Consider the ODE  $$y'=f(y(t)),y(0)=a\in\mathbb{R}$$ Where $$f:\mathbb{R}\rightarrow\mathbb{R}$$  Which of the following statements are necessarily true?  $A.$ There exist a continuous function $f:\mathbb{R}\rightarrow\mathbb{R}$ and $a\in\mathbb{R}$ such that the above problem does not have a solution in any neighbourhood of $0.$  $B.$ The problem has a unique solution for every $a\in\mathbb{R}$ where $f$ is lipscitz continuous.  $C.$ When $f$ is twice continuously differentiable, the maximal interval of existance for the above I.V.P. is $\mathbb{R}.$  $D.$ The  maximal interval of existance for the above problem is $\mathbb{R}$ when $f$ is bounded and continuously differentiable.  According to me $A$ is false by Picard existance theorem, and $B$ is true and for $C$ $y'=y^{2}$ is counterexample. I have no idea about option $D.$ Please help me to handle option $D.$ In answer key option $D$ is given correct. Thanks in advance.
Consider the IVP $$y'(t)=f(y(t)), \ \ \ \ y(0)=a \in \mathbb{R}$$ $$f : \mathbb{R} \rightarrow \mathbb{R}$$ Which of the following is/are true $(A)$  There exists a continuous function $f : \mathbb{R} \rightarrow \mathbb{R}$ and $a \in \mathbb{R}$ such that the above problem does not have a solution in any nbd of $0$. $(B)$  The problem has unique solution for every $a \in \mathbb{R}$ when $f$ is Lipschitz continuous $(C)$ When $f$ is twice continuously differentiable the maximal interval of existence for the above IVP is $\mathbb{R}$ $(D)$ The maximum interval of existence  for the IVP is $\mathbb{R}$ when $f$ is bounded and continuously differentiable.  It is obvious to me that $A \ \&amp; \ B$ are false from traditional existance and uniqueness theorems (viz Picards Theorem). I am not sure about $C \ \&amp; \ D$ and this is really bugging me. Please could anyone shed light on this. PS: Multiple correct options are allowed.
A pitcher contains $X + 1$ blue balls and $Y + 1$ red balls. It is known that $X, Y$ are independent random variables, and it is given that $X \sim \mathrm{Poisson}(n), Y \sim\mathrm{Poisson}(2n)$. Calculate the limit: $\displaystyle\lim_{n\to \infty}\mathbb P (X + n = Y) $   I tried to condition on the value of $Y$ and to use the Law of total probability but it did not lead to a solution. ( I tried to sum: $ \sum_{i=0}^\infty\mathbb P(X+n=Y|Y=i)\mathbb P(Y=i) $ and get an expression that I can calculate its limit as n goes to infinity) Trying to Get the Most Cooling Power out of TEC Peltier Element: Smaller or Larger Peltier? I am trying to achieve the strongest cooling effect using a TEC Peltier element.    Currently, my set up is two small, 20mm x 20mm peltier TES1-4902 run with a 5V 2.4A USB Battery Pack per peltier.   I need a way to reduce my costs while keeping the same cooling power.   Right now, it is expensive for me to run this set up due to the fact I need two batteries, and I am also looking to improve my efficiency while cutting out an entire battery and/or Peltier. The reason I have two separate battery packs is that I can not find a cheap, small, and thin battery pack that will output 4.8 Amps: however, putting two battery packs in a series may help eliminate one peltier.     So, I have a few branching question that center around a theme:  1) Would a larger Peltier allow for a stronger cooling effect as a result of a larger surface area, say like a 50MM x 50MM peltier, despite the fact (presumably) that it may be more difficult to cool one single peltier than 2 smaller, seperate ones?   2) Is there a Peltier size that would allow me to reduce my power consumption while maintaining a similar cooling effect?   3) Is there a way that I can output 5V 4.8A that is not in two separate USB battery packs that is small and relatively cheap, as battery price:output ability for small scale consumer is quite high and USB battery packs have been the cheapest source of sustainable power for me.    My problem stems from the fact that I am trying to pump as much power and achieve as much cooling power as possible without having to have, say, two battery packs and make the project too expensive. I need a way to reduce my costs while keeping the same cooling power. $2\cos^2x=1+\sin x$.  Determine the exact values of $x$ such that $0 \leq x \leq 2\pi$.    I am experiencing problems with factoring this question.  First I started by getting everything on to the left side so: $2\cos^2x-\sin x-1=0$.  Then seeing as there are two trig variable here ($\cos$ and $\sin$), I used the pythagorean identity and replaced $2\cos^2x$ with $1-2\sin^2x$ to make it into a one variable question: $1-1-2\sin^2x-\sin x-1=0$.   By collecting like terms it then becomes $-\sin^2x-\sin x=0$.  Now you could factor these with $0$ and $-1$, but then it creates uneven brackets so this musn't work.  I don't think the quadratic formula is necessary, but I am wondering if this could be treated as a difference of squares question.    If you replace the $\sin x$ and make the question into $-x^2-x=0$ it is a little easier to look at, however I am not sure if this is the right approach.  If someone could please clarify my answer, that would be appreciated! "Washroom", "restroom", "bathroom", "lavatory", "toilet" or "toilet room" I've always been confused by the terms washroom, restroom, bathroom, lavatory, toilet and toilet room. My impression is that Canadians would rather say washroom while Americans would probably say bathroom or Saint John's in the same situation.  I guess the difference here is not only in different kinds of English, but also in whether one is referring to a room in their house or in some public place.  Which do you usually use? Please specify the difference if you use more than two from those six with different meanings, and also where you are from (i.e. what type of English you speak). The main effect will be non-significant if the interaction is significant? Nouns can modify nouns: cat food, coffee cup, gold ring, laser surgery, flood insurance. It seems to me there are even cases where a noun sounds better than the corresponding adjective: sociology papers sounds a bit better than sociological papers.  Some noun-noun combinations don&rsquo;t work. You wouldn't say America gangsters or measurement spoons or greed crimes. If you spilled water on your keyboard you probably wouldn't call it a keyboard spill or even a water spill.  Are there rules that tell which combinations are acceptable? Is `sudo pip install` still a broken practice? Reading an Excel file in PHP I'm trying to read an Excel file (Office 2003). There is an Excel file that needs to be uploaded and its contents parsed.  Via Google, I can only find answers to these related (and insufficient topics): generating Excel files, reading Excel XML files, reading Excel CSV files, or incomplete abandoned projects. I own Office 2003 so if I need any files from there, they are available. It's installed on my box but isn't and can't be installed on my shared host.  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Here is the relevant portion of my code:          outFolderPath = r"U:\tempStuff"         outName = "Stuff.sde"         databasePlatform = "SQL_Server"         instance = "SQL Server"         arcpy.CreateDatabaseConnection_management(outFolderPath, outName,                                                   databasePlatform,                                                   instance)         print "Centroid of buidling:(%r,%r) Side length: %r" % (xCoordinate,                                                                 yCoordinate,                                                                 bob)         print "workspace is %r" % outName #outName is workspace          #Start edit session with an undo/redo stack then start         #edit operation         print "Starting editing"         edit = arcpy.da.Editor(outName) #outName is workspace Line 86         edit.startEditing(True, True)         edit.startOperation()          #stuff happens          cursor = arcpy.da.InsertCursor( fc, ["SHAPE@"])         cursor.insertRow([stuff])             #stop the edit operation and then save changes         edit.stopOperation()         edit.stopEditing(True)          del cursor         print "Building should be drawn"   This is the exact error I've been getting:  Traceback (most recent call last):   File "C:\Users\153289\AppData\Local\ESRI\Desktop10.3\AssemblyCache\{0BEF4E61-6332-4BCA-A908-959CA3C0B4E7}\leetScripts_addin.py", line 86, in onClick     edit = arcpy.da.Editor(outName)#outName is workspace RuntimeError: cannot open workspace   What's am I doing wrong in my code? 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