2
votes
1answer
515 views

Sum of series using integration

In certain special series, we can use the sigma notation to obtain the sum of $n$ terms. For example; $$1^3 + 2^3 + 3^3 + 4^3 + 5^3 +\cdots+ n^3 = \frac {n^2(n+1)^2}{4}$$ The sum can also be ...
1
vote
0answers
94 views

Question on O.D.E

Given three parameters $L,a$ and $\alpha$, we consider the differential equation : $$(E)\qquad x''+\alpha x' +a x + \sin x =L, \ t\geq0$$ Assume that $a>0$ and $\alpha\geq 0$ We consider the ...
1
vote
1answer
44 views

correct understanding mathematical question

suppose that we have following question,this question is not related to itself mathematics confusion,but language problem and please help me to clarify English language terms in mathematics. ...
6
votes
1answer
93 views

On Decompositions of Finite Group

Any finite non-cyclic abelian group $G$ can be written as product $HK$ of two proper subgroups. Here $HK=\{ hk\colon h\in H, k\in K\}$. A step further, if $G$ is a finite group such that the ...
0
votes
1answer
27 views

Symmetric $N\times N$ matrix, multiplicity $N-1$, for any $N$

The $N\times N$ matrix has $1-s$ along the diagonal and $s/(N-1)$ on the off diagonal. For $N=2,\dots,5$ the characteristic polynomial is $(X-1)(X+\frac{N}{N-1}s-1)^{N-1}$ where $X$ denotes ...
1
vote
1answer
73 views

Interesting related rates question

A circle C in the xy-plane is described as follows: A point P on the circumference of C traces out the graph of $f(x) = \sqrt{x}$; the center of C is the y-intercept of the tangent line of $f(x)$ at ...
1
vote
0answers
75 views

Weak topologies and direct product

In several places I have seen the embeddings of spaces with weak topology into direct products. More precisely, if $X$ is a Banach space and $X'$ is it's dual, then $X$ with weak topology can be ...
3
votes
1answer
236 views

Factoring $a^2+b^2+c^2$?

Is it possible to factor $a^2+b^2+c^2$ ? If we make this into only two factors, I know it has to look like this: $(a+b+c+\cdot \cdot \cdot )(a+b+c+\cdot \cdot \cdot )$ . But I don't know how to get ...
3
votes
1answer
311 views

Is there a reference for compact imbedding of Hölder space?

Suppose $0<\alpha <\beta$. Then, the Hölder space $C^\beta$ is compactly imbedded to $C^\alpha$. See the Wikipedia article Hölder condition. However, I could not find precise reference from ...
1
vote
1answer
98 views

Combination problem question.

I am working on a combination problem and I need to check if I'm doing this right. There is a deck of cards that consist of 20 cards. There are four different colors, including 2 Green, 6 Yellow, ...
5
votes
1answer
48 views

$p\nmid 2n-1,$ then $\sum_{k=1}^{p-1}\frac{1}{k^{2n-1}}\equiv 0 \pmod{p^3} \Leftrightarrow \sum_{k=1}^{p-1}\frac{1}{k^{2n}}\equiv 0 \pmod{p^2} $

Is it true that if $p$ is a prime and $p\nmid 2n-1,$ then $$\sum_{k=1}^{p-1}\frac{1}{k^{2n-1}}\equiv 0 \pmod{p^3} \hspace{12pt}\Leftrightarrow \hspace{12pt} \sum_{k=1}^{p-1}\frac{1}{k^{2n}}\equiv ...
1
vote
1answer
69 views

Is there a proof that no lower bound exists for the totient function?

I read here that there is no lower bound for the totient function. Is there a proof of that?
1
vote
3answers
1k views

proof about the sum of lim sup

I have questions about the solution below. I couldn't understand the red lines. What is $X_nN_1$? I'm not sure how it led to the contradiction. Thank you! Exercise 3: For any two real sequences ...
2
votes
1answer
118 views

Integral extensions of rings, when one of the rings is a field

The following is from page 61 of Introduction to Commutative Algebra by Atiyah & Macdonald: Proposition 5.7. Let $A\subseteq B$ be integral domains, $B$ is integral over $A$. Then $B$ is a ...
0
votes
1answer
53 views

$10$ Balls: Selecting Certain Numbers of Each Color

There exist piles of balls of the colors red, blue, and green. Each pile contains at least $10$ balls. How many ways can $10$ balls be selected if at least one red ball, at least two blue balls, and ...
3
votes
2answers
66 views

Largest square written as $p^2+pq+q^2$ where $p, q$ are primes?

I got this problem from the website Brilliant, but I have doubts about the solution presented there: $(p+q)^2-k^2=pq$ $(p+q+k)(p+q-k)=pq$ Now either $(p+q+k)=p$ and $(p+q-k)=q$ (which doesn't work), ...
0
votes
1answer
93 views

How to check if a set is orthogonal.

Hi guys Im stuck on a question. Given $\{u,v,w\}$ orthonormal set, prove that $\{u+2v+w,u-v+w,u-w\}$ is an orthogonal set. I know that im supposed to prove $$\langle u+2v+w,u-v+w\rangle=0$$ ...
5
votes
1answer
116 views

Existence of $j$ with strange sequence.

I define a sequence $(a_n)$ $$a_n= \begin{cases} 0 &\text{if $\cos{\left ( \dfrac{2^n\pi}{q}\right )}<-\dfrac12$} \\\\ 1 &\text{if $\cos{\left ( \dfrac{2^n\pi}{q}\right )}>-\dfrac12$} ...
1
vote
2answers
136 views

Analytic function and connected region.

We have the result. Let $G$ be an open connected set in $\mathbb{C}$, and let $f : G \rightarrow \mathbb{C}$ be an analytic function. Then the following statements are equivalent. *1. $f(z) = 0$ in ...
1
vote
2answers
180 views

Every bounded function has an inflection point?

Hello from a first time user! I'm working through a problem set that's mostly about using the first and second derivatives to sketch curves, and a question occurred to me: Let $f(x)$ be a function ...
0
votes
1answer
53 views

some integral and series whose value is $1$.

Give me some integral and series whose value is $1$. Where can I find a large number of these kinds of examples. I have two examples here, but I cannot think up more... This is geometry series, ...
11
votes
3answers
803 views

What is the motivation for differential forms?

I am that point in my mathematical career where I am learning differential forms. I am reading from M.Spivak's Calculus on Manifolds. So far I have gone over the tensor and wedge products and their ...
3
votes
2answers
165 views

Proof involving sum of squares of integers

I am required to prove that if $a$, $b$, and $c$ are integers such that $a^2 + b^2 = c^2$, then at least one of $a$ and $b$ is even. A hint has been provided to use contradiction. I reasoned as ...
1
vote
1answer
101 views

finding a generating function of a gambler question

Suppose a gambler starts with one dollar and plays a game in which he or she wins one dollar with probability p and loses one dollar with probability 1 - p. Let fn be the probability that he or she fi ...
1
vote
2answers
454 views

Putting $n$ Things in $m$ Boxes

A bicycle collector has $100$ bikes. How many ways can the bikes be stored in four warehouses if the bikes and the warehouses are considered distinct? What if the bikes are indistinguishable and the ...
0
votes
2answers
64 views

determine x in $x\log_\frac{1}{10}(x^2+x+1)>0$

I wanted to know, how can i determine the values of x for which $x\log_\frac{1}{10}(x^2+x+1)>0$ going to the question, we must have $x>0$ and $\log_\frac{1}{10}(x^2+x+1)>0$ or both must ...
1
vote
1answer
61 views

Prove normal of direct product of GL

Let $G=(\mathrm{GL}(2,\mathbb{R}) \oplus \mathrm{GL}(2,\mathbb{R}))$ and let $H = \{(A,B) \in G \mid\det(A)=\det(B)\}$. Prove that $H$ is normal in $G$. Mostly confused on what $G$ is.
0
votes
1answer
82 views

Coin Flip: “Exactly” and “At Most”

A coin is flipped $10$ times. How many outcomes have exactly three heads? How many outcomes have at most three heads?
0
votes
0answers
34 views

Regulated and semi-regulated functions with values in Banach spaces

In the following definitions, we assumed that $(X,\left\|\cdot\right\|)$ is a Banach space. Definition 1. $f:[a,b]\to X$ is regulated if it has one-sided limits at every point of $[a,b]$, i.e. for ...
1
vote
1answer
156 views

Prove $\sqrt{k}$ is not a rational number. [duplicate]

Suppose $k>1$ is an integer, and k is not a square number, then $\sqrt{k}$ is not a rational number. Proof: Let $\sqrt{k}=\frac{p}{q}$, and $(p,q)=1$,So $q^2|p^2$, $p\neq 1$, $k$ is not an ...
4
votes
1answer
234 views

On a $p$-adic unit and the existence of its $n$-th root

Let $\mathbb{Q}_p$ be the field of $p$-adic numbers. Let $\alpha$ be a $p$-adic unit, i.e. an invertible element of the multiplicative monoid $\mathbb{Z}_p$. Consider the set $S = \{n \in \mathbb{Z}, ...
2
votes
2answers
238 views

The union of a sequence of countable sets is countable.

While working on the theorem below, I constructed the following proof: Theorem. If $\left\langle E_{n}\right\rangle_{n\in\mathbb{N}}$ is a sequence of countable sets, then $$ ...
7
votes
1answer
130 views

Matrix + combinatorial or conditional probability: bit patterns

I'm trying to get my head around a problem, and it's not working. The problem: consider an NxN matrix that represents a binary number. For instance, a 4x4 matrix is a 16 bit number, a 6x6 matrix is ...
0
votes
1answer
148 views

Why in a space in which compact sets have empty interior, the closure of open sets is not totally bounded?

I'm trying to understand the proof of the theorem: The Baire space $\mathcal{N}$ is unique up to homeomorphism, non-empty zero-dimentional Polish space for which all compact subsets have empty ...
1
vote
1answer
184 views

Littlewood's 1914 proof relating to Skewes' number

From Littlewood's 1914 theorem (paraphrase): I propose to show there are arbitrarily large values of x for which successively $\psi(x) - x < - K\sqrt{x}\log\log\log x \tag{A}$ $ ...
1
vote
1answer
71 views

Basket of Fruits and Vegetables

A basket contains six distinct fruits and seven distinct vegetables. In how many ways can we select from the basket a collection of four items that has at least one vegetable?
1
vote
2answers
324 views

What does it mean to “compute” a generic formula without values?

Very elementary math question here in regards to my discrete math class. I've got a problem here that says... "Compute the following": $$\sum_{j=1}^n \frac{1}{j(j+1)}$$ What on earth does it mean ...
3
votes
2answers
163 views

Generalization of Jensen's inequality for integrals?

Jensen's inequality for sums says that for $f$ convex, $$f\left(\sum_1^n \alpha_i x_i\right)\leq \sum_1^n \alpha_i f(x_i), \,\,\,\,\text{for } \sum_1^n \alpha_i = 1.$$ I have read that a ...
3
votes
2answers
71 views

Prove: For all $n\geq 1$, $a_{n+1}-a_n<8^ka_n^{\left(1-\frac{1}{k}\right)^3}$.

Set $S=\left\{\left.x^k+y^k+z^k\right|x,y,z\in Z^+\cup \{0\}\right\}$, k is a positive integer, sort elements of $S$ increasingly, that $a_1<a_2<a_3<\text{...}<a_n<\text{...}$. Prove: ...
0
votes
1answer
121 views

Is this infinite series a Fourier series?

I have what looks like a Fourier series but I don't quite understand how (or if) it is possible to recover a function from this. ...
1
vote
3answers
60 views

the number of components working at a particular time

Suppose a system has $10$ components and that a particular time the $j$'th component is working with probability $1/j$ for $j=1,2,\dots,10$. How many components do you expect to be working at that ...
1
vote
2answers
4k views

Men and Women: Committee Selection

There is a club consisting of six distinct men and seven distinct women. How many ways can we select a committee of three men and four women?
0
votes
1answer
59 views

How to show that $\dim\left(\operatorname {Im}A\right)=\dim\left(\operatorname {Im}A^*\right)$?

Let $A:\mathbb{R}^m\to\mathbb{R}^n$ be a linear map and $A^*:\mathbb{R}^n\to\mathbb{R}^m$ be the adjoint of $A$ (that's $\langle Ax,y\rangle=\langle x,A^*y\rangle$ for all ...
1
vote
0answers
66 views

not geometric, not independent, kind of geometric probability problem

Select a card from a standard deck without replacement until you get an ace. Let $X$ denote the number of cards drawn prior to drawing the first ace. (a) Find the probability distribution ...
2
votes
1answer
177 views

Showing that two topologies on the unit circle are the same

Consider the unit circle, described two ways. The first is as a quotient space, as in What does it mean to "identify" points of a topological space?. (I'm using the first definition of its ...
28
votes
2answers
1k views

Limit of recursive sequence $a_{n+1} = \frac{a_n}{1- \{a_n\}}$

Consider the following sequence: let $a_0>0$ be rational. Define $$a_{n+1}= \frac{a_n}{1-\{a_n\}},$$ where $\{a_n\}$ is the fractional part of $a_n$ (i.e. $\{a_n\} = a_n - \lfloor a_n\rfloor$). ...
2
votes
1answer
48 views

$X$ is a complete metric space, $Y$ is compact. $X \times Y$ is Baire?

Requesting a hint or solution. X is a complete metric space and Y is a compact hausdorff space. Trying to show that $X \times Y$ is a Baire space.
2
votes
2answers
78 views

$\frac{\sin x}{x^5} - \frac{1}{x^4} \underset{x\to 0}{\approx} \frac{-1}{6} \cdot \frac{1}{x^2}$, right?

I was reading an set of notes about Taylor series, and I came across a part I think is a typo. I want to make sure, because I want to understand this stuff correctly. Here is the relevant page of the ...
11
votes
1answer
219 views

The evaluation of the infinite product $\prod_{k=2}^{\infty} \frac{k^{2}-1}{k^{2}+1}$

How does one show that$$ \prod_{k=2}^{\infty}\frac{k^{2}-1}{k^{2}+1} =\frac{\pi}{\sinh \pi} ?$$ My attempt: $$ \begin{align} \prod_{k=2}^{\infty}\frac{k^{2}-1}{k^{2}+1} &= \lim_{n \to \infty} ...
2
votes
0answers
90 views

Geometry of Lie Group Around Identity

Let $G$ be a continuous compact Lie group. And let $K,\ H$ be closed subgroups. How can we take $W$ which is a small open set around $e$ and satisfies the following : If $K\subset WH$ then $KH/H$ ...

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