0
votes
0answers
7 views

Are these definitions of a continuous random variable equivalent?

In a textbook I'm reading: A random variable $X$ is continuous if there exists a function $f_X$ such that $f_X(x) \le 0$ for all $x$, $\int_\infty^\infty f_X(x) dx = 1$ and for every $a \le b$, ...
0
votes
0answers
7 views

subtraction method

I have studied long ago different method for subtracting two numbers, for instance the borrow method or the Austrian method (I hope I am using the right names; I am not an english native speaker and ...
0
votes
0answers
6 views

Explication on how obtaining $\int \langle \nabla w, \nabla w\rangle = \lambda \int \langle w, w\rangle$

Could anyone is able to explain to me how to obtain $\int \langle \nabla w, \nabla w\rangle = \lambda \int \langle w, w\rangle$ related to user7530's comment in the question : Rayleigh quotient ...
0
votes
0answers
2 views

Separability and reduced tensor fields

Let $L/K$ be a finite field extension. I need to prove that if $L/K$ is separable, then $E\otimes_KL$ is reduced for every algebraic field extension $E/K$. I've readed that I need to use the ...
3
votes
0answers
20 views

Unable to remove the indeterminacy of a limit.

Evaluate $$\lim_{x\to 1} \frac{p}{1-x^p}-\frac{q}{1-x^q}$$ where $p,q$ are Natural Numbers. $$$$ I tried rationalization, but I wasn't able to get anywhere. I'm not being able to remove the ...
0
votes
3answers
7 views

Constructible $n$-gons

Let $\xi$ be the primitive root of unity. If $n=5$, then the minimal polynomial of $xi$ over rationals would have degree $5-1=2^2$ which is a fermat prime so $5$-gon is constructible. If $n=8$, ...
1
vote
1answer
8 views

Derivative of convolution type integral equation with respect to time

Consider the equation: $$x(t) = \int\limits_{0}^t \gamma^{t-s} u(s)ds$$ where $x, u$ are functions and $\gamma \in \mathbb{R}_{>0}$ I am trying to obtain the derivative of this equation with ...
0
votes
0answers
9 views

Total derivative using $(x,y)\mapsto f(x,y)$ notation

Related to What is the difference between $\frac{\mathrm{d}}{\mathrm{d}x}$ and $\frac{\partial}{\partial x}$? Is it possible to make sense of $(x,y)\mapsto f(x,y)$ when taking the total derivative ...
0
votes
0answers
8 views

Closed form for $\int\frac{\left((x + i) \beta\right)^\beta x^{\beta - 2}}{(x^2 + 1)^\beta} \exp\left(-\frac{\alpha}{A x}\right) \, \mathrm{d} x$

The following integral comes up in the solution of a differential equation when solved by Maple: $$ \begin{equation} \int\frac{\left((x + i) \beta\right)^\beta x^{\beta - 2}}{(x^2 + 1)^\beta} ...
0
votes
0answers
8 views

Do the paradoxes of naive set theory mean we shouldn't ever think of sets as collections?

I'm asking this because, it seems to me that all of my math professors employ naive set theory when they talk about sets. For example, in my algebra, analysis, and differential geometry classes, ...
1
vote
2answers
19 views

Find the arithmic series in the question

The situation in wich the questions are asked are the following: When gaining $32$m in depth the temperature gains $1°$C. when you are $25$m below the surface temperatures of $10°$ are measured. ...
0
votes
2answers
22 views

Prove the sequence of partial sums is monotonically increasing

Consider the series: $$\sum_{k=0}^{\infty}\frac{1}{k!}$$ Prove that the sequence of partial sums ($s_{n}=\sum_{k=0}^{n}\frac{1} {k!}$) $n>0$ is monotonically increasing. My approach: $$s_{1}= ...
1
vote
4answers
14 views

Prove that the $7$-th cyclotomic extension $\mathbb{Q}(\zeta_7)$ contains $\sqrt{-7}$

Prove that the $7$-th cyclotomic extension $\mathbb{Q}(\zeta_7)$ contains $\sqrt{-7}$ I thought that the definition of the $n$-th cyclotomic extension was: $\mathbb{Q}(\zeta_n)=\{\mathbb{Q}, ...
-1
votes
0answers
14 views

The map $S^n \to S^n \times S^n$ given by $x \to (x, -x)$ is a homotopy equivalence

I'm trying to get Homotopy chapter in my course of Topology I. I understood what is an Homotopy, but I'm having several difficulties in this problem, that I suppose is quite standard. Denote by ∆ = ...
0
votes
0answers
3 views

Concerning substitution and existential elimination in classic natural deduction using sequents

I am trying to prove $\exists x(P\lor Q)\vdash \exists x P \lor \exists x Q$, so I have: $$(1) \ \exists x (P \lor Q) \quad \mbox{[premise]} \\ (2) \ (P \lor Q)\lbrace u/x \rbrace \quad ...
-1
votes
3answers
43 views

Identifying $\sum\limits_{k=0}^\infty\frac{1}{k!}\frac{x^{k+6}}{k+6}$

I'm trying to prove this equality. $$\sum_{k=0}^\infty\frac{1}{k!}\frac{x^{k+6}}{(k+6)} {=} e^x(x^5-5x^4+20x^3-60x^2+120x-120)+120$$ posted by: http://math.stackexchange.com/q/832368
1
vote
1answer
9 views

square root commutes with multiplication for positive elements in a $C^*$ algebra?

Let $A$ be a unital $C^*$ algebra. If $z\in A$ is invertible, then so is $z^*$ and $z^*z$ and, furthermore, $z^*z$ is positive, so we can define using the functional calculus $|z|=\sqrt{z^*z}$. My ...
1
vote
3answers
25 views

Are there locally compact subsets of $\mathbb{R}$ which are not compact?

Are there locally compact subsets of $\mathbb{R}^n$ tht are not compact? A set $X \subset \mathbb{R}^n$ is compact if it is closed and bounded. Every infinite set $\{ x_n \}$ has a subsequence ...
2
votes
1answer
25 views

Is my proof correct that there are uncountably many sets of positive integers?

Let $\mathbb{N}$ be the set of natural numbers. Prove that $2^{\mathbb{N}}$ is uncountable. Proof: Suppose that $2^{\mathbb{N}}$ is countable then $2^{\mathbb{N}}=\{A_1, A_2, A_3,\dots\}$. We have to ...
0
votes
0answers
5 views

prove that the minimum number of trails in an odd graph is n/2

In my HW assignments I was asked to prove that If a graph G consists of only odd degree vertices, then the minimum number of trails that decompose it (without having any common edge between each two ...
0
votes
0answers
5 views

Solvability and representation of finite groups

Let $G$ be a finite solvable group, and let $G=G^{(0)}\unrhd G^{(1)}\unrhd...G^{(n)}=1$ be its derived series. Is it true that any irreducible representation of $G$ has dimension at most $n$?
0
votes
0answers
9 views

Convergence of a mixed distribution

Let $Y_n=Z+\sum_{i=1}^n \delta_{1/i^2}$, with $\delta$ a point mass and $L(Z)=N(0,\sigma^2)$. Show that $\lim_{n\to\infty} Y_n=Y$, where $L(Y)=N(\frac{\pi^2}{6},\sigma^2)$ The answer file uses ...
-1
votes
0answers
10 views

Inverse trigonometry problem4

Let $\tan^{-1}x + \tan^{-1}y=a+b\tan^{-1}(\frac{x+y}{1-xy})$ then what would be the sum of all possible values of a and b? Please explain how to solve this and these type of problems.
0
votes
0answers
5 views

Finishing off this Sturm-Liouville BVP

I'm looking at the Sturm-Liouville BVP $$\begin{cases} y'' + \lambda y = 0\\ y(0) + y'(0) = 0, y(1) = (0) \end{cases}.$$ For the case $\lambda > 0$ my working is as follows: Set $k^2 = \lambda$. ...
-1
votes
0answers
13 views

How do I describe this prime number sieve process with a simple equation or equations?

I have developed a new prime number sieve akin to Sieve of Eratosthenes and Sundaram. I can't describe it here due to space limits. What I am having a problem with is how to define the sieve and/or ...
0
votes
0answers
8 views

Convolution of matrix coefficients from inequivalent representations

Suppose $\delta_1,\delta_2$ are two inequivalent representations of a compact Lie group. Let $dy$ be the normalised Haar measure and define convolution for functions $f,g:G\rightarrow \mathbb{C}$ ...
0
votes
0answers
18 views

Identifying the iteration Scheme

The iteration $x_{n+1} = \frac{1}{2} (x_{n} + \frac{2}{x_{n}}) , n \ge 0$ for a given $x_{0} \ne 0$ is an instance of fixed point iteration for $f(x) = x^2 - 2$. Newton's method for $f(x) = x^2 ...
0
votes
0answers
12 views

composition with biholomorphism

Let $f:G_1 \rightarrow G_2$ be a local biholomorphism and $g:G_2 \rightarrow G_1 $ continious such that $(f\circ g)(z)=z$ Proof that $g$ is a biholomorphism I know that $g$ is biholomorphic if $g' ...
0
votes
1answer
3 views

Defining a subsegment from an ordered set using mathematical notation

I have a ordered set of time stamps events $T$ indexed by $t$;I also have a set of observations at each time stamps. $O_{T}$ is my set of observations indexed by $O_t$ where $O_t$ is a observation at ...
0
votes
2answers
26 views

Inverse trigonometry problem3

Let $\cos^{-1}x - \tan^{-1}\frac{\sqrt{1-x^2}}{x}=\pi $,then what are all the possible values of $x$? Please explain in detail.
1
vote
0answers
9 views

Logarithmic Spiral

Let's assume a particle starts on an equilateral triangle of side length "A" with some constant speed u. The particle goes on a logarithmic spiral around the centroid. Find the distance covered by the ...
1
vote
0answers
7 views

For a densely defined symmetric operator $A$, is $A^2$ also densely defined?

Let $A : D(A) \to H$ be a possibly unbounded, densely defined symmetric operator on a Hilbert space $H$ ($A$ being symmetric means that $(\varphi, A\psi) = (A\varphi, \psi)$ for all $\varphi, \psi \in ...
0
votes
3answers
25 views

How do I write n ∈ all of the known number sets

If I want to say that n ∈ all of the known number sets do I have to write n ∈ $\mathbb{N} , \mathbb{Z} , \mathbb{Q} , \mathbb{R} , \mathbb{C}$ or should I just leave it blank?
0
votes
0answers
10 views

Why is (x-xi)^n still a linear factor (Partial Fraction Decomposition)?

When we perform a Partial Fraction Decomposition and one of the solutions of the denominator is a multiple solution (let's say quadratic), we write: $$\frac{A_{1}}{(x-x_{i})} + ...
0
votes
2answers
17 views

Why aren't there uncountably many disjoint open intervals of $\mathbb{R}$?

I know that this can't be true given that $\mathbb{R}$ is separable, but I'm having a hard time coming to grips with why this is, exactly. In particular, can't I just take some uncountable strictly ...
-1
votes
2answers
34 views

Prove the function is continious.

If the function $f(x)$ is continious at $x=0$, using definitions show that $f(rx)$ is continious at $x=0$. Here $r$ is a real number.
2
votes
3answers
48 views

Degree of field extension $\mathbb{Q}\subseteq\mathbb{Q}(i,i\sqrt2)$

I have a field extension $\mathbb{Q}\subseteq\mathbb{Q}(i,i\sqrt2)$ that I want to find the degree of. Usually I find it easiest to find the minimal polynomial, but I can't start by saying ...
1
vote
1answer
33 views

Derivatives of the Dirac delta function

From what I understand the Dirac's Delta derivatives have the meaning $$\int_{-\infty}^{\infty}\delta^{(k)}(x)\phi(x)dx=(-1)^k\int_{-\infty}^{\infty}\delta(x)\phi^{(k)}(x)dx$$ Assuming, of course that ...
0
votes
2answers
27 views

Working with C++ for GF(2)

Pardon me if it is off topic.But, is there anyone who could suggest me some basics with how to get started with working with C++ for GF(2)?? I am new in C++.I am learning to working with arrays and ...
0
votes
1answer
19 views

Find all $(a,b) \in \Bbb Z^2$ such that $b \equiv 2a \pmod 5$ and $28a+10b=26$

I'm stuck with this exercise: Find all $(a,b) \in \Bbb Z^2$ such that $b \equiv 2a \pmod 5$ and $28a+10b=26$ It's from my algebra class, we are looking into diophantic and congruence equations. ...
3
votes
4answers
47 views

Intuitive explanation of $L^2$-norm

I have to play a lot with the $L^2$-norm defined as $\|w\|=\sqrt{\int_a^b <f,f>}$. However, I don't understand the interpretation of that norm. We know that the euclidean norm measure the length ...
0
votes
1answer
29 views

How many sequences $a_1,a_2,…,a_n$ in length $k$ so $a_i \in \{1,2,3,4…,n\}$ satisfy

I have the follow two questions : How many sequences $a_1,a_2,...,a_n$ in length $k$ so $a_i \in \{1,2,3,4....,n\}$ satisfy : 1) $a_1<a_2<....<a_k$ while $(a_i \neq a_{i+1}+1)$ 2) $a_1 ...
2
votes
3answers
33 views

Evaluating the integral of $\frac{\cos(x) - e^{-x}}{x}$ using contour integration

I am trying to evaluate the value of $$\int_0^\infty\frac{\cos(x) - e^{-x}}{x}dx$$. I am assuming I am supposed to use contour integration, as I was required just before to calculate the value of ...
0
votes
0answers
10 views

Probability of stopping a production line even when the failed ratio is not more than 6 percent

Imagine a company has someone who controls the production line and every time he does that, he choses 15 products and makes sure that no more than 1 product has failed according to the rules. If more ...
2
votes
5answers
36 views

Can someone help me with this question of finding x as exponent?

The equation is: $$6^{x+1} - 6^x = 3^{x+4} - 3^x$$ I need to find x. I forgot how to use logarithm laws. Help would be appreciated. Thanks.
-1
votes
1answer
15 views

Set of marginals is convex

Let $[n] = \{1,2,\cdot,n\}$ and $[m] = \{1,2,\cdot,m\}$. Let $Z_{1,2}$ denote the set of all probability distribution on the Cartesian product $[m]\times [n]$. Let $S_{1,2}$ denote a convex closed ...
0
votes
0answers
4 views

How to create a function for residual sum of squares in matlab

I need to write a matlab loss function for the residual sum of squares function. in particular: $$G(\beta)=(\sum_{i=0}^n y_i-X\beta)^2 $$ where y is a 60 x 1 vector x= matrix of 5 vectors ...
0
votes
0answers
3 views

Solving Binary Linear Programming Problem Using KKT

Execuse me, I know that if I searched a lot I could find the answer, However I have already did my research and I am running out of time. I need the detailed solution of the following linear problem ...
1
vote
0answers
15 views

Can you prove that modules over a field are vector spaces in a categorical way?

Let $\mathcal{C}$ be the category of abelian groups endowed with tensor product. It is a monoidal category with unit object the integers. Given a monoid $(R, \mu_R , \iota_R)= R$ in this category ...
1
vote
1answer
42 views

Find all positive integers that solve Mordell's equation $y^2=x^3+37$

Find all Mordell's equation: $$y^2=x^3+k$$ where $k=37$ positive integer numbers,I can't find the when $k=37$ the mordell equation solution with some result,and we can known this equation have ...

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