Double summation with infinite limit, change of index using Iverson bracket notation

I’m trying to understand the change of variable for this double summation:

$$sum_{n=0}^{infty} sum_{m=0}^{n} xi(m,n)=sum_{ell=0}^{infty} sum_{m=0}^{(ell/2)} xi(m,n=ell-m)$$

where $ell=m+n$ and $(ell/2)$ means summing up to the largest integer that is less or equal than $ell/2$. So based on this question Double summation, index change clarification. I was trying to use Iverson bracket notation but I’m not sure of my deduction.
What I have so far is

$$begin{align}sum_{n=0}^{infty} sum_{m=0}^{n} xi(m,n) &= sum_{n,m} (0leq n<infty)cdot (0leq m leq n) xi(m,n) \ &= sum_{n,m} (0leq m leq n < infty) xi(m,n) end{align}$$

From here I could write

$$sum_{n,m} (0leq m <infty)cdot (mleq n < infty) xi(m,n) = sum_{m=0}^{infty} sum_{n=m}^{infty} xi(m,n)$$

So to get my result I’m not sure if I can also write

$$begin{align}(0leq m leq n < infty) &rightarrow (0leq 2m leq n + m < infty) \ &rightarrow (0leq 2m leq ell < infty)cdot (ell=m+n) \ &rightarrow (0leq m leq ell/2)cdot(0 leq ell < infty)cdot (ell=m+n)end{align}$$

What also confuses me is if in the last line I should write the first bracket as either $2mleqell$ or $2m < infty$ to separate the conditions.

Limit of a piecewise function from $mathbb{R}^2_{ne0} to mathbb{R}$

Determine if the function $f:mathbb{R}^2_{ne0} to mathbb{R}$ has a limit at the origin, when $$f(x,y) = left{
begin{array}{ll}
x^2, & x<0 \
|y|, & x =0, yne0 \
-y^2, & x >0 \
end{array}
right.$$

How should I do this with piecewise functions? Usually with these it helps to look at $x=0$ and $y=0$ seperately, but that didn’t work here. Any hints?

google sheets – Limit number of rows

I have a Google Sheets document that keeps adding hundreds of rows and slowing down the project, and sometimes causing it to crash.

Is there a way to limit the number of rows to say 100. I do not want to remove all empty rows, because it is a log that constantly needs data added to so a new row is always needed. However, we only use around 100 rows for each document.

I am not sure if this is due to a script I have attached or if it is just part of google sheets.

   function setFormat() {

    var ss = SpreadsheetApp.getActiveSpreadsheet();
    var sheets = ss.getSheets();
for (var i in sheets) {
    var MaxRows = sheets(i).getMaxRows();
    var MaxColumns = sheets(i).getMaxColumns();

sheets(i).getRange(2, 1, MaxRows, MaxColumns)
.setFontFamily('Arial')
.setFontSize('9')
.setFontStyle('normal')
.setFontColor('black')
.setVerticalAlignment('center')
.setBorder(true, true, true, true, true, true, "black", SpreadsheetApp.BorderStyle.SOLID);

sheets(i).getRange(1, 1, 1, MaxColumns)
.setFontFamily('Comfortaa')
.setFontSize('9')
.setFontStyle('bold')
.setFontColor('black')
.setVerticalAlignment('center')
.setBorder(true, true, true, true, true, true, "black", SpreadsheetApp.BorderStyle.SOLID);

sheets(i).autoResizeColumns(1, 1);
sheets(i).setColumnWidths(2, 1, 80);
sheets(i).setColumnWidths(3, 1, 75);
sheets(i).setColumnWidths(4, 1, 90);
sheets(i).setColumnWidths(5, 4, 45);
sheets(i).setColumnWidths(9, 3, 75);
sheets(i).setColumnWidths(12, 2, 130);
sheets(i).setColumnWidths(14, 1, 95);
sheets(i).setColumnWidths(15, 1, 100);
sheets(i).setColumnWidths(16, 1, 95);
sheets(i).setRowHeights(2, MaxRows-1, 15);
sheets(i).setRowHeights(1, 1, 16);   }}

and

   function dataAlignment1() {

     var ss = SpreadsheetApp.getActiveSpreadsheet();
     var s = ss.getActiveSheet();
     var lr = s.getLastRow();
     var r = s.getRange(2, 1, lr, 2);
     var set = r.setHorizontalAlignment('left');
   }

   function dataAlignment2() {

     var ss = SpreadsheetApp.getActiveSpreadsheet();
     var s = ss.getActiveSheet();
     var lr = s.getLastRow();
     var r = s.getRange(2, 3, lr, 1);
     var set = r.setHorizontalAlignment('right');
   }

   function dataAlignment3() {

     var ss = SpreadsheetApp.getActiveSpreadsheet();
     var s = ss.getActiveSheet();
     var lr = s.getLastRow();
     var r = s.getRange(2, 4, lr, 1);
     var set = r.setHorizontalAlignment('left');
   }

   function dataAlignment4() {

     var ss = SpreadsheetApp.getActiveSpreadsheet();
     var s = ss.getActiveSheet();
     var lr = s.getLastRow();
     var r = s.getRange(2, 5, lr, 4);
     var set = r.setHorizontalAlignment('left');
   }

   function dataAlignment5() {

     var ss = SpreadsheetApp.getActiveSpreadsheet();
     var s = ss.getActiveSheet();
     var lr = s.getLastRow();
     var r = s.getRange(2, 9, lr, 2);
     var set = r.setHorizontalAlignment('right');
   }

   function dataAlignment6() {

     var ss = SpreadsheetApp.getActiveSpreadsheet();
     var s = ss.getActiveSheet();
     var lr = s.getLastRow();
     var r = s.getRange(2, 11, lr, 1);
    var set = r.setHorizontalAlignment('center');
   }

   function dataAlignment7() {

     var ss = SpreadsheetApp.getActiveSpreadsheet();
     var s = ss.getActiveSheet();
     var lr = s.getLastRow();
     var r = s.getRange(2, 12, lr, 4);
     var set = r.setHorizontalAlignment('left');
   }

   function dataAlignment8() {

     var ss = SpreadsheetApp.getActiveSpreadsheet();
     var s = ss.getActiveSheet();
     var lr = s.getLastRow();
     var r = s.getRange(2, 16, lr, 1);
     var set = r.setHorizontalAlignment('center');
   }

   function headerAlignment() {

     var ss = SpreadsheetApp.getActiveSpreadsheet();
    var s = ss.getActiveSheet();
     var lr = s.getLastRow();
     var lc = s.getLastColumn();
     var r = s.getRange(1, 1, 1, lc);
     var set = r.setHorizontalAlignment('center');
   }

questionnaire – Can using graded scale help limit number of “Don’t knows”?

Obviously the more options you present, the more likely it is that they will use them rather than just selecting “Don’t Know” – unless they actually don’t know the song of course. So directly answering your question, yes it will help if you have more options.

However, if you are asking a user to rate a song then you should opt for a more familiar approach. Allow the user to select a rating score. For example, let them select from 0-5 stars, or a 1-10 rating, etc. This easily covers the “grey areas” and nearly all users will be familiar with this concept.

With regards to handling the “don’t knows” in a rating system. Either allow the user to not select a rating and skip that song, or if you want explicit action to confirm it then just have a button next to the rating control for “I don’t know this song”.

real analysis – Find the answer of a limit

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complex analysis – Real roots of a uniform compact limit funtion, if the sequence of functions only have real roots

Let $(f_n)_n$ be a sequence of entire functions that converge uniformly on compact subsets of $mathbb{C}$ to some function $f$. Then $f$ is entire, by Morera’s theorem. The question is,

If $f_n$ has $n$ simple real roots (just say all of $f_n$‘s all have only real roots), does $f$ have only real roots as well?

What comes in my mind has something with the following theorem, what other call it as the Hurwitz Theorem,

Let $G$ be a region and suppose the sequence $(f_n)$ of holomorphic functions on $G$ converging to a limit function $f$ on $G$. If $fnotequiv 0$, $overline{B}(a;R)subseteq G$, and $f(z)neq 0$ for $zin partial B(a,R)$, then there is an integer $N$ such that for $ngeq N$, $f$ and $f_n$ have the same number of zeros in $B(a,R)$.

Correct me, if I am wrong. Assume contrary that $f$ has a simple non-real root $z_0in mathbb{C}$. Let $r>0$ be such that the closed disc $overline{B}(z_0,r)$ does not intersect the real axis at all. Then, for sufficiently large $n$, both $f$ and $f_n$ have the same number of roots in $B(z_0,r)$. This result indicates that $f_n$ has roots in $B(z_0,r)$, which is impossible, because it does not contain real roots.

My wordings may sound a bit confusing. What do you think? Are there some details I should mention or not mention?

Thank you

rom flashing – How to flash /system using twrp when system.img is 1.7GB and internal storage has 1.5GB max limit

I happen to have an old phone laying around with messed up system partition(a couple of files accidentally deleted). This phone has an internal storage of max 1.5 GB(total 4gb, after setting everything up leaves me with 1.5 GB). I downloaded the stock firmware for the phone and the system.img size is almost 1.7GB I have twrp recovery. Is it possible to flash with an external sdcard.
N.B: I unfortunately don’t have my pc with me atm.

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Bitcoin users limit

How bitcoin handle users limit if only 4000 transactions in every 10 minutes ? Let’s say I transfer 2 bitcoin to Anuj account then how much time maximum it will take to reflect in Anuj’s account and what will be the process running in background on bitcoin during this record.