0
votes
0answers
15 views

Number of zeros of $ f^n $

Let $f:\Bbb R\to \Bbb R$ be infinetly differentiable function that vanishes at $10$ distinct points in $\Bbb R$.suppose $ f^{n} $ denote $n$-th derivate of $f$, for $n \ge 1$. Then which of following ...
0
votes
1answer
31 views

Complex square matrices. Difficult proof.

$det(I+A\cdot\bar{A}) \ge 0$ Is it possible to prove the inequality is true for all complex square matrices $A$ where $I$ is the identity matrix and $\bar{A}$ is the complex conjugated matrix.
0
votes
0answers
8 views

Area between two functions

My question is from Apostol's Vol. 1: One-variable calculus with introduction to linear algebra textbook. Page 94. Exercise 16. Let $f(x)=x-x^2$, $g(x)=ax$. Determine $a$ so that the region above ...
0
votes
0answers
1 views

Permutations and inverzion of permutations

We have a permutation $a_1,a_2,...,a_n$ of the set {1,2,...,n}. For a pair $(a_i,a_j)$ we say it is an inversional permutation if $i<j$ and $a_i>a_j$. Find the number of permutations in which ...
2
votes
4answers
46 views

Calculate $ \lim_{n\to+\infty}\cos{\big{(}\pi \sqrt{n^2+n}\big{)}}$

I need to prove that $$ \lim_{n\to+\infty}\cos{\big{(}\pi \sqrt{n^2+n}\big{)}}=1$$ But this is something that seems highly unlikely to have a limit, but I am probably wrong. Wolfram Alpha says "$0$ ...
1
vote
3answers
25 views

Image of (0,1] under continous function

Let $f: \mathbb{R} \to \mathbb{R}$ be a continuous function. Which of the following sets cannot be image of $(0,1]$ under $f$. {$0$} $(0,1)$ $[0,1)$ $[0,1]$ My initial guess was using intermediate ...
-1
votes
0answers
28 views

Number of adjacent permutations.

What is the number of permutations for any number of adjacent elements swapping places at the same time in an array of length $n$? My solution: I think that we only need to count the number of ...
3
votes
1answer
24 views

Difficulty in finding a counter example

Finding difficulty in finding a counter example that if $f : (0,\infty) \to(0,\infty) $ is uniformly continuous then implying that $\lim_{x\to \infty} \frac{f(x+\frac{1}{x})}{f(x)} =1$.
2
votes
2answers
20 views

Limit and Fibonacci

How to prove $$\LARGE{\lim_{n \to \infty} \frac{F_{kn}}{{F_n}^k} = 5^{(k-1)/2}}$$ Non-induction method is prefered.
1
vote
0answers
26 views

Dimension of Polyhedra [on hold]

Can someone explain the question below? I'm pretty new in this area, and I did not understand anything. Question; Let $P$ be defined by the following $$\begin{align}x_1+x_2+x_3&\le ...
1
vote
1answer
10 views

Computing the Fourier series of $\lvert x\rvert$

I am getting very confused when trying to compute the Fourier series of $f(x) = \lvert x\rvert$, $x \in [-1/2,1/2]$. Normally I have no trouble with this because it is mindlessly integrating to get ...
2
votes
2answers
21 views

Evaluate the integral $\int_0^{\ln(2)} \sqrt{(e^x-1)}dx$

Evaluate the integral $\int_0^{\ln(2)} \sqrt{(e^x-1)}dx$ Why is it wrong to... $$\int_0^{\ln(2)} \sqrt{(e^x-1)} dx= \int_0^{\ln(2)} (e^x-1)^{1/2} dx= \frac{2}{3}(e^x-1)^{3/2} |_0^{\ln(2)}$$
0
votes
0answers
5 views

Help me with proof concerning functions

Problem: Let $X$ and $Y$ be non-empty sets and let $f: X \rightarrow Y$ be a function. We define $F: P(Y) \rightarrow P(X)$ by $F(B) = f^{-1}(B)$ for all $B \in P(Y)$. Proof that $F$ is injective if ...
0
votes
1answer
6 views

Question about Convergence Definition for Finite Difference Scheme

I have a question about the Convergence Definition for Finite Difference Scheme. The definition is given by Convergence: for one-step schemes approximating a IBVP to be convergent we compare ...
1
vote
0answers
34 views

Closed form for the summation $\sum_{k=1}^n\dfrac{1}{r^{k^2}}$

Is there any closed form for the finite sum $$\sum_{k=1}^n\dfrac{1}{r^{k^2}}$$ or infinite sum ( when $|r|<1$) $$\sum_{k=1}^\infty\dfrac{1}{r^{k^2}} ?$$ While solving this problem, I found this ...
0
votes
0answers
2 views

Given the sum of four Exp(1) distributed random variables, what is the conditional density of sum two of them?

Let T := X+Y+Z+K be indepedent and Exp($1$)- distributed random variables. What is the density of (X+Y) given {T = $1$} ? For M:= X+Y and N := Z+K given {M + N = $1$} The joint density is $ ...
1
vote
2answers
540 views

Roots of biquadratic equation

This question also was a part of my today's maths olympiad paper: If squares of the roots of $x^4 + bx^2 + cx + d = 0$ are $\alpha, \beta, \gamma, \delta$ then prove that: $64\alpha\beta\gamma\delta ...
0
votes
0answers
6 views

I need help to solve this problem

Let $m,n$ nenegative integers and $m>n$. Find polinom $g(x),r(x)$ from the ring $R[x]$ such that $x^m -1 =q(x)(x^n -1) + r(x)$ , $r(x)=0$ or $deg(r(x))<n$. In which case $x^n -1|x^m - 1$.
0
votes
1answer
4 views

Identify encryption scheme - possibly DSA or Diffie Hellman - only client shares key

I could use a hand in identifying the encryption scheme used in this scenario. (Its from source code known to work) There is a client connecting to a server such that: ...
0
votes
1answer
7 views

Transformation of an equation

How do you get from the left side to the right side in this equation? $$\frac{1+\sqrt{5}}{2} + 1 =\left(\frac{1+\sqrt{5}}{2}\right)^2$$
3
votes
2answers
53 views
+50

Limit superior of $\sum_{j=1}^n X_j$ with $\mathbf{P}[X_j = 1] = \mathbf{P}[X_j = -1] = 0.5$

This is Exercise 2.3.1 from Achim Klenke: »Probability Theory — A Comprehensive Course«. Exercise: Let $(X_n)_{n\in N}$ be an independent family of $\mathrm{Rad}_{1/2}$ random variables (i.e., ...
0
votes
0answers
7 views

Are $((0,1]\cap\mathbb Q)\times\mathbb Q$ and $\mathbb Q \times([0,1)\cap\mathbb Q)$ order isomorphic?

The ordering here isn't specified but I assume it's lexicographic. I think the answer is yes. My reasoning is the following: Neither sets have a least or a greatest element because the second ...
0
votes
0answers
37 views

Integer Factorization: Possible progress

I build an algorithm for solving Integer Factorization problem when the number to factor is selected by multiplication of two prime numbers with the same bit count. It can factor RSA1024 within ...
0
votes
0answers
4 views
0
votes
1answer
11 views

Is there a subset of natural numbers with a special property

Let set $A$ be an infinite big subset of the set $\mathbb{N}$ (set of natural numbers),it is not equal to $\mathbb{N}$ and it has the following property: For every $a$ that is not from the set $A$ ...
0
votes
1answer
11 views

Which one of given set is connected…

Which of the following are connected? (Notation: $c(a, r) =\{(x, y) \in\mathbb R^2: (x-a)^2+(y-b)^2=r^2\}$) $c(0,1) \cup c(0,2)$ $c(0,1) \cup c(1,3)$ $c(0,1) \cup c(1,1)$ $c(0,1) \cup c(2,1)$
0
votes
1answer
15 views

A connection between a matrix norm and a related matrix's largest eigen-value

I have been asked to prove that for $A\in M_n(\mathbb{C})$, with $||A||:=\sup_{x\in\mathbb{C}^n,|x|=1}|Ax|$, $$||A||=\sqrt{\lambda}$$ where $\lambda$ is the eigen value of largest modulus of $A^*A$. ...
0
votes
0answers
7 views

Using Intermediate value theorem and Rolle's theorem

Find how many solutions $2\ln x+2x^2+7=0$ has. Define: $f(x)=2\ln x+2x^2+7$, derive it and equate to $0$: $f'(x)=0 \\ 2+4x^2=0$ The discriminant is negative so there are no solutions, so from ...
6
votes
3answers
47 views

Proof: if p is prime, and 0<k<p then p divides ${p \choose k}$

Question : IF p is prime, and 0< k< p show that $ p | {p \choose k}$ ${p \choose k}$ can be rewritten as: $${p(p-1)(p-2)... (p-(k-1))(p-k)! \over (p-k)! k(k-1)(k-2)...3.2.1}$$ Now the (p-k)! ...
0
votes
1answer
24 views
+50

Transformation Ricker equation

The classical Ricker equation for modelling density-dependent population growth is: $N_{t+1} = a_t * e^{r * \left(1-\frac{N_t}{k}\right)}$ where $N_t$ is the initial number of individuals (starting ...
50
votes
21answers
4k views

What is your favorite application of the Pigeonhole Principle?

The pigeonhole principle states that if $n$ items are put into $m$ "pigeonholes" with $n > m$, then at least one pigeonhole must contain more than one item. I'd like to see your favorite ...
3
votes
1answer
30 views

Integer solutions to a two variable equation.

For $m, n \in \mathbb{Z}$, show the only integer solutions to $f(m,n) = \displaystyle \frac{3^m(2^n+1)-2^{m+n}}{2^{m+n}-3^{m+1}}$ are $f(1, 2) = -7$, $f(0, 1) = -1$, and $f(0, 2) = 1$. More ...
-1
votes
0answers
26 views

Radius of Convergence of Complex Series [on hold]

Please help me with this question, I've tried using D'Alembert Ratio but I don't understand when theres the complex z involved, thanks. Find the radius of convergence of: $$\begin{align}a)\ \ \ \ ...
1
vote
1answer
9 views

Proof that the finite product of nonempty sets is nonempty without axiom of choice from ZF [duplicate]

How do you prove that for $X_{i} \neq \emptyset$, $i \in \{1,...,n\}$ that $\prod_{i=1}^{n} X_{i} \neq \emptyset$ only using the ZF axioms but not the Axiom of Choice? I would like to see a rigorous ...
5
votes
1answer
67 views

Paul Erdős showed a simple estimate for $\pi(x) \ge \frac{1}{2}\log_2 x$; is it possible to tweak his argument to improve the estimate?

Paul Erdős gave a simple argument to show that $\pi(x) \ge \dfrac{1}{2}\log_2 x$. Is it possible to tweak the argument and get a better estimate? I am wondering how good an estimate for $\pi(x)$ can ...
0
votes
1answer
9 views

Proving that if $A\leq_c B$ then $\chi(A)\leq_o \chi(B)$

By Hartog's Theorem we knoe that for every set $A$ There is a definite operation $\chi(A)$ which associates with each set $A$, a well ordered set $\chi(A) = (h(A),{\leq_{\chi(A)}}),$ such that $h(A) ...
0
votes
0answers
6 views

Probability question involving drawing balls from an urn

Suppose there's an urn containing $r$ red balls and $b$ blue balls. At each trial, I'm drawing a ball at random from the urn, without replacement. Let $R$ denote the event of drawing a red ball, and ...
0
votes
0answers
5 views

Probability Theory - Showing something generates the Borel $\sigma$-algebra

If I have $\{(x,y] : x,y \in (0,1] \}$, how would I show that this generates the Borel $\sigma$-algebra on $(0,1]$? So we are just showing that we can form open sets through countable union, countable ...
1
vote
2answers
20 views

Using the distributive property to factor $(5^{-1}\cdot 5^x - 5^x - 5\cdot 5^x + 5^2\cdot 5^x)$

I can't seem to understand the distributive property. Take this: $$ 5^{-1}\cdot 5^x - 5^x - 5\cdot 5^x + 5^2\cdot 5^x$$ becoming this: $$ 5^x\left(\frac 15 - 1 - 5 +25\right) $$ Help? :D
2
votes
2answers
30 views

Is the composition function again in $L^2[a,b]$

Let $f \in L^2[a,b]$. 1- In what condition(s) on a function $g:[a,b]\rightarrow [a,b]$ we can get $$f \circ g \in L^2[a,b]?$$ 2- In what condition(s) on $g:[a,b]\rightarrow [a,b]$, the operator ...
1
vote
1answer
337 views

Prove that centroid, orthocenter and circumcenter are collinear

Prove that centroid, orthocenter and circumcenter are in the ratio 2:1.!! my attempt.. I could prove they are in the ratio 2:1.Assuming that they are collinear.But couldn't prove that they are ...
12
votes
2answers
208 views
+50

Is it possible to uniquly number faces of a hexagonal grid with consecutive numbers

You have a grid of regular hexagons. The aim of the game is to have each hex contain the numbers 1-6 on its edges. Each edge must also be connected to another edge that has a value one higher and ...
1
vote
0answers
14 views

How many solutions exist for a matrix equation $A^2=I$?

Let $A$ be a square matrix of order three or two, and $I$ be a unit matrix. How many solutions are possible for the equation $$A^2=I$$? In case the solutions are infinite, or very large, how do I ...
1
vote
0answers
20 views

Help me to solve this matrix problem, please.

Show that LHS = $$\begin{vmatrix}a_1+b_1t & a_2+b_2t & a_3+b_3t \\ a_1t+b_1 & a_2t+b_2 & a_3t+b_3 \\c_1 & c_2 & c_3 \\\end{vmatrix}$$ RHS = (1-t^2) ...
0
votes
0answers
18 views

Proof that there is no identity to integral operation on any set of functions

The statement is: Let $f\in F, f:x\mapsto f(x)$ be a function($F$ contains sufficiently non-trivial functions). Then $\not\exists I\in F$, so that $$\int_{-\infty}^\infty If=f(0)$$ What I am implying: ...
1
vote
3answers
28 views

Solve diophantine equation using modular arithemtic

Solve for integers, $x, y$ $4258x+147y=369 \implies 4258x \equiv 369 \pmod{147}$ I got this question from SE, but I want to try this approach. I suppose we will find the inverse modulus of $4258 ...
0
votes
3answers
31 views

Will someone explain this polynomial regression equation?

I am in high school and I need to write a program that does polynomial regression to any degree on a set of data for a personal project. I think that this Wikipedia Article has the equation that I ...
3
votes
0answers
78 views

Can the cube of every perfect number be written as the sum of three cubes?

I found an amazing conjecture: the cube of every perfect number can be written as the sum of three positive cubes. The equation is $$x^3+y^3+z^3=\sigma^3$$ where $\sigma$ is a perfectnumber (well it ...
4
votes
0answers
53 views
+50

How to find the inverse arc in the configuration space

The following Figure shows the function from configuration space (Torus) to operational space (Annulus). There is a naturally defined continuous function from configuration space $(\theta_A, ...
0
votes
1answer
21 views

Plotting Step Functions - connecting the steps?

How does one choose whether or not to draw vertical lines connecting the steps of a step function? Let's take the cdf of some discrete random variable as an example. My intuition tells me to graph ...

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