Are page numbers of a book a good analogy for a clustered index?

The title pretty much covers the question.

I think that the index of a book is a good analogy for a non-clustered index, as it demonstrates the extra storage and physical separation properties of a non-clustered index. I also think that the page numbers of a book represent well the physical ordering of data, similar to the structure of data with a clustered index.

However I’m a bit of a newbie to more advanced database theory. Does the analogy for page numbering make sense for a clustered index, or are there any properties of a clustered index where this analogy falls over?

Are the class numbers of $mathbb{Q}(cos(2pi / m))$ $O(m^n)$ for some fixed $n$?

Question: Are the class numbers of $mathbb{Q}(cos(frac{2pi}m))$ $O(m^n)$ for some fixed $n$?

Remark: If such $n$ exists, then $ngeq2$. By the paper unit groups and class numbers of real cyclic octic fields by Yuan-Yuan Shen, there exists infinitely many cyclic octic fields with conductor $32b$ and class numbers at least $c(epsilon)b^{3-epsilon}$ for every $epsilon>0$. As every number field extension $E/F$ satisfies $h(F)$ divides $(E:F)h(E)$, the class numbers of such $mathbb{Q}(cos(frac{2pi}{32b}))$s are at least $c(epsilon)b^{2-epsilon}$.

A key feature of Yuan-Yuan Shen’s octic fields is that they have regulators of size $log^6(b)$. If such an infinite family of number fields (i.e. real abelian with poly-logarithmic regulators) exists for every fixed degree, the answer for this question will be “no”.

TCP Sequence numbers and timeouts

How do sequence numbers and timeouts provide a reliable channel for application-layer data?

list manipulation – Correct way to format numbers

I have defined the following function:

randvec[mag_]:= mag Normalize[Table[RandomReal[], {j, 1, 4}]]

I would like to format the numbers in the list so that they are 4 d.p. Trying the following:

randvec[mag_] := NumberForm[mag Normalize[Table[RandomReal[], {j, 1, modes}]], 4]

casts each entry as a string. What is the appropriate way to format the list without changing the type of the entries?

magento2 – Generate Random numbers for multiple block call

I am trying to implement custom flash toy for website. To do this I have created custom module.

In custom module, I have created a block class. Block class contain the function to generate the Random digits.

Here is my block's Class Name:  WM_Jackson/Block/Index/Index

Random Id generator code in block:

/**
 * generate random number for player
 *
 * @return int
 */

public function playeridGenerator()
{   
     $ran = rand(10, 999999);
    return $ran;
}

This rando Id generator code works fine when I call my block in cms page using below code:

{{block class="WMJacksonBlockIndexIndex" template="WM_Jackson::flash_elem.phtml" }}

But when I add multiple blocks on the same page, it generates the duplicate ids for all the blocks.

for example if I add block code 3 times, It shows flash player 3 times but it shows same id for 3players. Hence when user try to interact with Flash show, only first player works.

  {{block class="WMJacksonBlockIndexIndex" template="WM_Jackson::flash_elem.phtml" }}
  {{block class="WMJacksonBlockIndexIndex" template="WM_Jackson::flash_elem.phtml" }}
  {{block class="WMJacksonBlockIndexIndex" template="WM_Jackson::flash_elem.phtml" }}

Above code renders output 3 times but block function playeridGenerator() is generating same number 3 times.

How can I generate Random numbers when calling blocks multiple times? Is there any native library in magento for this?

Any help would be appreciated.

prime numbers – How do I reproduce the animation in this video?

I don’t know much about the prime number theory and neither am aware of a lot of special mathematical functions. But am interested in having a general idea about how zeros of the Extended Harmonic Series are related to Prime Number distribution.

I was watching this video. It mentioned how non-trivial zeros of the extended Harmonic Series are related to the approximation of the modified version of the prime number counting function.

I figured out it is implemented as PrimePi and I tried to make it look like the modified version as follows:

PrimePsi(x_):=Sum(If(PrimePowerQ@n,Log@Surd(n,PrimeOmega@n),0),{n,1,x});

But after some digging around I found a function MangoldtLambda that does exactly that (which is surprisingly not on the function page of PrimePi). So I modify as follows:

PrimePsi(x_):=Sum(MangoldtLambda(n),{n,1,x});

Now I can get the step function shown in the video as follows:

Plot(
    {PrimePsi(x)},{x,0,32.5},
    PlotLegends->"Expressions",
    PlotRange->{{0,32.5},{0,34}},
    ImageSize->Large,
    AspectRatio->1/2
)

But how do I get that green function on the graph here? I know WL implements Extended Harmonic Series as Zeta and its non-trivial zeros can be grabbed by Im(ZetaZero(...)) but what is that green function in the graph shown in the video and how do I make it in WL?

css – Mask that accepts numbers and letters

I have the following mask:

$('.form_fornecedor input#fornecedor_razao_social').mask('SSSS');

Currently the mask only accepts 4 letters… I need the mask to accept many letters and many numbers, without accepting special characters such as double and single quotes

Is there a pair of tuples N, M of prime numbers which (a) have the same product, and (b) whose partial products have the same sum?

If N is a finite list of numbers, let $p(N)$ be the product of the numbers in N, that is, $$p(N)=Pi_{i=1}^{|N|}N_i$$ and let $s(N)$ be the sum of the partial products of the numbers in N, that is $$s(N)=Sigma_{k=1}^{|N|}left(Pi_{i=1}^kN_iright)$$

Is there a pair of lists $Nneq M$ for which $p(N)=p(M)$ and $s(N)=s(M)$?

What if they only contain odd numbers 3 or higher?

What if $N$ and $M$ only contain prime numbers? Only odd primes?

What if they only contain elements of ${3,7,19}$? (This is the case I’m most interested in)

It seems possible that there’s no such (distinct) $N$ and $M$ for the last case, but I’ve no idea how to prove that.

How to repeat consecutive numbers (1234…12 1234…18) on the x-axis of an excel chart

I have conducted two independent experiments (Exp-A and Exp-B), measuring the same parameter (NO3) in both.

Exp-A took 12 days(days 0-12) consisting of 6 independent sequenced batches (2 days each).

Exp-B took 18 days (days 0-18) consisting of 6 independent sequenced batches (3 days each).

Now I am trying to present the results of both experiments on the same chart, show the data (NO3 on the Y axis) for each day (X axis) separately. The closest I got is when I use “Line with Markers” chart type:
The numbers at the beginning and end of every batch are repeated

I tried to use XY Scatter with the same day format, but the values of Exp-B overlap Exp-A as they are redrawn from the same Exp-A X-axis origins:Overlapped results

The only way to avoid repetitive days for sequenced batches is when I add up the numbers of the days (day 0-Day 30) in an “XY Scatter” chart which looks like I had a single 12+18=30 days experiment which is not correct. I could separate the experiment by different background colors but the day numbers are still wrong.Data are shown in one long experiment format but the shape of the graph is correct.

I’m using Office 365. I would be very grateful to if you could please some tips and help. It’s a very important report for my thesis and I have to get it right.

co.combinatorics – Does the ordinary generating function of Bell numbers converge?

I am working in a field not really based on combinatorics, therefore I appologize if my question is in any kind invalid. Nevertheless, in my calculations, the Bell numbers appeared. I need to find some $x$ such that the ordinary generating function
$$B(x) = sum_{n=0}^{infty}B_n x^n$$
converge. I haven’t found the answer nowhere in the literature. On the opposite, there are quite a lot of results concerning $B(x)$, but none of them is questioning for which $x$ it has some sense. It is evident that the case $x>1$ lead to a divergent series, which is not much of an interest. But what about $x<1$? I suppose there must be such $x$, otherwise it is nonsense to study such series, is it not?

One other thing suprised me. There is a nice representation in Klazar of $B(x)$ such that
$$B(x) = sum_{n=0}^{infty}frac{x^n}{(1-x)(1-2x)cdots(1-n x)}.$$
But what if $x$ equals to $1/k$ for some $k$ natural? Would that make the series divergent? I am sorry, but is has been bugging me for some time that there is no explanation in the literature that I have been looking into. Does anyone have some relevant source of information?

Thank you, I would appreciate any help.