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3answers
89 views

Making a space complete

If I have a homeomorphism between $X$ and $Y$. Suppose $X$ is complete and $Y$ is incomplete under the same metric, how can I make $Y$ complete using the fact that it is homeomorphic to $X$. Consider $...
0
votes
1answer
1k views

Find number of trials of the binomial distribution

The probability of a man hitting the target is 1/4, how many times he should shoot so that his probability of hitting the target at least once becomes 3/4.
2
votes
1answer
179 views

Representation of a subset of a finite affine space as a variety

It is simple to see that every subset of a an affine space over a finite field is a variety - for example, it follows from the fact that finite subsets are closed in the Zariski Topology of every ...
0
votes
1answer
222 views

What is the X, Y, Z “resolution” of a three-dimensional grid of points?

I came accross a software which requires the X, Y and Z resolution of a three-dimensional grid of points as Integer. What is a "3D grid resolution" and how do I find it? From what I understand, the ...
1
vote
1answer
209 views

What is the sufficient global/local convergence condition of inverse quadratic interpolation?

Given a root-finding function f(x)=0, what is the sufficient global/local convergence condition of inverse quadratic interpolation?
4
votes
3answers
1k views

Riemann sphere and Maps

Could somebody please clarify the following for me? I am not too clear about the relationship between the Riemann sphere and Möbius maps. I know that we can through projection make some Möbius maps ...
14
votes
1answer
712 views

What are the branches of the $p$-adic zeta function?

I'm reading the book $p$-adic Numbers, $p$-adic Analysis, and Zeta-Functions by Neal Koblitz. In it, Koblitz wants to iterpolate the Riemann Zeta function for the values $\zeta_p(1-k)$ with $k \in \...
1
vote
2answers
813 views

Find outline of $N$ points in a plane

If I have $N$ point coordinates $P_i = ( x_i, \, y_i ) $ and I want to draw the outline connecting only the points on the "outside", what is the algorithm to do this? This is what I want to do: Not ...
1
vote
1answer
59 views

Calculating interest-like problem

I'll make this short as this is a silly question. I have a number say 500. And a number n, which represents how many times I apply to my first number a transformation(add 20% to current value). This ...
0
votes
1answer
94 views

enumeration of subsets

Good evening. Let $\lambda>\omega$ a cardinal. We know that there is a bijection $\pi$ between $\lambda^+$ and $\lambda\times\lambda^+$. I don't understand in Remark 1 of the paper Shelah's proof ...
1
vote
2answers
149 views

Rolle's theorem for showing that $(x-a)^k$ divides $p(x)$ . . .

Have the following I'm stuck on: Suppose $p(x)=p_0+p_1 x+p_2 x^2+\cdots+p_n x^n$ is a polynomial of degree $n \geq 1$. Show that if $(x-a)^k$ divides $p(x)$ for some $a\in\mathbb R$ and some ...
5
votes
2answers
485 views

Analytic versus Analytical Sets

Browsing MathOverflow I came across a question about analytical sets. Through the discussion following a comment made by our very own Asaf, I learned that bold face $\mathbf{\Sigma^1_1}$ and light ...
1
vote
0answers
57 views

Are tempered representations unitarizabile?

Let $G$ be a locally compact, unimodular group and $Z$ be its center Clearly, square integrable representations with central unitary character is unitarizabile, since their matrix coeffecient imbed ...
5
votes
3answers
19k views

Find equation of a plane that passes through point and contains the intersection line of 2 other planes

Find equation of a plane that passes through point P $(-1,4,2)$ that contains the intersection line of the planes $$\begin{align*} 4x-y+z-2&=0\\ 2x+y-2z-3&=0 \end{align*}$$ Attempt: I found ...
7
votes
4answers
669 views

Evaluating $ \int_{-\infty}^{\infty}x\exp\left(-b^{2}\left(x-c\right)^{2}\right)\mathrm{erf}^{2}\left(a\left(x-d\right)\right)\,\mathrm{d}x $

I have big difficulties solving the following integral: $$ \int_{-\infty}^{\infty}x\exp\left(-b^{2}\left(x-c\right)^{2}\right)\mathrm{erf}^{2}\left(a\left(x-d\right)\right)\,\mathrm{d}x $$ I tried to ...
1
vote
1answer
2k views

Why are persistent states of a Markov chain on a finite state space non-null?

i would like to understand the following statement about Markov chains on a finite state space S: "If S is finite, then one state ist persistent and all persistent states are non-null." It is more ...
1
vote
1answer
407 views

closest point on a plane to another point in $\mathbb{R}^3$

Given $4$ points in $\mathbb{R}^3$: $A(0,2,4);B(-2,6,-2);C(2,-4,8);D(10,2,0)$, find the line equation $AK$ when $K$ is the projection of $D$ on the plane $ABC$. The first thing I did was find the ...
2
votes
1answer
3k views

Direction of arrows in a phase portrait.

If I have a system like: $\frac{dx}{dt} = x, \frac{dy}{dt} = -y+x^2$ and I am asked to draw a phase portrait, once I have found the type of portrait (saddle point, node, spiral, etc.) from the ...
6
votes
2answers
2k views

Proving the non-existence of a limit

Here's a homework question I'm trying to solve: Prove or disprove: if $\lim_af$ and $\lim_ag$ do not exist, then $\lim_a(f \cdot g)$ do not exist either. So I know that $$(\forall l\in\mathbb{R})(...
1
vote
1answer
164 views

Does division of polynomials give an increasing function?

How can I show that \begin{equation} f(a)=\frac{\sum_{i=1}^{k^*-1} \left(\begin{array}{c} K \\ i \\ \end{array} \right) \left(-1-\frac{1}{ar}\right)^i+1}{\sum_{i=1}^{k^*-1} \left(\begin{array}{c} K \\ ...
-1
votes
1answer
58 views

Loss estimation for replacement of stolen item [closed]

If an item of price $x$ is stolen from me or lost (assume that it is never going to be found again or returned to me), and I replace it by buying it again, but at its current price $y$, what is my ...
3
votes
1answer
208 views

Categorification of the (co-)induced topology

In second semester analysis we learned about the product topology which is quite easy to categorify using limits. However, we also learned about the coinduced topology $\mathfrak{V}$ induced by $f: X →...
11
votes
2answers
918 views

Showing that some symplectomorphism isn't Hamiltonian

I have the next symplectomorphism $(x,\xi)\mapsto (x,\xi+1)$ of $T^* S^1$, and I am asked if it's Hamiltonian symplectomorphism, i believe that it's not, though I am not sure how to show it. I know ...
3
votes
4answers
283 views

Factor 90301 without the aid of the computer

Computer break down easily know $90301=73\cdot1237$ Is there any way I want, without the aid of the computer to determine 90301 is a prime number or Composite number
10
votes
2answers
420 views

Example of a functor which preserves all small limits but has no left adjoint

The General Adjoint Functor Theorem (Category Theory) states that for a locally small and complete category $D$, a functor $G\colon D \to C$ has a left adjoint if and only if $G$ preserves all small ...
7
votes
3answers
418 views

Interesting puzzle about a sphere and some circles

Suppose I have a sphere and I choose a point $P$ on it. Then I draw $N\ge 3$ circles on the sphere passing through that point in a manner such that all the intersection points of the final result ...
7
votes
2answers
2k views

vector/tensor covariance and contravariance notation

As I looked over the Wikipedia article on covariance and contravariance of vectors and $\mathbf{v}=v^i\mathbf{e}_i$ is said as a contravariant vector while $\mathbf{v}=v_i\mathbf{e}^i$ is said as ...
5
votes
1answer
449 views

How to find $\int{\left(\frac{\sqrt{x+1}}{x-1}\right)^x}dx$?

I have tried to find $$\int{\biggl(\dfrac{\sqrt{x+1}}{x-1}\biggr)^x}dx$$ but I don't know how to do it, because it combines $u^x$ and $\dfrac{u}{v}$.
8
votes
1answer
3k views

number of combination in which no two red balls are adjacent.

given x spaces(you can fit 1 ball in 1 space) and unlimited number of identical red and white balls, find the total number of combinations in which no two red balls are adjacent to each other. i ...
12
votes
3answers
6k views

divergence of a vector field on a manifold

I've been asked to show the following: For a vector field $V$ on a semi-Riemannian manifold with metric $g$ that $$Div \cdot V = \frac{1}{\sqrt{\det(g)}}\partial_i\left(\sqrt{\det(g)}V^i\right)$$ I ...
1
vote
2answers
232 views

Taylor's formula

Taylor's Formula Write taylor's formula for $F(x,y)= \sin(x)\sin(y)$ using $a=0$, $b=0$, and $n=2$. $$\sin(h)\sin(k)=hk−\frac 16h(h^2+3k^2)\cos\theta h\sin\theta k−\frac 16 k(3h^2+k^2)\sin\theta h\...
7
votes
2answers
365 views

Sex distribution

Suppose there are N male and N female students. They are randomly distributed into k groups. Is it more probable for a male student to find himself in a group with more guys and for female student ...
0
votes
0answers
88 views

I'm interested in different meanings of “normal”~ [duplicate]

Possible Duplicate: What is it to be normal? I've learned in algebra class that "normal" means a linear operator is commutative with its adjoint; also we say that $H$ is a normal subgroup of $...
1
vote
2answers
3k views

Calculate whether two objects collide given their movement equations

$x_1 (t) = 2t +1$ and $y_1 (t) = 4t^2$ $x_2(t) = 3t$ and $y_2 (t) = 3t$ How to calculate whether $x$ and $y$ is collide? Or, in which way I can calculate this? (I do not need the actual ...
0
votes
1answer
127 views

Simple question about an asymptotic equality

Could someone please explain the second equality in Conjecture 1.1: http://arxiv.org/pdf/math/0501313v2.pdf ? (reproduced below) $(1+o(1))n^22^{1-n}=\left(\frac{1}{2}+o(1)\right)^n$ Initially, I ...
11
votes
6answers
14k views

Book recommendations from basic algebra to precalculus?

I graduated high school a while ago, hardly remember anything and have no idea where to begin relearning.
1
vote
0answers
57 views

cubic forms with a rank condition

Let $C(\boldsymbol{x})$ be a cubic form in $n$ variables $\boldsymbol{x}=(x_1,...,x_n)$ over $\mathbb{Z}$. We can write $C(\boldsymbol{x})=\sum_{i,j,k=1}^{n}c_{ijk}x_ix_jx_k$, where the $c_{ijk}$ are ...
5
votes
2answers
251 views

How can I write an algorithm to perform the following calculation exactly? (references accepted)

Given natural numbers $N, K, m, C$, with $3^{m/3}K>C$, I want to be able to write an algorithm to exactly compute the number $$ \left\lceil \log_3 \left(\frac{N}{3^{m/3}K-C}\right) \right\rceil $$ ...
3
votes
1answer
385 views

Expectation of $QQ^T$ where $Q^TQ=I$

It's exercise 1.1 on p.2 of this book. The goal is to is to show that, for some random matrix $Q \in \mathbb{R}^{n\times k}$ where $k<n$ and the columns of $Q$ are orthogonal (i.e. $Q^T Q = I$; ...
1
vote
1answer
10k views

Coterminal Angles?

I understood coterminal angles as angles that have the same terminal angle value. By this logic, why aren't 135 and 315 coterminal? They both have a terminal angle of 45. Is my interpretation of ...
4
votes
0answers
524 views

Can a rectangle be written as a finite almost disjoint union of squares?

Given a closed rectangle $R$ are there closed squares $(S_{i})_{1\leq i\leq n}$ such that $R=\underset{1\leq i\leq n}{\cup}S_{i}$ and $S_{i}^{\circ}\cap S_{j}^{\circ}=\varnothing$ for $i\neq j$ ?
11
votes
1answer
87k views

Calculate the vector normal to the plane by given points

How can one calculate the vector normal to the plane that is determined by given points? For example, given three points $P_1(5,0,0)$, $P_2(0,0,5)$ and $P_3(10,0,5)$, calculate the vector normal to ...
1
vote
1answer
819 views

The exponent of self-adjoint operator

If $X$ is a Hilbert space and $A$ is an unbounded self-adjoint operator on $X$, is it necessarily that $A^2$ is self-adjoint as well?(admittedly, $A^2$ is densely defined)
1
vote
2answers
2k views

Tensor calculus - Christoffel symbol of the second kind

and I understand these parts up there, but I cannot understand how the second formula of the last equality leads to the third formula. Can anyone show me what relabeling indices rules are used to ...
1
vote
1answer
72 views

5-variable polynomial, constant in 1 variable

I have a polynomial function $f(x_1,x_2,x_3,x_4): \mathbb{C}^4 \to \mathbb{C}$, which obeys the equality $f(x_1+tx_3,x_2+tx_4,x_3,x_4) = f(x_1,x_2,x_3,x_4)$ for all $t \in \mathbb{C}$. My question ...
11
votes
1answer
326 views

Existence of an entire function with algebraically independent derivatives

Let $\mathbb{A}$ be the algebraic closure of $\mathbb{Q}$ in $\mathbb{C}$. A collection of functions $F=\lbrace f_i:X \rightarrow\mathbb{C}\rbrace$ is said to be algebraically independent over $\...
11
votes
3answers
968 views

Ramanujan Summation

It seems that under the light of Ramanujan Summation the following is plausible: $$1 + {2^{2n - 1}} + {3^{2n - 1}} + \cdots = - \frac{{{B_{2n}}}}{{2n}}(\Re)$$ Alas, I can't really find any ...
2
votes
0answers
2k views

Two Disk/Washer Method Problems (given a diagram)

Given a diagram from Calculus of a Single Variable by Larson and Edward (9th edition): I am interested in finding the volume of various regions when rotated about various lines. Specifically, I am ...
1
vote
0answers
703 views

mean of an arbitrary distribution

Is it possible to find the mean or center of a continuous arbitrary distribution. Assuming that an object O is arbitrarily distributed within arbitrary shape, can we find its mean or center ...
6
votes
1answer
271 views

Determining the Length of a Curve Using Partitions

I have encountered the following problem: Let $f$ be continuous on $[a,b]$. Define the length of $f$ on $[a,b]$ by $$l=\sup_P[\lambda_P(f)],$$ where $$\lambda_P(f)=\sum_{k=1}^N\sqrt{(x_k-x_{k-...

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