# All Questions

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### 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 ...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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, ...
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### 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 ...
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### 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 ...
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### 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, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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.
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### 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?
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### 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 ...
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### 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 ...
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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 $S = \{n \in \mathbb{Z}, ... 2answers 238 views ### The union of a sequence of countable sets is countable. While working on the theorem below, I constructed the following proof: Theorem. If$\left\langle E_{n}\right\rangle_{n\in\mathbb{N}}is a sequence of countable sets, then $$... 1answer 130 views ### Matrix + combinatorial or conditional probability: bit patterns I'm trying to get my head around a problem, and it's not working. The problem: consider an NxN matrix that represents a binary number. For instance, a 4x4 matrix is a 16 bit number, a 6x6 matrix is ... 1answer 148 views ### Why in a space in which compact sets have empty interior, the closure of open sets is not totally bounded? I'm trying to understand the proof of the theorem: The Baire space \mathcal{N} is unique up to homeomorphism, non-empty zero-dimentional Polish space for which all compact subsets have empty ... 1answer 184 views ### Littlewood's 1914 proof relating to Skewes' number From Littlewood's 1914 theorem (paraphrase): I propose to show there are arbitrarily large values of x for which successively \psi(x) - x < - K\sqrt{x}\log\log\log x \tag{A} ... 1answer 71 views ### Basket of Fruits and Vegetables A basket contains six distinct fruits and seven distinct vegetables. In how many ways can we select from the basket a collection of four items that has at least one vegetable? 2answers 324 views ### What does it mean to “compute” a generic formula without values? Very elementary math question here in regards to my discrete math class. I've got a problem here that says... "Compute the following":$$\sum_{j=1}^n \frac{1}{j(j+1)}$$What on earth does it mean ... 2answers 163 views ### Generalization of Jensen's inequality for integrals? Jensen's inequality for sums says that for f convex,$$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.$$I have read that a ... 2answers 71 views ### Prove: For all n\geq 1, a_{n+1}-a_n<8^ka_n^{\left(1-\frac{1}{k}\right)^3}. Set S=\left\{\left.x^k+y^k+z^k\right|x,y,z\in Z^+\cup \{0\}\right\}, k is a positive integer, sort elements of S increasingly, that a_1<a_2<a_3<\text{...}<a_n<\text{...}. Prove: ... 1answer 121 views ### Is this infinite series a Fourier series? I have what looks like a Fourier series but I don't quite understand how (or if) it is possible to recover a function from this. ... 3answers 60 views ### the number of components working at a particular time Suppose a system has 10 components and that a particular time the j'th component is working with probability 1/j for j=1,2,\dots,10. How many components do you expect to be working at that ... 2answers 4k views ### Men and Women: Committee Selection There is a club consisting of six distinct men and seven distinct women. How many ways can we select a committee of three men and four women? 1answer 59 views ### How to show that \dim\left(\operatorname {Im}A\right)=\dim\left(\operatorname {Im}A^*\right)? Let A:\mathbb{R}^m\to\mathbb{R}^n be a linear map and A^*:\mathbb{R}^n\to\mathbb{R}^m be the adjoint of A (that's \langle Ax,y\rangle=\langle x,A^*y\rangle for all ... 0answers 66 views ### not geometric, not independent, kind of geometric probability problem Select a card from a standard deck without replacement until you get an ace. Let X denote the number of cards drawn prior to drawing the first ace. (a) Find the probability distribution ... 1answer 177 views ### Showing that two topologies on the unit circle are the same Consider the unit circle, described two ways. The first is as a quotient space, as in What does it mean to "identify" points of a topological space?. (I'm using the first definition of its ... 2answers 1k views ### Limit of recursive sequence a_{n+1} = \frac{a_n}{1- \{a_n\}} Consider the following sequence: let a_0>0 be rational. Define$$a_{n+1}= \frac{a_n}{1-\{a_n\}},$$where \{a_n\} is the fractional part of a_n (i.e. \{a_n\} = a_n - \lfloor a_n\rfloor). ... 1answer 48 views ### X is a complete metric space, Y is compact. X \times Y is Baire? Requesting a hint or solution. X is a complete metric space and Y is a compact hausdorff space. Trying to show that X \times Y is a Baire space. 2answers 78 views ### \frac{\sin x}{x^5} - \frac{1}{x^4} \underset{x\to 0}{\approx} \frac{-1}{6} \cdot \frac{1}{x^2}, right? I was reading an set of notes about Taylor series, and I came across a part I think is a typo. I want to make sure, because I want to understand this stuff correctly. Here is the relevant page of the ... 1answer 219 views ### The evaluation of the infinite product \prod_{k=2}^{\infty} \frac{k^{2}-1}{k^{2}+1} How does one show that$$ \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} ... 0answers 90 views ### Geometry of Lie Group Around Identity LetG$be a continuous compact Lie group. And let$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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