0
votes
0answers
9 views

Set theory with multiple countable infinities

In set theory, all sets that are countably infinite are generally considered to have the same size since there is a bijection between them. Has anyone tried formalising set theory in a way which ...
0
votes
0answers
5 views

Notation used in this Ring theorem

Lemma: F is a field only if $F\left [ x \right ]$ is a Principal Ideal Domain. This is a theorem from Ring; divisibility of integral domain. What does $F\left [ x \right ]$ means? Thanks in ...
0
votes
0answers
9 views

Showing that $\sum_{n=0}^{\infty}\frac{2^{n+2}}{{2n\choose n}}\cdot\frac{n-1}{n+1}=(\pi-2)(\pi-4)$

Showing that (1) $$\sum_{n=0}^{\infty}\frac{2^{n+2}}{{2n\choose n}}\cdot\frac{n-1}{n+1}=(\pi-2)(\pi-4)$$ see here (2) ...
1
vote
3answers
26 views

Where is $x^x$ continuous?

The idea of continuity of a function is something I come across quite regularly, but I've never really understood it well. I'm trying to fix that by looking at some interesting functions. What ...
0
votes
1answer
14 views

Simple but hard 2 by 2 system in $x$ and $y$

Is there a systematic way of solving this system, analytically? $$\begin{cases} x \ + \ y^2=11\\ x^2+y\ \ =\ 7\\ \end{cases} $$ I mean, other than brute-force.
0
votes
1answer
10 views

Does there exists an additive group homomorphism between two $K$-vector space that is not $K$-linear

My question is: Give me a field $K$. Can we always find two $K$-vector space $V_{1}$, $V_{2}$ and a map $f:V_{1}\rightarrow V_{2}$ such that: (1) If we view $V_{1}$, $V_{2}$ as additive group, then ...
0
votes
0answers
3 views

How to prove that linear transformations transforms lines into lines

I want to answer this: Given $T(x,y)=(ax+by,cx+dy)$ prove that: T transforms straight lines into straight lines T transforms parallel lines into parallel lines I have no clue in ...
0
votes
0answers
8 views

What is a transitive relation on set S

MY answer: Given r,s,t$\in S$ a transitive relation on the set $S$ is when the elements $rRs$ and $sRt$ then $rRt$ i.e., $rRs\land sRt\rightarrow rRt$ Does my definition words correct?
0
votes
1answer
22 views

Calculate $\lim \limits_{x \to \infty} x\int_{0}^{x}e^{t^2-x^2}dt$

I'm trying to find this limit $$\lim \limits_{x \to \infty} x\int_{0}^{x}e^{t^2-x^2}dt$$ From the graph I can see that it equals $1/2.$ I've looked into making substitution in order to modify the ...
0
votes
1answer
11 views

How to find $\sum_{r=0}^n \left(\frac{(-1)^r}{\binom{n}{r}}\right)$?

If n is an even natural number, then find $$\sum_{r=0}^n \left(\frac{(-1)^r}{\binom{n}{r}}\right)$$ I tried to solve the question using conventional method, by trying to use calculus, but I ...
0
votes
0answers
11 views

Finding a bijection and using the Schröder-Bernstein to prove same cardinality

I've been asked to prove that $\mathbb{R}$ and the interval $(-\infty,0)$ have the same cardinality using two methods, one being find a bijection and the other to use the Schröder-Bernstein theorem. ...
0
votes
0answers
8 views

Logic of Set Theory & Partially Order (Informative Discussion)

My final exam passed but, honestly I want to understand what this (Question 4) problem means because I don't know what it is asking for. I am a undergraduate, so it would be most helpful if the ...
0
votes
0answers
4 views

Weight Modification for Computationally-Efficient Nonlinear Least Squares Optimization

There was a time where I could figure this out for myself, but my math skills are rustier than I thought, so I have to humbly beg for help. Thank you in advance. I am solving a weighted nonlinear ...
-1
votes
0answers
4 views

Gauss- Legendre and Simpson in double integral MATLAB

I'm trying to resolve $$f(x,y)=\int_{0}^{1}\int_{0}^{2}e^{-xy}dxdy$$ with Gauss- Legendre quadrature and Simpson's rule, but being honest I have no idea how to write it on MATLAB. So, that's my ...
0
votes
1answer
11 views

compound interest amount

Find the compound interest on a principal of: $6540 at 6% p.a. for 4.5 years. I did: T=6540(1+6%)^4.5 to get 8500.69 I have been told the answer is wrong though. What have I done which is ...
1
vote
1answer
7 views

How to express the following statement with Quantifiers and Predicates

Use quantifiers and predicates with more than one variable to express this statement: There is a student in this class who has taken every course offered by one of the departments in this school ...
0
votes
1answer
15 views

Least squares without solution

Talking about simple linear regression (k=1), in which cases the Normal Equations have unique solution? And infinite? And when the Normal equations have no solution?
1
vote
2answers
30 views

How to turn Riemann sum into definite integral? [on hold]

I'm having trouble with turning this Riemann sum into a definite integral. I'm not sure as to where to start. $$\lim_{n \to \infty}\dfrac{3}{n}\sum_{k = 1}^{n}\dfrac{3k/n}{(3k/n+3)^2}$$ Original ...
2
votes
3answers
39 views

Limit of ($\sqrt{x^2+8x}-\sqrt{x^2+7x}$) as $x$ approaches infinity

I've been stuck on this one problem for 3 days now, I don't know how to proceed. Any help would be appreciated. The problem is asking for the $$\lim_{x\to\infty} (\sqrt{x^2+8x}-\sqrt{x^2+7x}) $$ ...
0
votes
0answers
17 views

My attempt to show the conditions where if $\lim_{x\to a}f(x)=b\land \lim_{y\to b}g(y)=c\implies\lim_{x\to a}(g\circ f)(x)=c$

I want to check if this way to prove the theorem about limits of compositions of functions is correct. Im equating logical statements but Im not sure if this formal proof is correct or I need to ...
0
votes
0answers
15 views

$D=\{2^a3^b|a,b \in \mathbb N\}$. Then $D^{-1}\mathbb Z \cong \mathbb Z[1/6]$

Question $D=\{2^a3^b|a,b \in \mathbb N\}$. Then $D^{-1}\mathbb Z \cong \mathbb Z[1/6]$ even though $D \neq \{1,6,6^2,..\}$ I can't make isomorphism since there are too many things to consider. For ...
1
vote
1answer
11 views

From brownian bridge to brownian motion proof

Let $B_t$ be a brownian motion. and let $\{W_t=B_t-tB_1:0\le t\le 1\}$ be a brownian bridge. Now let $Y_t=(1+t)W_{t\over 1+t}$. Proof that $Y_t$ is a brownian motion in $[0, \infty)$ My attempt: 1) ...
0
votes
0answers
10 views

Dominant morphism on affine varieties

Let $X,Y\in \mathbb{A}^{n}_{k}$ affine varieties, I know that a morphism $f:X\rightarrow Y$ is dominant iff the correspondent morphism $\phi:k[Y]\rightarrow k[Y]$ is injective. How can I show from ...
-2
votes
2answers
97 views

Prove $\pi+e$ or $\pi e$ is transcendental.

I understand to prove at least one of them irrational you would compose a function by which $\pi$ and $e$ are roots $((x-\pi)(x-e))$, and show that at least one coefficient cannot be rational because ...
1
vote
2answers
16 views

Number of discontinuous values

We have to find the number of values of $x$ at which the function $$ f(x) = \frac{2x^5-8x^2+11}{x^4+4x^3+8x^2+8x+4}$$ is discontinuous. I thought that since both numerator and denominator are ...
1
vote
3answers
31 views

If $\gamma :[a,b]\rightarrow \mathbb{R}^3$ is smooth then $\gamma(t)=x$ has finite number of solutions

Let $\gamma :[a,b]\rightarrow \mathbb{R}^3$ be a smooth curve ($\gamma$ is differentiable with $\gamma'(t)\neq \mathbf{0}$ for all $t\in[a,b]$). Show that, for $x\in\mathbb{R}^3$, the equation ...
1
vote
2answers
23 views

What does a norm of a polynomial space mean?

When talking about polynomial vector space, the following example was provided. A polynomial of degree $n$ in two variables is $$p(X)=\sum_{0\leq k+j \leq n} a_{j,k}x_1^jx_2^k$$ where $k+j=n$ and ...
0
votes
0answers
9 views

Finding the upper tight bound of a mathematical function. (Big O)

I am trying to understand Big-$O$ notation through a book I have and it is covering Big-$O$ by using functions although I am a bit confused. The book says that $O(g(n))$ where $ g(n)$ is the upper ...
1
vote
1answer
24 views

Representations of an abelian group

Let $V$ be an $F$-vector space, and let $f:G\to GL(V)$, where $G$ is a group. For $g\in G$, how can we show that if $G$ is abelian then the eigenspace of $f(g)$ is a $G$-invariant space? Moreover, ...
3
votes
2answers
38 views

If $f: X \rightarrow Y$ is continuous, and $X$ is connected, then $f(X)$ is connected

I'm trying to understand the proof of: If $f: X \rightarrow Y$ is continuous, and $X$ is connected, then $f(X)$ is connected. What are we trying to do in the following proof (are we proving the ...
2
votes
0answers
10 views

Bending a horizontal from 0 to infinity real number line, ninety degress counter-clickwise at 1.

Can the real number line from 0 to infinity, which of course is often represented as a horizontal straight line, also be represented as being bent ninety degrees counter-clockwise at 1? I.e., if such ...
0
votes
1answer
28 views

Curve sketching without a computer program

How to sketch the curve x^6 + y^6 = (x^4)*y without using a computer program ? Could someone give me the step by step ?
0
votes
0answers
3 views

Finite element method - master element

What is supposed to be a quadratic master element with 3 degrees of freedom? I think I have to consider a 3D case, with x,y,z directions...is it? And I think it is quadratic because I have to go ...
1
vote
0answers
16 views

Compactness theorem

Let $\Sigma$ be a set of theorems, such that for every $\varphi\in\Sigma$ exists a finite modell $\mathcal{M}$, with $\mathcal{M}\models\varphi$. Show: It exists an infinite modell ...
2
votes
0answers
11 views

Universal $C^*$-algebra of countable family of self-adjoint operators have boundedly complete standard Schauder basis

Let $A = C^*(T_1,T_2,T_3,... | T_i=T_i^*, ||T_i|| \leqslant 1)$ - universal enveloped $C^*$-algebra of countable family of self-adjoint operators. $A$ have standard Schauder basis, which contains all ...
0
votes
2answers
34 views

$df_n(x)/dx$ exists even if $df(x)/dx$ does not.

Question For any function f(x) continuous over the reals, define the sequence $f_n(x)=n\int_x^{x+1/n}f(y)dy$ for n=1,2,3,... Show that $df_n(x)/dx$ exists even if $df(x)/dx$ does not, that ...
0
votes
0answers
23 views

Isomorphism between multiplicative and additive groups of reals

Consider: • $(\mathbb{R}\setminus\{ 0\}, \cdot)$ • $(\mathbb{R}_+, \cdot )$ • $(\mathbb{R} \times \mathbb{Z}_2, +)$, where for all $x, y \in \mathbb{R}$ and all $a, b \in \mathbb{Z}_2$ we define ...
2
votes
1answer
24 views

If $2^n-1$ is prime, then n is prime - proof involving the Mersenne primes by counterexample

Let $2^n-1$ be prime. Suppose that $n=p_1p_2\cdots p_s$ is composite. Then we have $2^{p_1p_2\cdots p_s}-1$; call it $k$. If $k$ is prime, then its only divisors are $k$ and $1$. But consider the case ...
0
votes
1answer
32 views

How to make a comparison between two variables like $4\sqrt{\frac{b}{a}}$ and $\frac{2}{a}+2b$?

Both a and b are positive numbers. And how to make a comparison between two variables like $4\sqrt{\frac{b}{a}}$ and $\frac{2}{a}+2b$? I tried subtraction and division, but both failed.
-1
votes
1answer
17 views

Matrix for rounding to the nearest whole number

http://imgur.com/nwYFNhz Having trouble with part b. Is there a way to get a matrix calculation to round to the nearest integer?
1
vote
3answers
33 views

Solve the following equation for $x$,$\left(\frac{\frac{1}{2}\cdot(n-x^2)}{x}\right)^2 =\frac{1}{2}\cdot(n-x^2)$

I am not great at transposition and wolfram alpha confused me so I would like to see the steps in solving for x. $$\left(\frac{\frac{1}{2}\cdot(n-x^2)}{x}\right)^2 =\frac{1}{2}\cdot(n-x^2)$$ Wolfram ...
0
votes
1answer
11 views

If (X,T) is perfect and A is a dense subset of X, then A has no isolated points.

If $(X,T)$ is perfect and $A \subseteq X$ is a dense subset of X, then A has no isolated points. Since $A$ is dense $\Rightarrow (\forall U \in T)(A \cap U \neq \emptyset)$ and since $(X,T)$ is ...
1
vote
2answers
17 views

Is $Q(5^{1/4},√11,i)/Q $ normal

I think all roots of $x^4-5$ and $x^2-11$ and $x^2+1$ are in the field but it seems impossible to find a irreducible polynomial that contains all those roots. How can we check if it is normal? I ...
1
vote
1answer
22 views

How to calculate this integral $\frac{1}{2} \int_0^1\ 1.5 e^{-ik\pi \ t} \ \ dt, \, k \in \mathbb{Z} $

$$\frac{1}{2} \int_0^1\ 1.5 e^{-ik\pi\ t} \ \ dt, \, k \in \mathbb{Z} $$
0
votes
0answers
16 views

Exponential-ish function from 0,0 to 1,1: how to push the turning point of the curve

I am trying to find a weighting function to map $x$ values $0 < x < 1$ to a $y$ values $0 < y < 1$, following something similar to an exponential curve. So far, I have been using the ...
2
votes
2answers
72 views

Is the reverse statement to “open” in Munkres topology true?

There is a problem: Given $(X, \tau), A \subseteq X, \forall x \in A, \exists U \in \tau, x \in U \text{ s.t. } U \subset A \implies A$ is open in $X$ So what I did was to show that $A$ is in ...
2
votes
0answers
35 views

Is there a general method for solving this type of recurrence?

Edit: Here is the original problem; it is possible that my recurrence for the stationary distribution $\pi$ is incorrect. Consider a single server queue where customers arrive according to a ...
0
votes
0answers
8 views

How to Derive Softmax Function

Can someone explain step by step how to to find the derivative of this softmax loss function/equation. \begin{equation} L_i=-log(\frac{e^{f_{y_{i}}}}{\sum_j e^{f_j}}) = -f_{y_i} + log(\sum_j e^{f_j}) ...
0
votes
0answers
17 views

Generalized Mean Value Theorem: parametric equation interpretation

My intuition behind the generalized mean value theorem is similar to that elaborated in this post. For example, if you have two runners running for 10 seconds, at some point $c$ the ratio of the ...
0
votes
1answer
28 views

Show that $\overline A \cap \overline B \not\subseteq \overline {A \cap B}$ using definition

It is well known that $\overline {A \cap B} \neq \overline A \cap \overline B$ I wish to show that $\overline A \cap \overline B \not\subseteq \overline {A \cap B}$ by using the definition ...

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