All Questions

0
votes
0answers
2 views

topology and metrizability

Let X be a nonempty set and Y is a proper subset of X. Set τ =2^Y U {X}. a) Show that τ is a topology on X. b) Find all nonempty subsets X such that ...
0
votes
0answers
2 views

How to solve propositional logic problems by induction

I'm trying to solve a bunch of problems like this one and every time I get stuck. So I don't need an actual solution but to understand how you solve this kind of problems. I know they're usually ...
-1
votes
0answers
2 views

I don’t understand this,can you explain it simply

What is the condition that the cubic y=ax^3+bc^2+XX+d shall have two extremes?
0
votes
0answers
9 views

Why we can safely treat objects like mathematical entities?

In the study of number systems we learn the axioms of real numbers. For example: The commutative axiom x.y = y.x The distributive axiom x.z + y.z = (x+y).z Well, the thing is that we are working ...
0
votes
0answers
4 views

Proper class if and only if there is a bijection with the class of ordinals.

I am looking for a way to formalize the sentence "$A$ is a proper class", and so I was wondering if I could write it like "a proper class is something that is in bijection with the class of all ...
0
votes
0answers
5 views

Branch Points and Branch Cuts for cube root

Having trouble with finding the branch points/branch cuts of this function: f(z)= $3\sqrt\frac{(z-2)(z+1)}{(z+2)}$ ive tried using the equation $z=re^{i\theta +2\pi n}$ but then I don't really know ...
1
vote
0answers
9 views

Solving $\cos(t\cos\theta)\cos(t\sin\theta) \sin 2\theta = \sin(t\cos\theta)\sin(t\sin\theta)$

I am trying to find all $\theta\in\mathbb{R}$ such that $$\cos(t\cos\theta)\cos(t\sin\theta) \sin 2\theta = \sin(t\cos\theta)\sin(t\sin\theta)$$ holds for all $t\in\mathbb{R}$. Just by looking at the ...
0
votes
1answer
6 views

Reduction formula for integral $\int 1/(a^2+x^2)^n dx$

I want to solve the integral $\int 1/(a^2+x^2)^n dx$. I guess this can be solved using the reduction formula as in the case of $\int x^2/(a^2+x^2)^n dx$. However, I am not able to do it.
0
votes
0answers
5 views

How to show that the following sets are equinumerous

This is my first question on this website, so please do give me feedback where nessecary. I got stuck on a question in Computability and Logic 5th edition by Boole i.a. It's question 2.10 and the ...
1
vote
1answer
9 views

Limit with 3 variables approaching infinite

I'm having some problems resolving this limit. The fact is that I'm finding little information about limits in three variables online and in my manuals, and even less information about three variables ...
0
votes
0answers
11 views

Find m(A), where m denotes Lebesgue measure

Let A be the set of numbers in the interval [0, 1] that have the digit “0” in the first, second or third place in their decimal expansion. Find m(A), where m denotes Lebesgue measure. My knowledge:...
-5
votes
1answer
14 views

Hey need help . . . . . . . . . .

can someone help me solve this 2 question 1:Let X and Y be sets. Prove that X × Y = Y × X iff X= ∅ or Y= ∅ or X=Y 2:Let X,Y,Z be sets. Prove or refute: • X ∪ (Y × Z) = (X ∪ Y) × (X ∪ Z) • (X × X)(Y ...
-1
votes
0answers
6 views

Statistic Question about Neurons

Question Hi, I want to solve this question and I know what variance, mean, covarianceö however, I am struggling to understand what question asking for. Can anyone help me to clarify what is expected ...
1
vote
0answers
7 views

Proving the convergence of $\int e^{x^2/2} dF_{1,2}(x)$ in Cramers proof of the

In Cramer 1936 the proof of the Cramer decomposition theorem contains proving the following integrals are finite $$\int e^{x^2/2} dF_{1}(x), \quad \int e^{x^2/2} dF_{2}(x)$$ to later use in finding a ...
0
votes
1answer
19 views

$\int_0^1\frac{x^2\ln x}{\sqrt{1-x^2}}dx$

$\int_0^1\frac{x^2\ln x}{\sqrt{1-x^2}}dx$ I tried putting $x=\sin \theta$ and changing the limits from $0$ to $\frac{\pi}{2}$and i got $\int_0^\frac{\pi}{2}\sin^2\theta\ln \sin \theta d\theta$ i ...
-3
votes
0answers
22 views

Let $M$ be a $3\times 3$ matrix satisfying $M^3=0$

Let $M$ be a $3\times 3$ matrix satisfying $M^3=0$. Prove that $$\left|\frac{M^2}{2}\pm M+I\right|\neq 0$$
-1
votes
0answers
9 views

Successive differentiation 30

question : $p^2=a^2cos^2\theta+b^2sin^2\theta$ , prove that $p+y''=\frac{a^2b^2}{p^3}$ $p^2=a^2cos^2\theta+b^2sin^2\theta$ $2py'=-2a^2(cos\theta)(sin\theta)+2b^2(sin\theta)(cos\theta)=(cos\theta)(sin\...
0
votes
0answers
8 views

For some positive definite function $\sigma(x)$ does there always exist $\psi(x)$ such that $\sigma(x)=|\psi(x)|^{2}$???

Ok, this seems like common sense to me but I just had to check. Given some positive definite function $\sigma(x)$ can we always find $\psi(x)$ such that we can represent it as $$\sigma(x)=|\psi(x)|^{...
1
vote
0answers
5 views

Sufficient and neccesary condition for almost convergence of $\max (X_1,X_2,\cdots,X_n)/a_n$

$X_1,X_2,\cdots,X_n$ are an iid sequence with standard normal distribution. Suppose that $a_n\rightarrow \infty$, find the sufficient and neccesary condition for $a_n$ such that $$\frac{\max (X_1,X_2,...
-9
votes
0answers
20 views

$3 \times 3$ Determinant

Without expanding the determinant to prove that $$\begin{vmatrix} 0 & 2 & 3 \\ -2 & 0 & 4\\ 3 & -4 & 0\end{vmatrix}= 0$$
-2
votes
2answers
16 views

Show that vectors are linearly independent

Let $\{u_1, u_2, u_3\}$ be a set of linearly independent vectors in $V$. Assume that $v_1 =a_{11}u_1 + a_{12}u_2 + a_{13}u_3$ $v_2 =a_{21}u_1 + a_{22}u_2 + a_{23}u_3$ $v_3 =a_{31}u_1 + a_{32}u_2 + ...
0
votes
0answers
8 views

Nonsingular projective curve corresponding to $y^2 = x^4+1$

Consider the affine curve $C_1 = V(y^2 - (x^4+1)) \subset \Bbb A^2_k$. In the answers to this question, they claim that there is a (unique?) nonsingular projective curve $C_2$ corresponding to $C_1$ (...
0
votes
0answers
4 views

Show that $\alpha f + \beta g$ is measurable when $f,g$ are measurable.

Consider a measure space $(\Omega, \mathcal{F}, \mu)$ and two integrable, measurable functions $$f,g: \Omega \to [-\infty, + \infty]$$ I.e., $$\int fd\mu, \int g d \mu \in \mathbb{R}$$ I proved ...
0
votes
1answer
15 views

Rational numbers in irrational bases

If you take the base-$b$ expansion of a rational number where $b$ is irrational, do you get a non-terminating sequence of digits (assuming you pick the right(?) digits)? More informally, do rational ...
1
vote
0answers
14 views

Matrix exponentials, computing the product of $\exp(-iBt)$ and $\exp(-iB^{-1}t)$.

Suppose I have some matrix exponential $U(t)=\exp(-iAt)$ where $t$ is some real valued number, $A$ is a hermitian matrix (so $U(t)$ is unitary) where $A=B+B^{-1}$ and $B$ is unitary. Because $B$ and $...
-1
votes
0answers
10 views

confusion over probability problems,

A) Some cereal companies say that they sell one of four kinds of toys in a cereal box for sales promotion, and give a free box to those who collect all four kinds of toys. When a person buys 10 boxes, ...
1
vote
0answers
17 views

How to find P(Y1 < Y2 <… <Yn)

Suppose that we have set of independent and identically distributed (iid) variables: Y1 , Y2 , Y3 ,…, Yn. Find the probability that Y1 < Y2 < Y3 <…< Yn. I am quite clueless about this ...
0
votes
2answers
26 views

An. Trignometry

Let $x$ and $y$ be positive real numbers and θ an angle such that $\theta \ne n\frac{\pi}2$ for any integer $n$. Suppose $ \dfrac{\sin\theta}x = \dfrac{\cos\theta}y$ and $ \dfrac{\cos^4\theta}{x^4} + ...
1
vote
2answers
12 views

If C, D, E, and K is matrices, and CKD=E, is there any other way to determine matrix K beside expanding that equation with matrix multiplication?

If C is 3 x 2 matrix, D is 2 x 3 matrix, and E is 3 x 3 matrix, that K should be 2 x 2 matrix if CKD = E. With specific entries of each matrices, I could get the elements of K with expanding the ...
-3
votes
1answer
20 views

Meaning of and /or in the set theory

Winnie has 165 workers in process X,110 in process Y,97 in process Z.Out of this workers 281 are skilled in the activities of x and/or why ,269 are skilled in the activities of y and/or z ,241 are ...
0
votes
0answers
11 views

Sum of a multiplicative arithmetic function

Prime factorization of $n$ is $\prod p_i^{e_i}$ Then radical of $n$ is defined as $\text{rad}(n)=\prod p_i$ Let $S(N) = \sum_{n=1}^{N}\text{rad}(n)$ I want to calculate $S(N)$ for very large value ...
1
vote
0answers
29 views

Show that the sequence $y_n = 1+\frac{1}{2^k}+\frac{1}{3^k}+…+\frac{1}{n^k}$, where $n\in \mathbb{N}$ and $k\geq 2$ is convergent. [duplicate]

Let $y_n = 1+\frac{1}{2^k}+\frac{1}{3^k}+...+\frac{1}{n^k}$, where $n\in \mathbb{N}$ and $k\geq 2$. I need to show that this sequence is convergent. I did $y_{n+1}-y_n$ and I got that it is ...
3
votes
3answers
37 views

Need some help solving this equation

The equation is $$\sqrt{\frac{4-x}x}+\sqrt{\frac{x-4}{x+1}}=2-\sqrt{x^2-12}$$ I tried squaring both left side and right side then bringing them to same numerator but got lost from there ... any ...
0
votes
0answers
8 views

Variance random variables

someone can explain me why the difference of 2 random variables indipendent is the sum of 2 random variables and not the difference? So, why Var(X-Y) = Var(x) + Var(y) ?
1
vote
2answers
11 views

Problem computing Lie algebra of unitriangular matrices

Let $G$ the Lie group of the upper unitriangular matrices, i.e. \begin{align} G : = \{ A= (a_{ij})_{ij} \, \, |\, a_{ii} = 1 \, \, \, \forall \, i \, \, \text{and} \, \, a_{ij} = 0 \, \, \forall \, i&...
0
votes
1answer
14 views

Bernoulli's inequality and secuences

How can I use Bernoulli's inequality to prove c^{1/n}->1 for $c>0$ ? I have to use $c^{1/n}=1+Xn$. Thank you!
-3
votes
1answer
16 views

Power set in measure theory

Let X be any set and A = P(X) be the power set of X. Let x1,...,xn be distinct points in X and let α1, . . . , αn be positive real numbers. Show the measure on A. I'm uncertain if μ = α1δx1 + α2δx2 +...
0
votes
0answers
7 views

Calculating subdifferential through proximal mapping

I was wondering if it was possible to characterize the elements in a subdifferential of a seminorm, $\partial \|\cdot\|$, through means of the proximal mapping. I know of therelation $$u = Prox_f (v) ...
0
votes
0answers
6 views

group homomorphism from a profinite group continuous iff kernel open

I have a question regarding a (probably simple) fact. However I am lacking some basic topological knowledge. Let $G$ be a locally pro finite group, i.e. ever open neighborhood of $1_G$ contains a ...
0
votes
0answers
6 views

How to prove this equation by relating stratified random sample to simple random sample?

Given a stratified random sample, $y_{hi}$,$i = 1,2,..,n_h$, $h = 1, 2, ..,L$, and$\sum_{h=1}^L = n$: Prove $$V(\bar{y}_{srs}) = \frac{1-f}{n} S^2 = \frac{N-n}{n(N-1)}[\frac{1}{N}\sum_{h=1}^L\sum_{j=1}...
0
votes
2answers
12 views

how the following ways of selection differs from each other

Suppose I want to select $2$ elements out of $6$ elements. Then I get $6C2 = 15$ combinations. Here $C$ represents the standard formula of ($N C R$). Now my question is what differences arise when ...
1
vote
0answers
13 views

Prove upper bound inequality for the dimension of the space of symmetric tensors

I want to check that the dimension of the space of symmetric tensors $N(n,m) := dim(Sym^m(\mathbb{R}^n))$ satisfies $N(n,m) \leq \frac{n^m}{m!}(1+\frac{2m^2}{n})$. Thus I need prove inequality. If $...
-1
votes
2answers
15 views

If A, B, C, and D are sets, can I proof that (A-B)-(C-D)=(A-C)-(B-D) with membership table?

I tried to figure the problem out with membership table that, A-B=A $\cap$ ~B, but it wasn't proved that (A-B)-(C-D)=(A-C)-(B-D).
0
votes
0answers
12 views

how to isolate this properly

I'm doing an induction proof and for some reason, I don't know how to go about this I'm trying to isolate $10^n-1$ from $(10^n*10^1)-1=9a$
2
votes
1answer
15 views

Showing that $Y^c$ is dense in $X$, if $ Y \subset (X, \| \cdot \|)$

Exercise : Let $X$ be a normed space $(X, \| \cdot \|)$ and $Y$ be a proper subspace of $X$, $Y \subset X$. Show that the complement set $Y^c$ is dense in $X$. Question : I'm totally at loss on ...
1
vote
0answers
17 views

Integers which are squared norm of 3 by 3 integer matrices

Question: Which integers are of the form $\Vert A \Vert^2$, with $A \in M_3(\mathbb{Z})$? For $2$ by $2$ matrices, it is proved here that it is exactly the integers which are sum of two squares. So ...
1
vote
1answer
24 views

Linear Algebra Statements

Which of these statements are true? (I) If $A$ is a matrix of full rank, then $A$ is invertible. (II) If $A$ can be expressed as a product of elementary matrices, then $A$ is of full rank. (III) If ...
2
votes
0answers
18 views

Show that a specific map is a submersion of $O(3)$ in $S^2$

Denoting the components of the $3\times3$ matrix $A \in O(3)$ as $a_{ij}$, show that $$ F: O(3) \rightarrow S^2, a_{ij} \mapsto a_{1j} $$ is a submersion. (The map is well defined since for $A \in O(...
3
votes
3answers
39 views

Prove all terms of quadratic pattern is positive.

I recently did a Mathematics exam from a previous year, and I stumbled across a question's answer I struggled to fully understand. It is given: The quadratic pattern 244 ; 193 ; 148 ; 109; ... I've ...
0
votes
3answers
22 views

Is it true that $P(A|B) = P(A|C) \cdot P(C|B) $?

I think that $$P(A|B) = P(A|C) \cdot P(C|B) $$ is True. You are just transforming the information from $B$ through $C$. Is this correct and if it is, what's the name for this property?

15 30 50 per page