1
vote
1answer
135 views

continuum and aleph one

We have symbols of cardinal numbers. The most known are aleph zero and continuum. Somewhere I've noticed the sequence of cardinal numbers as aleph zero, aleph one, aleph two... where $\aleph_n$ = ...
0
votes
1answer
76 views

Integral of $\frac{x}{x}, \frac{2}{x}, \frac{x}{2}$, and how they relate.

I'm studying for my diploma of higher studies (i.e. the diploma which gives me access to university) and I have a bit of trouble with building intuiton around integrals. Derivatives were relatively ...
1
vote
1answer
41 views

With $X, Y$ vector fields and $f$ a smooth function, show that $X(gY) = (Xg)Y + gXY$

I know that $X = \sum_i a_i \frac \partial\partial_{x_i}$ and $Y=\sum_j b_j \frac \partial\partial_{x_j}$ but I'm not sure how to proceed. The only approach I can think of is something to do with the ...
1
vote
1answer
29 views

proving $E$ is $\nu$-null iff $|\nu| (E)=0$

I am having trouble proving the converse of the statement below. So far I have that $\nu (E)=0$, but that doesn't mean that $E$ is necessarily $\nu$-null. I can't seem to find a way to prove that if ...
1
vote
1answer
40 views

Are the following two definitions of Borda winner equivalent?

The Borda count is a method used to determine the winner object where people rank objects. For instance, imagine each person ranking 3 objects. The highest ranked object gets 2 points, the second gets ...
2
votes
3answers
116 views

Question about the boudary of a set $A \subseteq \mathbb{R}^n $.

let $A \subseteq \mathbb{R}^n$. Let $X = \{ x \in \mathbb{R}^n : \forall \epsilon > 0, \; \; B(x, \epsilon) \cap A \neq \varnothing \; \; and \; \; B(x, \epsilon) \cap ( \mathbb{R}^n \setminus A ) ...
3
votes
1answer
116 views

Are positive semi-definite matrices always covariance matrices?

This may be trivial. While covariance matrices of random variables are positive semi-definite, does the converse hold true as well, that positive semi-definite matrices are also valid covariance ...
0
votes
2answers
114 views

Writing a diffEQ as $P(x,y)dx + Q(x,y)dy = 0$ instead of in terms of $dy/dx$

I'm reading in Tenenbaum and Pollard's Ordinary Differential Equations where they introduce the concept of the differential. Suppose $y=f(x)$ is differentiable. He defines the differential by $dy(x, ...
0
votes
1answer
85 views

Tips for integrating on a symmetric domain?

One of the problems of my homework consists in integrating $\iint_D(x^2y^2+sin(xy)e^{{x^2}y^2})dA$ on the quadrilateral domain $D$ formed by (1,0), (0,2), (-1,0) and (0,-2). This domain is symmetric ...
1
vote
2answers
159 views

Distribution of sum of jointly normal random variables with given covariance matrix

Assume that $(X_1, X_2, X_3)$ are jointly normal random variables with the mean vector $(a,b,c)$ and the covariance matrix: $$\left( \begin{array}{ccc} \sigma_1^2 & \alpha & \beta \\ \alpha ...
1
vote
1answer
94 views

A proof of the Noether Normalization Lemma

Look at the following proof of the Noether Normalization Lemma taken from Qing Liu's book "Algebraic Geometry and Arithmetic Curves": I don't understand the highlighted part. To be more ...
0
votes
1answer
59 views

Cardinality of a Monoid and Constant Functions

Let $X$ be a set. Show that $((M(X),\circ)$ has an absorbing element iff $|X|\leq 1$ iff $M(X)$ is commutative. In this problem $((M(X),\circ)$ is a monoid and M(X) is the set of all maps from X to ...
0
votes
1answer
39 views

Difference between Galois and other automorphisms

With respect to the definition of Galois field, for $E$ an extension of $F$ ($E$ and $F$ are finite fields) $\mathrm{Gal}\,(E/F)$ is the set of automorphisms of $E$ which fix $F$ pointwise. So I think ...
5
votes
2answers
518 views

At least one prime between N and N-(sqrtN)

I don't know if this is an already existing conjecture, or has been proven: There is at least one prime number between $N$ and $N-\sqrt{N})$. Some examples: $N=100$ $\sqrt{N}=10$ Between and 90 and ...
0
votes
1answer
393 views

Prove the Countable additivity of Lebesgue Integral.

Let $E\subset\mathbb{R}$ a measurable subset, $f\in L^1(E)$ and $\{E_n\}$ a disjoint countable union of measurables sets such that $\bigcup E_n=E$. Show that $$ \int_Ef=\sum_{n=1}^\infty\int_{E_n} ...
0
votes
1answer
2k views

Find an expression for the area under the graph of f(x) as a limit?

$f(x) = \frac{2x}{x^2 +1}, 1 \leq x \leq 3$ Basically, I need to find an expression for the area under the graph within these intervals for the function as a limit. I understand the concept of the ...
1
vote
2answers
72 views

Let $f(x, y)$ differentiable function $\forall (x,y)$. Let $g(u, v) = f(u^2 - v^2, u^2v).$ Known $\nabla f(-3, 2) = (2, 1).$ Find $ \nabla g(1, 2)$

I need an explanation for this solution. I tried to solved the following exercise: Let $f(x, y)$ differentiable function $\forall (x,y)$. Let $g(u, v) = f(u^2 - v^2, u^2v).$ Known $\nabla f(-3, ...
3
votes
0answers
84 views

Buchberger's criterion to show Grobner basis for linear forms

Let $k$ be a field. A polynomial of the form $l=a_1x_1+\cdots+a_nx_n$ is called a linear form ($a_i\in k$), and its support is the set of all variables $x_i$ such that $a_i\neq 0$. Let $L\subseteq ...
0
votes
2answers
215 views

Comparing two decibel values

For my son's science fair project, we are measuring wi-fi signal strength in decibels, a logarithmic scale. We want to determine the relative strength of two values. I think that a value of -60 is ...
3
votes
0answers
44 views

Size of the collection of morphisms of a category

Suppose we use Grothendieck'universes, at least 2 (named U and W). U has elements called (small-) Sets and its subcollections are called Classes. W has elements called Classes and its subcollections ...
2
votes
2answers
71 views

Prove that $r_1 = r_2$ iff $n | (b - a)$

I need to know if I'm clear in my proof since I will have to present the answer to my class. Here's the full question: Let $n$ be a fixed positive integer. Then for any integers $a$ and $b$, let ...
0
votes
1answer
89 views

The elements of finite order in an abelian group form a subgroup: proof check

If G is an abelian group, show that the set of elements of finite order is a subgroup of G. Proof: Let G be an abelian group and H be the set of elements of finite order. (1) nonempty Now e ∈ H, ...
0
votes
1answer
51 views

About proof by induction

Proof by induction consists in following scheme: Proof by induction or intuitively, let be a predicate $ P(n) $ with $ n \in \Bbb{N} $: if $P(0) $ is true $P(k)\to P(k+1), \forall k \in \Bbb{N} $ ...
2
votes
4answers
799 views

Find integral when $dx$ is in the numerator

Can someone please walk me through the steps to find the following integral? I'm not sure what to do when $dx$ is at the top. $$ \int \frac{ x^{2}dx }{ (x^{3} + 5)^{2}} $$
1
vote
2answers
169 views

Understanding Euler's paper on curvature

We've recently discussed, in the course on differential geometry which I am taking, Euler's theorem regarding curvature of sections of surfaces in $\mathbf{R}^3$. Being curious, and knowing that ...
1
vote
2answers
56 views

Arc Length problem, not sure how to go about it.

if $$4x^2 - y^2 = 64$$ show that: $$ds^2 = \frac4{y^2}(5x^2 - 16)dx^2$$ I'm not sure what to do. Could someone explain it to me? I tried solving for y and then plugging it into the Arc Length ...
0
votes
1answer
100 views

Linear mapping between a non-orthogonal basis and an orthogonal basis?

Consider a set of $n$ linearly independent $d$-dimensional vectors $\left\{\vec{a}_i\right\}_{i=1}^{i=n}$ that span the vector space $V$ and that are not in general orthogonal with respect to the ...
5
votes
3answers
239 views

Is the center of the universal enveloping algebra generated by the center of the lie algebra?

Let $\mathfrak{g}$ be a Lie algebra over a field $k$, and let $U(\mathfrak{g})$ be its universal enveloping algebra. $\mathfrak{g}$ is canonically embedded in $U(\mathfrak{g})$; identify it with its ...
1
vote
2answers
196 views

Prove this is a subspace of V

Let T: V $\to$ W be a linear map between vector spaces and let N be a subspace of W. Define $T(N) := {v∈V : Tv ∈ N}$. Prove that T(N) is a subspace of V. I know the properties that a subspace must ...
19
votes
2answers
1k views

If every real-valued continuous function is bounded on $X$ (metric space), then $X$ is compact.

Let $X$ be a metric space. Prove that if every continuous function $f: X \rightarrow \mathbb{R}$ is bounded, then $X$ is compact. This has been asked before, but all the answers I have seen prove the ...
0
votes
2answers
117 views

The problem of x = ln(x)

I am trying to find x values for points along the normal distribution curve, and I ended up with a problem that goes back to the method of solving $x = \ln x$. Right now, I have $\ln(a \mu) - \ln(10) ...
2
votes
1answer
75 views

Dual of polynomial ring

Consider the free $k$-algebra $k[x_i]_{i \in I}$ indexed by $I$. Then is $Hom_{k-Mod}(k[x_i]_{i \in I},k) \cong k[x_i]_{i \in I}$?
0
votes
1answer
176 views

What does it mean for a Turing machine $M$ to accept $\epsilon$

Suppose $B_{TM}$ = $\{ \langle M \rangle$ | $M$ is a Turing machine over $\{0, 1\}$ and $M$ accepts $\epsilon\}$. I do not understand what it means for $M$ to accept $\epsilon$. Can someone explain ...
2
votes
1answer
144 views

Direct sum of eigenspaces of a compact operator has finite codimension

In an infinite dimensional Hilbert space the orthogonal complement of the (closure) of the direct sum of eigenspaces of a compact normal operator is finite dimensional. Why is this the case? thanks.
4
votes
1answer
922 views

Is Cartesian Product same as SQL Full Outer Join?

Is Cartesian Product same as Full Outer Join found in Relational Database SQL? I ask because I am taking a Discrete Mathematics course and I just want a better understanding of how what I am studying ...
1
vote
3answers
56 views

Is it possible to have a bounded continuous function f:(a,b) to R such that the derivative of f tends to infinite as we approach b?

CLAIM:Is not possible to have a bounded differentiable function f:[a,b) to R such that it's derivative tends to infinite as we approach b. Is the claim correct? How can i prove it? I tried the ...
2
votes
1answer
115 views

Having birthday at the same day

There are 17 people. We assume that a year has 365 days. a) What is the probability that at least two of them have birthday at the same day of the year? b) What is the probability that exactly two ...
0
votes
1answer
37 views

Problem solving question with average

Johnny had to take a test a day late. His 96 raised the class average from 71 to 72. How many students, including Johnny, took the test? I tried to do trial and error to see how many students there ...
0
votes
0answers
48 views

Is $X$ measurable and $h\circ X$ integrable?

The Markov-inequality says: Let $h\colon\mathbb{R}_{\geq 0}\to\mathbb{R}_{\geq 0}$ be monotonously increasing function, so that $h\circ X\in\mathcal{L}_{\mu}^1$ for a non-negative random ...
1
vote
1answer
553 views

How do you find angular velocity given a pair of 3x3 rotation matrices?

Let's say I have two 3x3 rotation matrices R1 and R2, each signifying rotation from the global frame to the local frame. I am also given the time difference t between these two matrices. How would I ...
12
votes
1answer
199 views

Why is $e$ the Identity?

Some authors use $e$ to be the identity element of a group instead of $1$. What is the origin of this notation? Was this before or after we used $e$ to represent the base of the natural logarithm? ...
1
vote
1answer
118 views

Prove that a field automorphism sends a root into a root

I came across the following problem: If $E$ is an extension of $F$ and if $f(x)\in F[x]$ and if $\phi$ is an automorphism of $E$ leaving every element of $F$ fixed, prove that $\phi$ must take a ...
1
vote
1answer
50 views

Splitting up bracket terms

I found a statement saying: Let $\circledast $ be an associative binary operation on a set $\mathbb{X}$. A bracket term of length n, consisting of n elements $a_1, ..., a_n$ and arbitrary brackets, ...
0
votes
1answer
148 views

Combinations Integer Solutions to Inequalities

How do I find the number of integer valued solutions to the following? \begin{equation} x_1 + x_2 + x_3 < 27 \text{ for all }x_i > 0\end{equation} \begin{equation} x_1 + x_2 + x_3 = 27 ...
2
votes
1answer
73 views

Multiply (as a Babylonian): 141 times 17 1/5

How do we multiply 141 times 17 1/5 as a Babylonian? I wasn't sure the space between 17 and 1/5, now I see that 17 1/5 is 17.2 in our notation. Is there a formula that I can solve this? Any hint, ...
1
vote
1answer
177 views

Proving the transitive property of an equivalence relation

I have to prove an equivalence relation.. $x$ is related to $y$ in the reals if $|x-y|\le3$ Reflexivity was easy. Symmetry was just a matter of breaking up the +ve and -ve case and it worked out. ...
1
vote
2answers
60 views

solve the following recurrence exactly.

$$t(n)=\begin{cases}n&\text{if }n=0,1,2,\text{ or }3\\t(n-1)+t(n-3)+t(n-4)&\text{otherwise.}\end{cases} $$ Express your answer as simply using the theta notation. I don't know where to go ...
0
votes
2answers
89 views

Combination and probability problem. GMAT related.

The Carson family will purchase three used cars. There are two models of cars available, Model A and Model B, each of which is available in four colors: blue, black, red, and green. How many different ...
0
votes
1answer
240 views

Dummit & Foote 13.2.18

I am having a hard time trying to understand why $k(x)$ is an extension of $k(t)$? Let $k$ be a field and let $k(x)$ be the field of rational functions in x with coefficients from $k$. Let $t \in ...
4
votes
4answers
277 views

Alternative ways to solve this trigonometric inequality?

The inequality is: $$ \frac{\sin{\theta}+1}{\cos{\theta}}\leq 1 \text{ with } \cos{\theta}\neq0 \land 0\leq \theta\lt 2\pi$$ I've tried splitting it up into cases of $\theta$ that make ...

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