1
vote
2answers
63 views

fibered knots in $ S^3$

Given a fibered knot $k$ in $S^3$, we have the decomposition of $S^3$ as union of $M$ and $S^1\times D^2 $, where M is a fiber bundle over $S^1$, with fiber $F$ such that its boundary is the knot $k$. ...
4
votes
1answer
79 views

How do topologists count infinite dimensional holes?

For example, it seems like there "should" be an infinite dimensional hole (or perhaps many) in $S^1 \times S^1 \times \ldots$. (Or perhaps none...) Is there an invariant that would count it? What ...
4
votes
1answer
760 views

Kronecker product and outer product confusion

I have two column vectors: \begin{equation} u = \left[\matrix{ 1 \cr 2\cr }\right] \end{equation} \begin{equation} v = \left[\matrix{ 4 \cr 4\cr }\right] \end{equation} I'm trying to ...
1
vote
3answers
46 views

I need help understanding a simple concept that I can't seem to get with a certain type of inequality

I've got this inequality: $$\dfrac{1}{x}-\dfrac{x}{2x-1}\geq1$$ And the solutions are supposed to be $(0,1/2)$. But why? Whenever I make my inequality table, it ends up telling me the solution is ...
2
votes
2answers
187 views

Matrix equation $X^2 + E = 0$

Find all real-valued $n\times n$ matrices $X$ such that $$ X^2 + E = 0, $$ here $E$ --- identity matrix, $n$ --- odd.
0
votes
1answer
42 views

Equivalence to being a scattered space.

Let $X$ be a topological Hausdorff space with Cantor-Bendixson rank $\lambda$. A paper I am reading claims that these two are equivalent: $X$ is scattered. $X = \cup_{\alpha < \lambda} ...
3
votes
3answers
44 views

Integral properties

**Hi, someone knows how make the procedure to reach, this, I dont undestand how integration limits are change **
0
votes
1answer
416 views

Addition and Multiplication table for Ring/Ideal

I'm not sure if it's possible to show it here, but how would the addition and multiplication table look like for R/I (where R is rings with ideal I) when $$ R = Z_{12} \text{ and } I = \{0,3,6,9\} ...
1
vote
2answers
64 views

integral of a product of functions being $0$

Suppose we have a continuous function $f$ on $[a,b]$ such that for all integrable functions $g$ such that $\int_{[a,b]}g=0$, $\int_{[a,b]}fg=0 $. Show that $f$ must be constant. Well, it's clear ...
1
vote
3answers
79 views

Integral $\int\frac{e^x \sin x}{\sinh x+\cosh x}dx$

$$\int\frac{e^x \sin x}{\sinh x+\cosh x}dx$$ I know that the answer simplifies to $-\cos x$, but I have no idea how to start this question. Would I use integration by parts? u-substitution?
0
votes
2answers
87 views

Evaluate the sum $\sum_{0\leq j < k\leq n}\binom{n}{j}\binom{n}{k}$

Could someone give me a hint on how to do this? I believe I know what the answer to be (I computed some low values and checked on OEIS). However, I was hoping someone would be able to explain to me ...
0
votes
2answers
137 views

Eliminate xy term of conic

For the following problem, I am trying to eliminate xy, and I've tried numerous times to solve if with no luck. I need to find the general form of the equation rotated to eliminate the xy term. $ ...
1
vote
1answer
304 views

Distance between any two points in a compact metric space

I am given the following problem: Show that if a metric space (X,d) is compact (meaning X is compact with respect to the metric d), then there exist points a,b ∈ X such that d(a,b) = ...
0
votes
1answer
39 views

limit of supremums

define $\limsup s_n = \limsup \{s_n : n > N\}$ i'm trying to figure out why lim sup $s_n^2 = (\limsup s_n)^2$ is false but lim sup $s_n^3 = (\limsup s_n)^3$ is true. can anyone give me a nudge in ...
0
votes
1answer
43 views

How to prove a function is in $L_2(G)$

For $G$ is open bounded set in $\mathbf{R}^n$, $L_2(G)$ is defined as $$L_2(G) = \{f:G\rightarrow \mathbf{R} \text{ measurable}: \int |f|^2<\infty\}$$ Now, I want to prove that $f:\mathbf{R} ...
1
vote
2answers
67 views

How can I convert $\dfrac{(x – 1)^{2}}{16} +\dfrac{(y + 2)^{2}}{9} = 1$ to polar form?

How can I convert $\frac{(x – 1)^{2}}{16} + \frac{(y + 2)^{2}}{9} = 1$to polar form? This is an ellipse centered at $(1, -2)$. I understand the general approach for converting between the coordinate ...
0
votes
1answer
102 views

Breaking even: is there a point where you can always profit?

If say a tennis match is paying 7:1 on Player X and 2:1 on Player Y can I place certain amounts on each to break even? Or even profit? How can I calculate that? EDIT Sorry, X is playing Y.
0
votes
2answers
64 views

Understanding a Similarity Transformation with a Fixed Point from the Definition of the Derivative for Complex Functions

My book has given me the definition of the derivative for complex functions in several ways but points to one in particular for its geometric aid, I quote: In this formulation ...
0
votes
1answer
101 views

Solutions of SDE do not explode when drift term is zero.

Suppose we have $dX_t = \sigma(X_t) dW_t$ where $\sigma : \mathbb{R} \rightarrow \mathbb{R}$ is Borel and $W_t$ is a standard one-dimensional Brownian motion. I am trying to show that $X_t$ cannot ...
1
vote
2answers
642 views

What is the geometric meaning of the transformation of R2/R3 when every vector is multiplied by −1? Is it a rotation?

I'd imagine a sphere with the center at the origin and all length of the vectors equals the radius. But I can't imagine what would happen if all the vectors is multiplied by -1, what would it be? I ...
5
votes
1answer
119 views

Every metrizable Toronto space is discrete.

$X$ is a Toronto space if for every $Y \subseteq X$ such that $|Y|=|X|$ then $Y$ is homeomorphic to $X$. I am trying to prove that every metrizable Toronto space is discrete. I have the following ...
1
vote
1answer
81 views

Finding extrema with multiple constraints without Lagrange multipliers

Find the maximums and minimums of $z = 15x+14y$ with constraints $0 \leq x \leq 10, 0 \leq y \leq 5, 3x+2y \geq 6$ I obviously can't take the partial derivatives of inequalities, so I'm at a loss ...
0
votes
1answer
54 views

Why do we define cosets of the kernel this way?

My book defines cosets of kernel of a homomorphim $\phi$ as $$ aH = Ha = \{ x \in G: \phi(x) = \phi(a)\}.$$ When I tried to define the coset myself before looking at the proof, I defined it as $$aH ...
2
votes
3answers
66 views

Solving cubic equation?

I am trying to figure out the following cubic root thing. $ax^3+bx^2+cx+d=0$ I set up $x=y-\frac{3}{ba}$ Then I plug in for x $a(y-\frac{3}{ba})^3+b(y-\frac{3}{3a})^2+c(y-\frac{3}{ba})=0$ The ...
3
votes
1answer
75 views

Calculating Integral help

I am currently trying to calculate this integral but it is giving me quite a bit of trouble. Would you guys mind giving me suggestions? $$ \int_1^n \int_1^{\sqrt{y}} \frac{1}{x^4 + y^2} dx \, dy$$ ...
4
votes
2answers
161 views

Solve numerical system of nonlinear equations?

I need to solve a nonlinear system of equations that looks like this ...
3
votes
2answers
124 views

Universal cover of a torus “pillow”

I was thinking today, what is the universal cover of a torus with the "donut hole" shrunk to a point? I am certain it must include a sphere, but that can't be enough because of the point at the ...
1
vote
2answers
48 views

Adjoint problem eigenvalue

Here is a problem I am stuck on even though I suspect the solutions is fairly straightforward: Show that if $\lambda$ is an eigenvalue of the problem $$ Ax = \lambda x $$ for some square matrix $A$ ...
2
votes
1answer
92 views

If $\Gamma^k_{ij}(p)=0$, then $\nabla_{E_i}E_j (p)=0?$

I'm having the same problem as it was questioned here. I can't get throught the step where I need to show that $\nabla_{E_i}E_j (p)=0$. It only leads to $$ \nabla_{E_i}E_j(p)=\sum_{lk}^n ...
-1
votes
2answers
47 views

What is the value of x?

I tried solving it but I just don't get the answer, does anybody know what the answer is or how to solve it?
1
vote
1answer
34 views

Prove that if (X, $\mathcal{A}$, $\mu$) is a measure space, $f$ is measureable $/iff$ $f^+$ and $f^-$ are measurable

Prove that if (X, $\mathcal{A}$, $\mu$) is a measure space, $f$ is measureable $\iff$ $f^+$ and $f^-$ are measurable$\mathcal{A}$ Where $f^-(x) = \left\{ \begin{array}{ll} 0 & \mbox{if } ...
0
votes
3answers
241 views

A field is a nonzero commutative ring …

I was confused when I read this statement. I thought a Ring must have the additive inverse $0$. Does the statement imply that there is no zero in a field?
0
votes
1answer
71 views

Adaptation of this proof of spectral theorem to the complex case

My question is quite simple, I would like to know why we can't use this proof to the complex case, i.e., the operator $T$ is self adjoint on a complex n-dimensional inner product space $V$. Can we ...
0
votes
1answer
14 views

Confidence intervals review

It is known that monthly spending by individuals in a community is normally distributed. To estimate the average monthly spending, a statistician asked 4 people randomly about their monthly spending. ...
1
vote
1answer
40 views

Approximating zeros on an interval

I'm writing a program for my AP Calculus class, and I'm trying to write an equation solver that approximates the zeros of functions. Right now it can take symbolic derivatives and evaluate functions. ...
1
vote
4answers
116 views

How to prove U•V = |U|•|V|cos(θ), if θ is the angle between |U| and |V| [duplicate]

This is a snippet from my book. How did they get from |U|$^2$ = U • V = |U|•|V| |U|/|V| ?
2
votes
2answers
115 views

Proving $\prod_{i=1}^np_i+1$ is not a perfect square

Let $m=\displaystyle{\prod_{i=1}^np_n}$ be the product of the first $n$ primes $(n>1)$. prove that $m+1$ cannot be a perfect square. I think that the opposite it correct: $m+1$ is not a ...
0
votes
3answers
82 views

PUGS is a rectangle. If the equation of PU is $y=\frac{2}{3x} + 4$. What is the slope of SP?

I don't get the answer to this problem, can somebody please tell me what the answer is.
1
vote
0answers
54 views

Prove that eventually Hannah and the other swimmer will settle into a pattern where they pass each other (Please refer to the context in my question)

From the 2014 Mathcamp quiz: Hannah is about to get into a swimming pool in which every lane already has one swimmer in it. Hannah wants to choose a lane in which she would have to encounter the other ...
0
votes
1answer
50 views

Change of Basis - Homework Question

Please help me understand what is being asked, I feel I am missing something. Compute the change of basis matrix for each of the bases, and use it to find the coordinate vector v with respect to B ...
6
votes
1answer
162 views

Is $\operatorname{Hom}_\mathbb{Z}(\mathbb{Q},\mathbb{Q}/\mathbb{Z})\cong\bigoplus_p\mathbb{Q}_p$?

Is $\operatorname{Hom}_\mathbb{Z}(\mathbb{Q},\mathbb{Q}/\mathbb{Z})\cong\bigoplus_p\mathbb{Q}_p$? Or maybe $\prod_p\mathbb{Q}_p$? I know $\mathbb{Q}/\mathbb{Z}\cong\bigoplus_p \mathbb{Z}_{p^\infty}$, ...
0
votes
1answer
44 views

Proving group is $p$-group by contradiction

http://www.proofwiki.org/wiki/Group_is_P-Group_iff_All_Elements_have_Order_Power_of_P Is $k$ a prime or a prime power? Sorry for this stupid question but I can't tell what $k$ is in this context ...
1
vote
1answer
53 views

Uniform distribtion: clarification of $f_X(x)$

I have $Y=2(X-1)^2 -1$ where $X$ is uniform distributed on $[0,2]$ I want to find the pdf of $Y$ and expected value of $Y$. My question is just: Does $X$ have pdf $f_X(x)= \frac{1}{2}$?
5
votes
1answer
116 views

$\lim_{x\to 2} \, \sqrt{x-2}$

$$\lim_{x\to 2} \, \sqrt{x-2}$$ When you take the right hand limit for this expression, you get $0$. However, if you take the left hand side it gives an imaginary number. However, do you consider ...
1
vote
1answer
48 views

Question on Normal Coordinates

I'm having a hard time trying to understand something that I'm suspicious is pretty stupid. I'll refer to Wikipedia to settle the term's I'll refer to. ...
1
vote
6answers
72 views

Proving that all eigenvalues are $0$ bar one.

A symmetric $n\times n$ matrix is given by $\mathbf{A}=\lambda \mathbf{ee}^{\text{T}}$ where $\mathbf{e}$ is a unit vector. Show that $\mathbf{A}$ has an eigenvector of $\mathbf{e}$ with ...
1
vote
0answers
57 views

Do signs matter in SVD?

I have written an algorithm to compute the SVD of a 2x2 matrix. I was checking against a Mathematica query, and I noticed that the signs in the $U$ and $V$ matrices do not match those from my ...
1
vote
1answer
104 views

Topology of $GL_n(K)$

I need to show any of the following results: Consider $K=\mathbb{R}$ or $\mathbb{C}$, then, 1) The compact-open topology and the usual topology of $GL_n(K)$ are the same. 2) Taking inverses and ...
1
vote
3answers
108 views

How to find sum of factors of $2^{2012}$?

This question really is confusing me and I was wondering if there was a simple way this could be achieved. I've come up with this so far after skimming through a few articles on the net. I assumed ...
0
votes
2answers
30 views

How to get **the product** of two matrices AB if they are placed into a **determinant** ?

I just got stuck with a question if there is : det(AB) .. and det(A)= 4 .. det(B)=5 the question is how to get the product of the two matrices AB inside the determinant ?

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