0
votes
0answers
7 views

Calculate and Forecast Churn Rate of Members

I need to estimate the total number of times users logged in to an app, cumulatively at the end of a year. If I have a mobile app, and 100,000 people download it. And they only use it once, then the ...
1
vote
0answers
18 views

Some questions concerning continuity and relations

A lot of equivalent conditions for functions between topological spaces $$ X\overset f\longrightarrow Y $$ is proved on this site. Here some of them formulated from the perspective of 'relations': ...
0
votes
1answer
35 views
1
vote
2answers
30 views

Naive proof that $\sum_{n=1}^{N-1}\cos(2\pi\frac{n}{N})=-1$ [duplicate]

As part of a larger proof, I must show that: $$\sum_{n=1}^{N-1}\cos(2\pi\frac{n}{N})=-1$$ I have thought about this but can't figure out how to get my hands on the value since I don't know any ...
-2
votes
0answers
14 views

use logical way to calculate the least percentage [on hold]

If 70 per cent. have lost an eye, 75 per cent. an ear, 80 per cent. an arm, 85 per cent. a leg q1: what is the least percentage lost all four q2: what is the least percentage lost one of them q3 what ...
-2
votes
0answers
16 views

Impuls after collision [on hold]

I am trying to understand collision and collision response for two spheres. This is my current knowledge background Collision Response The linear momentum is no problem. But I am stuck with the ...
0
votes
2answers
18 views

Proportionals and squares

Let $a,b,c,d \in \mathbb{Z^+}$ and $a:b::c:d$ and $ac$ is a square. Can we prove that $bd$ is a square too?
0
votes
2answers
40 views

A identity relating a infinite series and a definite integral

Prove that, $$ \sum_{n=1}^{\infty} \frac{1}{n^n} = \int_{0}^{1} x^{-x}dx$$ I made no significant progress, I'm looking for hint/ideas to approach this problem. Thanks!
0
votes
0answers
22 views

Question on notation - real analysis

"There exists an open interval J containing a such that f(x) > 0 and g(x) > 0 for x in J \ {a}." What does the notation J \ {a} mean? Thanks
0
votes
1answer
24 views

What is the Singular Value Decomposition for the Zero Matrix?

I am interested in the singular value decomposition of a matrix: $\mathbf{M} = \mathbf{U} \mathbf{S} \mathbf{V}^T$. Suppose $\mathbf{M} = \mathbf{0}$ (zero matrix) and square. Clearly, $\mathbf{S} = ...
1
vote
2answers
24 views

Prove $\sum_{cyc} \frac{\sqrt{xy}}{\sqrt{xy+z}}\le\frac{3}{2}$ if $x+y+z=1$

if $x,y,z$ are positive real numbers and $x+y+z=1$ Prove:$$\sum_{cyc} \frac{\sqrt{xy}}{\sqrt{xy+z}}\le\frac{3}{2}$$ Additional info:I'm looking for solutions and hint that using Cauchy-Schwartz ...
1
vote
2answers
31 views

Why can I not write every $N \times M$ matrix as multiplication of an $N \times 1$ and a $1 \times M$ matrix?

My intuition says I simply can't express $N \times M$ independent variables in terms of $N+M$ variables but how can I show that?
0
votes
1answer
24 views

Determinants of Matrices det(4A) equals?

Suppose A is a 4 x 4 matrix such that det(A) = 1/64. What will det(4A^-1)^T be equal to? Here's my thinking, det(A^T) = det(A) I has no effect on the determinant. And det(A^-1) = 1/det(A) so ...
2
votes
1answer
96 views

Proof that group is commutative if every element is its inverse (feedback wanted)

This is one of my first proofs about groups. Please feed back and criticise in every way (including style & language). Axiom names (see Wikipedia) are italicised. $e$ denotes the identity element. ...
2
votes
4answers
35 views

If $n > 0$ is an even composite integer, then $\varphi(n)$ is even? [duplicate]

If $n > 0$ is an even composite integer, is the corresponding totient $\varphi(n)$ also even? I found that it is not the case for $n$ odd; for $\varphi(15) = 8$.
0
votes
2answers
24 views

Matrix in $\mathbb{Z}_5$

Let $A=\begin{bmatrix}3&2\\3&3\end{bmatrix} \in M_2(\mathbb{Z}_5).$ Then if I calculate $A^{105}$ like $105 \equiv 0 \pmod 5$ , $A^{105} = Id_2$ ? Thank you.
1
vote
1answer
10 views

Power of a matrix, given its jordan form

Can someone please explain how to find the power of a matrix $A$, given $A=MJM^{-1}$ where the matrix $J$ is in the Jordan canonical form? Or else please explain how to find the powers of a matrix ...
0
votes
0answers
22 views

Teacher/student exam assignment matching problem - equivalent problem?

I have a sort of matching problem. I am wondering if you know if this problem reduces to a familiar one. It arises from my friend's job, and something we were wondering about this morning on the ...
1
vote
0answers
23 views

A combinatorial enumeration problem on graph

Let $G$ be a complete graph of order $n$, we now delete $i$ edges from it, then how many complete subgraphs are there with order $m$ in the rest graph? (You can assume $m\ll n$ and $i\ll m$ if ...
1
vote
1answer
24 views

How much above the cost price must he marks his goods above the cost price to make a profit of 17%

A businessman allows a discount of $10$% on written price. How much above the cost price must he mark his goods to make a profit of $17$% ? note: $x\times\frac{90}{100}=x\times\frac{117}{100}$ ...
0
votes
3answers
38 views

Equivalence class for the relation $R$ (work shown)

For an integer $n\in \mathbb{N}$ define $P(n) = \{p : p \mid n \text{, where $p$ is a prime} \}$. For example $P(12)=\{2,3\} $ and $P(1)=\emptyset$. Question: Consider the relation $R$ on ...
0
votes
0answers
11 views

Which matrix norm gives the minimal variation of eigenvalues?

This is a follow-up of this question. The original question is intentionally as general as possible, because I was interested in the most general possible answer. I am now trying to understand its ...
2
votes
1answer
24 views

Using Taylor series with remainder

Arfken and Weber expand $(1+n^{-2})^{-1}$ as $$(1+n^{-2})^{-1}=1-n^{-2}+n^{-4}-\frac{n^{-6}}{1+n^{-2}}$$ However, if I use Taylor/Maclaurin expansion $$(1+x)^m=1+mx+ \frac{m(m-1)}{2!} x^2 + \cdots + ...
-2
votes
0answers
17 views

Riemann's sum inequality problem [on hold]

Iam having touble with a certain question on my assignment. I dont know how to replicate the math symbols on this site so I have jst put down a link to the full assignment: ...
1
vote
0answers
13 views

English translation of Minkowski's Geometry of Numbers

Is there an English translation of Minkowski's Geometry of Numbers? I have searched it but have found nothing.
1
vote
1answer
10 views

Convex Combination of 3 point in R2 and Triangle

I am new to convex combination, and I am quite amazed by some easy result. I know that convex combination of 2 points($P_1P_2$) in $R^2$ is all points in the line segment $P_1P_2$. And then I see a ...
0
votes
0answers
24 views

Subring of $\mathbb{Z}[i]$ and an infinite set $X$ such that $\exists x \forall y \in X \,\,x^2\mid y^2$ but $\forall x \forall y \,\,x \not\mid y$

This is a question derived from A subring of the ring of Gaussian integers such that $a^2 \mid b^2$ does not lead to $a\mid b$ in infinitely many such cases. Is there a subring $R$ of Gaussian ...
1
vote
1answer
9 views

How to compute the group inverse of $M+aI_n-\frac{a}{b}J_{n\times n}$?

For a square matrix $M$, the group inverse of $M$, denoted by $M^\#$, is the unique matrix $X$ such that $MXM=M$, $XMX=X$ and $MX=XM$. Given an $n \times n$ matrix $M$, let $I_n$ denote the identity ...
0
votes
1answer
26 views

In how many days $Y$ alone finish the work

$X$ and $Y$ do a piece of work in $30$ days .They work together for $6$ days and then $X$ quits and $Y$ finishes the work in $32$ more days.In how many days $Y$ alone finish the work. note:$X+Y=1/30$ ...
1
vote
2answers
40 views

Proving that $A$ is diagonalizable

Let $A\in\mathbb{C}^{n\times n}$ be a $n$ by $n$ matrix such that $A^k = I$ for some natural number $k$. Find a nonzero annihilating polynomial of A and prove that A is diagonalizable. I will say ...
0
votes
1answer
44 views

Integrate $\int \sqrt{1+\cos(t/2)} dt$

I am looking for a neat and smart way to do this. I tried by substituting $u = 1+\cos(t/2)$ But I think its not the simplest way
3
votes
2answers
30 views

If $a_n\to0$, there exists $\pm$ such that $\sum\limits_n\pm a_n$ converges [duplicate]

Our Analysis I lecturer in his last lecture for the course gave us a problem to think about. I've been thinking about it for a while and has been bothering me for some time. It looks like a ...
1
vote
1answer
14 views

Showing unitary similarity of these two matrices

Let $A \in B(H)$ for a Hilbert space $H$, and $\alpha \in \sigma_{p}(A)$, the point spectrum of $A$. Suppose ker$(\alpha I-A)$ is not a reducing subspace of $A$ then $A$ has the following matrix ...
2
votes
0answers
26 views

Probability: the average times to make all the balls the same color

Suppose there are n balls with different colors with each other in a bag. In one loop, One take two balls in sequence out of the bag and replace them with two balls with the same color of the first ...
1
vote
0answers
32 views

Solution of a linear system equation when $A^2=\mathcal I_n$ [on hold]

Please help me to find an answer for this question. Let $A$ be an $n\times n$- matrix such that $A^2=\mathcal I_n$ and $y$ an $n$-dimensional vector. The Linear system equation $Ax=y$ has how many ...
2
votes
1answer
37 views

$\int \sqrt{1+\sin ^2 x} dx$ an elliptic integral?

It seems to be an elliptic integral of the second kind, but when $k=i$? This is going by the definition that $E(\theta,k)=\int_{0}^{\theta} \sqrt{1-k^2 \sin^2x}dx$. That seems a bit off. Or is this ...
0
votes
2answers
38 views

How to graph functions without a calculator?

How do you graph a function such as $$f(x)=\frac{x^2+3x+2}{x+1}$$ and find its limits $\lim_{x\to-1^-}f(x)$, $\lim_{x\to-1^+}f(x)$, $\lim_{x\to-1}f(x)$? Thank you!
0
votes
0answers
11 views

How many combinations can be made with this scenario? [on hold]

So I wrote a program that uses arrays. I want to know how many combinations it could make. Array one has 18 options. Array two has 15 options. Array three has 15 options. Array four has 30 ...
1
vote
2answers
27 views

Prove $a^3+b^3+c^3\ge a^2+b^2+c^2$ if $ab+bc+ca\le 3abc$

if $a,b,c$ are positive real numbers and $ab+bc+ca\le 3abc$ Prove:$$a^3+b^3+c^3\ge a^2+b^2+c^2$$ Additional info:I'm looking for solutions and hint that using Cauchy-Schwartz and AM-GM because ...
0
votes
1answer
22 views

Proof of trigonometric identity sin(A+B)=sinAcosB + cosAsinB

All the proofs I've seen are geometrical, assuming that A+B is less than 90 degrees. How can you prove this identity for A+B greater than 90 degrees, or more generally, any arbitrary value?
2
votes
1answer
58 views

List of functions $f(cx) = C\cdot f(x)$

I was looking for some complex functions f(x), which satisfies the condition: $$\exists (c, C) \in \Bbb C^2 \backslash\{(1,1)\}, \forall x \in \Bbb C, f(cx) = C\cdot f(x)$$ Till now I have got ...
-1
votes
0answers
12 views

Prove this is markov chain [on hold]

Let $Y_i = i$ with probability $p_i$. Let $X_i = max[Y_i , Y_i-1, ...]. $ We have $i={0,1,2,3,4,...}. $ Prove that $X_i$ is a Markov chain and write down its matrix.
0
votes
1answer
38 views

A question on short exact sequences.

The following is an excerpt from Atiyah-Macdonald on short exact sequences. I don't understand the part where the author says "Then $d(x'')$ is defined to be the image of $y'$ in Coker ($f'$)". Is ...
1
vote
0answers
22 views

Calculating the error [on hold]

I am doing corrections to controls for a UI application. Here is the requirement... The expected width and height as per specification is 490 x 340 The visual studios(where i develop User ...
1
vote
0answers
41 views

What do double vertical lines mean?

I am reading a paper on computer graphic and having hard time to understand this formula: What is the double vertical lines means? Do they always go with power of 2? If I want to learn further ...
2
votes
1answer
38 views

Why does the Fourier sine series of $x^2$ on $[0,l]$ converge to 0?

When expanding, for example, $x^2$ on $[0,l]$ as a sine series, we get $f(x) = \sum_1^{\infty}b_n sin(\frac{n\pi x}{l})$ If we plug in $x=l$ to this expansion, we get $f(x)=0$. Why aren't we getting ...
0
votes
1answer
27 views

Recursive definition of the set of odd numbers

Show that the following is another recursive definition of the set ODD (keep in mind you’ll need say something about even numbers too): Rule 1: 1 and 3 are in ODD. Rule 2: If x is in ODD, then so is x ...
0
votes
0answers
20 views

How many sequence of functions are there to converge pointwise to a given function on $E\subseteq \mathbb R$?

yesterday night, I was studying sequence of functions in $\mathbb R$ and then this question came to mind. When a sequence of real valued function is given, we can find out it's pointwise limit ...
2
votes
2answers
35 views

Is there any infinite quantity small enough to be affected by finite changes?

Hilbert's paradox of the Grand Hotel shows us, among other useful things, that the cardinality of any infinite set is a quantity equal to n more than itself for any finite n. I am interested in ...
0
votes
1answer
46 views

Identities involving the floor function

Are either of these statements false? if so what is the counter example? $⌊x − 2⌋ = ⌊x⌋ − 2$ or for any odd integer n, $⌊(n^2/4) + 1⌋ = (n^2+3)/4$ also I'm struggling to make a proof of either if ...

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