## 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..

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
$$begin{equation} int_a^1 F(x)~dx=I_1 ~text{and}~int_b^1 F(x)~dx=I_2, end{equation}$$
where $$a and $$I_2le I_1le I_2+(b-a)F(b)$$. Consider the integral
$$begin{equation} G(t)=int_t^1 F(x)~dx, ~tin(a,b). end{equation}$$

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

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 $$0 , so $$y> 0$$ always, and similarly we have $$0, 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
$$begin{equation} sum_{i=1}^n frac{1}{a_i} geq frac{n^2}{m} end{equation}$$
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?