Can comments be used in JSON? Having recently reviewed a few suggested edits, I've found that not being able to see the comments on the post is hindering my understanding of the context of some edits.  In particular, there have been a few posts where someone has made a substantial edit to a question or answer which significantly alters its meaning. Typically I'd reject that as "too radical", since it can't be known that the new meaning of the post was what the original author intended.  However, in some cases it is clear from the comments that the original author is aware of the intended change, and as such the editor is not in fact misrepresenting the author's intentions. In these cases, it would have been useful to see the comments when making the review decision.  As an example, a common cause of this is when an inexperienced user is asked to clarify their question, and ends up writing a comment that would be better off as an edit to the question (due to length, excess code content, etc), presumably due to lack of awareness that the question can be edited. Often, other users will helpfully edit the question to include the information the OP put in the comment (see ,  and  for examples). In this situation, it is clear that the edit reflects the OP's intentions, but this information is only present in the comments which are not visible in the review interface.  Following is the suggested edit screen showing the suggested edit. It seems indentation is increased and one line is moved a few lines upwards and some empty lines added to increase readability. But it look like an invalid edit since changing the related line may cause alteration on the runtime. So not looks lie a valid edit. Better open the answer and examine it        Checking the Q&amp;A itself reveals that a talk had taken place between the OP and the answerer and revision done by the OP with the permission of the answerer. So it was indeed a valid edit.        In short: is there any particular reason comments are not displayed in the suggested edit review interface? I think it would add useful contextual information when making review decisions, and presumably this would be trivial to implement.  Note: this is not a dupe of , as it does not concern comments posted after the suggested edit. Looking to identify a short story I read in a sci-fi anthology as a child, about an angry, bullied somewhat misanthropic boy who is given immense near-omnipotent powers by an alien intelligence. I read this in English, sometime in the late 1990s/early 2000s. He goes on to use this power to destroy the human race, which causes him to age terribly in a very short amount of time. At the end of the story he is a lonely old man stripped of the powers he held briefly, and is alone on an isolated Earth. The aliens then proceed to invade Earth. How can I enable Hardmode? Relation between rank and number of distinct eigenvalues of a matrix Let $T : V\to V$ be a linear transformation such that $\dim\operatorname{Range}(T)=k\leq n$, where $n=\dim V$. Show that $T$ can have at most $k+1$ distinct eigenvalues.   I can realize that the rank  will correspond to the number of non-zero eigenvalues (counted up to multiplicity) and the nullity will correspond to the 0 eigenvalue (counted up to multiplicity), but I cannot design an analytical proof of this.  Thanks for any help . In Back to the Future, how did Marty get the timing right with the lightning strike? So, in the first movie, presumably the Doc and Marty know exactly when the lightning will strike the clock tower. This is because Marty has the flyer with the picture of the stopped clock on it. In order to power the DeLorean, the Doc cooks up this contraption to conduct the lightning through a hook into the DeLorean. All Marty has to do is be driving at 88 mph and hit the wire at the exact right moment.  Here's the question though - if you recall, the Doc sets a timer and tells Marty to drive back to a pre-designated starting point. When the timer goes off, Marty is supposed to accelerate to 88 mph. My assumption is that the Doc has calculated the maximum acceleration of the DeLorean, and using this value has determined the exact starting point that the car needs to be at in order to hit the lightning at the right time.  Unfortunately, the car stalls. It takes Marty a while, but he's finally able to start it up. Being that he started late though...shouldn't he have missed the lighting? How do I prepare best for my first wedding photography event? I am going to shoot a wedding this weekend, and I want to do my best for my friend's big day.    What can I do to prepare for the day in order to minimize failures and mistakes? How do I avoid missing important shots? What backup plan should I have?   I have been shooting for 6 years or so, and I know wedding is one of the hardest thing to do. I have told my friend what I end up producing will be of no match with the work of dedicated wedding photographers, not even the ones that are about to fall out of business.  They are on a tight budget, they said some photos are still better than none, they really must minimize the cost as much as they can. I exaggerated again that my photos will be worse off than the cheapest wedding photographer you can find.  She has been a close friends for years, I want to do everything I can to help her.  Their wedding is at 12:30pm coming Sunday, I have visited the venue last Sunday at 12:30pm to check traffic condition and also the lighting.  The setup is mainly outdoor (no church), the indoor area will not have a good ceiling for bounce. Being at the middle of the day, this can mean some tricky fill flash. I have jot down a list of must-have shots. I am also asking for the seating plan and the event run-down.  Gear:   Canon EOS 60D , EF-S 15-85mm f/3.5-5.6 IS USM Canon EOS 450D (or I can borrow a 550D), EF 50mm f/1.4 Speedlite 580 EXII Speedlite 270 EXII Yongnuo 460 (a cheap but powerful manual flash as backup) Samsung EX-1 f/1.8 point and shoot (backup of a backup...?)   Other lenses that I have but I don't think will be useful:   50mm f/1.8 EF-S 50-250mm f/4.0-5.6 IS II EF-S 18-135mm f/3.5-5.6 IS (I have the better 15-85 on 60D, should I put this on the 450D as a backup allrounder?) EF-S 60mm f/2.8 Macro (should I replace 50mm f/1.4 with this for added macro? You can hardly use f/1.4 under direct sunlight if you want to have fill-flash)   I have two batteries for all cameras (60D 450D and EX1). I have like in total 64GB SD card (I am really inclined to get more, or bring a laptop for on-the-fly backup, or do both). I recently repaired my 580EXII, it has a brand new flash head so the recycle time is as good as it can be, I just need a tonne of batteries. No I don't have external power pack or battery grip for my cameras.  I have a great monopod, but a cheap tripod.  I am happy to bring everything, just putting them in the boot of my car. The carpark is on-site, walking from venue to my car is under 1 minute.  I will have my girlfriend carry the 450D and shoot details / non-portraits.  One concern that I have is fill flash, I really do not have a large surface to bounce off  from. I have shot events before but they are not as dynamic/critical as a wedding.  How do I prepare for differing weather conditions?  If its sunny, fill flash is probably going to be in every single shot. If its cloudy, I probably will turn off the flash. If it rains, my 60D should be fine I have used it in slight rain before, but I really don't know their plan and if they have extendable shelters etc.  What else should I prepare for? Having no experience I want to be as prepared as I can be. Maybe I have overlooked something, maybe there is something else I can do to ensure the shooting will go smoothly. Your help is greatly appreciated! Experimental Fractional scaling makes fonts dirty in Ubuntu 17.10 I updated to 17.10 from 17.04.   I activated the fractional scaling by running the following command.   gsettings set org.gnome.mutter experimental-features "['scale-monitor-framebuffer']"   Then, I changed 125% scale. The scale was changed, but the fonts and icons are so dirty.    The launcher icon and font in Google Chrome and other application except the system setting are dirty.  How can I resolve this problem?  For some reason, font in the system setting is not dirty... Gauss's law in a uniform charge distribution extending infinitely in all directions Let us assume the universe filled with positive charge. About a particular point, all the positive charged particles will be symmetrical. Now consider a sphere of radius $r &lt; \infty$ and apply Gauss law. According to Gauss law the electric flux through the sphere will be some finite value as it contains some positive charge. But by symmetry the electric field through the sphere and hence electric flux will be zero. This is a contradiction. Smoke Generator does not work When I try to run the smoke simulator in Blender 2.62 I get no smoke at all. When I attempt to render, the only output is a black screen. What am I doing wrong here that is making the smoke simulation not render?  Is it a mistake I made, or a problem with Blender version 2.62? .NET Attributes: Why does GetCustomAttributes() make a new attribute instance every time? So I was playing around a little more with attributes in .NET, and realized that every call to Type.GetCustomAttributes() creates a new instance of my attribute. Why is that? I would think that the attribute instance would basically be a singleton-per-MemberInfo, with 1 instance bound to the Type, PropertyInfo, etc...  Here is my test code:  using System;  namespace AttribTest { [AttributeUsage(AttributeTargets.Class)] class MyAttribAttribute : Attribute {     public string Value { get; set; }      public MyAttribAttribute()         : base()     {         Console.WriteLine("Created MyAttrib instance");     } }  [MyAttrib(Value = "SetOnClass")] class MyClass { }  class Program {     static void Main(string[] args)     {         Console.WriteLine("Getting attributes for MyClass.");         object[] a = typeof(MyClass).GetCustomAttributes(false);         ((MyAttribAttribute)a[0]).Value = "a1";          Console.WriteLine("Getting attributes for MyClass.");         a = typeof(MyClass).GetCustomAttributes(false);         Console.WriteLine(((MyAttribAttribute)a[0]).Value);          Console.ReadKey();     } } }   Now if I were to implement attributes, I would expect the output to be:  Created MyAttrib instance Getting attributes for MyClass. Getting attributes for MyClass. a1   Where the "class loader" (sorry, I have more of a Java background, not 100% sure how .net loads its types) would compile MyClass, and create an instance of MyAttribAttribute, and store them together somewhere. (probably the Perm Gen in the heap if this were Java) The 2 calls to GetCustomAttributes() would then just return the same earlier created instance.  But the actual output is:  Getting attributes for MyClass. Created MyAttrib instance Getting attributes for MyClass. Created MyAttrib instance SetOnClass   So... why? It seems like creating a new instance of all these objects for every call is a bit excessive, and not good for performance/memory management.  Is there any way to always get the same instance over and over?  Anyone have any ideas why it was designed this way?  The reason I care at all is because I made a custom attribute that internally holds some validation information, so in the Attribute I basically have a "private bool Validated" that I set to true. The validation stuff takes a while, so I don't want to run it every time. Now the problem is that since it creates a new instance of the attribute each time I get the attributes, Validated is always "false". How to use the SSD for caching in Windows without Intel Smart Response? My motherboard is Asus P8B75-M. It doesn't have Intel Smart Response. But recently I bought a SSD. Since I mostly use linux I have Linux installed in it. But can I use it to speed up Windows also? How? I don't have Smart Response... (I run Windows 8.1). The SSD is only 64GB. What hinge is this? Bit of a newbie to DIY here.  We've moved in to a house that came with a summer house but the double doors were a bit wonky/wouldn't open easily without breaking in the door. I need to get 4 new hinges for the doors but I'm not sure where I can find these ones. I've looked online on UK sites such as Screwfix but as I'm not sure of the particular name I'm not having much luck.  I've attached a drawing (sorry it's the best I can do as my camera isn't working). My Paint skills aren't the best, it's actually a cylinder shape and half of it spins/moves (this is what is broken/doesn't move anymore so it breaks in to the door).    Cheers, Jack A common good practice in Machine Learning is to do feature normalization or data standardization of the predictor variables, that's it, center the data substracting the mean and normalize it dividing by the variance (or standard deviation too). For self containment and to my understanding we do this to achieve two main things:   Avoid extra small model weights for the purpose of numerical stability. Ensure quick convergence of optimization algorithms like e.g. Conjugate Gradient so that the large magnitude of one predictor dimension w.r.t. the others doesn't lead to slow convergence.   We usually split the data into training, validation and testing sets. In the literature we usually see that to do feature normalization they take the mean and variance (or standard deviation) over the whole set of predictor variables. The big flaw I see here is that if you do that, you are in fact introducing future information into the training predictor variables namely the future information contained in the mean and variance.  Therefore, I do feature normalization over the training data and save the mean and variance. Then I apply feature normalization to the predictor variables of the validation and test data sets using the training mean and variances. Are there any fundamental flaws with this? can anyone recommend a better alternative? Why can't I use switch statement on a String?