0
votes
0answers
18 views

question about gcd

I encountered the following question: Find a natural number x which satisfies the following: 12345 mod 54321 = 6 mod 54321 I tried using the extended Euclidean algorithm, but failed to solve the ...
0
votes
1answer
24 views

Help understanding a particular proof of the compactness theorem for Propositional Calculus.

I've reading through this proof, I don't understand the last part: the claim $\tau \models \Sigma$. Note: I'll use $AP(\varphi)$ and $\text{Var}(\varphi)$ interchangeably, to mean the variables that ...
0
votes
0answers
26 views

Books or website about solving IMO problems

Hey I want to solve IMO problems like the problem in the image below, but I cannot solve the problem or any of the problems in the IMO, so do you guys have some good website or books that teach how to ...
1
vote
0answers
8 views

Does there exist a $k$ such that for all $n \ge 3$, $\text{gpf}(\lfloor n^{(\log{n})^k} \rfloor) \gt n$?

Does there exist a $k \in \mathbb{R}$ such that for all $n \in \mathbb{N}, n \ge 3$, $\text{gpf}(\lfloor n^{(\log{n})^k} \rfloor) \gt n$, where $\text{gpf}(x)$ is the greatest prime factor of $x$? I ...
0
votes
0answers
10 views

Conditional probability,two conditions

A doctor operates on patient with a certain disease if he is 80% sure that he has it.We have a patient for whom the doctor is 60% sure that he has the disease,so he makes him do another test which is ...
-1
votes
0answers
13 views

$f(a)\leq g(a)\in (o(a))^{\frac{1}{2}}$ implies $f(a)\in o(a)$

Consider the real-valued functions $f,g$ such that $f(a)\leq g(a)\in (o(a))^{\frac{1}{2}}$ as $a \rightarrow 0$, where $o(\cdot)$ denotes little o notation explained here. In order to show $f(a)\in ...
3
votes
1answer
28 views

A formula for length of representation of a number in a “base” without zeros

If you had 2 items the sequence would go like this: $$1,1,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5, \ldots$$ This is $\lfloor\log_2(n+2)\rfloor$. What if I ...
5
votes
3answers
237 views

Proving the existence of a proof without actually giving a proof

In some areas of mathematics it is everyday practice to prove the existence of things by entirely non-constructive arguments that say nothing about the object in question other than it exists, e.g. ...
0
votes
0answers
8 views

Upper bound on the product of independence number and transversal for graph

I am trying to prove if $G$ is an $n$ vertex graph such that $|E(G)| \leq \alpha(G)\tau(G)$, then $|E(G)| \leq \frac{n^2}{4}$ where $\tau(G)$ is the smallest transversal in $G$. A transversal is a ...
1
vote
0answers
20 views

Does the order of a finite group divide the product of degrees of a system of parameters of the invariant algebra?

Let $V$ be a vector space of dimension $n$ over a finite field $\mathbb{F}$, and let $G$ be a subgroup of the finite group $\operatorname{GL}(V)$. Then $G$ acts on the graded algebra $\mathbb{F}(V)$ ...
9
votes
4answers
108 views

Why isn't $e^{2\pi xi}=1$ true for all $x$?

We know that $$e^{\pi i}+1=0$$and $$e^{\pi i}=-1$$ So$$(e^{\pi i})^2=(-1)^2$$$$e^{2\pi i}=1$$ Because $1$ is the multiplicative identity,$$(e^{2\pi i})^x=1^x$$$$e^{2\pi xi} =1$$should also hold ...
2
votes
2answers
20 views

Differential equation$ (x^2-x)y' = (y^2+y)$

Can i get help solving the differential equation $$y' = \frac{y^2+y }{x^2 -x}$$ I tried searching but could not find anything similar. Thank you!
3
votes
3answers
48 views

Basic algebra problem: $ \frac{\frac{1}{x}+\frac{1}{y}}{\frac{1}{x^2}-\frac{1}{y^2}} $

Basic algebra problem I can't seem to figure out: $$ \frac{\frac{1}{x}+\frac{1}{y}}{\frac{1}{x^2}-\frac{1}{y^2}} $$ $x,y \in \mathbb{R}, x^2 \neq y^2, xy\neq0$. Now I know the result is: ...
2
votes
2answers
41 views

Factorial Summation Definition

A while back I found the series $$\sum_{k=0}^n \binom n k (-1)^k (x+k)^n = (-1)^n n!$$ while messing around in Algebra class (specifically when $n$ is any natural number and $x$ is any real number) I ...
0
votes
2answers
37 views

Flaw in the technique I am using to find the area between line and curve

I am asked to find the area between ${y = 7}$ and ${x^2 -5x + 13}$ Combining these equations together I get ${-x^2 - 5x + 6 = 0}$. Factorising into ${(x - 3)(x - 2)}$ I am taking ${y = 7}$ to be ...
0
votes
0answers
13 views

Where should I start from if I want to learn Mathematical Modeling?

I am a student of political science and have basic knowledge of mathematics. But I need to learn Mathematical Modeling if I want to continue a research of mine. I'm willing to spend even 1 or 2 years ...
0
votes
1answer
9 views

Test predictability with Bayes' Theorem

Say we have a disease and a test for it. P(A :=a person has the disease)= 0.01 ( example) P( B:=test is positive | A )=0.95 Is this enough information to calculate the probability that a person has ...
1
vote
0answers
46 views

A method of writing all primes

I've recently noticed a method of describing primes. As an example: $13=5*11-2*3*7$. This pattern must follow these rules: $x-y$ such that $x*y$ is the product of all previous primes (allowing powers ...
0
votes
1answer
10 views

Maximally extended solution of this ODE.

So I am asked to find the positive, maximally extended solutions to this ODE. $$u'(x) = \frac{x}{u(x)}$$ Now a solution is given by $$u(x) = (\int_{y_0}^y t dt )^{-1}\circ \int_{x_0}^x s ds = ...
2
votes
4answers
34 views

Prove that $16\cos^5A-20\cos^3A+5\cos A=\cos5A$

Prove that $$16\cos^5A-20\cos^3A+5\cos A=\cos5A$$ My solution begins here; $$ \begin{align} \text{RHS} & =\cos5A \\ & =\cos(A+4A) \\ & =\cos A\cos4A-\sin A\sin4A \\ & =\cos ...
0
votes
1answer
16 views

Pick out a polynomial such that ideal $J=q(x)R$ , where $q(x)$ is polynomial and $R$ is ring

In the ring of polynomials $R =\mathbb Z_5[x]$ with coefficients from the field $\mathbb Z_5$, consider the smallest ideal $J$ containing the polynomials, $p_1(x) = x^3 + 4x^2 + 4x + 1$ $p_2(x) = ...
0
votes
1answer
17 views

What is the maximum of the following function?

Let $f(x,y) = \frac{xy^\alpha}{x+y},\alpha\in(0,\infty)$. How to compute $$\sup_{(x,y)\in[a,b]\times [0,c]}\frac{xy^\alpha}{x+y},$$ with $b>a>0$?
2
votes
2answers
26 views

given $-\pi < \theta \leq \pi$ prove $f(z) = z^{1/3}$ is not entire.

I don't want the solution at all, but I'm incredibly stuck, and I really need some (hopefully not much) help. What I've considered: Liouville's Theorem Not applicable because f is not bounded. ...
3
votes
3answers
35 views

Simplify $(k +1)! > (k + 1)^2$ to prove true for $k ≥ 4$

I am trying to prove this statement is true for $k ≥ 4$. I don't know how to work with $k + 1$ factorial, so I'm a little lost on proving this.
0
votes
0answers
16 views

Solve $A \partial_t w + B \partial_t\partial_x^4 w + C \partial_x^4 w + \partial_t^2 w = 0$

a non-mathematician wants me to solve a PDE. The problem is that I don't know a lot of theory to solve PDE's except the fouriertransform. This is the PDE $$A \partial_t w + B \partial_t\partial_x^4 w ...
-2
votes
0answers
18 views

del operator and einstein's index notation

How can I prove this equation using Einstein's index notation? https://en.wikipedia.org/wiki/Del we are using this operator. I think we can prove this if we simplify RHS. But, I would like to ...
0
votes
0answers
14 views

Approaches to stability of newtonian systems

I am having some difficulties figuring out how to approach "Test stability problems". I usually test the linearization of the system (since it is very straightforward and easy), and if that doesn't ...
1
vote
2answers
9 views

Prove equivalence ratio and find class of $B\subseteq A$ , $X \sim Y \Leftrightarrow X \cap B = Y \cap B$

Prove equivalence ratio and find class of $B\subseteq A$ , $X \sim Y \Leftrightarrow X \cap B = Y \cap B$. Well, I've proved really easily that it is reflexsive, symmetrical and transitive. But I'm ...
0
votes
1answer
21 views

Proof of $a \le 0$ $\Leftarrow \Rightarrow$ $\forall \epsilon > 0$ $a < \epsilon$

In analysis class I saw a proof but I would like see another Proof: $\Rightarrow$ Suppose that $a \le 0$ $\land$ $\epsilon > 0$ if $a=0$ by hypothesis $a < \epsilon$ if $a<0$ $\Rightarrow$ ...
0
votes
0answers
25 views

Number of solutions of equation with natural numbers

Given natural numbers $s, n, k$. How to find number of solutions to equation $a_1 + a_2 + \ldots + a_s = n-s$ where $0 \leq a_i \leq k-1$ and $a_i \in \mathbb{N}$?
1
vote
3answers
27 views

Number of Likes and Dislikes to Star Rating system

I have a 5 star rating system that outputs a recommendation. A user inputs a rating based on a Like or Dislike. What I would like to know is how to convert the number of likes and dislikes into a ...
2
votes
3answers
50 views

Can I further simplify $5^k \cdot 5 + 9 < 6^k \cdot 6$ to prove this is true

I am trying to prove this statement, but I'm not sure where to go from here. Is don't think this is sufficiently reduced to conclude the statement is true, but I'm not positive. $k ≥ 2$ Can I ...
2
votes
0answers
38 views

Einstein notation

I'm confused about a specific issue that I have with the Einstein notation (for tensor fields on manifolds). I want to write the following thing: Let $X$ be a smooth manifold. Choosing local ...
3
votes
1answer
38 views

How to find $z$-score

I have some probabilities, but I have to find the $z$-score. I am not sure how do to this when I am told I have to use slope-intercept. Where do I plug the numbers in exactly? Here is one of my ...
0
votes
1answer
14 views

Laplace transform and value in x(0)

Somebody told me that if i have something like this: $x''(t) + x'(t) = -2x(t) + u$ $x(0) = 7$ and use laplace transform on it i will get $s^2X(s) + sX(s) = -2X(s) + U(s)$ next i'm getting ...
0
votes
1answer
23 views

What does three bars and “def” mean in a partial derivate problem?

I'm reading the book "Mathematical Models in Biology" by Leah Edelstein-Keshet and in page 70 the following explanation appears. Here, F(x,y) is a function with P = F(X0 + Y0) and the idea is to ...
2
votes
2answers
42 views

Proof on Functions /Set Theory

Let $S$ be the set of all numbers of the form $a + b\sqrt 2$ where $a$ and $b$ are rational. Let $f : S \to R$ be a function such that $f(x+y)=f(x)+f(y)$ for all $x$ and $y$ in $S$. Then $f(x)=f(1)x$ ...
0
votes
1answer
14 views

Show that $p = u \cdot (\zeta -1)^{p-1}$, where $u$ is an invertible element of $Z[\zeta]$

Show that $p = u \cdot (\zeta -1)^{p-1}$, where $u$ is an invertible element of $Z[\zeta]$. This outcome is the result of this link. So I think I have to use the previous result and the ...
1
vote
1answer
28 views

Find $|CM|$, if $|CA|=a$ and $|CB|=b$.

Let $O$ be a center of a circle, circumscribed over $\triangle ABC$. Perpendicular, drown from the point $A$ on the line $CO$, cross the line $CB$ in the point $M$. Find $|CM|$, if $|CA|=a$ and ...
0
votes
1answer
40 views

Does $2^{k+1} = 2^k * 2^1$?

I'm not sure how to deal with an exponent like this. Can I simplify it into terms that are easier to work with? I know that $2^3 · 2^4 = 2^{3+4} = 128$, but I don't know about $2^{k + 1}$
0
votes
0answers
8 views

different definitions of a subnet

The classical definition of subnet seems to be that $\Psi: J\to X$ is a subnet of $\Phi: I\to X$ if there exists a monotone, final map $h: J\to I$ s.t. $\Psi = \Phi\circ h$. I found another definition ...
1
vote
0answers
36 views

Conjecture: $\int_0^{\infty}dx\frac{e^{i\alpha\sqrt{x^2+1}}}{\sqrt{x^2+1}}J_1(Qx)=\left(e^{i\alpha}-e^{i\sqrt{{\alpha}^2-Q^2}}\right)/Q$

Here $\alpha>0$, $Q>0$, and $J_1$ is a Bessel function. I'm fairly certain the closed form in the title is accurate for a couple of reasons. First, I've evaluated the integral numerically in ...
0
votes
1answer
44 views

Find the following limits without using l'Hospital's rule

I need some help solving the following limits. I am not allowed to use l'Hospital's rule. $$\lim_{x\to\frac\pi4}\left(tg2x\cdot tg\left(\frac\pi4-x\right)\right)$$ $$\lim_{x\to1}(1-x)\cdot ...
1
vote
4answers
42 views

How does computing the determinant of a matrix with unit vectors give you the Cross Product?

Say you had $(a_x,a_y,a_z)\times(b_x,b_y,b_z)$, you would set up a matrix like the following: And the resulting would be your Cross Product or the coordinates of an orthogonal vector. My question ...
0
votes
0answers
11 views

Stationary process vs stationary increments

Am I right that these are not the same, i.e. a stationary process need not have stationary increments and vice versa? example: Brownian motion is not a stationary process but it has stationary ...
0
votes
0answers
39 views

Why is that if $z^n = |z|^2$, then $|z| = 1$?

We have $z^{n-1} = \bar{z}\ \forall\ n > 2$ which gives us $z^n = |z|^2$, but I dont see why that means $|z| = 1$?
0
votes
0answers
21 views

Number of field homomorphisms [on hold]

Let $E$ be a finite field extension over $K$. Show there are at most $[E:K]$ $K$-homomorphisms $E \to F$, where $F$ is an algebraic closed field extension of $E$.
0
votes
0answers
15 views

Calculate the discrete density of the variables of a Markov chain

$X$ and $Y$ are independent random variables of Bernoulli with parameter $\frac{2}{3}$. $Z=X+Y$ $\{X_n\}_{n \in \mathbb{N}}$ with values in {0,1,2} having $Z$ such as initial law and the transition ...
1
vote
1answer
12 views

Implications of $n a_n=O((n^{\frac{1}{\alpha}}|b_n|)^\alpha)$

Consider two sequences of real numbers $\{a_n\}_n$, $\{b_n\}_n$. Suppose $n a_n=O((n^{\frac{1}{\alpha}}|b_n|)^\alpha)$ where $\alpha \in \mathbb{R}$, $na_n\geq 0$ and big $O$ notation is explained ...
0
votes
0answers
10 views

what happens to the wave equation?

What happens to the wave equation when the solutions are $$v(x,0)=f(x)=x $$$$v_t(x,0)=g(x)=0.$$ I have a homework question that asks to discuss the solutions of wave equations and this is the first ...

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