3
votes
1answer
35 views

Is the matrix form of the cross product related to bilinear forms.

The cross product of two vectors $\mathbf{x}, \mathbf{y} \in \mathbb{R}^3$ can be represented as a matrix product as follows, if $\mathbf{x} = (x_1, x_2, x_3)^{\top}$ then $\mathbf{x} \times ...
1
vote
2answers
30 views

A consistent set of formulas

In a logic system, a set $\Sigma$ of formulas is said to be inconsistent if $\Sigma \vdash \False$, and consistent otherwise. Does it mean that $\Sigma$ is consistent if and only if $\Sigma \vdash ...
0
votes
0answers
30 views

Question about proving a real number

If we know that for any, $\alpha \in \{0, 2\}^\mathbb{N}$ that $0 \le \sum\limits_{k = 0}^\infty\frac{\alpha(k)}{3^k} \le 3$, then what property of real numbers do we have to use to prove that ...
1
vote
2answers
25 views

Projective varieties and irreducibility

The "modern"(schematic) definition of a projective variety is the following: Let $k$ be an algebraically closed field. A projective variety over $k$ is a closed subscheme of $\mathbb ...
6
votes
1answer
89 views

Why is adding Cohen reals so “uninteresting”?

I read the following in this paper (Otmar Spinas, Proper products. Proceedings of the AMS, 137 (8), (2009), 2767–2772): $ \mathbb{L}^2 $ adds a Cohen real. Thus it was considered uninteresting ...
6
votes
1answer
114 views

Can the product of $n$ factorials be $n$ factorial?

Are there any solutions to the equation $a_1!\cdot a_2!\cdots a_n!=n!$ with all variables being integers greater than or equal to $2$?
3
votes
2answers
63 views

Book for Undergrad Differential Geometry

I am soon going to start learning differential geometry on my own (I'm trying to learn the math behind General Relativity before I take it next year). I got the sense that a good, standard 1st book ...
2
votes
0answers
25 views

Composing a smooth even function and square root

Let $f:\mathbb{R}\to\mathbb{R}$ be smooth and satisfy $f(-x)=f(x)$ for all x. Define $g:[0,\infty)\to\mathbb{R}$ by $g(x)=f(\sqrt{x})$. Is $g$ necessarily smooth at $0$? I guess the answer is ...
3
votes
2answers
50 views

Proof $e^n*n!$ is an asymptote of $(n+1)^n$

I would like to prove $\lim_{n\to \infty}e^nn!-(n+1)^n=0$. All I have really done is show $(n+1)^n=\sum_{i=0}^n\frac{n!}{(n+1)^i(i!)(n-i)!}$
3
votes
3answers
58 views

$x^y < y^x$ for $y\ll x$?

Sorry if this is a naive question; I am not very good at mathematics. It seems obvious that for many $x$ and $y$, $x^y < y^x$ if $y \ll x$, e.g. $2^{10} > 10^2$. If $x$ and $y$ are very close ...
0
votes
1answer
23 views

Calculate the supremum of $\frac{e^{|\gamma_n+1|}-1}{|\gamma_n+1|}$

If $\{\gamma_n\}$ is a sequence of real number and $\exists M>0$, finite, such that $|\gamma_n|\leq M$, find the supremum of the following sequence: $$\frac{e^{|\gamma_n+1|}-1}{|\gamma_n+1|}$$
1
vote
0answers
11 views

Basis representation for non-negative, compact support, reasonably smooth spectral function

I was wondering if anyone has ideas on representing a non-negative, compact support (from x=-1 to 1 on the real axis) spectral function as a superposition of basis elements. Ideally, the basis ...
1
vote
0answers
21 views

How do I show the probability of failure in my program? (using a lock-free stack)

I've got a question about code I'm writing here, and am looking for a way to see into the math of what's happening. Thought you guys could help :) I'll see what I can do about describing the ...
2
votes
1answer
33 views

Does differentiability imply absolute continuity? [duplicate]

Suppose $f:[a,b] \rightarrow \mathbb{R}$ is a function which is (i) differentiable at all $x \in (a,b)$ (ii) the right-derivative at $x=a$ exists and the left-derivative at $x=b$ exists. Does it ...
2
votes
1answer
70 views

Probability Question: Who's right, me or the book?

I'll be giving some classes on probability theory later this year, and so I've been going through the textbook to check that I'm up to speed. I came across the following question: The discrete random ...
1
vote
1answer
60 views

Roots of $\tan x - x$

The function $\tan x - x$ has exactly one root $x_n$ in the interval $(n\pi, (n + \frac{1}{2})\pi)$. Show that $$x_n = n\pi + \frac{\pi}{2} - \frac{1}{n\pi} + r_n$$ where $\lim_{n\rightarrow \infty} n ...
2
votes
0answers
35 views

What do rank-2 tensor entries represent? What's their geometrical meaning?

I just followed a course on tensor analysis and I think I understood almost everything. Except the following: given a vector (with, say, 3 dimensions), if I multiply that with a second-rank tensor, I ...
0
votes
2answers
27 views

Question about partial sum

I'm confused on the partial sums formula Why is $$\sum_{i=m+1}^\infty \frac{2}{3^i}=\frac{1}{3^m},$$ if $$\sum\limits_{k = 0}^\infty\frac{2}{3^k} = 3.$$
5
votes
1answer
45 views

Class group of $\mathbb{Q}(\sqrt[4]{-2})$

I would like to show directly that $C(K)$ is trivial, where $K = \mathbb{Q}(\sqrt[4]{-2})$. Write $\delta = \sqrt[4]{-2}$. It is pretty easy to see that $\mathcal{O}_K = \mathbb{Z}[\delta] = R$. Then ...
2
votes
1answer
52 views

What does this paper mean by “$f(x)$ is practically a rational function”?

The paper "Infinite integrals involving Bessel functions by contour integration" by Qiong-Gui Lin gives a method to solve integrals of the form $\intop_{0}^{\infty}x^{v}f(x)J_{v}(qx)\, dx$. One of the ...
3
votes
0answers
39 views

If $f_n(t):=f(t^n)$ converges uniformly to continuous function then $f$ constant

Let $f \colon [0,1] \to \mathbb R$ be continuous and let $f_n$ be defined by $$ f_n(x):=f(x^n), \qquad x \in [0,1], \,\, n \in \mathbb N. $$ Suppose there exists a continuous function $g$ on ...
1
vote
5answers
107 views

Limit of factorial function: $\lim\limits_{n\to\infty}\frac{n^n}{n!}.$ [duplicate]

I am studying for a test and I am given this problem: $$\lim_{n\to\infty}\frac{n^n}{n!}.$$ How do I go about solving this limit? Intuitively I see how the numerator is growing much faster, but how ...
0
votes
1answer
28 views

Linear Transformation from V to W (bijective) Show that T(v) is a basis of W if B is a basis of V.

$V, W$ two vector spaces and $T: V \to W$ is a bijective linear transformation. $B$ is a basis of $V$. Prove that $\{T(\mathbf{v}) | \mathbf{v} \in B\}$ is a basis of $W$. I started by doing ...
0
votes
1answer
64 views

Quick question about $\epsilon -\delta$ proofs

There is one step in $\epsilon - \delta$ proofs that I hope somebody could bring clarity to for me. Say we wanted to show $\displaystyle \lim_{x \to 2} x^2 = 4 $. Somewhere along the proof we would ...
0
votes
5answers
44 views

What is the probability that Brian makes money on his first roll

Brian plays a game in which two fair, six-sided dice are rolled simultaneously. For each die, an even number means that he wins that amount of money and an odd number means that he loses that ...
0
votes
2answers
20 views

Conditional joint probabilities with CDFs

Let X and Y be independent continuous random variables with marginal CDFs given by $F_X(x) =\begin{cases} 0 & \text{for }x < 0\\ x/3 & \text{for }0 \leq x \leq 3\\ 1 & \text{for }x ...
0
votes
1answer
42 views

Chain rule application in fundamental Theorem of Calculus

I have attached a question that I came across in trying to understand the fundamental theorem of calculus. The solution (highlighted with an arrow). I have difficulty understanding why the chain rule ...
2
votes
1answer
18 views

Can I convert between a rotation about an axis and a rotation according to two angles (all in 3D) without solving a system of nonlinear equations?

I am writing a program that needs to be able to switch between a rotation described by 2 angles to a rotation described an axis and one angle. I found one way to do this from this question, which ...
2
votes
3answers
155 views

Is there any theorem about figures of equal area and perimeter being congruent?

I had an idea, that all geometric objects, that are different, as they're not a translation, rotation, and a reflection of one another cannot have the same area AND perimeter, as compared to ONE ...
1
vote
3answers
53 views

Transformation of two independent uniform random variables

Suppose $X,Y \sim \text{Uniform} \left(0,1 \right)$ are independent. Then I need to find the PDF for $W=X/Y$. By the CDF technique this is seen to be : $$F_W( w)=\int_{0}^1 \int_{0}^{wy} ...
0
votes
1answer
23 views

How to get polarised electromagnetic TE wave differential equation from Maxwell's Equations?

I wish to understand how the following equation: $\frac{\partial^2 E_x}{\partial y^2} + \frac{\partial^2 E_x}{\partial z^2} + n^2 k_0 E_x = \frac{\text{d} (\ln \mu)}{\text{d}z}\frac{\partial ...
2
votes
1answer
18 views

Rounding Percentages

Okay so I have been having a problem and would really appreciate any help. I have posted a table below with the aim of making this as simple/precise as possible. ...
1
vote
0answers
30 views

Discontinuous linear operator on $\ell^{2}$

Let $e_{n} = (0, 0, \ldots, 0, 1, 0, \ldots)$ where $1$ is in the $n$th position. Then $\{e_{n}\}$ is an orthonormal basis for the Hilbert space $\ell^{2}(\mathbb{N})$. Does there exists a linear ...
5
votes
3answers
98 views

Find the value of this infinitely nested radical (it appears to obtain multiple values)

Find the value of $$\sqrt{1-\sqrt{\frac{17}{16}-\sqrt{1-\sqrt{\frac{17}{16}-\cdots}}}}$$ This is not as simple as it looks for one reason - there are $2$ real solutions to the equation ...
3
votes
2answers
37 views

The union of all the open sets in a family of topologies

I'm starting studying topology for the first time and my teacher just wrote this. I just don't understand the last line: Let $\{\tau_\alpha\}$ be a family of topologies on X. [...] To say that ...
1
vote
2answers
68 views

Polynomial Division - “Define the largest natural number…” [on hold]

Would someone mind helping me with this question? The more detailed possible so I can have 100% of understanding. Thanks. Question: Define the largest natural number m such that the polynomial ...
2
votes
5answers
47 views

How to show if A is denumerable and $x\in A$ then $A-\{x\}$ is denumerable

My thoughts: If $A$ is denumerable then it has a bijection with $\mathbb{N}$ So therefore $A\rightarrow \mathbb{N}$. Then x is a single object in A and A is infinite. So if a single object is ...
1
vote
2answers
32 views

P/1 Actuary Exam Question

I was doing problems and came across this one and was wondering why the P[1<=x<=2] is F(2) - lim (x->1-) F(x) rather than F(2)-F(1)? Could someone please explain this for me?
0
votes
0answers
49 views

What is the inverse Fourier transform of $|k|^{-\alpha}$?

What is the inverse Fourier transform, $\mathcal{F}^{-1}\{|k|^{-\alpha}\}$? I am specifically interested in the case where $1<\alpha<2$. To do this, I need to compute the integral ...
-1
votes
1answer
38 views

Find the volume generated! [on hold]

Find the volume generated by revolving the curve bound by $y=4-x^2$, the $y$-axis and $x$-axis about the $x$-axis using the disk method.
-1
votes
2answers
18 views

55/45% Partnership distribution [on hold]

Partner 1 owns 55% of a business and Partner 2 owns 45% percent of the business. If they sell the business for $ \$500,000$, how much more will Partner 1 make than Partner 2 of the $\$500.000$? What ...
0
votes
0answers
4 views

Legendrian isotopy

I think the definition of Legendrian isotopy is that you can find an isotopy of the ambient manifold such that it takes a Legendrian knot to another through Legendrian knots. What I cannot figure out ...
2
votes
1answer
43 views

How to prove that solution to ODE in spherical coordinate is equivalent to the ODE in cartesian coordinates if it is a thin shell

Solving a diffusion-type ODE across a spherical shell, the equation is: $$\frac{d}{dr}\left(r^2\frac{df}{dr}\right)=0\tag{1}$$ with boundary conditions $f(r_1)=f_1$ and $f(r_2)=f_2$. The solution is: ...
1
vote
1answer
24 views

For what values of $a > 0$ does $f(x,y)=(x^{2}+y^{a})^{-1} $ belong to $ L^{1}([0,1]^{2})?$

I am trying to understand for what values of $a>0$ the function $$f(x,y) = \frac{1}{x^2+y^a}$$ belongs to $L^1([0,1]^2)$. I think $a \geq 2$ should work. But how to show that it is not the case ...
3
votes
1answer
34 views

How to calculate the number of integer solution of a linear equation with constraints?

If an equation is given like this , $$x_1+x_2+...x_i+...x_n = S$$ and for each $x_i$ a constraint $$0\le x_i \le L_i$$ How do we calculate the number of Integer solutions to this problem?
0
votes
1answer
24 views

Linear operator on a hilbert space

Let $\{e_{n}\}_{n = 1}^{\infty}$ be an orthonormal basis of a Hilbert space $H$. Suppose $T: H \rightarrow H$ is a linear operator. For each $x \in H$, then $x = \sum_{n}\langle x, e_{n}\rangle e_{n}$ ...
1
vote
1answer
41 views

How many ways it can be done?

There are 5 chairs. Bob and Rachel want to sit such that Bob is always left to Rachel. How many ways it can be done ? I have solved this in the following way: seats: * * * * * ...
3
votes
1answer
42 views

Double integral proof, where is my mistake?

The bounds are 0 < x < b , 0 < y < b. $$ \int_0^b \int_0^b e^{-(x^{2}+y^{2})} dxdy $$ Since it is a square, x=y so we can write: = $$ (\int_0^b e^{-(x^{2}+x^{2})} )^{2} dxdy $$ = $$ ...
0
votes
1answer
21 views

Marginal densities of a joint distribution

Let $X$ and $Y$ be continuous random variables having joint density $f$ given by $f(x,y) = \lambda^2 e^{-\lambda y}, 0 \leq x \leq y$ and $f(x,y) = 0$ elsewhere. Find the marginal densities of $X$ ...
1
vote
1answer
23 views

Is a ruled surface of degree>2 always singular?

Let $X=\mathbb{C}\mathbb{P}^3$ and let $V\subset X$ be a closed algebraic sub variety. By V-is ruled, I mean that for every point in $V$ there is a line passing through it which also lies in V. ...

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