1
vote
0answers
88 views

Criteria for a Regular Singular Point

My book on differential equations says that $x_{0}$ is a regular singular point of the differential equation $P(x)y'' + Q(x)y' + R(x)y = 0$ if $(x - x_{0})\frac{Q}{P}$ and $(x - ...
4
votes
1answer
634 views

finding the multiplicative inverse in a field

Let $L/K$ be a field extension. Let $a\in L$ and $K[a]=\{p(a)\;|\; p\in K[x]\}$; then $K[a]$ is clearly an integral domain. I want to show that when $a$ is algebraic over $K$, then $K[a]$ is a field. ...
2
votes
3answers
142 views

The equation $(x-2xy-y^2)\frac{dy}{dx}+y^2=0$

Can one give a hint how to solve the following equation? $(x-2xy-y^2)\frac{dy}{dx}+y^2=0$ Thanks in advance.
3
votes
1answer
78 views

Samples in the convex body vs. samples on the convex surface

Let $K$ be a bounded convex body in $\mathbb{R}^n$. Suppose we have a sampler $\mathcal{S}_1$ that can generate points uniformly distributed in $\mathrm{int}K$, and another sampler $\mathcal{S}_2$ ...
7
votes
1answer
119 views

How to prove $\sum\limits_{k=1}^{\infty}|\alpha_{k}|\lt \infty$, given that $\sum \limits_{k=1}^{\infty}\alpha_{k}\phi_{k}$ converges …?

I am solving some problems in a text, I come across this question. I thought I will not take much of my time on it, but that is not the case. Question: Prove that if $\sum ...
5
votes
2answers
205 views

About joint probability divided by the product of the probabilities

Let $X$ and $Y$ be two events. So $P(X)$ is the probability of $X$ happens, and $P(Y)$ is the probability of $Y$ happens. So $P(X,Y)$ is probability of both $X$ and $Y$ happen. So what is the ...
1
vote
1answer
55 views

Automorphisms of a certain variety

Which is the group of automorphisms of the variety $(\mathbb{A}^1\setminus\{0,1\})^{\times 3}$ ? ($0,1$ are two points of $\mathbb{A}^1$, say the points corresponding to $0,1$ of the base field ...
2
votes
1answer
198 views

How do I determine the measure for a volume integral?

If $I = \int r^2 dm$, how do I set up an integral over the volume of any object? I can't use any assumptions about symmetry or shortcuts because the goal is to rotate around an arbitrary axis. $m = ...
0
votes
0answers
213 views

Rotating an object around a tilted axis

I have the coordinates of a centre point . I also have an array called the asteroid normal which I assume is the relative rotation of the axis (its 3 numbers between zero and one). How can I make an ...
0
votes
2answers
192 views

Can someone give me a counterexample to disprove this statement?

Claim : For any even number $n$ there is at least one prime number of the form : $$p=k\cdot2^{n}-1$$ with following properties : $k=2^{a-n}+1 , n\leq a < 2n , $ and $a,n\in ...
4
votes
2answers
305 views

Different standards for writing down expressions in a formal way

What are standard ways to write mathematical expressions in a (semi)formal way ? In different posts of mine concerning similar question I have encountered for a generic expression of the type "for all ...
3
votes
1answer
714 views

The field of fractions of a field $F$ is isomorphic to $F$

Let $F$ be a field and let $\newcommand{\Fract}{\operatorname{Fract}}$ $\Fract(F)$ be the field of fractions of $F$; that is, $\Fract(F)= \{ {a \over b } \mid a \in F , b \in F \setminus \{ 0 \} \}$. ...
5
votes
1answer
268 views

At most finitely many (Hamel) coordinate functionals are continuous - different proof

If $X$ is a vector space over $\mathbb R$ and $B=\{x_i; i\in I\}$ is a Hamel basis for $X$, then for each $i\in I$ we have a linear functional $a_i(x)$ which assigns to $x$ the $i$-th coordinate, ...
0
votes
1answer
83 views

Decompose boolean function of multiple variables into multiple functions of one variable

say I have a function $$f(x, y) : bool$$ of two variables x and y - whose type can be anything - returning either true or false. I would like to create two functions of one variable each $g(x)$, ...
1
vote
1answer
52 views

What does this quantifier evaluates to?

I need help solving the following question, Suppose U=Z. Simplify the quantifer, and say that if it is true or false with the help of and example, ~$\exists$x $(x| x|5 \Rightarrow x|15)$ Note ...
3
votes
4answers
357 views

What does $\mathbb{Z}_{7429}$ mean?

What exactly does $\mathbb{Z}_{7429}$ mean? Is it the set of all integers up to and including 7429?
7
votes
1answer
397 views

Convergence of a bounded sequence with bounded L2 variation

I have the following question, that a friend of mine asked me yesterday: Let $H$ be a Hilbert space, with norm $\|\cdot\|$. Let $(x_k)_{k \ge 0}$ be some sequence in $H$. Assume that $x_k$ is bounded ...
3
votes
1answer
123 views

Elementary matrices

Is there a way to visualize the action of elementary matrices? (Or perhaps matrices in general). Perhaps someone could give an intuitive view of the effects of elementary row operations. Actually I ...
2
votes
2answers
177 views

How formal or informal should math texts (written for different purposes) be?

When writing math articles (or just math text), do you write down mathematical expression in a formal way or describe it in words, e. g. "Let $X$ be a normed vector space. Then $X$ is called a ...
0
votes
1answer
45 views

Bessels and initial conditions

I'd like to know if I have got the following ideas right: 1) $f(r,\theta,t)=\sum\limits_{n=-\infty}^\infty \sum\limits_{k=1}^\infty a_{nk}J_n(j_{nk}r)\exp[in\theta-j^2_{nk}t]$ subjected to initial ...
0
votes
2answers
142 views

Show that there exists an invertible $S$ for $A^2=0$ such that $S^{-1}AS=\begin{bmatrix} 0 & 0\\ 1 & 0 \end{bmatrix}$

$A$ is a nonzero $n \times n$ matrix such that $A^2=0$. If $n=2$, is it possible that I can show that there exists an invertible $2 \times 2$ matrix $S$ such that $S^{-1}AS=\begin{bmatrix} 0 & ...
1
vote
1answer
2k views

One container contains 6 red and 4 white balls, while a second container contains 7 red and 3 white balls

One container contains 6 red and 4 white balls, while a second container contains 7 red and 3 white balls. A ball is chosen at random from the first container and placed in the second. Then a ball is ...
0
votes
0answers
119 views

Can any two straight lines lay on a common plane?

With respect to vector analysis cross-product Is it possible to place two straight lines of any orientation on a common plane?
0
votes
1answer
78 views

Help needed with tensors [duplicate]

Possible Duplicate: An Introduction to Tensors Recently I came across the concept of tensors and heard it is very difficult to understand. Is there a ...
0
votes
1answer
79 views

Effect of $k$ on turning point?

In the function $$y=(k-x)e^x ,$$ What is the effect of $k$ on the turning point of the function? I can't see any clear pattern when I change the variable. What are some real-life scenarios to which ...
3
votes
1answer
136 views

$n$-th homotopy group of subsets of $\mathbb{R}^n$

Let $X$ be a compact, path-connected subset of $\mathbb{R}^n$. I need a reference for the fact that the $n$-th homotopy group $\pi_n(X)$ is trivial. EDIT: Quite embarrassing. Indeed this is false, ...
0
votes
1answer
67 views

Natural space to consider solution to polynomial equations

Why is the complex projective planes the most natural place to look to consider solutions of polynomial equation? Why is the complex plane $\mathbb{C}$ adequate for polynomial equations of one ...
1
vote
1answer
192 views

Solution of Diophantine equation

I have seen $3^x$ + $3^y$ = $6^z$ and $4^x$ + $18^y$ = $22^z$ on lecture series of Prof. Gandhi. In my own study, I have constructed the following theorem (I am not sure about solvability) and I am ...
0
votes
0answers
119 views

Is it true that $n> a^2\Rightarrow n!>a^n$, $n\in\mathbb{N}, a\in\mathbb{R}$?

If so, how can it be proven? (I have evaluated it up to $n=25$.) If not, does there exist a $k\in\mathbb{R}$ such as that $n> a^k\Rightarrow n!>a^n$, with $n\in\mathbb{N},a\in\mathbb{R}$? It ...
4
votes
1answer
427 views

Category theory, a branch of abstract algebra?

In Steve Awodey's book on category theory, he claims the latter is a branch of abstract algebra. I've never seen such a classification before. Is this really correct?
2
votes
1answer
206 views

Category theory for describing systems?

In Rosetta Stone quantum description is interpreted from the category theory point of view. Systems (Hilbert spaces of wave functions) are objects and processes (linear operators) are arrows. But the ...
2
votes
0answers
123 views

If I chop my apple into smaller pieces, can I fit more apple in my lunchbox?

Well, that's how my little sister asked it, and I couldn't provide a quick answer. If we ignore any changes in shape from the cutting process, then my first thought was that the scaling wouldn't make ...
1
vote
2answers
133 views

Linear Algebra: Subspaces

In my text it's not very clear as to what the procedure is for determining when a vector is a subspace in say the subset $R^3$ in this example: Consider the vector of the following form ...
4
votes
1answer
191 views

Is $k^2+k+1$ prime for infinitely many values of $k$?

Let's define an infinite sequence of positive integers as : $a_n=k^2+(2n-1)k+2n-1 $ , where $ k,n \in \mathbf{Z^{+}}$ Suppose that one can prove that this sequence contains infinitely many ...
2
votes
1answer
111 views

How to solve $x^x=k$? [duplicate]

Possible Duplicate: $x^x=y$. How to solve for $x$? If we have $x^x=4$ it's easily solved by substituting $x$ with $2$. But for general equation like $x^x = k$, how we can find the solution? ...
1
vote
1answer
186 views

Prove $\|T(x)\| = \|x\|$ iff $\langle T(x),T(y)\rangle = \langle x,y\rangle$

Let T be a linear operator on an inner product space $V$. Prove that $\lVert T(x)\rVert = \lVert x\rVert$ for all $x$ in $V$ iff $\langle T(x), T(y)\rangle = \langle x,y\rangle$ for all $x,y$ in $V$. ...
3
votes
2answers
2k views

Central Limit Theorem and sum of squared random variables

This is a two-part question. Suppose I am drawing random variables $X_i\sim A$, $1\leq i \leq n$ where $A$ is a zero-mean, finite variance $\sigma_A^2$, symmetric probability distribution having ...
2
votes
2answers
367 views

Series: 1 to infinity vs. 1 to n as n approaches infinity

$$ f(x) = \frac{\sum_{i=1}^{\infty}x^i}{1+\sum_{i=1}^{\infty}x^i} = \frac{\sum_{i=1}^{\infty}x^i}{\sum_{i=0}^{\infty}x^i} = x $$ I haven't taken many math classes (so bear with me if I'm wrong), ...
1
vote
2answers
2k views

Finding a polynomial with given roots, degree, and specific coefficient

Find a degree 4 polynomial having zeros -6, -3, 2 and 6 and the coefficient of $x^4$ equal 1 The first step is something like $p(x)=c(x-6)(x+3)(x-2)(x-6)$ as those are all the 4 zeros. The ...
1
vote
1answer
41 views

Distance between $N$ sets of reals of length $N$

Let's say, for the sake of the question, I have 3 sets of real numbers of variate length: ...
8
votes
3answers
628 views

Limit $\lim\limits_{n\rightarrow\infty}\left({2\sqrt n}-\sum\limits_{k=1}^n\frac{1}{\sqrt k}\right)$

How to find $\displaystyle\lim_{n\rightarrow\infty}\left({2\sqrt n}-\sum_{k=1}^n\frac{1}{\sqrt k}\right)$ ? And generally does the limit of the integral of f(x) minus the sum of f(x) exist? How to ...
2
votes
2answers
119 views

A question about an identity involving Dirichlet characters

Let $\chi$ be a Dirichlet character $\bmod q$. We have $$\sum_{n=0}^{\infty} (-1)^{n-1} \chi(n) n^{-s}=\prod_p ...
2
votes
1answer
114 views

Proof for $\sqrt[n]{n!}\le{{n+1}\over2}$ using induction

Prove $\sqrt[n]{n!}\le{{n+1}\over2}$, $n \in N^+$, using induction. This is how far I got, but then I got stuck: $$\sqrt[n+1]{(n+1)!}\le{{n+2}\over2}$$ $$((n+1)!)^{1\over{n+1}}\le{n\over2}+1$$ ...
2
votes
2answers
1k views

Is $f'$ continuous at $0$ if $f(x)=x^2\sin(1/x)$

Let $f(x)=x^2\sin(1/x)$ for $x≠ 0$ and $f(0)=0$ for $x=0$. Is $f'$ continuous at $0$? My attempt: $f'(x)=2x\sin(1/x)-\cos(1/x)$. Since when $x$ goes to $0$, the limit of $\cos(1/x)$ does not ...
9
votes
2answers
464 views

How to find $\zeta(0)=\frac{-1}{2}$ by definition?

I would like to know how we can find the following result: $$\zeta(0)=-\frac12$$ Is there a way, using the definition, $$\zeta(s)=\sum_{i=1}^{\infty}i^{-s}$$ to find this?
1
vote
0answers
126 views

Binary sequences in primes

Is anything known about these problems? If we make a string S of 0's and 1's with 1 in n'th position if the the nth prime $p_n$ is of the form $1+m 2^{9^{9^{9^{9}}}}$, else 0, does every finite string ...
4
votes
1answer
263 views

What are some examples of applications of König's lemma?

König's lemma states that given an infinite tree, an infinite path exists, where of course by tree we mean a full binary tree. I found some examples in Logical Labyrinths by Smullyan where he ...
4
votes
2answers
416 views

A concrete example of a choice function

I'm trying to understand a bit what lies behind the Axiom of Choice, and I was wondering, are there concrete examples of a choice function on the Borel set? The Borel set seems nice enough for a ...
2
votes
1answer
93 views

Decomposition of sets without arithmetic progressions

This is consequence of my earlier post. I am not very happy with earlier replies of my post. I hope, I will get good response for this post. With lot of hope, I am sending the following problem and I ...
2
votes
2answers
142 views

Alphametic-like fraction equaling 1/2; uniqueness of solutions

This problem is kind of like those alphametics puzzles. The challenge is to assign each whole number from 2 to 9 to the letters in $$\frac{10^3A+10^2B+10C+D}{10^4+10^3E+10^2F+10G+H}$$ such that the ...

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