My math teacher claimed that he could guess any polynomial with non-negative coefficients given two values that he asked for. For example, he asked me to write down a function of which I wrote down (x^5 + 3x^2) and didn't tell him. Simply by knowing the values of f(1) and f(5), he was able to guess the function with accuracy. Is this pure guessing or is there a mathematical explanation for it? Need to know a minimum of two values to guess a polynomial with non-negative coefficients.
Suppose that you are given a polynomial $p(x)$ as a black box (i.e. some oracle, to which you feed $x$ and it returns $p(x)$). It is known that the coefficients of $p(x)$ are all positive integers. How do you determine what $p(x)$ is in the quickest way possible?  (There are 2 metrics for quickness: the number of calls to the oracle and total number of operations. The relationship between the two is not given so we try to minimize both.)
When do I have to use 'will' instead of 'going to'? Does going to only express an intention and will some kind of prediction that doesn't necessarily happen?  EDIT: Thanks for all your answers. I asked this question because I always fail to complete exercises like this from an ESL book:   We  will fly to Venice in June. (Correct: But, how can I be sure that I'm not sick in June?) Philipp will be 15 next Wednesday (Correct: Since his birthday is a fixed day, this will happen). They will get a new computer. (Wrong: Why?, If I need a new computer, I simply get a new one) In 2020 people are going to buy more hybrid cars.  (Wrong, Why? Who can tell for sure). Determining application path in a Python EXE generated by pyInstaller I have an application that resides in a single .py file.  I've been able to get pyInstaller to bundle it successfully into an EXE for Windows.  The problem is, the application requires a .cfg file that always sits directly beside the application in the same directory.  Normally, I build the path using the following code:  import os config_name = 'myapp.cfg' config_path = os.path.join(sys.path[0], config_name)   However, it seems the sys.path is blank when its called from an EXE generated by pyInstaller.  This same behaviour occurs when you run the python interactive command line and try to fetch sys.path[0].  Is there a more concrete way of getting the path of the currently running application so that I can find files that are relative to it? After creating some hi-poly model, i'm always creating low-poly model by approximating that hi-poly mesh with triangle-by-triangle LP mesh - with snap to face mode turned on.   So the process looks like CTRL + RMB (x times), and hitting F on keyboard, so the new face was made.  Then I'm using this low poly model to texture baking, mainly for gamedev purposes.  This method is very time consuming, so how to make retopology process easier and quicker? How do I tell Finder to always show my Library folder on Mavericks? I thought there was a checkbox to suppress the hiding of my user Library folder on Mavericks, but I can't seem to find it. I appreciate hiding Library by default, but  have missed it being easily accessible.  How can I get the library folder to be visible in Finder, the Finder go menus / file save dialogs without needing to hold down shift? How can I assign different random values to each row in a SELECT statement? Please look at this code:  create table #t1(   id int identity (1,1),   val varchar(10) );   insert into #t1 values ('a'); insert into #t1 values ('b'); insert into #t1 values ('c'); insert into #t1 values ('d');   Now, whenever you execute this  select *,      ( select top 1 val from #t1 order by NEWID()) rnd  from #t1 order by 1;   you will get a result with where all rows have the same random value. e.g.  id          val        rnd ----------- ---------- ---------- 1           a          b 2           b          b 3           c          b 4           d          b   I know a way using a cursor to loop throw the rows and get different random values, but that is not performant.   A clever solution to this is   select t1.id, t1.val, t2.val from #t1 t1     join (select *, ROW_NUMBER() over( order by NEWID()) lfd from #t1) as t2 on  t1.id = t2.lfd    But I simplified the query. The real query looks more like  select *,      ( select top 1 val from t2 where t2.x &lt;&gt; t1.y order by NEWID()) rnd  from t1 order by 1;   and the simple solution doesn't fit. I'm looking for a way to force repeated evaluation of  ( select top 1 val from #t1 order by NEWID()) rnd    without the use of cursors.  Edit: Wanted output:  perhaps 1 call  id          val        rnd ----------- ---------- ---------- 1           a          c 2           b          c 3           c          b 4           d          a   and a second call  id          val        rnd ----------- ---------- ---------- 1           a          a 2           b          d 3           c          d 4           d          b   The value for each row just should be a random value independent from the other rows  Here is the cursor version of the code:  CREATE TABLE #res ( id INT, val VARCHAR(10), rnd VARCHAR(10));  DECLARE @id INT DECLARE @val VARCHAR(10) DECLARE c CURSOR FOR SELECT id, val FROM #t1 OPEN c FETCH NEXT FROM c INTO @id, @val WHILE @@FETCH_STATUS = 0 BEGIN     INSERT INTO #res     SELECT @id, @val, ( SELECT TOP 1 val FROM #t1 ORDER BY NEWID()) rnd      FETCH NEXT FROM c INTO @id, @val END CLOSE c DEALLOCATE c  SELECT * FROM #res There is a hint to show that if there were points in $\mathbb{Q}$, then there would exist $r,s, t \in \mathbb{Z}$ with $\gcd(r, t) = 1$ such that $2s^2 = t^4 - 17r^4$ , and then show that any prime dividing $s$ is a quadratic residue modulo $17$.  I'm stuck on the first bit - if you suppose $\left(\frac{a}{b}, \frac{c}{d}\right)$ is a point on the curve in $\mathbb{Q}$ (so $a, b, c, d \in \mathbb{Z}$) then you get to $2a^2d^4 = b^2c^4 - 17b^2d^4$, but I'm not sure where to go from here since you don't know that $b$ is a square? Does a logistic regression maximizing likelihood necessarily also maximize AUC over linear models? # ~/.profile: executed by the command interpreter for login shells. # This file is not read by bash(1), if ~/.bash_profile or ~/.bash_login # exists. # see /usr/share/doc/bash/examples/startup-files for examples. # the files are located in the bash-doc package.  # the default umask is set in /etc/profile; for setting the umask # for ssh logins, install and configure the libpam-umask package. #umask 022  # if running bash if [ -n "$BASH_VERSION" ]; then     # include .bashrc if it exists     if [ -f "$HOME/.bashrc" ]; then     . "$HOME/.bashrc"     fi fi  # set PATH so it includes user's private bin if it exists if [ -d "$HOME/bin" ] ; then     PATH="$HOME/bin:$PATH" fi   Can anyone explain the contents of the ~/.profile file? So when you enter into the ~/.profile file what does all the writing mean? I have my own AppleID ("Bob") but I don't use this email addy "Bob" anymore. I want to add my current, daily used email addy ("Jim") to my ID, or better yet, make it my only AppleID and delete "Bob". BUT... I already used "Jim" for an AppleID for an older friend who had no email address, so set it up for him on mine, "Jim".  My older friend now has an email address ("Ivan") so can I   add "Ivan" to his AppleID, make it the primary email address, and then delete "Jim" from it? Or... delete his AppleID and create a new one for him "Ivan" knowing he would lose his minor purchases   And most importantly, this would all be pointless unless: - will deleting/removing "Jim" as linked to my friend's AppleID then make "Jim" available for me and no longer registered to another account? I want to know, if this bug is fixed, because all previous versions of Ubuntu didn't work for me. So I just want to know, should i leave Windows 10 and go for Ubuntu 17.10. Thanks! Relative log file matching with linux? Overriding some DNS entries in BIND for internal networks A ring of a planet has an inner radius of approximately 52,000 km (measured from the center of the planet) and a radial width of 19 km. Use differentials to estimate the area of the ring. (Round your answer to the nearest thousand.)   I hate asking questions where I have nothing to contribute but I have absolutely no Idea on how to approach this question. A step by step solution to this problem which I could apply to similar problems would be greatly appreciated, thanks in advance. I am trying to evaluate the following:  $\displaystyle \int_{0}^{\infty} \frac{1}{1+x^a}dx$, where $a&gt;1$ and $a \in \mathbb{R}$  Any help will be much appreciated. I realize that I lack any numerical intuition for logarithms. For example, when comparing two logarithms like $\log_78$ and $\log_89$, I have to use the change-of-base formula and calculate the values with a calculator in order to determine which of the two is larger.  Can anyone demonstrate an algebraic/analytic method to find out which of the two is larger without using a calculator? Is every discrete absolute value non-archimedean? I've been reading Milne's notes on absolute values and local fields (Chapter 7 of the book on algebraic number theory), and a possibly minor point is confusing me.   On page 126, Milne writes, "$K$ is a field complete with respect to a discrete absolute value $|\cdot|$..." and then he talks about things that only make sense for non-archimedean absolute values, like the ring of integers and its maximal ideal.   Is every discrete absolute value on a field $K$ also non-archimedean? This would explain a lot, but I also suspect that it is not true, because I can't find any statement of this fact anywhere.  EDIT: An absolute value is discrete if the value group $|K^\times|$ is a discrete subgroup of the multiplicative group $\mathbb{R}_{&gt;0}$. An absolute value is non-archimedean if it satisfies the stronger triangle inequality $$ |x+y| \le \max (|x|,|y|) $$ or equivalently if the set $\{|1|,|1+1|,|1+1+1|,\ldots\}$ is a bounded subset of $\mathbb{R}$. Does this parallel structure need coordinated prepositions to avoid being a faulty parallelism? Let he who believes in this prophet speak now what he knows.       Let him who believes in this prophet speak now what he knows. Trackpad not working after update to Ubuntu 14.10 on Dell Inspiron 15R When I updated from 14.04 to 14.10 I noticed my track-pad cursor and left and right mouse clickers were not working. I have tried looking this issue up and can not find any solutions anywhere. If anyone could help me that would be amazing, I am using a wired mouse to my USB port for a temporary fix. I have a Dell Inspiron 15R. . If you filter out a range of wavelengths, how can you calculate the energy from the remaining wavelengths?   To make it concrete, say we filter everything below 450nm or alternatively above 700nm.