I need to proof that for any nature number $n$ such that $30<n$: $$\pi(4n-3)<n.$$ In this inequality, $\pi(x):\mathbb{N}\to \mathbb{N}$ is the defined as follows: $$\pi(x):=\lbrace p \ ...
How would I go about finding a solution to the following system?$x\equiv 1\text{ mod }12\ \ \ \ \ \ \text{(1)}\\x\equiv4\text{ mod }21\ \ \ \ \ \ \text{(2)}\\x\equiv18\text{ mod } 35\ \ \ \ \text{(3)}$It is a problem because $\gcd(12,21,35)\neq1$.I c
I'm really confused on how to do modular reduction. I understand we're supposed to take the factor of the exponent?for example how would I go about doing modular reduction on: $5^{17}$ mod 16
I am having a test in few days and I saw an interesting question while I was skimming through the book problems. The problem is concerned about initial-boundary value problem of 2nd order PDEs. To be specific, a IBVP problem of the diffusion (heat) e
Is there an easy way to tell if a convex polygon is cyclic? I was told that if the vertices of the $n$-gon are $A_1,A_2,\ldots,A_n$, it is enough to prove that $A_1A_2A_3A_i$ is cyclic for each $i$ in $4 \leq i \leq n$. Does any know or can prove thi
Problem Statement: Determine the equlibrium temperature distribution inside a circular annulus $r_1\leq r \leq r_2$. If the outer radius is at temperature $T_2$ and inner radius at temp $T_1$. So my teacher told us to use the formula $\Delta u=\frac{
Let T(a,b) = (a+b,2a-b,3a)a)Find the matrix representing T. b)Find the image of T (as a span of vectors)So I found that T is a linear transformation. Now would the matrix just be? $A$= $\begin{bmatrix}1 & 1\\2 & -1\\ 3 &0\end{bmatrix}$ I feel
$X$ and $Y$ are independent random variables identically exponentially distributed with $\lambda$. Take $Z=X+Y$.Show that $(X|Z=z)$ is uniformly distributed over $(0<x<z)$.Then, find $$E[X^k|Z=z]$$.I need help getting started.$$P[X|Z=z] = \frac{P[X,
An umbrella is made by stitching 10 triangular pieces, of cloth of two different colors, each piece measuring 20 cm long, 50 cm and 50 cm. How much cloth of each color is required for the umbrella?You can count the number of triangles of each color,
first time on this site and I need help with this, I got 2 stands 1 is longer and goes top of the other. The top of the stand will have 70 lbs on it and I want to know how much pressure (if ...
I am a little baffled by this question. Is it safe to assume that since $f$ is an isomorphism, $f (1) = 1$ ? And, if it is safe to assume this, could I construct a proof by induction, by using the fact that $f(1) =1$,$f(-1)=-f(1)$, $f(1+1) = f(1) +f(
Given a Markov chain $\{X_n \mid n \in \{0, 1, \ldots\}\}$ with states $\{0, \ldots, N\}$, define the limiting distribution as $$ \pi = (\pi_0, \ldots, \pi_N) $$ where $$ \pi_j = \lim_{n \to +\infty} \mathbb{P}\{X_n = j \mid X_0 = i\} $$I am confused
I have a random walk Markov chain that has states from $0$ to $N$. The conditions are that when the chain is at $0$, the chain will go to state $1$ with probability $1$. When the chain is at state $N$, it will go to state $N-1$ with probability $1$ a
Is the following proof valid? (Note: I know there is a post discussing this problem, but I am curious to see if my argument works).Let $S \subseteq \mathbb{R}$. Prove that if $x$ is an isolated point of $S$, then $x$ is a boundary point of $S$.Suppos
Let $v: \mathbb{R^n} \rightarrow \mathbb{R^m}$ be a function such that $v(y) \neq 0,\forall y \in \mathbb{R^n}$ that is differentiable at $ x \in \mathbb{R^n}$. (a) Show that the value-wise magnitude $|v|: \mathbb{R^n} \rightarrow \mathbb{R}$ defined
How would you connect each black box once to each colored box without any lines overlapping, this is racking my brain so please help.Note that you can move the boxes where ever you want.Mayb ...
I want to proof this specifically using the Compactness Theorem from propositional logic (this is an exercise from Model Theory, Hodges).$G$ orderable means there is a total ordering s. t. $g\leq h$ implies $gk\leq hk$ and $kg\leq kh$ for all $h$.My
I am reading the book. On page 244, the formula (9.2.3.4). I would like to compute the bracket on g^* induced from the Poisson bracket on C[G] explicitly in the example of $G=SL_2$. The formula is: \begin{align} [\xi_1, \xi_2]_{g^*}(X) = \frac{d}{dt}