Exercise 11 of section 3.2.2(The art of computer programming)

I’m asking about its part a

a) If $ f(z),a(z),b(z)$ are polynomials with integer coefficients,let us write $ a(z)\equiv b(z) (\operatorname{mod} f(z)\space and\space m)$ if $ a(z) = b(z) + f(z)u(z) + mv(z)$ .Prove that the following statement holds when $ p^e>2,f(0)=1$ :

If $ z^\lambda \equiv 1 (\operatorname{mod} f(z)\space and\space p^{e})$ and $ z^\lambda \not\equiv 1 (\operatorname{mod} f(z)\space and\space p^{e+1})$ then $ z^{p\lambda} \equiv 1 (\operatorname{mod} f(z)\space and\space p^{e+1})\space but\space z^{p\lambda} \not\equiv 1 (\operatorname{mod} f(z)\space and\space p^{e+2})$

Here’s my reasoning:
We have $ z^\lambda$ has the form of $ $ f(z)u(z) + p^ev(z) + 1$ $ where $ p^{e+1}|f(z)u(z),f(z)|p^ev(z)$ and $ v(z)\not\equiv 0(\operatorname{mod} p)$ (*)
Or we can just simplify it as:$ z^\lambda = Z + 1$ ,where Z is a multiple of $ f(z)$ and $ p^e$ but not $ p^{e+1}$ ‘s one. So:$ $ z^{p\lambda} = 1+{p\choose 1}Z+{p\choose 2}Z^2+…$ $
$ z^{p\lambda}-1$ is certainly divisable by f(z).Also,we can see that $ {p\choose k}Z^k(1\leqslant k\leqslant p-1)$ is a multiple of $ p^{e+1}$ and $ p^{e+2}|Z^p$ (due to the fact $ p^e>2$ ,$ p^{pe}|Z^p$ implies $ p^{e+2}|Z^p$ ).We now can conclude: $ $ z^{p\lambda} \equiv 1 (\operatorname{mod} f(z)\space and\space p^{e+1})$ $

Next,it’s easy to establish:$ p^{e+2}|p^{ek}$ ,where $ k\geqslant$ 3.And that shows: $ $ z^{p\lambda} \equiv 1 + p^{e+1}v(z) + v(z)^2p^{2e+1}(p-1)/2 (\operatorname{mod} p^{e+2}\space and\space f(z))$ $ $ p^e>2$ is applied again and we have: $ $ z^{p\lambda} \equiv 1 + p^{e+1}v(z) (\operatorname{mod} p^{e+2}\space and\space f(z))$ $ Due to (*),$ 1+ p^{e+1}v(z) \not\equiv 1(\operatorname{mod} p^{e+2})$ ,completing the proof


My proof is based on the answer of excercise 11.And the difference between it and mine makes me confused.

If $ p^{e+1}v(z) \equiv 0 (\operatorname{mod} f(z)\space and\space p^{e+2})$ ,there must exist $ a(z)$ and $ b(z)$ such that $ p^{e+1}(v(z) +pa(z)) = f(z)b(z)$ .Since $ f(0)=1$ ,this implies $ p^{e+1}|b(z)$ (by Guass’s lemma 4.6.1G);hence $ v(z) \equiv 0 (\operatorname{mod} f(z)\space and\space p)$ ,a contradiction.

So,my question:Why is the above argument necessary?

UK tourist visa refused under section 4.2 (a) and (e) is there any hope in reapplying using my dad as a sponsor [duplicate]

This question already has an answer here:

  • UK visa refusal on V 4.2 a + c (and sometimes 'e') 1 answer

enter image description here So I was recently denied under paragraph 4.2 a and e . As to proof that I will return I am currently studying and my course is 6 years ( medicine). I am currently in my 5th year so there’s no chance of me wasting 5 years of my life and not returning to finish my course. I provided a letter from my university that I am a legitimate student with proof of my duration of study. I also have previous travel history within the last 6 months. I also provided my bank statement which had a good amount of money in there to support me as I planned to stay in the UK for only one week . The money was also not put recently. Is there any point in reapplying using my dad or sister as my sponsor ? And when I am reapplying how do I prove the question of origin of my own funds? My parents typically send me a large sum once a year to cover my school fees and living costs for that year so they don’t have to bother with monthly payments. This was done last year and i indicated this in my cover letter. Please help as I don’t want to reapply get denied again and decrease my chances of ever getting approved. I also applied for tourist visas to Schengen countries with the same bank statement and there was no issue. I was approved.

Comment section on my blog disappears

When there are no comments in a post, you can still see the comment section, for example: https://duonghoimanga.blogspot.com/2018/11/takane-no-ran-san-chap-14.html

However, as soon as someone comments on a post, the comment section in that post disappears, for example: https://duonghoimanga.blogspot.com/2018/09/takane-no-ran-san-chap-12.html

I am using a third party blogger template, so asking this question on the Blogger Help Forum did not help.

This is really frustrating, because even though I can still see the comments in my admin page, I cannot reply to them and other people visiting my blog cannot see them.

Can anyone help me? Thanks in advance.

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In View panel, adding fields not showing up “Configure: field” and the field is not added. i.e. The red-boxed is not shown as below (see step2):

Normal behavior step1: enter image description here

Normal behavior step2: enter image description here

The event happened after my

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  3. Apache upgraded from 2.2.12 to 2.4.23

No application nor server-level’s error log shown. Any ideas on how can I debug or trace the problem? Many thanks for yours help.

Loop through all posted articles and print specific section

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I more or less achieved this using views. Basically {{ page.content }} in twig, but found it to be really inflexible and I didn’t really get the results that I wanted. What I want to do is just to iterate through all the articles and print each section of that article “manually”. For instance:

{% for page in pages %}   {{ page.content.title }}   {{ page.content.datePosted }}   {{ page.content.body }} {% endfor %} 

So that I can have more control of what is happening and not making a lot of configuration in the views module when deploying. What is the best soltuion to achieve this? Thanks!

How to calculate the volume of a section of a convex body?

The following is essentially a partial case for my previous question.

Let $ B\subset\mathbb{R}^m$ be the unit ball with respect to a concrete norm on $ \mathbb{R}^m$ , say $ l^p$ -norm, $ p\in (1,\infty)$ . Let $ v_1,…,v_n\in \mathbb{R}^m$ be linearly independent.

How to calculate the $ n$ -dimensional volume of $ B\cap span\{v_1,…,v_n\}$ ?

I need to express this volume through the coordinates of $ v_1,…,v_n$ , or perhaps through some distances between certain combinations of them. I know that there is extensive literature on related matters, but I hope that this specific question has a specific answer..