# All Questions

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### 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 $k$. ...
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### 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 ...
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### Kronecker product and outer product confusion

I have two column vectors: $$u = \left[\matrix{ 1 \cr 2\cr }\right]$$ $$v = \left[\matrix{ 4 \cr 4\cr }\right]$$ I'm trying to ...
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### 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 ...
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### 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.
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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} ... 3answers 44 views ### Integral properties **Hi, someone knows how make the procedure to reach, this, I dont undestand how integration limits are change ** 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 $$R = Z_{12} \text{ and } I = \{0,3,6,9\} ... 2answers 64 views ### integral of a product of functions being 0 Suppose we have a continuous function f on [a,b] such that for all integrable functions g such that \int_{[a,b]}g=0, \int_{[a,b]}fg=0 . Show that f must be constant. Well, it's clear ... 3answers 79 views ### Integral \int\frac{e^x \sin x}{\sinh x+\cosh x}dx$$\int\frac{e^x \sin x}{\sinh x+\cosh x}dx$$I know that the answer simplifies to -\cos x, but I have no idea how to start this question. Would I use integration by parts? u-substitution? 2answers 87 views ### Evaluate the sum \sum_{0\leq j < k\leq n}\binom{n}{j}\binom{n}{k} Could someone give me a hint on how to do this? I believe I know what the answer to be (I computed some low values and checked on OEIS). However, I was hoping someone would be able to explain to me ... 2answers 137 views ### Eliminate xy term of conic For the following problem, I am trying to eliminate xy, and I've tried numerous times to solve if with no luck. I need to find the general form of the equation rotated to eliminate the xy term. ... 1answer 304 views ### Distance between any two points in a compact metric space I am given the following problem: Show that if a metric space (X,d) is compact (meaning X is compact with respect to the metric d), then there exist points a,b ∈ X such that d(a,b) = ... 1answer 39 views ### limit of supremums define \limsup s_n = \limsup \{s_n : n > N\} i'm trying to figure out why lim sup s_n^2 = (\limsup s_n)^2 is false but lim sup s_n^3 = (\limsup s_n)^3 is true. can anyone give me a nudge in ... 1answer 43 views ### How to prove a function is in L_2(G) For G is open bounded set in \mathbf{R}^n, L_2(G) is defined as$$L_2(G) = \{f:G\rightarrow \mathbf{R} \text{ measurable}: \int |f|^2<\infty\}$$Now, I want to prove that f:\mathbf{R} ... 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} = 1to polar form? This is an ellipse centered at (1, -2). I understand the general approach for converting between the coordinate ... 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. 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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? $$\int_1^n \int_1^{\sqrt{y}} \frac{1}{x^4 + y^2} dx \, dy$$ ... 2answers 161 views ### Solve numerical system of nonlinear equations? I need to solve a nonlinear system of equations that looks like this ... 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 ... 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$A$... 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 ... 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? 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 } ... 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? 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 ... 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. ... 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. ... 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| ? 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 ... 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. 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 ... 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 ... 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}, ... 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 ... 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}? 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 ... 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. ... 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 ... 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 ... 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 ... 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 ...