Does energy have a sign?
In physics sometimes we find energy that is negative. What does the negative sign indicate?
Proof of Russian Peasant Multiplication I'm studying a combinatorics text () in preparation for taking discrete math this upcoming semester. I've taken a fair amount of math in college, through Calculus II, but this is my first 500+ level math course.    In chapter 2 of the  book, the author introduces an algorithm for multiplication called Russian Peasant Multiplication. I want to prove that this algorithm will always result in the product of two numbers.  I've been chewing on the proof for a while now, and just can't seem to get any traction. Thanks in advance for any help.  Description of Algorithm  I'll try to explain the algorithm clearly. The goal of the algorithm is to find the product of two numbers $m$ and $n$. The below table is filled out for $m$ and $n$:  $$\begin{array}{c} m&amp;&amp;n&amp;\\ \hline \bigl\lfloor \frac{m}{2} \bigr\rfloor&amp;&amp; 2n \\ \bigl\lfloor \frac{m}{4} \bigr\rfloor&amp;&amp; 4n \\ \bigl\lfloor \frac{m}{8} \bigr\rfloor&amp;&amp; 8n \\ \bigl\lfloor \frac{m}{16} \bigr\rfloor&amp;&amp; 16n \\ \bigl\lfloor \frac{m}{32} \bigr\rfloor&amp;&amp; 32n \\ \end{array}$$  The sequence of numbers in both columns is the sequence $a_n = 2a_{n-1}\text{ where }a_1=2$. The columns are extended down until the value in the left column is equal to the number one. After filling out the table, the product is found by adding the values of the right column where the left column is an odd number.  For example, to multiply 18 and 37 with the algorithm:  $$\begin{array}{c} m&amp;&amp;n&amp;\\ \hline 18 &amp;&amp; 37 \\ 9 &amp;&amp; 74 \\ 4 &amp;&amp; 148 \\ 2 &amp;&amp; 296 \\ 1 &amp;&amp; 592 \\ \end{array}$$  To find the product, cross out values of the left column where the value of the right column is even:  $$\begin{array}{c} m&amp;&amp;n&amp;\\ \hline \require{enclose} \enclose{horizontalstrike}{18} &amp;&amp; \enclose{horizontalstrike}{37} \\ 9 &amp;&amp; 74 \\ \enclose{horizontalstrike}{4} &amp;&amp; \enclose{horizontalstrike}{148} \\ \enclose{horizontalstrike}{2} &amp;&amp; \enclose{horizontalstrike}{296} \\ 1 &amp;&amp; 592 \\ \end{array}$$  And add the remaining numbers in the right column to get the result: $18\times37=74+592=666$  Alternate Description of Algorithm  As an aside, here is a shorter way to describe the algorithm, repeat the below steps until the left column == 1:  • Halve the last number in the first column • Write the result below (discard the remainder) • Double the last number in the first column. • Write the result below.   Now, for every even number in the first column, delete the adjacent number in the second column. Add the remaining numbers in the second column.  What I've done so Far  I haven't been able to get much of anywhere on the proof. One thing I did was create a bit of a different way of arriving at the same outcome:  $$\begin{array}{c} m&amp;&amp;n&amp;&amp;\text{index}&amp;&amp;\text{remainder}&amp;&amp;\text{if remainder == 1, }n\times2^{Index}&amp;\\ \hline 18 &amp;&amp; 37 &amp;&amp; 0 &amp;&amp; 0 \\ 9 &amp;&amp; 74 &amp;&amp; 1 &amp;&amp; 1 &amp;&amp; 74\\ 4 &amp;&amp; 148 &amp;&amp; 2 &amp;&amp; 0 \\ 2 &amp;&amp; 296 &amp;&amp; 3 &amp;&amp; 0 \\ 1 &amp;&amp; 592 &amp;&amp; 4 &amp;&amp; 1 &amp;&amp; 592\\ \end{array}$$  Which brings to light the relationship between this algorithm and binary numbers (the fourth column in the above table is the product in binary, read bottom to top).   I also wrote some Python functions to help me explore the problem, Cant install via Ubuntu Software- "apt transaction returned result exit-failed" I'm using 17.04 ( Clean Install ), and as far as I remember, every try to install any sofware- it fails with that error code.  I tried sudo apt-get update &amp;&amp; apt-get upgrade and problem persists.  any idea why?   edit: Using software Updater gave this result :   edit : output of update  guy@LG-LAPTOP:~$ sudo apt-get update &amp;&amp; apt-get upgrade [sudo] password for guy:  Ign:1 http://il.archive.ubuntu.com/ubuntu zesty InRelease Ign:2 http://il.archive.ubuntu.com/ubuntu zesty-updates InRelease               Ign:3 http://il.archive.ubuntu.com/ubuntu zesty-backports InRelease             Err:4 http://il.archive.ubuntu.com/ubuntu zesty Release                           503  Service Unavailable Get:5 http://security.ubuntu.com/ubuntu zesty-security InRelease [89.2 kB]      Err:6 http://il.archive.ubuntu.com/ubuntu zesty-updates Release                   503  Service Unavailable Get:7 http://archive.canonical.com/ubuntu zesty InRelease [10.2 kB]             Err:8 http://il.archive.ubuntu.com/ubuntu zesty-backports Release                 503  Service Unavailable Get:9 http://archive.canonical.com/ubuntu zesty/partner Sources [1,416 B]       Get:10 http://archive.canonical.com/ubuntu zesty/partner amd64 Packages [1,616 B] Get:11 http://archive.canonical.com/ubuntu zesty/partner i386 Packages [1,612 B] Get:12 http://archive.canonical.com/ubuntu zesty/partner Translation-en [708 B] Get:13 http://security.ubuntu.com/ubuntu zesty-security/main amd64 DEP-11 Metadata [11.7 kB] Get:14 http://security.ubuntu.com/ubuntu zesty-security/main DEP-11 64x64 Icons [9,219 B] Get:15 http://security.ubuntu.com/ubuntu zesty-security/universe amd64 DEP-11 Metadata [14.4 kB] Get:16 http://security.ubuntu.com/ubuntu zesty-security/universe DEP-11 64x64 Icons [31.0 kB] Reading package lists... Done                                          E: The repository 'http://il.archive.ubuntu.com/ubuntu zesty Release' does no longer have a Release file. N: Updating from such a repository can't be done securely, and is therefore disabled by default. N: See apt-secure(8) manpage for repository creation and user configuration details. E: The repository 'http://il.archive.ubuntu.com/ubuntu zesty-updates Release' does no longer have a Release file. N: Updating from such a repository can't be done securely, and is therefore disabled by default. N: See apt-secure(8) manpage for repository creation and user configuration details. E: The repository 'http://il.archive.ubuntu.com/ubuntu zesty-backports Release' does no longer have a Release file. N: Updating from such a repository can't be done securely, and is therefore disabled by default. N: See apt-secure(8) manpage for repository creation and user configuration details. guy@LG-LAPTOP:~$    output of apt get  guy@LG-LAPTOP:~$ sudo apt-get install putty Reading package lists... Done Building dependency tree        Reading state information... Done The following additional packages will be installed:   putty-tools Suggested packages:   putty-doc The following NEW packages will be installed:   putty putty-tools 0 upgraded, 2 newly installed, 0 to remove and 0 not upgraded. Need to get 661 kB of archives. After this operation, 2,713 kB of additional disk space will be used. Do you want to continue? [Y/n]  Err:1 http://il.archive.ubuntu.com/ubuntu zesty/universe amd64 putty-tools amd64 0.67-3   503  Service Unavailable Err:2 http://il.archive.ubuntu.com/ubuntu zesty/universe amd64 putty amd64 0.67-3   503  Service Unavailable E: Failed to fetch http://il.archive.ubuntu.com/ubuntu/pool/universe/p/putty/putty-tools_0.67-3_amd64.deb  503  Service Unavailable E: Failed to fetch http://il.archive.ubuntu.com/ubuntu/pool/universe/p/putty/putty_0.67-3_amd64.deb  503  Service Unavailable E: Unable to fetch some archives, maybe run apt-get update or try with --fix-missing? guy@LG-LAPTOP:~$ Decode HTML entities in Python string? Rotate whole animation by 90 degrees I make my animation and now I realized that its pointed in wrong direction I used -y (native blender) as direction but I need to be pointed at x axis so what would be the easiest way to rotate whole animation, to switch -y with x :) Proof of $1+x\leq e^x$ for all x? Python function attributes - uses and abuses Not many are aware of this feature, but Python's functions (and methods) can have . Behold:  &gt;&gt;&gt; def foo(x): ...     pass ...      &gt;&gt;&gt; foo.score = 10 &gt;&gt;&gt; dir(foo) ['__call__', '__class__', '__delattr__', '__dict__', '__doc__', '__get__', '__getattribute__', '__hash__', '__init__', '__module__', '__name__', '__new__', '__reduce__', '__reduce_ex__', '__repr__', '__setattr__', '__str__', 'func_closure', 'func_code', 'func_defaults', 'func_dict', 'func_doc', 'func_globals', 'func_name', 'score'] &gt;&gt;&gt; foo.score 10 &gt;&gt;&gt; foo.score += 1 &gt;&gt;&gt; foo.score 11   What are the possible uses and abuses of this feature in Python ? One good use I'm aware of is 's usage of the docstring to associate a syntax rule with a method. But what about custom attributes ? Are there good reasons to use them ? How do you tell what brand/make/model of bike I have? I have a bike I'm trying to sell because it's just been sitting in my basement. It looks as if someone primered it and the only thing I found was a serial number. Is there like a website I can look up more info on the bike using just that number? Is there a limit to the number of web properties one can create in Google Analytics? What can I do when getting "We are no longer accepting questions/answers from this account"? I am curious about the practical implementation of a binary split in a decision tree - as it relates to levels of a categorical predictor $X{j}$.  Specifically, I often will utilize some sort of sampling scheme (e.g. bagging, oversampling etc) when building a predictive model using a decision tree - in order to improve its predictive accuracy and stability. During these sampling routines, it is possible for a categorical variable to be presented to a tree fitting algorithm with less than the complete level set.  Say a variable X takes on levels {A,B,C,D,E}. In a sample, maybe only levels {A,B,C,D} are present. Then, when the resulting tree is used for prediction, the full set may be present.  Continuing from this example, say a tree splits on X and sends {A,B} to the left and {C,D} to the right. I would expect the logic of the binary split to then say, when faced with new data: "If X has value A or B, send to the left, otherwise, send this case to the right". What seems to happen in some implementations is "if X has value A or B, send to the left, if X has value C or D send to the right". When this case takes on value E, the algorithm breaks down.   What is the "right" way for a binary split to be handled? It seems the much more robust way is implemented often, but not always (see Rpart below).  Here are a couple examples:  Rpart fails, the others are ok.  #test trees and missing values  summary(solder) table(solder$PadType)  # create train and validation set.seed(12345) t_rows&lt;-sample(1:nrow(solder),size=360, replace=FALSE) train_solder&lt;-solder[t_rows,] val_solder&lt;-solder[-t_rows,]  #look at PadType table(train_solder$PadType) table(val_solder$PadType) #set a bunch to missing levels(train_solder$PadType)[train_solder$PadType %in% c('L8','L9','W4','W9')] &lt;- 'MISSING'   #Fit several trees, may have to play with the parameters to get them to split on the variable  ####RPART mod_rpart&lt;-rpart(Solder~PadType,data=train_solder) predict(mod_rpart,val_solder) #Error in model.frame.default(Terms, newdata, na.action = na.action, xlev = attr(object,  :  #factor 'PadType' has new level(s) D6, L6, L7, L8, L9, W4  ####TREE mod_tree&lt;-tree(Solder~PadType,data=train_solder,split="gini") predict(mod_tree,val_solder) #works fine  ####ctree mod_ctree&lt;-ctree(Solder~PadType,data=train_solder,control = ctree_control(mincriterion = 0.05)) predict(mod_ctree,val_solder) #works fine Wait until all jQuery Ajax requests are done? The muffins were rock hard, what did I do wrong? Here is the recipe I used, I did not mix it for long, as that make the muffins very hard.   The muffins were very hard, and had a bad after taste and smell. I missed adding some oil, but what else could be the reason ?   Red Millet Flour, 50 g Whole Wheat flour, 50 g Toasted Oat Bran, 40g Whole Green Mung Beans - Sprouted, 100 g Domino - Light Brown Sugar 50g Baking Soda, 2 tsp Salt, 1 Tablespoon Liquid Egg White, 4 eggs 100% Whey (Vanilla), 75g Chobani - Greek Yogurt - 0% Vanilla, 5.3 oz (small packet) Carrots - Raw, 1 cup, grated Bananas - Raw, 1 cup, mashed White, all-purpose, enriched, bleached, 40 g   Also I wanted to try yeast instead of baking soda, but due to eggs and other veggies I cannot let is rise overnight (any ideas ?). I was trying to make good protein muffins which are an ideal breakfast, grab one with a glass of milk and done!   They run approximately 160 calories per muffin 25g carbs, 1g fat, 12g protein. Fundamentally, what does a friction force do for a rolling object? I am very confused by the way my textbook explains it. In my textbook , it says:  A wheel rolling on a horizontal flat surface at a constant velocity experiences no friction force. Why?   On the same surface, there is an acceleration of the wheel pointing to the right (probably caused by a force), so the ball is angularly accelerating in the clockwise direction. In this case, a friction force appears, and it is also pointing to the right. How come? What does the friction do for this wheel?  On an inclined plane, a ball freely rolls down the surface. The direction of friction is up the ramp, which confuses me because in the previous example the friction force was in the direction of the wheel's acceleration.   And there is a difference if a wheel is freely rolling and if there is a torque acting on the ball's center of mass. WHY???  What does friction do in these cases? How does it cause an object to roll? How can I get the list of all fields(API name,Label,type) of any sObject? I'm trying to build a Dynamic list(table) of records by selecting the Sobject and its fields.So I need to get all the fields with its API name,label,type.But I can't get the label and type.(Tried DescribeSObjectResult Class methods to get label,but it didn't help to get the label dynamically)  I tried it with the following code.  System.debug('Selection-----'+selectedObject); Map&lt;String, Schema.SObjectType&gt; gd = Schema.getGlobalDescribe();  Schema.SObjectType ctype = gd.get(selectedObject);  Map&lt;String, Schema.SobjectField&gt; fmap = ctype.getDescribe().fields.getMap();    List&lt;FieldWrap&gt; strList = new List&lt;WrapMap&gt;(); for(String s: fmap.keySet()) {     WrapMap wmp = new WrapMap();     wmp.name = s;     wmp.label = String.valueOf(fmap.get(s));     strList.add(wmp);  }   Where FieldWrap is my wrapper class with name and field of String types. I'm getting the List of API name of fields,but not the Label and type of it.Is there a way to get the label and type of field.If so, please help me. Proving that two systems of linear equations are equivalent if they have the same solutions I've just begun to work learn Linear Algebra on my own through Hoffman and Kunze's book and the first problem set already has a question that I can't solve:  Prove that if two homogeneous systems of linear equations in two unknowns have the same solutions, then they are equivalent.  I can't seem to figure out how to prove this without resorting to case work where you account for the cases where one of the coefficients are zero and when both are.  Is there an elegant way to prove in general that when two systems of linear equations have the same solutions, they are equivalent? The converse is obvious enough though.  Definition of equivalence from the text :     Let us say that two systems of linear equations are equivalent if each equation in each system is a linear combination of the equations   in the other system. Have red shifted photons lost energy and where did it go? I think the title says it. Did expansion of the universe steal the energy somehow? CLI path to ftp network connection I've setup a connectio to an FTP server in Nautilus / Nemo / Caja, the file manager. Now I can browse the contents via this file manager.  How can I browse these files via the command line? What is the path to this network connection? Is this possible, is this a system connection, or specific to the file manager, and do I have to setup a separate FTP connection to do that? If I open this connection in Caja, I see it in Nemo as well, so it seems like a system connection. Site hosted on EC2 / Amazon held hostage with private key I have a new client whose previous developer has left her high &amp; dry with her site on Amazon/EC2. He holds the private key to the instance, and though she has full access to the Amazon control panel (she's paying $300/mo for a site that has like 3 visitors per week) she doesn't have any way to get her content off and move to a new host.  I'm not familiar with AWS and am not sure how to help her. The site's group/file permissions are screwed up on the server so exporting the content/database via her WordPress control panel isn't working.  Any ideas on how to help her? Thanks! Why is my mesh inversed when Imported to Unity? So I modeled this sword and all the parts were separate meshes. I used a mirror modifier when I was modeling and then I applied the modifier when I was done to make the parts into solid meshes. Put a UV map on everything and it all looked good in Blender.  Then I put it into Unity and I got what you see in the picture. Part of the sword is inversed. Oddly, the inversed part actually maintains the proper silhouette. It maintains the correct shape, but then is shaded like the front half is missing.  Does anyone have an idea of what's going on and how to fix it?  Thanks, James Use chown to set the ownership of all a folder's subfolders and files? How can I use the chown command to change the ownership of all a folder's subfolders and files?