Sunday, May 8, 2022

Recent Questions - Mathematics Stack Exchange

Recent Questions - Mathematics Stack Exchange

Probability of a sum of squared integers knowing if they are 0 or not

Posted: 08 May 2022 09:31 AM PDT

are you aware of any result on the following problem ? I have $M$ integers $x_1, x_2, \dots, x_M \ge 0$ summing to $M$, i.e. $$\sum_{i=1}^M x_i = M$$ I am looking for the value of the following probability: $$P\left(\sum_{i=1}^M x_i^2 = j \; \Big{|} \sum_{i=1}^M y_i = k\right)$$ where $y_i = 1$ if $x_i > 0$ and $y_i = 0$ if $x_i = 0$. Any indication would be welcome.

Minimising surface area of a cuboid with a surface area $S$ and volume $V$

Posted: 08 May 2022 09:27 AM PDT

I am able to chose a pair of values $(V,l)$ where V represents the volume of a cuboid and $l$, one of its side lengths. This cuboid must have two sides that are in the ratio $1:l$. What pair of value for $(V,l)$ are such that the cuboids volume and surface are equal and that the surface area of the cuboid is the least possible.

I started of by finding an expression for the surface area of this cuboid in therms of $V$ and $l$ only and minimising by differentiation which led me to:

$$V = l^2$$

Am I taking the right approach? Any help would be appreciated.

operations research

Posted: 08 May 2022 09:24 AM PDT

A company must complete three jobs. The processing time required (in minutes) for each job is shown in Table 1. A job must move through the machining stations in sequence. In essence, this means that, Job one must first be processed on machine 1 before it can move to machine 3, and then only can the job move to machine 4 and after that can it go to machine 5 for completion. Job two will start on machine 1, then it can go to machine 2 and only then be able to move to machine 4 for completion. Finally, Job three starts on machine 3 and then it can move to machine 4 and lastly to machine 5 for completion. Table 1: Processing times of the three jobs Machine Job 1 2 3 4 5 1 26 – 22 17 36 2 39 15 – 27 – 3 – – 28 28 12 Once a job begins processing on a machine it must finish processing on that machine without interruption, and at most one job can be processed per machine at any given time. If the flow time for a job is the difference between its completion time and the time at which the job begins its first stage of processing, formulate an Integer Linear Program (ILP) whose solution can be used to minimize the average flow time of the three jobs.

Sierpiński Constant

Posted: 08 May 2022 09:30 AM PDT

If we note $r_2(n)$ the number of representations of n as sum of two square we have that $\sum_{k = 1}^n \frac{r_2(k)}{k} = K + \pi \ln(n) + O(n^{\frac{1}{2}})$ in $+ \infty$

(the number $K$ is a constant called Sierpiński constant)

I search a complete proof of this result. Or any document/link/book to prove this.

Someone can help me ?

Thank you

How does $ \frac{\partial u}{\partial x}\Big{|}_{x+dx} = \frac{\partial u}{\partial x}\Big{|}_{x} + \frac{\partial ^2 u}{\partial x^2}{dx} $

Posted: 08 May 2022 09:20 AM PDT

I am currently studying some equations related to a flexible beam bending problem, but I don't fully comprehend how this relation is being derived below:

$$ \frac{\partial u}{\partial x}\Big{|}_{x+dx} = \frac{\partial u}{\partial x}\Big{|}_{x} + \frac{\partial ^2 u}{\partial x^2}{dx} $$

I don't think its derived as a result of something specific to the beam bending problem, but I've attached an image below of what the diagram shows just in case:

Flexible Beam.

Can anyone explain to me how the differential on the left hand side is transformed into the right hand side? Maybe I'm missing something fundamental, I haven't looked at calculus in a while.

Dijkstra's algorithm confusion

Posted: 08 May 2022 09:19 AM PDT

I have this graph:

and I am trying to apply Djiktra's algorithm to find the shortest path from A to E.

Here is my progress:

However I havent understood what I have found.Okay I have some values for A,B,C,D,E but how does that help me find the shortest path from A to E? And what about the last step?I havent done the last step because I didnt know what to do.(E isnt written out from the unvisited nodes).How should I continue?

Separability of the Frobenius morphism

Posted: 08 May 2022 09:13 AM PDT

We are in an algebraically closed field $k$ with positive characteristic $p>0$. Let $X\subseteq \mathbb{P}^2$ be an irreducible algebraic curve and $\phi:X\to\phi(X),\ (x_0,x_1,x_2)\mapsto (x_0^p,x_1^p,x_2^p)$ the Forbenius morphism. Show that $\phi$ is inseparable.

The way to do it is to show that the extension of the function field of $X$ and $\phi(X)$ is separable. So let us calculate the function fields: \begin{equation} X=\{x\in P^2: f_1(x)=\dots=f_k(x)=0\},\ Y=\{x\in P^2:f_1(\phi^{-1}(x))=\dots=f_k(\phi^{-1}(x))=0\} \end{equation} As we are in characteristic $p$, we have $(a+b)^p=a^p+b^p$ in general and we get for $Y$: \begin{equation} Y=\{x\in P^2:f_1^{1/p}(x)=\dots=f_k^{1/p}(x)=0\} \end{equation} This means for our function fields: \begin{equation} k(X)=Frac(k[x,y,z]/(f_1,\dots,f_k)),\ \ \ k(Y)=Frac(k[x,y,z]/(f_1^{1/p},\dots,f_k^{1/p})) \end{equation} What is left to show is that $k(Y)/k(X)$ has inseparability degree $>$ 1. This, however, I do not know how to do. Can someone please hlep me?

Help Recognizing Certain Big-O Runtime Complexities

Posted: 08 May 2022 09:08 AM PDT

I seem to understand basic Big-O estimates such as O(n) and O(n^2), but I am having trouble comprehending certain trickier Big-O estimates.

How can we recognize each of the following Big-O estimates? If one or more examples could be provided as well, it would be very helpful:

  1. O(log(n))
  2. O(nlog(n))
  3. O(2^n)
  4. O(n!)
  5. O(sqrt(n))

I have question about extension of the semigroup for a Cauchy problem

Posted: 08 May 2022 09:29 AM PDT

please I have a question about semigroup theory, Let $(A, D(A))$ be the generator of a strongly continuous semigroup $(S(t))$ on $X$. Then the following Cauchy problem $$ \left\{\begin{array}{l} \dot{u}(t)=A u(t) \quad \text { on}\,\,\, (0,T) \\ u(0)=x \end{array}\right. $$ has a unique mild solution for all $x\in X$. This is a well-known result. My confusion comes when they talk about the value of the semigroup $S(t)$ at $t=T$. I can't see why $S(T)$ has a meaning!! (cause the Cauchy problem we solve is only on $(0,T)$). If anyone can explain it to me please.

A permutation problem regarding number of ways of a given permutation

Posted: 08 May 2022 09:01 AM PDT

Given a permutation of N length. Lets say the permutation is : p1,p2,....,pn. How many tuples [a,b,c,d] such that: pa < pc and pb > pd. Example: 5 3 6 1 4 2 for this permutation of 6 length, there are 3 tuples which satisfy the condition. [1,2,3,4], [1,2,3,6], [2,3,5,6].

Determining existence of ODE solution around irregular singular point

Posted: 08 May 2022 08:59 AM PDT

I am trying to determine the existence of solutions of the second order ODE

$t^2x'' - x' - cx = 0$ Where c < 1 around the irregular singular point t = 0.

My intuition says there are no solutions defined in a neighbourhood of 0, and I can see that the only solution found using power series methods is the trivial solution (the series is divergent unless the first term is 0).

I am unsure how to proceed and show whether other solutions exist.

Thanks!

Gronwall inequalityy

Posted: 08 May 2022 09:04 AM PDT

Let $\phi \in L^1((0,T),\mathbb{R})$ non-negative, $a\in C^0([0,T],\mathbb{R})$ monotonous non-decreasing and non-negative. For $u\in C^0([0,T],\mathbb{R})$ it holds $$u(t)\leq a(t) + \int_0^t\phi(s)u(s) \forall t\in [0,T] (1)$$ Then $$u(t)\leq a(t)e^{\int_0^t\phi(s)ds}=:z(t) \forall t\in[0,T] (2)$$

$\textbf{Instructions}$ were given for the proof: (a) show $$a(t)+\int_0^t\phi(s)z(s)\leq z(t) \forall t\in [0,T] (3)$$ with $a$ monotonous increasing and differentiable and $\phi$ continous. Then show the general case by smoothing approximation.

(b) show $((1) \text{and} (3))\Rightarrow (2)$ by contradiction: define $z_{\delta}$ with $a_{\delta}(t)=a(t)+\delta, \delta>0$ and show $u(t)<z_{\delta}(t)$. Suppose $a<z_{\delta}$ on $[0,\tilde t)$ and $a(\tilde t)=z_{\delta}(\tilde t)$.

$\textbf{My idea}$: (a) I tried to differentiate (3) with respect to t: $\frac{d}{dt}(a(t)+\int_0^t\phi(s)z(s))=a'(t)+e^{\phi(t)}\int_0^t\phi(s)a(s) + e^{\int_0^t\phi(s)ds}\phi(t)a(t)$ by product rule and $z'(t)=\underbrace{a'(t)}_{>0}e^{\int_0^t\phi(s)ds}+a(t)e^{\phi(t)}$. Now I need to show $a(t)\leq \int_0^t\phi(s)a(s)ds$. Since $a$ is not differentiable in general my idea was to consider $a_{\epsilon}=a\ast \psi_{\epsilon} \rightarrow a$ for $\epsilon\rightarrow 0$ with $\psi \in C_c^{\infty}(0,T)$.

(b) Define $z_{\delta}(t):=(a(t)+\delta)e^{\int_0^t\phi(s)ds}$. Suppose $\exists \epsilon>0, \tilde t \in [0,T]$: $u(\tilde t) \geq z_{\delta}(\tilde t)$. It is for all $t\in [0,\tilde t)$: $u(t)<z_{\delta}(t)$ and $u(\tilde t)=z_{\delta}(\tilde t)$, so it holds $$z_{\delta}(\tilde t)=u(\tilde t) \overset{(1)}{\leq}u(\tilde t)+\int_0^{\tilde t}\phi(s)u(s)<u(\tilde t)+\int_0^{\tilde t}\phi(s)z_{\delta}(s)\overset{(3)}{\leq}z_{\delta}(\tilde t).$$ So $z_{\delta}(\tilde t)<z_{\delta}(\tilde t)$ contradiction! So it has to be $u(t)<z_{\delta}(t)\overset{\delta>0}{\Rightarrow} u(t)\leq z(t)$?

Thanks for any help proving this!

If $m,n \in \mathbb N$ and $c>1$, then $c^m>c^n$ if and only if $m>n$.

Posted: 08 May 2022 09:08 AM PDT

I read a very nice math book and there is one exercise in this book. I have to prove that

If $m,n \in \mathbb N$ and $c>1$, then $c^m>c^n$ if and only if $m>n$.

The proof is:

If $m>n$ then $k:=m-n \in \mathbb N$. But $c^k>c>1$ (we can use this fast). Since $c^k=c^{m-n}$ this implies that $c^m>c^n$.

My question is WHY? Why does $c^k=c^{m-n}$ imply that $c^m>c^n$?

Thank you for help!!

Why is this set a field? [duplicate]

Posted: 08 May 2022 08:51 AM PDT

Given the following set $Z_3[i]=\{a+bi | a,b \in Z_3\}$, which is a ring, prove that it is also a field.

To be a field, every nonzero element of the ring has to be invertible for the multiplication. So:

$(a+bi)^{-1} = \frac{1}{a+bi} = \frac{a-bi}{(a+bi)(a-bi)} = \frac{a}{a^2+b^2}-\frac{bi}{a^2+b^2}$.

We can consider the case in which $a=b=1$, so we have the elements $(1+i)$ and $(\frac{1}{2}-\frac{1}{2}i)$. Both elements are the inverse of each other, but, in the second element, $a=\frac{1}{2} \notin Z_3$.

We can conclude that $Z_3[i]$ is not a field, which contradict what I watch in a video in which the youtuber says that this set is a field (she didn't show the proof).

What is the correct result and why?

Finding a weak solution to an SDE

Posted: 08 May 2022 09:08 AM PDT

Consider the SDE $$dX_t=\text{sign}(X_t)dB_t$$ with $X_0=0$ and where $$\text{sign}(x)=\begin{cases}-1&\text{if }x\leq0\\1&\text{if }x>0\end{cases}.$$ I am asked to find a weak solution to this SDE. From the SDE, if $X_t$ is positive then it evolves as the Brownian motion $B$ and if non-positive then evolves as the reflection of the Brownian motion $B$. As $X_0=0$, the evolution will start as the reflection of $B$, and then evolve as the above, which makes me tempted to say that $X=-B$ is a weak solution.

However, in addition to not being sure how to show this, I am also asked to show that there is no strong solution - but I think $X=-B$ would be a strong solution as it would be adapted to the filtration generated by $B$?

Solution verification: what are the solution of the differential equation $y^{(6)}-\alpha^6 y =0$?

Posted: 08 May 2022 09:15 AM PDT

Let $a\in\mathbb{R}$. I need to find the solution of the differential equation $$y^{(6)}-\alpha^6 y =0.$$ I consider characteristic polynomial $p(\lambda)=y^6-\alpha^6 $. I need to find the solutions of $p(\lambda)=0$, i.e. I need to solve $$\lambda^6-\alpha^6 =0.$$

Now, as $\alpha\neq 0$, thus $\lambda = \pm\alpha$ are $2$ solutions both of multiplicity $6$. Thus, the solution of the differential equation corresponding at $\lambda =\alpha$ is $$y= e^{\alpha x}(c_1 +c_2 x+ c_3 x^2 +c_4 x^3 +c_5 x^4 +c_6 x^5),$$ while the solution of the differential equation corresponding at $\lambda =-\alpha$ is $$y= e^{-\alpha x}(c_1 +c_2 x+ c_3 x^2 +c_4 x^3 +c_5 x^4 +c_6 x^5).$$

On the other hand, when $\alpha=0$, the solution is $$y= c_1 +c_2 x+ c_3 x^2 +c_4 x^3 +c_5 x^4 +c_6 x^5.$$

Could someone please tell me if my solution is correct?

Thank you in advance.

Reference for Hausdorff metric

Posted: 08 May 2022 09:12 AM PDT

Consider $\mathbb{R}^n$ with the Hausdorff metric, $$d(A,B) = \max(\sup_{a\in A}\inf_{b \in B} ||a-b||,\sup_{b\in B} \inf_{a\in A}||a-b||).$$ I'm looking for a reference containing the statement that $\lim_{i\to 0} A_i = A$ if and only if $A$ is the set of all limits of convergent sequences $\{x_i\}_{i\in \mathbb{N}}$ with $x_i\in A_i$.

Is a usual open ball in a complex algebraic variety Zariski dense?

Posted: 08 May 2022 09:26 AM PDT

Let $X$ be an affine variety $\operatorname{Spec} \mathbb C[x_1,\dots, x_n]/(f_1,\dots, f_m)$. Suppose the set of closed points gives a smooth complex analytic variety in $\mathbb C^n$. Pick any $p\in X$, and let $U$ be a small open ball in $\mathbb C^n$.

Question: Is $U\cap X$ Zariski dense in $X$?

I think this is correct but would like more careful thinking. A baby example may be as follows: the open unit disk in $\mathbb C$ is Zariski dense.

Inequality of positive integers [duplicate]

Posted: 08 May 2022 08:58 AM PDT

I have to prove the following inequality for positive integer $n$: $$ 2^ {n(n+1)}> (n+1)^{n+1} \left(\frac{n}{1}\right)^{n} \left(\frac{n-1}{2}\right)^{n-1}...\left(\frac{2}{n-1}\right)^{2} \left(\frac{1}{n}\right) $$ I tried using weighted $AM > GM > HM $ for $ (n+1), \left(\frac{n}{1}\right), \left(\frac{n-1}{2}\right),...\left(\frac{2}{n-1}\right), \left(\frac{1}{n}\right)$ . Any sort of hint/help is appreciated. Thank you.

Simplifying $\frac{\tan(\alpha+\frac32\pi)-\cot(\alpha+\frac{13}2\pi)}{\cot(\alpha+\frac32\pi)+\tan(\alpha+\frac{13}2\pi)}$

Posted: 08 May 2022 09:14 AM PDT

Simplify $$\dfrac{\tan\left(\alpha+\dfrac{3}{2}\pi\right)-\cot\left(\alpha+\dfrac{13}{2}\pi\right)}{\cot\left(\alpha+\dfrac{3}{2}\pi\right)+\tan\left(\alpha+\dfrac{13}{2}\pi\right)}$$ We have $$\tan\left(\alpha+\dfrac{3}{2}\pi\right)=-\cot\alpha,\\\cot\left(\alpha+\dfrac{3}{2}\pi\right)=-\tan\alpha$$ What else?

Is the Mandelbrot set path-connected if and only if it is locally connected?

Posted: 08 May 2022 09:09 AM PDT

This question mentions that it's an open question whether the Mandelbrot set is path-connected and the answer conflated it with the more famous open question of whether the Mandelbrot set is locally connected. But locally connected and path-connected are different notions and it's not obvious that they are equivalent in this case. Are they equivalent for the Mandelbrot set?

Here's a possible proof for the Mandelbrot set being path-connected: There is a homeomorphism between the complement of the Mandelbrot set and the complement of a closed disk. This homomorphism can be used to define a continuous function from a circle onto the boundary of the Mandelbrot set, which witnesses that the boundary of the Mandelbrot set is path connected. Any set with path-connected boundary has path-connected closure, and the closure of the Mandelbrot set is the Mandelbrot set itself since it is closed. Where does this argumant go wrong?

Counterexample : composition of power series

Posted: 08 May 2022 09:16 AM PDT

I'm looking for some counterexample for the following situation : let $S$ et $T$ be two power series, with respective positive radius $R_S$ and $R_T$, with $T(0)=0$. Therefore there is $\rho>0$ such that $\rho<R_T$ and for all $z\in\mathbb C$ such that $|z|<\rho$, $\left|T(z)\right|<R_S$.

We know that the power series $S\circ T$ has a radius at least $\rho$, and for all $z\in\mathbb C$ such that $|z|<\rho$, $(S\circ T)(z) = S(T(z))$.

To be more precise, $S\circ T$ is the formal power series $\sum_n s_nT^n$, and $S(T(z)) = \sum_n s_n(T(z))^n$.

What I'm looking for is some example of $S$ and $T$ such that there is $z\in\mathbb C$, $|z|<R_T$ and $(S\circ T)(z) \ne S(T(z))$.

I'm at a loss to find such an example, so any help would be appreciated. As my english is a bit poor, I'm sure you'll want some more explanation, feel free to ask me :-)

Proving an identity involving the discrepancy function

Posted: 08 May 2022 08:51 AM PDT

Let $A\in R^{m\times n}$, $y\in R^m$, and let $x_{\delta}\in R^n$ be the Tikhonov regularized solution of $Ax=y$, i.e., $x_{\delta}$ minimizes the functional $||Ax-y||^2+\delta||x||^2$, $\delta>0$

Define the 'discrepancy function' $f:R_+\to R_+$ by $f(\delta)=||Ax_{\delta}-y||^2$.

Prove that $f^{'}(\delta)=2\delta\langle x_{\delta},(A^TA+\delta I)^{-1}x_{\delta}\rangle=2\delta x_{\delta}^T(A^TA+\delta I)^{-1}x_{\delta}$.

So I started by writing the norm squared as the product of the inner product $\langle Ax_{\delta}-y\rangle$ with itself: $||Ax_{\delta}-y||^2=\langle Ax_{\delta}-y\rangle\langle Ax_{\delta}-y\rangle$. But I don't know how to differentiate the inner product.

A following up question on Weierstrass function approximation

Posted: 08 May 2022 08:58 AM PDT

This is a following-up question of

Proof — Weierstrass Approximation Theory for derivatives

and $f \in C^{\infty}$

Basically I wanted to explore whether there exists a sequence of polynomials $p_n$ such that $p^{(k)}_n$ converges uniformly on $[0, 1]$ to $f^{(k)}$ (the k-th derivative) for $ k = 0, 1, 2, . . . $

My idea is as follows:

Given any $k$, there exists a sequence of polynomial $q$ such that as $n\gt N$ we have $\|f^{(k)} - q_n\| \lt \epsilon.$ This is a direct consequence of Weierstrass Theorem.

We define $g(x) = f^{(k)}(x)$, and let $p_n(x) = \underbrace{\int_{0}^{x}... \int_{0}^{x}}_\text{$k$ times}q_n(s)ds + f(0)$ and we are done.

Given that such $k$ is an arbitrary positive integer, the result holds for all $k$.

Is this proof valid?

Prove the following property for Euler's totient function

Posted: 08 May 2022 09:10 AM PDT

I have been asked to prove that $\phi(n) > \dfrac{9n}{50}$ for all $n$ that have at most seven prime factors. I'm trying to think of how this relates to any theorems/ properties I have learnt regarding the totient function, but I have drawn a blank. Could anyone help to explain how the property holds?

Why is the function $||\mathbf{J}||_{\infty}$ is $1/n$-Lipschitz w.r.t to the Euclidean norm?

Posted: 08 May 2022 08:55 AM PDT

Assume that $\mathbf{J}=\{J\}_{ij}$ are centered independent standard Gaussian with variance $1/n$ random variables for $i, j=1,\dots, n$. Why is the function $||\mathbf{J}||_{\infty}$ is $1/n$-Lipschitz w.r.t to the Euclidean norm, that is $$ |\sup_{||u||=1}<u, \mathbf{J}u>-\sup_{||v||=1}<v, \mathbf{J}v>|\le \frac{1}{n}\|u-v\|_2 $$ Thus, it has a sub-Gaussian tail which is $$ P(||\mathbf{J}||_{\infty}-E[||\mathbf{J}||_{\infty}]\ge x)\le e^{-nx^2/2}. $$

The original statement is as follows. enter image description here

Let $S$ be the set of all positive integers which are _not_ the sum of 8 or fewer sixth powers of non-negative integers. Is S finite or infinite? [closed]

Posted: 08 May 2022 09:13 AM PDT

Let $S$ be the set of all positive integers which are not the sum of 8 or fewer sixth powers of non-negative integers. Is S finite or infinite?

On the function $n \mapsto |a_n|^{\frac 1n}$ for a given power series $\sum_{n} a_n z^n$

Posted: 08 May 2022 09:18 AM PDT

I am currently doing research involving power series on the unit disk in $\Bbb C$: precisely I am studying the properties of converging power series of a standard form $$ f(z)= \sum_{n=0}^\infty a_n z^n \label{1}\tag{1} $$ where (assuming conventionally for the discourse that $0\notin\Bbb N$)

  1. $\limsup_{\substack{n\in\Bbb N \\n\to\infty}} |a_n|^{\frac 1n}=1$, and
  2. $z\in \Bbb D=\{z\in\Bbb C : |z|<1\}$.

From the basic properties of $\limsup$, condition 1 implies the boundedness of the set $\big\{|a_n|^{\frac 1n}\big\}_{n\in\Bbb N}$ , i.e. it implies the existence of $\sup_{\substack{n\in\Bbb N \\n\to\infty}} |a_n|^{\frac 1n}$, and this has some important consequences on the structure of \eqref{1}, namely an estimate of the size of zero free regions, the size of the minimal univalence radius etc., described in an old paper by Milos Kössler [1].

The question

The paper [1] focuses on what a finite value of $\sup_{\substack{n\in\Bbb N \\n\to\infty}} |a_n|^{\frac 1n}$ implies: does there exist other studies trying to analyze the implications of a given behavior for $n \mapsto |a_n|^{\frac 1n}$, $n\in \Bbb N$ or for the related $n \mapsto \sup_{k>n}|a_k|^{\frac 1k}$?

References

[1] Milos Kössler, "O významu čísla $\sup |a_n|^{\frac 1n}$ v teorii mocninných řad (The signification of the number $\sup |a_n|^{\frac 1n}$ in the theory of power series)", (Czech, English summary), Časopis Pro Pěstování Matematiky a Fysiky 74, No. 1, 47-53 (1949), MR0034833, Zbl 0033.26504

Find coordinates of a 2D plane within a 3D plane

Posted: 08 May 2022 09:05 AM PDT

I'm not sure this is the right place to ask this question, if not I do apologies and I will move on. I am asking this question as a programmer, however it seemed entirely maths based.

Image one is sitting still and holds in front of their face a sheet of graph paper with grid lines. This piece of paper represents a 2D plane, however although it exists inside of another 3D environment, it can be move about such that it's axis may not line up in any meaningful way.

Given 3 points where both the real world X,Y,Z values are known and the relative X,Y coordinates of their position within the paper drawn grid and given that two of these points share the same virtual 2D Y value but differ on X and the third point shares neither the same X nor Y value but does indeed exist on the same virtual 2D plane.

How can I then transform any given coordinate of one set to the other. Say I want to know where square (12,20) on the graph paper exists in the real world and vice versa.

Also, say I have a set of real world coordinates that is not on my 2D plane but directly above it. Is there a way and if so, how would i: traverse "downwards" through 3D space perpendicular to the find the 2D point directly below.

Why is this statement a corollary of Fermat's little theorem?

Posted: 08 May 2022 09:14 AM PDT

I know that if $p$ prime and $p\nmid a$, then $a^{p-1}\equiv 1\pmod p$ and I know also that $a^{p}\equiv a \pmod p$ using the fact $a\equiv a \pmod p$ and multiplying the members.

What I couldn't understand is why in the Fermat little theorem we have $a^{p}\equiv a \pmod p$ for all integer $a$.
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