Terminology – What does "a set of polynomials / number of operations" mean in these contexts?

I am reading the book "The Outer Limits of Reason" and came across a description that I am very confused about. I am afraid to say that this may be because I am not a native English speaker.
These are the contexts from The Outer Limits of Reason:

On p.136 it says:

After seeing some examples, let's find a nice definition. We
denotes with NP the class of all decision problems, the 2n, n! or less require
Operations to be Solved.6 A problem in NP is called an “NP problem”. There is P
the class of problems that can be solved in a polynomial set of operations, and
Polynomials grow more slowly than exponential or faculty functions, we have
P is a subset of NP. This means that every "simple" problem is an element of
the class of all "hard and simple" problems.

and on p.142:

We don't want the transformer to arbitrarily change an instance of Problem A.
an instance of problem B. We require that the instances have the same answer. in the
In other words, we will insist that the transformer accept inputs with a "yes" answer for
Problem A to inputs that answer "Yes" for problem B. If an entrance to problem A gets a
Answer “No” from the problem A decision maker, the transformer should issue an instance
that will have a "no" answer. We have another requirement: this transformer
should do its job in a polynomial set of operations, The need for it
The determination quickly becomes obvious.

on p.169:

Once we have shown that a particular problem cannot be solved, it is not difficult to show
that there is another problem too. The method used is to reduce one problem to another.
or a reduction.5 Suppose there are two decision problems: problem A and problem
B. Also assume that there is a way to transform a problem instance
A into an instance of problem B so that an instance of problem A has a yes
The answer is forwarded to an instance of Problem B with the same answer, for
No answers. (We do not make the requirement, as we did in the last chapter, that the
Transformation in a polynomial number of operations, We don't have any here
Interest in how long such a transformation takes, only whether it can be carried out.) We
could imagine this transformation as in Figure 6.6.

on p.179:

First some definitions. P was defined as the set of problems that can be solved by
a normal computer in a polynomial number of operations, We generalize. Consider
each oracle X. Define PX as the set of X oracle problems that can be solved in a
Polynomial number of operations. NP was defined as the set of all problems that it can
be solved by a normal computer in at most an exponential or factorial number of
NPX is the set of X oracle problems that can be solved in at most exponential or factorial numbers of operations.

These are repeated several times, but I think it's enough. My problem is in bold parts. What does it mean "in a poly set / number of operations"? because I always thought it was "in a lot / number of polynomial operations. Please keep it simple.

Python – Difficulty updating the number of rounds in the championship function !! Neem game – urgent help

I can't find the bug in the part when the game rounds need to be updated, i.e. the game in the championship part has to have 3 rounds and save how many times the PC and the user have won, only that the number is not updated by rounds and saves consequently not the frequency with which the participants won.

      def partida(n,m):    i = 1    while i <= 3:
           n = int(input('Quantas peças? '))
           m = int(input('Limites de peças jogadas: '))
           if n % (m + 1) == 0:
               print('Você começa')
               usu = 0
               n = usuario_escolhe_jogada(n,m)
               i += 1

            print('Computador começa')
            pc = 0
            n = computador_escolhe_jogada(n,m)
            i += 1

def campeonato(n,m):
    quantpartida = 1
    pc = usu = 0

    while quantpartida < 4:
        print(f'**** Rodada {quantpartida} ****')
        n = partida(n,m)
        if usuario_escolhe_jogada(n,m) == n:
            usu += 1
            pc += 1
        quantpartida += 1

    print('**** Final do campeonato! ****')
    print(f'Placar: Voce {usu} x {pc} Computador')

def usuario_escolhe_jogada(n,m):
    a = 1
    usu = 0
    x = int(input('Quantas peças você vai tirar? '))
    if x != m:
        print('Oops! Jogada inválida! Tente de novo.')
    elif x == 1:
        n = n - x
        print('Voce tirou uma peça')
        print(f'Agora restam {n} peças no tabuleiro')
        n = computador_escolhe_jogada(n,m)
    elif x != 1:
        n = n - x
        print(f'Voce tirou {x} peças')
        print(f'Agora restam {n} peças no tabuleiro')
        n = computador_escolhe_jogada(n,m)
    elif n==m or n < m:
        print(f'Voce tirou {n} peças')
        print(f'Voce venceu essa rodada!')
        usu == 1
        n = campeonato(n,m)

def computador_escolhe_jogada(n,m):
    b = 2
    x = 1
    pc = 0
    escolha = False
    while x <= m and not escolha and n != 1:
        if (n - x) % (m + 1) == 0:
            # a escolha do computador vai ser igual a x
            n = n - x
            print(f"O computador tirou {x} peças.")
            print(f"Agora restam {n} peças no tabuleiro.")
            n = usuario_escolhe_jogada(n, m)
            escolha = True
    if n == m or n < m:
        print(f'O computador tirou {n} peças')
        print('Computador venceu essa rodada')
        pc == 1
        n = campeonato(n,m)

#Programa principal
print('Bem-vindo ao jogo NIM! Escolha:')
perg = int(input('1- para jogar uma partida isolada n2- para jogas um campeonato'))
if perg == 2:
    print('Voce escolheu um campeonato')
    print('Voce escolheu uma partida isolada')

p adic number theory – conjugation of maximal algebraic tori

Accept $ G $ is a connected, reductive algebraic group over a non-Archimedean local field $ F $which is divided over a finite extent $ E / F $,

I often see a result that says "everything is maximum $ F $-Tori are conjugated over $ E $", by which I understand the following: Let $ G (E) $ denote the $ E $-Dots of the algebraic group $ G $;; then for each maximum $ F $-tori $ T, T $ $ of $ G $is there $ x in G (E) $ so that $ T (E) = xT & # 39; (E) x ^ {- 1} $,

In addition, the definitions show that if $ T, T $ $ are maximum $ F $-tori from $ G $then there is an isomorphism of $ T (F) $ on to $ T & # 39; (F) $ which is defined via $ E $,

My question is: Can the isomorphism be assumed to be conjugation in the second statement (as in the first statement)? That means: it follows from these results that if $ T, T $ $ are maximum $ F $-tori in $ G $then it exists $ x in G (E) $ so that $ T (F) = xT & # 39; (F) x ^ {- 1} $?

Any help (including proof of the first statement) is greatly appreciated!

real analysis – equivalence between finite number of points and subset of measure zero of compact sets

$ f () $ is continuously differentiable $ (0.1) $, The set of $ x $ so that $ f (x) = 0 $ is of measure zero. Can we conclude that there is a finite number of points, so that $ f (x) = 0 $ ?

I think, if $ f () $ is only continuous, this would not be true: e.g. $ f (x) = x sin (1 / x) $ if $ x neq 0 $ and $ f (0) = 0 $, This function is continuous, but has an infinite number of separate zeros (for a set of measures zero). The "dimension zero" property only excludes the "flatness" of $ f (x) $, but does not exclude the possibility of an infinite number of separate points.

For continuously differentiable functions, I think this is correct. The continuous differentiability implies a certain smoothness of the function. And since I know that the set of $ x $ so that $ f (x) = 0 $ is of zero measure, there are some implicit restrictions $ f ’(x) $ (i.e. the amount so that $ f (x) = 0 $ and $ f ’(x) = 0 $ is empty or at least zero). In this case I should get back the finite number of zeros after more traditional lines (e.g. this question).

However, I am not sure if this is correct. And I can't find a counterexample.
And maybe I need something less restrictive than "continuously differentiable" to get to the property (e.g. only Lipschitz continuously?).

wp update post – change language of human time difference number

I added function

/* Function which displays your post date in time ago format */

function human_time_diff_nepali( $from, $to = '' ) {
  if ( empty( $to ) ) {
    $to = time();

  $diff = (int) abs( $to - $from );

  if ( $diff < HOUR_IN_SECONDS ) {
    $mins = round( $diff / MINUTE_IN_SECONDS );
    if ( $mins <= 1 )
      $mins = 1;
    /* translators: min=minute */
    $since = sprintf( _n( '%s मिनेट', '%s मिनेट', $mins ), $mins );
  } elseif ( $diff < DAY_IN_SECONDS && $diff >= HOUR_IN_SECONDS ) {
    $hours = round( $diff / HOUR_IN_SECONDS );
    if ( $hours <= 1 )
      $hours = 1;
    $since = sprintf( _n( '%s घण्टा', '%s घण्टा', $hours ), $hours );
  } elseif ( $diff < WEEK_IN_SECONDS && $diff >= DAY_IN_SECONDS ) {
    $days = round( $diff / DAY_IN_SECONDS );
    if ( $days <= 1 )
      $days = 1;
    $since = sprintf( _n( '%s दिन', '%s दिन', $days ), $days );
  } elseif ( $diff < MONTH_IN_SECONDS && $diff >= WEEK_IN_SECONDS ) {
    $weeks = round( $diff / WEEK_IN_SECONDS );
    if ( $weeks <= 1 )
      $weeks = 1;
    $since = sprintf( _n( '%s हप्ता', '%s हप्ता', $weeks ), $weeks );
  } elseif ( $diff < YEAR_IN_SECONDS && $diff >= MONTH_IN_SECONDS ) {
    $months = round( $diff / MONTH_IN_SECONDS );
    if ( $months <= 1 )
      $months = 1;
    $since = sprintf( _n( '%s महिना', '%s महिना', $months ), $months );
  } elseif ( $diff >= YEAR_IN_SECONDS ) {
    $years = round( $diff / YEAR_IN_SECONDS );
    if ( $years <= 1 )
      $years = 1;
    $since = sprintf( _n( '%s वर्ष', '%s वर्ष', $years ), $years );

  return apply_filters( 'human_time_diff_nepali', $since, $diff, $from, $to );

Now I want to change the numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 into my own language. How can I change numbers there into my own language?

Minutes and dates words are already translated, but I have to translate the number from 0 to 9.
Example: translate to 1 hour ago