All Questions
0
votes
0answers
11 views
Simultaneous equations for more unknowns than equations
I have a system of equations. This system is 8 equations for 16 unknowns. Is it possible to solve the equations for some answer? By that, I mean, it's highly likely that they will have several ...
2
votes
1answer
13 views
Solving matrices for certain variables using Cramer's rule
I have the following matrix equality:
$$\left(
\begin{array}{ccc}
u_{11} & u_{12} & -p_1 \\
u_{21} & u_{22} & -p_2 \\
p_1 & p_2 & 0
\end{array}
\right).\left(
...
0
votes
1answer
12 views
Probability of random shuffling of cards
I have a pack of cards and use the following method to shuffle them
Pick a random card from the deck and replace the first card with it
Put the first card back in the deck
Move to the second card ...
0
votes
1answer
15 views
Trigonometric limit solution
Why does the following limit equals 2:
$$\lim_{x \to 0}\frac{2x^2}{\sin^2 x}=2$$
I can't find a trigonometric conversion to get that result.
0
votes
0answers
10 views
is the following correct for tensor products?
Let's say that I have a three dimensional tensor $A = ((B \times_1 C_1) \times_2 C_2) \times_3 C_3$. where $B$ is $n \times n \times n$ tensor, $C_i$ are $n \times n$ matrices and $A$, as a result, is ...
0
votes
2answers
23 views
In what case this equation has a solution(s)?
I am trying to solve this problem saying, for what values of $a$ and $b$ (in which they are not zero), the equation $a(1-\sin(x))=b(1-\cos(x))$ has any solution(s)? Thank you very much. :)
1
vote
1answer
21 views
What does it mean when a function is finite?
When someone says a real valued function $f(x)$ on $\mathbb{R}$ is finite, does it mean that $|f(x)| \leq M$ for all $x \in \mathbb{R}$ with some $M$ independent of $x$?
1
vote
0answers
11 views
Algorithm and solver for large, dense, positive-semidefinite integer QP
I am interested in the solutions of a very large quadratic programming (QP) problem
\begin{align}
\min_{x \in \mathbb{R}^n} & x^T Q x\\
\mathrm{subject\ to} & A x = b\\
& x \in \{0,1\}^n
...
3
votes
1answer
50 views
Algebraic Solution to $\cos(\pi x) + x^2 = 0$
Today I was fiddling about with a TI-89 calculator, attempting as usual to confuse it. I figured that making it solve an equation with a periodic function would be fun, so I tried the following:
...
1
vote
0answers
9 views
Solving two-parameter linear recurrence with different initial values
I'm looking for a method to solve the well-known recurrence ralation with different initial values. The one-parameter linear recurrence is easier. Let's take a peek:
$a_0 = \alpha$; $a_1 = \beta$; ...
4
votes
1answer
25 views
Set of accumulation points is closed for every subset
I was trying to prove the following statement:
Let $(X,\mathscr{T})$ be a topological space. If the set of accumulation points of $\{x\}$ is closed for every $x\in X$, then the set of accumulation ...
0
votes
0answers
16 views
Functors preserve isomorphisms, sections and retracts. Do they reflect these properties?
Functors preserve isomorphisms, sections and retracts. Which of these properties, if any, do functors reflect?
Are there other fundamental properties preserved and or reflected by functors?
Please ...
3
votes
1answer
23 views
Dance couples riddle
Imagine there are $5$ women and $5$ men on a disco. Two of the gals have two brothers within the guys.
(image shows sibling relations)
In how many different ways can female-male couples be ...
3
votes
1answer
44 views
How would I go about proving this?
Sorry, this is very much 'can you do my homework' but I have a little competition at work that requires me to solve (and prove) the following.
Find all positive integers $L$, $M$, $N$ such that ...
0
votes
0answers
5 views
The Jacobson Radical of a Matrix Algebra
I am trying to solve the following question.
Let $A$ be the algebra over $\mathbb{C}$ consisting of matrices of the form
\begin{pmatrix}
* & * & 0 & 0 \\
* & * & 0 & 0 \\
* ...
0
votes
1answer
24 views
Closure of a nontrivial normed vector subspace that is equal to the whole space
Can you show me an example of a normed vector subspace $S$ strictly included in a normed vector space $V$ whose closure is equal to the whole $V$?
3
votes
2answers
49 views
Question on mathematical writing
I am now writing my graduate thesis, it includes some basics mathematical theorems/propositions.
I got a trouble in writing, more concretely, I do not know when can I state a mathematical claim as a ...
0
votes
0answers
40 views
Why don't humans perceive depth in 2D images? [closed]
Is there a mathematical difference between a 2D image that contains depth and a real 3D perspective?
For example:
Although the image is a picture of an object with depth, the human eye still ...
1
vote
1answer
37 views
Prove that if a matrix A is symmetric, then it is diagonalisable
I need to prove that if a matrix $A_{2 \times 2}$ is symmetric, i.e., $A^t = A$, then it is diagonalisable.
I know that a matrix $M_{n \times n}$ is diagonalisable, if and only if there is a basis of ...
2
votes
1answer
48 views
Number of Ones Puzzle
f(n) is a function counting all the ones that show up in 1, 2, 3, ...,
n. IE f(1)=1, f(10)=2, f(11)=4 etc. Discounting the trivial case f(1) = 1, when is the first time f(n)=n?
I found this on ...
0
votes
0answers
7 views
Estimating integral of decaying function over cubes
In $\mathbb{R}^d$, I would like to estimate (to within some universal constant, if possible)
\begin{equation*}
\int_{\mathcal{C}} \|x\|^{-\alpha}_2 dx,
\end{equation*}
where $\alpha>0$, and ...
1
vote
1answer
25 views
Looking for clarification on this definition: $K_1K_2 = K_1(K_2)$.
In my algebra book they define for field extensions
$L/K_1/F$ and $L/K_2/F$ the field $K_1K_2 = K_1(K_2)$.
The formal definition I have for this is
$$ K_1(K_2) = \bigcap_{K_2 < E < L} E \quad ...
1
vote
1answer
12 views
Generators for the radical of an ideal
I am interested in finding a generating set of the radical of an ideal given a set of generators for the ideal itself, but after a lot of thought I cannot figure out a good way to do it. Specifically:
...
0
votes
0answers
32 views
How to graph $x\sin(x)$
Graph $x\sin(x)$
I am having problem with the period section of the graph. I can see that the function is even, and that as $x \rightarrow \pm\infty$, the amplitude $\rightarrow \infty$.
I am ...
0
votes
0answers
16 views
Map restricted to subset
$A,B,C,D$ are Banach spaces with $A$ continuously injected in $B$ and $C$ continuously injected in $D$. If $T$ is a continuous linear map from $A$ to $C$ which maps $B$ to $D$, then is $T:B \to D$ ...
1
vote
2answers
27 views
Gaussians going towards delta “functions”
We have a sequence of random variables $x_1, x_2, x_3,...$ that are independent and are $N(0, 1/n)$ random variables. We want to show that $(x_1)^2 + (x_2)^2 + (x_3)^2 +...$ converges in probability ...
0
votes
1answer
23 views
Why do spectral projections give norm approximations?
First off, I'd like to ask:
If $H$ is a Hilbert space, and we have $A$ a bounded operator from $H$ to itself, $A$ being self adjoint (or normal), then if $A$ is compact there is a eigenspace ...
0
votes
0answers
6 views
Showing S is Jordan Measurable and Calculating the Volume
If S is the solid obtained by intersecting the ball $x^2+y^2+z^2=<4$ and $x^2+z^2=<1$
1) How do I show that S is jordan measurable? Can I simply say the following: "Clearly S is bounded, and ...
0
votes
1answer
29 views
Graph of the Dirichlet Function
we know, the Dirichlet function as: $$ f(x) = \begin{cases} 1, &\text{if } x \text{ is irrational; and } \\ 0, &\text{if }x \text{ is rational}. \end{cases} $$ R.A. Silverman in his book ...
1
vote
1answer
30 views
Proving $|w\overline{z}+\overline{w}z|\leq 2|wz|$
I want to prove $|w\overline{z}+\overline{w}z|\leq 2|wz|$.
My attempt:
$$\begin{array}{c c}|w\overline{z}+\overline{w}z| & =|(c+id)(a-ib)+(c-id)(a+ib)| \\ & =|2(ac+bd)| \\ & ...
0
votes
1answer
53 views
Logic translation involving the existential quantifier and “such that”
A: "There exists an integer greater than 5 such that it is less than
10"
B: "There exists an integer such that it is greater than 5 and less
than 10."
C: "There exists an integer less ...
3
votes
1answer
28 views
C* algebra inequalities
I need help on proving the following. If $A$ is a C* algebra with unit, not necessarily commutative, then why is it that $(b^*)b\leq||b||^21$ wrt the partial order induced by the cone $A_+$? Also, ...
3
votes
4answers
119 views
Prove or disprove: $(\mathbb{Q}, +)$ is isomorphic to $(\mathbb{Z} \times \mathbb{Z}, +)$?
Prove or disprove: $\mathbb{Q}$ is isomorphic to $\mathbb{Z} \times \mathbb{Z}$. I mean the groups $(\mathbb Q, +)$ and $(\mathbb Z \times \mathbb Z,+).$ Is there an isomorphism?
1
vote
2answers
48 views
Roots of unity?
The $n$th roots of unity are the complex numbers: $1,w,w^2,...,w^{n-1}$, where $w = e^{\frac{2\pi i} {n}}$. If $n$ is even:
The $n$th roots are plus-minus paired, $w^{\frac{n}{2}+j} = ...
0
votes
1answer
18 views
Example where the $\sup$-function not integrable
Let $\mu(\cdot)$ be a probability measure on $W \subseteq \mathbb{R}^m$, so that $\int_W \mu(dw) = 1$.
Consider a locally bounded function $f: X \times W \rightarrow \mathbb{R}_{\geq 0}$, with ...
1
vote
1answer
25 views
Fourier transform in Lp
Let the $f$ be a function in $L^s$ where $s \in [1,\infty) $. For which $r$ Fourier transform $\hat{f}$ belongs to $L^r$?
I'd be grateful for any kind of help including providing a literature or ...
0
votes
1answer
14 views
Runge-Kutta 4 for systems of equations
This question is part of an assignment in numerical methods class. I am supposed to find the position and velocity of a spaceship flying around the Earth and Moon. I am given initial values of the ...
1
vote
1answer
13 views
Proving well-definedness of “valuation of $f$ at $p$”?
Let $X$ be an irreducible variety, $p \in X$. Define $\mathcal{O}_{X,p}$ and $\mathcal{m}_{X,p}$ as usual. We have the following theorem: $\mathcal{m}_{X,p} = (\pi)$ is a principal ideal and $\bigcap ...
4
votes
2answers
106 views
Is the infinite root of any number equal to $1$?
I was messing around in IRB and I decided to make a $n^{th}$ root function and noticed that for very large roots of numbers, the answer always converges to $1$. It has been a while since I have done ...
0
votes
0answers
7 views
Next asymptotic term of the average order of sigma
$$
\sum_{k=1}^n\sigma(k)=\frac{\pi^2}{12}n^2+O(n\log n).
$$
Is the next asymptotic term known? That is, is there a monotonic increasing function $f(x)$ such that
$$
...
0
votes
2answers
15 views
Lower bound on a minimum of maximum of a sequence of standard normal random variables
Let $X = (x_{ij}) \in \mathbb{R}^{n \times p}$ be a matrix with independent $N(0,1)$ entries.
We know that $\max_j x_{ij} < \sqrt{2\log(p/\delta)}$ with probability at least $1-\delta$.
I would ...
1
vote
0answers
13 views
Reference: Qualifiers for defining trees
What is a good authoritative (i.e. has standard usage) reference accessible on the Net for checking the definitions of the multitude of terms used to describe trees in combinatorics?
Examples: ...
2
votes
2answers
43 views
is there a tensor that does the following?
I want a tensor (in the multi-linear algebra sense) which takes as an input a matrix $A$ of size $n \times n$ and returns as output an $n \times n$ matrix which is diagonal (zero off-diagonal), and on ...
1
vote
1answer
10 views
How to find/parameterize vector perpendicular to circle of constant $\ell_p$ norm
This should be very easy, but I can't get my head around it: given $1\leq p < \infty$, and a point x with $\|x\|_p = 1$, how do I get a (or the unit, or any) vector which is perpendicular to the ...
0
votes
0answers
14 views
First moment of area & second moment of area
This topic is still under development on wolframalpha and I've read some resources from wiki, youtube and yahoo answers and my basic understanding as of current is..
1st moment of area is area ...
2
votes
2answers
66 views
Finding $f'(x)$ when $f(x)=\int^1_0 e^{xy+y^2}dy$
If $f(x) = \int^1_0 e^{xy+y^2}dy$, find $f'(0)$.
I understand that this is function defined by an integral, and $e^{y^{2}}$ does not integrate into an elementary function. So, I will need to take ...
0
votes
2answers
26 views
$f:[-r,r]\to M$ continuous iff $f\circ \pi$ continuous in $B[-r,r]$
Let $\pi: \mathbb{R}^2\to \mathbb{R}$ defined by $\pi(x,y)=x$. Let $M$ be a metric space. Prove that $f:[-r,r]\to M$ is continuous if and only if $f\circ \pi: B[0,r]\to M$ is continuous on the ...
0
votes
0answers
14 views
Given $f(x,y)$ integrable in $[-1,1]\times [-1,1]$, are the following necessarily integrable?
Given that the function $f(x,y)$ is integrable in $K =[-1,1]\times [-1,1]$, can we derive either of the following:
$g(x,y)=2f(y,x)-f(-y,x)$ is integrable
$h(x,y)=f(\frac{x+y}{2},|x|-|y|)$ is ...
1
vote
0answers
49 views
A problem on the space of sequences
Consider the following sequence of sequences :
$x_{0}:=\{0,0,0,0......\}$,$x_{1}:=\{0,\frac{1}{2},0,\frac{1}{2}.......\}$ (i.e $0,\frac{1}{2}$ repeated infinitely), ...
-3
votes
1answer
81 views
Show that the series$\sum_{n=0}^\infty \frac{1}{1+n^2x}$ converges uniformly on $[a,\infty)$ for any $a>0$ but not on $(0,\infty)$.
Show that the series $$\sum_{n=0}^\infty \frac{1}{1+n^2x}$$ converges uniformly on $[a,\infty)$ for any $a>0$ but does not converge uniformly on $(0,\infty)$.
