0
votes
0answers
7 views

doubt with sequence and series question

show that the series ((1+x)/(1-x))^3=1+ summation n=1 to infinity (4n^2+2)x^n I tried with the partial frationaising the expression that gives me -6/(x-1) -12/(x-1)^2 -8/(x-1)^3 -1 how to ...
0
votes
0answers
4 views

Showing that the $\lim s_n\neq2/3$ when $s_n=\dfrac{2}{3n}$

I'm trying to verify if I what I did to show that limit does not exist is valid using the negation of the definition: $\exists \epsilon>0, \forall N \in \mathbb{N}$ such that $n>N \implies ...
1
vote
0answers
11 views

Constants for the Domain and Range of a Function

We're currently learning about this in high school, and I was looking ahead in the textbook. Since we just had finals and we're on break, I wasn't able to ask anyone about this problem. It would be ...
0
votes
3answers
31 views

How is a metric space a topological space?

I learned about metric spaces and topological spaces but I don't see how they correlate. How does a metric space follow the properties of a topological space.
2
votes
0answers
21 views

Learning concepts in mathematics

Apologies for the soft question, but I was wondering whether it is a good idea, in mathematics, to learn/study things simply for the sake of studying it. A very good example comes from category ...
0
votes
1answer
43 views

Spirit of the holidays.

My teacher gave us this homework problem to do over the break, and said the solution should be incredibly obvious, but I can't seem to figure out where I should start...can anyone help?! ...
1
vote
1answer
22 views

Determining a basis for a space of polynomials

Determine a basis from the following set of second degree polynomials. Does this basis span the space of the second degree polynomials? What is the dimension of the (sub)space that it spans? ...
1
vote
0answers
9 views

Sequence with Conditions and Possible Answers

Given that $a_1$, $a_2$, $a_3$, . . . $a_n$ is a sequence of positive real numbers such that: For all positive integers $m$ and $n$, $a_{mn}$ = $a_m$$a_n$, AND there exists a positive real number $B$ ...
0
votes
2answers
22 views

Pre Algebra book Recommendation

Can anyone suggest pre algebra book for beginner. Would like to see something more than the worksheets offered online. I would prefer a book which would teach strong fundamentals concepts about ...
1
vote
1answer
14 views

Can a Condorcet winner be generally dispreferred on an individual basis?

Suppose that a Condorcet winner exists in an election. Certainly it is possible that an individual voter prefers some other candidates to the Condorcet winner. They might even prefer most, or all, ...
0
votes
0answers
15 views

Elliptic curve- Component of point

If we have $y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6$,then if $P_1=P_2$ we have that $$\lambda=\frac{3x_1^2+2a_2x_1+a_4-a_1y}{2y_1+a_1x_1+a_3}, v=\frac{-x_1^3+a_4x_1+2a_6-2_3y_1}{2y_1+a_1x_1+a_3}$$ and then ...
1
vote
0answers
16 views

Unique Path lifting of covering map

Let $p:E\rightarrow B$ be a covering map (in particular $p$ is a fiber bundle with discrete fiber). We want to prove the following: Given a commuting diagram of the following form: $\{0\}\rightarrow ...
1
vote
4answers
37 views

Discrete Mathematics: $mn + 2m + 2n + 2 = n$ proof of uniqueness of $m$, $\forall n \in \mathbb{Z}$

Prove: There exists a unique integer $m$ such that for every integer $n$: $$mn + 2m + 2n + 2 = n$$ However I am not sure if my proof is correct. How do I prove uniqueness of $m$? I prove it by ...
0
votes
2answers
16 views

How to solve this inequality and sketch this graph?

I tried squaring and simplifying and got a solution set different to the one it says in the answer so I'm not really sure what I'm doing wrong. Also not sure how to sketch the graph any help?
3
votes
6answers
38 views

Proving the convergence of $\sum_{n\geq 2}\log\left(1-\frac{1}{n^2}\right)$

How to prove that the series $$\sum_{n\geq 2}\log\left(1-\frac{1}{n^2}\right)$$ is convergent? What about finding the sum? My attempt: $$\ln (1-1/n^2)= \ln(n-1) -2\ln n + \ln(n+1)$$ The term in the ...
2
votes
0answers
16 views

Normal Cone of $\mathbb{R}^n_+$ and $S^n$?

I'm trying to solve the problem $\min_x \{f(x) + \delta_X(x)\}$ where $f$ is a differentiable function and $\delta$ is the indicator function $\delta_X(x) = \begin{array}{l}0, x \in X \\ ...
1
vote
1answer
19 views

representation for Banach algebra

How we can represent any Banach algebra as a subspace or Subalgebra of Cb(X)?( in isometrically isomorphic concept)
2
votes
3answers
42 views

Tips on constructing a proof by induction.

So right now I'm working on a discrete mathematics course and I've been having a bit of trouble figuring out how to prove certain equations using mathematical induction. I have very little trouble ...
3
votes
2answers
29 views

How do I prove the circumference of the Koch snowflake is divergent?

How do I prove that the circumference of the Koch snowflake is divergent? Let's say that the line in the first picture has a lenght of $3cm$. Since the middle part ($1cm$) gets replaced with a ...
0
votes
0answers
27 views

Example closed 1-form on $\mathbb{R}^3 -\{0\}$

It's maybe a silly question but I was wondering if there exists a closed 1-form $\alpha$ on the manifold $\mathbb{R}^3 -\{0\}$ of the form $$\frac{1}{x^2+y^2+z^2}(adx+bdy+cdz)$$ with $a, b$ and $c$ ...
0
votes
0answers
12 views

Functorial Properties Preserved by Natural Transformation

This question was born from (and is in a sense a continuation of) this one, about functorial properties preserved by natural isomorphisms. What functorial properties are preserved by natural ...
2
votes
1answer
33 views

A matrix version of L'Hopital's Rule?

Is there a version of L'Hopital's Rule for matrix calculus? For example: let $A$ be a symmetric $n\times n$ positive definite matrix and $b$ be an $n\times 1$ vector. As $b$ converges to $0_{n\times ...
-2
votes
1answer
25 views

What are some applications of real analysis? [duplicate]

What are some applications of real analysis? Can someone post a simple example of how real analysis can solve such problems?
4
votes
4answers
30 views

Four balls with differnt colors in a box, how many times do I need to pick to see all four colors?

I have one white ball, one yellow ball, one red ball, one black ball. I put the four balls in a nontransparent box. I pick a ball from the box to see its color and put it back to the box. Assuming ...
2
votes
1answer
35 views

Limit of a function

I am trying to find the limit (If it does exist) $\lim_{n\rightarrow\infty}\left(1-|\mathcal{X}|^{-\alpha n}\right)^{2^{nC}\left(1-|\mathcal{X}|^{-\alpha n}\right)}$, where $0<\alpha<1$, ...
2
votes
1answer
27 views

What is the ideal of a point in algebraic geometry?

I found a problem as follows: Find the ideal of a point $z$, denoted by $\mathfrak j_z\subset\mathbb Q[X,Y]$, and its conjugates in $\mathbb C^2$ as $z=(\sqrt{2},\sqrt{3})$. I tried to Google but ...
2
votes
2answers
31 views

Evaluate fourier coefficient of $f(t)=t$.

Evaluate the Fourier coefficient of $f(t)=t$. $$\hat{f}(n) = \frac{1}{2\pi}\int_0^{2\pi} te^{-int}dt$$ I'd be glad for help with this calculation. My integration skills need an improvement. My ...
0
votes
0answers
9 views

How to choose a proper numerical optimisation method

Given a problem in numerical analysis in finance/econometrics, how to decide whther to choose Monte Carlo, Newton Raphson , Finite Difference , Gradient descent? I had this silly misconception that ...
1
vote
1answer
24 views

Prove that $(\{[0],[a],[2a],…[(b-1)a]\},+)$ is a subgroup of $(\mathbb{Z}_n,+)$.

$n$ is not a prime number. Then $n=ab$ for some $a,b\in \mathbb{N}$ : $1<a<n$ and $1<b<n$. It seems that I did the proof, but I'm not sure that everything is correct. My solution: Suppose ...
2
votes
1answer
36 views

To show that $\operatorname{Rank}(\mathbf{A}-\mathbf{I})=\operatorname{Nullity}(\mathbf{A})$

Problem is: Let $\mathbf{A}$ be $n\times n$ matrix with real entries such that $\mathbf{A}^{2} = \mathbf{A}$. If $\mathbf{I}$ denotes the identity matrix, then how do I prove the result: ...
-4
votes
3answers
53 views

How can you use the digits 2 0 1 5 to equal 28 [on hold]

you can use the numbers only once and have to use them all.
0
votes
2answers
40 views

Solve $x''(t) x(t)=x^4(t)$ with $x'(0)=\frac{1}{\sqrt 2},x(0)=0$.

Solve the Cauchy problem: $$x''(t) x(t)=x^4(t)$$ with $x'(0)=\frac{1}{\sqrt 2},x(0)=0$. I would appreciate some help with this problem. Thank you very much.
1
vote
1answer
30 views

if $f(z),\overline {f(z)}$ are analytic then they are constant

I'm trying to prove this "theorem": if $f(z),\overline {f(z)}$ are analytic in some open set $\Omega \subseteq \mathbb C$, then $f(z)$ is a constant. Hint: Use Cauchy-Riemann equations to show that ...
0
votes
1answer
17 views

Mean-value formulas

From PDE Evans, 2nd edition, pages 25-26. THEOREM 2 (Mean Value Formulas for Laplace's equation). If $u \in C^2(U)$ is harmonic, then $$u(x)=\def\avint{\mathop{\,\rlap{-}\!\!\int}\nolimits} ...
1
vote
0answers
18 views

System of linear diophantine modular inequalities

How can we best find a numerical solution to a system of $m\ge2$ linear diophantine modular inequalities $$\big((a^j x+b_j)\bmod n\big)<c\;\text{ for }1\le j\le m$$ where $x$ is the only unknown, ...
1
vote
1answer
30 views

Powers of adjacency matrix doesn't seem to correspond to observed number of paths on graph

I would really appreciate some help on this! $A^n$ represents $n^{th}$ power of the adjacency matrix of a graph. I keep reading that the $A^n_{ij}$ entry equals "the number of paths of length n ...
3
votes
0answers
32 views

Integers Placed On A Circle

My problem is such: On a circle there are $9$ distinct positive integers aranced in such a way that the product of two non-adjacent numbers in the circle is a multiple of $n$ and the product of any ...
1
vote
1answer
21 views

Integral of the convolution of two functions: Manipulating the integrals and their arguments

There is this proof for the integral of convolution between two functions: $$\begin{align}\int_{-\infty}^{\infty} (f*g)(x)dx&=\int_{-\infty}^{\infty}\left [ ...
3
votes
2answers
40 views

Why does the radius come before the angle?

Based on my understanding, when delineating two variables (for a coordinate system or otherwise) convention is to label the 'independent variable' first, then the 'dependent variable'. So for a ...
1
vote
2answers
40 views

What is the probability of sinking ships in a simplified game of battleship?

Consider a a simplified game of battleship. We are given a 4x4 board on which we can place 2 pieces. One destroyer which is a 1 × 2 squares and a submarine that is 1 × 3 squares . The pieces are ...
1
vote
1answer
57 views

Philosophers who became mathematicians — how did they do it? And who were they?

(I hope this is not too personal. If you want to get to the point scroll down to the end, where my questions are.) I'm a philosopher who's been -- gradually -- coming around to mathematics. I have ...
0
votes
1answer
40 views

If $(a,b)=1$ and $p \mid a^{2}+b^{2}$ why can one assume that $|a| < \frac{p}{2}$

There is a part of Euler's infinite descent proof I can't seam to get; If $(a,b)=1$ and $p \mid a^2+b^2$ why can one assume that $|a| < \frac{p}{2}$ and $|b| < \frac{p}{2}$ ?
0
votes
1answer
24 views

$P(\min(X_1,\dots,X_n) > t) = P(X_1>t,\dots, X_n>t)$

One step in the my solutions book shows... $P(\min(X_1,\ldots,X_n) > t) = P(X_1>t, \ldots, X_n>t)$, where $X_1, \ldots, X_n $ are independent and $X_j \sim \mathrm{Expo}(\lambda) $ Why is ...
-7
votes
0answers
24 views

Finding the sum [on hold]

How to find the sum of $\sum_{n=1}^{\infty}\frac{(-1)^n}{2^nn}$ ?
0
votes
3answers
25 views

If $T : F^{2 \times 2} \to F^{2\times 2}$ is $T(A) = PA$ for some fixed $2 \times 2$ matrix $P$, why is $\operatorname{tr} T = 2\operatorname{tr} P$?

I am asked to prove that if $T$ is a linear operator on the space of $2 \times 2$ matrices over a field $F$ such that $T(A) = PA$ for some fixed $2 \times 2$ matrix $P$, then ...
2
votes
0answers
24 views

about the ratio of the coefficients [duplicate]

Let $f$ be holomorphic on an open disk containing the unit circle, except in a pole $w$ on the unit circle. Let $\displaystyle \sum_{k=0}^{\infty} a_n z^n$ be its expansion. Show that ...
0
votes
4answers
55 views

How to prove that the sum of binomials equals $\begin{pmatrix}2n\\n\end{pmatrix}$ [duplicate]

I've stumbled upon this lemma a few times in my textbook: $$\sum_{k=0}^{n}\begin{pmatrix}n\\k\end{pmatrix}=\begin{pmatrix}2n\\n\end{pmatrix}$$ I've been trying to prove it, but I simply can't seem to ...
1
vote
0answers
33 views

How to simplify this inequality

I have the following inequality where $i$, $N$ and $p$ are constants, $j$ is a variable and $p_j$ is the chance that 'event' $j$ is happening: $$i\geq -pi+((1-p)\cdot \sum ^N _{j=0}(j\cdot p_j))+\sum ...
2
votes
4answers
70 views

Mathematical Christmas Anecdotes, Stories and Problems [on hold]

Since Christmas is coming soon, I'm curious if there are any math related christmas stories or anecdotes (no matter if real or fiction). Moreover, I would love to hear about some kind of christmas ...
0
votes
1answer
32 views

How many primes can be represented in JavaScript?

In JavaScript, the largest odd positive number representable is $2^{53}-1$. All integers between 1 and $2^{53}-1$ can be represented without loss of precision. How many prime numbers can be ...

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