# All Questions

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 $\aleph_n$ = ...
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 ...
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 ...
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 ...
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 ...
116 views

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 ...
159 views

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 ...
72 views

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 ...
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 ...
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 ...
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, ...
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 $P(k)\to P(k+1), \forall k \in \Bbb{N}$ ...
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. $$\int \frac{ x^{2}dx }{ (x^{3} + 5)^{2}}$$
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 $\ln(a \mu) - \ln(10) ... 1answer 75 views ### Dual of polynomial ring Consider the free$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}$? 1answer 176 views ### What does it mean for a Turing machine$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$. Can someone explain ... 1answer 144 views ### Direct sum of eigenspaces of a compact operator has finite codimension In an infinite dimensional Hilbert space the orthogonal complement of the (closure) of the direct sum of eigenspaces of a compact normal operator is finite dimensional. Why is this the case? thanks. 1answer 922 views ### Is Cartesian Product same as SQL Full Outer Join? Is Cartesian Product same as Full Outer Join found in Relational Database SQL? I ask because I am taking a Discrete Mathematics course and I just want a better understanding of how what I am studying ... 3answers 56 views ### Is it possible to have a bounded continuous function f:(a,b) to R such that the derivative of f tends to infinite as we approach b? CLAIM:Is not possible to have a bounded differentiable function f:[a,b) to R such that it's derivative tends to infinite as we approach b. Is the claim correct? How can i prove it? I tried the ... 1answer 115 views ### Having birthday at the same day There are 17 people. We assume that a year has 365 days. a) What is the probability that at least two of them have birthday at the same day of the year? b) What is the probability that exactly two ... 1answer 37 views ### Problem solving question with average Johnny had to take a test a day late. His 96 raised the class average from 71 to 72. How many students, including Johnny, took the test? I tried to do trial and error to see how many students there ... 0answers 48 views ### Is$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$for a non-negative random ... 1answer 553 views ### How do you find angular velocity given a pair of 3x3 rotation matrices? Let's say I have two 3x3 rotation matrices R1 and R2, each signifying rotation from the global frame to the local frame. I am also given the time difference t between these two matrices. How would I ... 1answer 199 views ### Why is$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? ... 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 ... 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, ... 1answer 148 views ### Combinations Integer Solutions to Inequalities How do I find the number of integer valued solutions to the following? $$x_1 + x_2 + x_3 < 27 \text{ for all }x_i > 0$$ x_1 + x_2 + x_3 = 27 ... 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, ... 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. ... 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 ... 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 ... 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 ...
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 ...