game development – What back-end language should I learn/use for a real time competitive gaming web app?

I hope your day is going well.

I want to make a competitive gaming web app that will hold many players (100,000 to 1,000,000 different users/players) in the same match all at the same time.

What I want from the back-end language:

  • Real-time updates on all user’s screens without needing to refresh
  • Being able to have hundreds of thousands or even millions of players
    playing on the same game (meaning the same exact match) all at
    the same time with the real-time updates from every player on
    everyone’s screen.
  • Good performance on games that have many people playing it all at the
    same time.

I do not care how hard or easy the language is as long as it meets the criteria mentioned above.

Thanks 😊 !

The functions $f,g$ are non constant, differentiable, real valued functions..

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I have found that $cos{x}$ and $sin{x}$ satisfies the given condition. But how to prove in general. Please give some hint

real analysis – Is this set connected in $mathbb{R}^3$

The following question was part of my analysis quiz any I was unable to solve 1 option . So, I am posting it here.

Let V be the span of (1,1,1) and (0,1,1) . Let $u_1 =(0,0,1) , u_2 =(1,1,0) $ and $u_3 =(1,0,1) $ . Then Is $(mathbb{R}^3$V )$bigcup$ { $t u_1 + (1-t) u_3 : 0leq t leq 1$} connected?

I think It will be connected because If I remove V from $mathbb{R}^3$ then also it is connected amd adding anything makes it connected . But answer is disconnected !

Why so?

real analysis – Subsets $Y$ of a partially ordered set $L$ need not have least upper bounds nor greatest lower bounds

I am currently studying the textbook Principles of Program Analysis by Flemming Nielson, Hanne R. Nielson, and Chris Hankin. Appendix A Partially Ordered Sets says the following:

Partially ordered set. A partial ordering is a relation $sqsubseteq : L times L rightarrow { text{true}, text{false} }$ that is reflexive (i.e. $forall l : l sqsubseteq l$), transitive (i.e. $forall l_1, l_2, l_3 : l_1 sqsubseteq l_2 land l_2 sqsubseteq l_3 Rightarrow l_1 sqsubseteq l_3$), and anti-symmetric (i.e. $forall l_1, l_2 : l_1 sqsubseteq l_2 land l_2 sqsubseteq l_1 Rightarrow l_1 = l_2$). A partially ordered set $(L, sqsubseteq)$ is a set $L$ equipped with a partial ordering $sqsubseteq$ (sometimes written $sqsubseteq_L$). We shall write $l_2 sqsupseteq l_1$ for $l_1 sqsubseteq l_2$ and $l_1 sqsubset l_2$ for $l_1 sqsubseteq l_2 land l_1 not= l_2$.

A subset $Y$ of $L$ has $l in L$ as an upper bound if $forall l^prime in Y : l^prime sqsubseteq l$ and as a lower bound if $forall l^prime in Y : l^prime sqsupseteq l$. A least upper bound $l$ of $Y$ is an upper bound of $Y$ that satisfies $l sqsubseteq l_0$ whenever $l_0$ is another upper bound of $Y$; similarly, a greatest lower bound $l$ of $Y$ is a lower bound of $Y$ that satisfies $l_0 sqsubseteq l$ whenever $l_0$ is another lower bound of $Y$. Note that subsets $Y$ of a partially ordered set $L$ need not have least upper bounds nor greatest lower bounds but when they exist they are unique (since $sqsubseteq$ is anti-symmetric) and they are denoted $bigsqcup Y$ and $sqcap Y$, respectively.

It is this part that I am unsure about:

Note that subsets $Y$ of a partially ordered set $L$ need not have least upper bounds nor greatest lower bounds but when they exist they are unique (since $sqsubseteq$ is anti-symmetric) and they are denoted $bigsqcup Y$ and $sqcap Y$, respectively.

This isn’t obvious to me. Can someone please explain this / make it clear ?

fa.functional analysis – The best bound of the integral of a nondecreasing real function in a closed interval

Let $F:(0,1)to (0,1)$ be a nondecreasing function. Given the definite integrals
int_a^1 F(x)~dx=I_1 ~text{and}~int_b^1 F(x)~dx=I_2,

where $a<b$ and $I_2le I_1le I_2+(b-a)F(b)$. Consider the integral
G(t)=int_t^1 F(x)~dx, ~tin(a,b).

How to find the best upperbound and lowerbound (represented by $I_1$ and $I_2$) of $G(t)$?

“The best bound” means that for given values $I_1$ and $I_2$, we can find an admissible function $F$ such that the bound of $G(t)$, $G(a)=I_1$ and $G(b)=I_2$ are achievable.

Personally, I think solving this problem will be useful in estimate the integral of the distribution function.

real analysis – How would this (x,y) function look like? (+integral)

let $g:B=(0,1)^2 rightarrow mathbb{R}$ be

$g(x,y)=begin{cases} frac{1}{y^2} & 0 <x<yleq 1\
-frac{1}{y^2} & 0<y<xleq 1 \
0 &else end{cases}$

I can not imagine at all how this function could look like, the values of x and y would only be from (0,1) so would there be any “problems” because of the fractions? I would think not, since the conditions are $0 <x<yleq 1$, so $y> 0$ always, and similarly we have $0<y<xleq 1$, so $x$ strictly bigger than zero.

Could someone show me how this would look like?

I also need to calculate the integrals

I have $int_{0}^{1} int_{0}^{1} f(x,y) dx dy = 1 neq -1 = int_{0}^{1} int_{0}^{1} f(x,y) dy dx$

would this mean that the integral $int int_B f(x,y) d(x,y)$ does not exist?

sequences and series – Lower bound for sum of reciprocals of positive real numbers

I am reading an article where the author seems to use a known relationship between the sum of a finite sequence of real positive numbers $a_1 +a_2 +… +a_n = m$ and the sum of their reciprocals. In particular, I suspect that
sum_{i=1}^n frac{1}{a_i} geq frac{n^2}{m}

with equality when $a_i = frac{m}{n} forall i$. Are there any references or known theorems where this inequality is proven?

This interesting answer provides a different lower bound. However, I am doing some experimental evaluations where the bound is working perfectly (varying $n$ and using $10^7$ uniformly distributed random numbers).


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real analysis – Prove $f(1)-f(-1)>f’left(-frac{1}{sqrt{3}}right)+f’left(frac{1}{sqrt{3}}right)$.

Suppose the 5th order derivative function of $f(x)$ ,say, $f^{(5)}(x)>0$ for $xin (-1,1)$.Prove $f(1)-f(-1)>f’left(-dfrac{1}{sqrt{3}}right)+f’left(dfrac{1}{sqrt{3}}right)$.

I know this can be done directly by Gauss-Legendre formula, but does there exist a more elementary proof?