2
votes
0answers
222 views

Is the product of Borel spaces a Borel space?

Let $(S_i,\mathbf{S}_i)$ be a sequence of Borel spaces, i.e. such that for all $i$ there is a 1-1 bimeasurable map $\varphi_i:S_i\to T_i$, where $T_i$ is a Borel subset of [0,1]. Is $\prod_{i=1}^n ...
3
votes
0answers
233 views

Understanding orientability of vector bundles

I'm having trouble understanding how orientability of vector bundles work. The book I'm reading, Spivak's A comprehensive introduction to differential geometry, is not very clear on this. Edit: ...
2
votes
2answers
324 views

is any subset of a manifold a submanifold?

by definition a submanifold is a subset of a manifold which is itself a manifold. consider $A$ a subset of an $n$-manifold $M$. a neighborhood of $x\in A$ is $\mathbb R^n$ since $x$ is an element of ...
5
votes
1answer
773 views

Career advice for MSc [closed]

I got into an interesting situation and I would love if you could help me with an advice, opinion or something. I have an offer for an MSc study (one year) in applied mathematics (with a scholarship) ...
4
votes
1answer
554 views

What is the connection between the definition of complete intersection variety and complete intersection ring?

An algebraic variety is called a complete intersection if its defining ideal is generated by codimension many polynomials. A Noetherian local ring $R$ is called a complete intersection if its ...
0
votes
1answer
325 views

Complexity of $T(n)=\sqrt{n}T(\sqrt{n})+n$

I tried to find the complexity of this recursion equation: $T(n)=\sqrt{n}T(\sqrt{n})+n$, by doing couple of iterations and getting a general idea, but I completely got lost. I'd really love your ...
56
votes
2answers
4k views

Open problems in General Relativity

I would like to know if there are some open mathematical problems in General Relativity, that are important from the point of view of Physics. I mean is there something that still needs to be ...
2
votes
4answers
221 views

Need help in Taylor series expansion

In this question, I have to write Taylor's series expansion of the function $f(x) = ln(x+n)$ about x = 0, where n ≠ 0 is a known constant. I have done the following: But my professor handed me back ...
0
votes
1answer
277 views

non transitive subgroups of the symmetric group

How to show that any non transitive subgroup of the symmetric group $S_n$ is up to conjugation contained in a Young subgroup $S_k\times S_{n-k}$? Take $\mathbb Z_3$ the subgroup of $S_3$ generated by ...
0
votes
1answer
64 views

Conditions for some inequality

Suppose I have $f(x)A+g(x)B+h(x)C \ge 0$. Here $A,B,C$ can be positive or negative and $f,g,h$ are nonnegative. I would like to obtain a condition for $f,g,$ and $h$ such that ...
1
vote
1answer
121 views

$T_{3}=\Theta(n^{0.99}) ,T_{2}=\Theta(n^{\log\log n}),T_{1}=\Theta\left(\frac{n}{\log n}\right)$

$T_{3}=\Theta(n^{0.99}),\quad T_{2}=\Theta(n^{\log\log n}),\quad T_{1}=\Theta \left(\frac{n}{\log n}\right)$ I need to decide what is the relation (ratio?) between $ T_{1},\, T_{2},\, T_{3}$? So by ...
10
votes
3answers
740 views

Proof that $\dim(U \times V) = \dim U + \dim V$.

The following theorem in Serge Lang's Linear Algebra is left as an exercise, namely, Let $U$ and $V$ be finite dimensional vector spaces over a field $K$, where $\dim U = n$ and $\dim V = m$. Then ...
10
votes
1answer
560 views

Prove that minimum of $\lambda \sin \theta + (1 - \lambda) \cos \theta \le -\dfrac{1}{\sqrt 2}$

I need a little nudge to the finish for the last bit of this problem. Express $\lambda \sin \theta + (1 - \lambda) \cos \theta$ in the form $R \sin (\theta + \phi)$, where $R(R>0)$ and $\tan ...
0
votes
2answers
158 views

Combinations Help

I have an application where I iterate through all k-combinations of a set of size n. For example here I have listed all k-combinations for when n is 4. Also I have separated each list of combinations ...
8
votes
1answer
396 views

Numb3rs Challenge

I am by no means a mathmatician. I guess you could say that I am a mathematically inclined individual, but never made anything of it until I was in my 30's and became a software engineer. Although ...
1
vote
1answer
84 views

Simple estimation $e^{a\sqrt{r}} > r$

I want to prove a simple theorem about contour integration via residues and I need the following estimation: $e^{a\sqrt{r}} > r$ for any real a > 0 and r >> 0. Is this true? If so, what is an ...
4
votes
2answers
428 views

True, false, or meaningless?

Are the following two assertions always true, always false or meaningless? $\exists i \in \emptyset$ $\forall i \in \emptyset$ Because it seems that one encounters expressions of this kind fairly ...
3
votes
2answers
8k views

Sum of n consecutive numbers [duplicate]

Possible Duplicate: Proof for formula for sum of sequence $1+2+3+\ldots+n$? Is there a shortcut method to working out the sum of n consecutive positive integers? Firstly, starting at $1 ...
0
votes
2answers
197 views

Convert number from one base to another

I have the next number: $$5.2880001*10^{-4}$$ Now I want to convert this number to a format of $X*2^y$. How I do it? Thank you.
4
votes
2answers
276 views

Implications of non unique factorization in diophantine equations

I am after a concrete example in which under the below conditions, $x+k\sqrt{-d}$ in $Z[-\sqrt{d}]$ is not a cube or an associate of a cube. Suppose that $Z[-\sqrt{d}]$ ($d$ squarefree) is a non ...
1
vote
2answers
145 views

Finding a random vector exactly yay far from another point in 3D space

So I am trying to find a vector a certain distance away from another point ( the distance varies based on an input ) and I've figured out that ...
0
votes
1answer
149 views

Compositions of prime numbers

This question is related to numbers found in the OEIS sequence A191837. In this sequence, $a(2) = 48 = 5 + 7 + 17 + 19$, where the summands of 48 are all prime numbers that are less than or equal to ...
0
votes
2answers
163 views

If X is Erlang$(k_1,\lambda)$ and Y is Erlang$(k_2,\lambda)$, then is X+Y Erlang$(k_1+k_2,\lambda)$?

If X is Erlang$(k_1,\lambda)$ and Y is Erlang$(k_2,\lambda)$, then is X+Y Erlang$(k_1+k_2,\lambda)$? Do X and Y need to be independent?
10
votes
5answers
2k views

Motivating linear algebra for economics students?

I'm a tutor for the introductory linear algebra course at my school; this course is required for most upper division economics classes, so a lot of my tutees are economics majors. This is a typical ...
2
votes
1answer
89 views

Arrangements of congruent rectangles

I have stumbled on an interesting problem. How many congruent rectangles on a plane can be arranged in such a way that they all touch each other but never overlap? I pondered this problem for a few ...
4
votes
3answers
283 views

Is it practical to use infinite continued fraction to generate random numbers?

I observed the pattern of this irrational number: $$\sqrt{1 + \sqrt{2}}$$ and realized that each element $a_i$ occurred very randomly. For the first 100 elements, this is the result: ...
2
votes
1answer
240 views

An exercise for weak convergence

Recently, I found an exercise in Hunter's Applied Analysis(last page in the link), which may be closely related to the question I raised two months ago. Consider heat flow in a rod with rapidly ...
4
votes
1answer
150 views

Geometry of number

This question seems not very hard, but it is starting to embarrass me. So I thought I can use your ideas to solve it, and I would be thankful in advance. Let $K/\mathbb{Q}$ be a number field of ...
2
votes
1answer
157 views

Need help in deriving condition for quartic to have only one double root

Given a polynomial of degree four: $ax^4+bx^3+cx^2+dx+e$, with $a,b,c,d,e$ real and $a\neq 0$, how do I derive the condition for there to be exactly distinct 3 real roots (i.e., one root is repeated)? ...
9
votes
2answers
818 views

Integration of powers of the $\sin x$

I have to evalute $$\int_0^{\frac{\pi}{2}}(\sin x)^z\ dx.$$ I put this integral in Wolfram Alpha, and the result is ...
23
votes
4answers
4k views

Teaching Introductory Real Analysis

I am currently helping teach an introduction to real analysis course at UC Berkeley. The textbook we are using in Rudin's "Principles of Mathematical Analysis" (aka baby rudin). I am trying to find ...
1
vote
2answers
406 views

Engineering Economics - Annual Cost Question

My question is as follows: 1.) An earth compactor costs $38,000 and has an economic life of 9 years. However, the purchaser needs it for only 1 project that will be completed in 3 years. At the end ...
2
votes
2answers
294 views

Closed-form Expression for $\sum_{j=0}^{k-1}(2j+2)\sum_{i=1}^j \frac 1 {i^2}$? (problem with Mathematica)

I need to calculate a closed-form expression for $\sum_{j=0}^{k-1}(2j+2)\sum_{i=1}^j \frac 1 {i^2}$. This isn't particularly difficult, and I do it by hand pretty much routinely. However I found out ...
1
vote
1answer
117 views

$(a_{1}+ a_{2} + …+a_{k})^{n}$ where $k >2$, what does it generate?

Binomial expansion generates the Pascal triangle but what does it generate when you have different amount of terms there? You can see here the geometric generation with only 2 terms. I am interested ...
3
votes
3answers
440 views

calculus essay assistance

I am writing an essay for my calculus class and one of the requirements to meet within the essay is to demonstrate an understanding of integration by explaining a metaphor that uses integration. ...
2
votes
2answers
153 views

How do I calculate $I = \int\limits_{t_a}^{t_b}{\left(\frac{d{x}}{d{t}}\right)^2dt}$?

$$I = \int\limits_{t_a}^{t_b}{\left(\frac{d{x}}{d{t}}\right)^2dt}$$ $x_a = x(t_a)$ and $x_b = x(t_b)$ I haven't integrated anything like this since a long time. Lost my powers of integration. How ...
8
votes
2answers
729 views

Characterize entire functions $f$ such that $|f(z)| \leq |\sin(z)|$

I want to determine all entire functions $f$ such that $|f(z)| \leq |\sin(z)|$. I searched around on MathSEx and I found the following question from which I tried to get inspired but I think it ...
1
vote
1answer
144 views

Are satellite knots prime?

Which satellite knots are prime? I do know that connected sum of knots is a satellite operation, but I found this statement: "the satellite knots all have structures which are well known and ...
1
vote
2answers
150 views

how do we prove this calculus statement?

Prove that if we write $z = re^{i\theta}$, then $d$ for derivative, $$dz=e^{i\theta}\,dr + ire^{i\theta}\,d{\theta}$$ (reference/context)
7
votes
0answers
165 views

$(\mbox{Sh,Sh-map})$ represents the category of sheaves on a stack

I'm trying to understand the following theorem, and I think I don't understand the definitions. Let $(\mathcal{C},J)$ be a site (with a subcanonical topology). Write $\mathcal{C}/X$ for the groupoid ...
1
vote
2answers
232 views

What is an efficient way to get blur from source and blurred images?

I'm doing little program to get blur from source image and blurred image. But I haven't learned so much things about math in school yet. The equation used for blurring the image A into B: ...
1
vote
2answers
131 views

Solving linear equations using matrix

I'm doing little program, where I need to solve multiple linear equations like: $$\begin{align*} B_1 &= A_{11}C_1 + A_{12}C_2 + A_{13}C_3 + A_{14}C_4\\ B_2 &= A_{21}C_1 + A_{22}C_2 + ...
1
vote
1answer
181 views

What is meant by “direct summand in a tensor product”?

I am currently working on the topic of Lie - Algebras and I have stumbled a few times over the expression "direct summand in a tensor product". The text says that $\ V(\lambda) $ as an ...
0
votes
1answer
179 views

What is meant by $\sum_{p + q = v + w} {\dim V_p * \dim W_q}$?

I am currently working on the topic of Lie - Algebras. What is meant by $\displaystyle\sum_{p + q = v + w} {\dim V_p * \dim W_q}$ ? $\ V_v $ and $\ W_w $ denote weight spaces I don't know how to ...
2
votes
1answer
491 views

Use of parenthesis in lambda calculus

As a summer project I am trying to learn lambda calculus. I am not that good with math but I have learned myself several programming languages and somehow got the idea that learning lambda calculus ...
3
votes
1answer
128 views

What to call the initial members of an ordered set?

If I have an ordered set X = {a, b, c} and another ordered set Y = {a, b}, I know that that ...
0
votes
1answer
448 views

convolution of random variables

I need to compute the pdf of the sum of a bunch of random variables $$\sum_{i=0}^{k-1} c_i X_i $$ where $X_i \sim 2\Omega x e^{-\Omega x^2}$, $\Omega > 0$ is a parameter and $c_i$ are positive ...
11
votes
2answers
1k views

The Hessian of the Determinant

It is well known how to take the derivative of the determinant: let $A(s)$ be a family of square matrices smoothly parametrised by the variable $s$ (in other words, $A:\mathbb{R}\to \mathbb{R}^{N^2}$ ...
9
votes
2answers
508 views

What's the difference between “duality” and “symmetry” in mathematics?

Motivated by the answer to this question--"What kind of “symmetry” is the symmetric group about?", I read the article about dual graph. It is said in this article that "the term 'dual' is used because ...
19
votes
4answers
877 views

Homotopy groups of $S^2$

I'd like to understand higher homotopy groups better and I guess there's no simpler way than understanding them for as simple spaces as possible; therefore $S^2$. My question essentially has two ...

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