# All Questions

1answer
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### existing of limiting point and delta exist

If $\lim_{x \to p} f(x) = L$ and If p is the limited point and p $\in R$, is this imply there exist of $\delta$? and $$|p-x| < \epsilon$$ $$|f(x)-L| < \delta$$ or condition(limited point ...
1answer
60 views

### Language concatenation

We learned in class that the regular languages are closed under concatenation (e.g $L_1L_2 =\{ w_1w_2 : w_1 \in L_1,w_2 \in L_2\}$ is a regular language if $L_1$ and $L_2$ are also regular ...
2answers
43 views

### Mathematics Equality Proof

Positive numbers a, b, c satisfy $1/a + 1/b + 1/c = 3$. a) Prove that $abc \ge 1$. b) Prove that $(a+b)(a+c)(b+c) \ge 8$. When does the equality hold? I want to say that I will be using conjugates ...
1answer
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1answer
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### Introductory books on Sieve methods

I want to start learning sieve methods, but the books I have come across is quite advanced, anyone know of easier books? Thanks.
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### Fibonacci… Easier by induction or directly via Binet's formula

I have tried both for several of them and haven't been able to get anywhere in 3 hours of work. It seems to not matter which method I choose, I end up in the middle of a HUGE mess of algebra. Could ...
0answers
61 views

### Understanding the matrix pertubation.

I would like to ask a question about perturbation in matrix, as I understand from this tutorial. Main task of perturbation is to determine what amount of minimum change is necessary in given ...
0answers
57 views

### Help with local coordinates computations-differential forms

I am reading a lecture notes on differential forms and tangent bundles on complex manifolds and I got stuck on one line where the author does not explain how he did the computation: Let ...
1answer
57 views

### Chances of winning a raffle when winning tickets are returned to bucket each time.

first time posting here - I like the site. I have a raffle odds question. I'm doing a raffle where I give away 365 prizes. Every winning ticket is returned to the barrel after each drawing (and ...
1answer
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2answers
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### Closed subspace of continuous functions

Is the subspace of continuous functions $C[0,1]$ with $f(0)=0$ closed with the metric $$d(f,g) = \max|f(x)-g(x)|?$$ Why or why not?
3answers
81 views

### Having a hard time counting this one .

A coin is tossed $12$ times . We have to find the number of ways that two heads do not occur consecutively. The solution which was provided in the book : Let $a_n$ denote the number of outcomes in ...
2answers
44 views

### show that $f$, given by $[f(x)]^2 = 2 \int_0^x f$, is differentiable

The full question reads Let $f$ be continuous on $[0,\infty)$. Suppose that $f(x) \neq 0$ for all $x > 0$ and that $[f(x)]^2 = 2 \int_0^x f$ for all $x \ge 0$. Prove that $f(x)=x$ for all ...
1answer
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### Extending $\mathbb R$ for the benefit of the unitary pulse function

The unitary pulse function (or sample function) is defined as follow: Let's $\newcommand\R{\mathbb R}d_1:\R\to\R$ be a positive intrgrable function such that $$\int_{-\infty}^{\infty}d_1(x)dx=1.$$ ...
1answer
60 views

### Cardinal of the set of all dense and enumerable subsets of $\mathbb R^2$

The problem statement: Calculate the cardinal of the set consisting of all the dense and enumerable subsets of $\mathbb R^2$ The attempt at a solution: I couldn't go farther than this: If $A$ is the ...
1answer
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1answer
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### How many outcomes of a coin being flipped 12 times have exactly 4 heads?

I know that there are a total of 4096 possible outcomes of tossing a coin twelve times, but I do not know how to calculate the number of possible outcomes with exactly 4 heads, with at least 2 heads, ...
0answers
49 views

### functions $\sin(x)/x$ is between $1$ and $2/\pi$ [duplicate]

We have to prove that for $0 < x < \pi/2$ ; $1 > \frac{\sin(x)}{x} > \frac{2}{\pi}$ . This is a simple problem . I just want to know what I am doing is correct or not . At $0$ function is ...
2answers
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### Solving $x\; \leq \; \sqrt{20\; -\; x}$

This is how I tried to solve it: By squaring both sides: $x^{2}\; \leq \; 20\; -\; x$ $x^{2}\; +\; x\; -\; 20\; \leq \; 0$ Thus $-5\; \leq \; x\; \leq \; 4$ However, it seems that values less ...
2answers
180 views

### Best way to approach a Multiple Choice exam

I have an exam tomorrow with 100 multiple choice questions. Each question has 4 options, only 1 is correct. If I answer the question and get it right, I get 1 mark. If I leave it blank, I get 0. If I ...
2answers
172 views

### Are spaces with isomorphic fundamental groups homotopically equivalent?

I know that the converse of this statement is true but I am not sure how to go about finding out the answer to this question.
2answers
505 views

Find the general solution of: $$\sin^3 \theta - \sin \theta = 0$$ Working out: (Factorise out) $$\sin \theta (\sin^2 \theta - 1) = 0$$ Solve for $\sin \theta$ and $\sin^2 \theta - 1$: For $\sin ... 1answer 446 views ### convexity of log of moment generating function Why is log of a moment generating function of random variable Z is convex? that is$\log \mathbb{E}[\exp(\lambda.Z)]$My logic says since expectation is linear so it is in particular convex and ... 2answers 497 views ### Quasicomponents and components in compact Hausdorff space Let$X$be a compact Hausdorff space,$x,y\in X$and$\mathcal{A}$a colection of closed subspaces of$X$such that for every$A\in \mathcal{A}$then$x$and$y$are in the same quasicomponent of$A$. ... 1answer 41 views ### System of ODEs with products How can we solve the system of differential equations$\dfrac{df(t)}{dt}=-f(t)h(t), \dfrac{dg(t)}{dt}=-g(t)h(t), \dfrac{dh(t)}{dt}=1-(h(t))^2$The system does not fall to standard ODE methods. 2answers 38 views ### Why if the rank of matrix$M=\begin{pmatrix}2x&2y&0\\1&0&1\\ \end{pmatrix}$is less than$2$,then$x=y=0$? The rank of matrix$M=\begin{pmatrix}2x&2y&0\\1&0&1\\ \end{pmatrix}$is less than$2$if and only if$x=y=0\$. I can't understand the only if part, that is why "if the rank of ...

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