1
vote
1answer
28 views

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 ...
2
votes
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 ...
0
votes
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 ...
1
vote
1answer
87 views

Infinite series challenging problem

Let $\sum_{k=1} ^\infty a_k$ be a convergent series of positive terms and let $t_n = \sum_{k=n} ^\infty a_k$ for each integer n. I'm trying to prove that $\sum_{k=m} ^n {a_k \over t_k} > 1 - {t_n ...
1
vote
1answer
31 views

The Quadratic Residues/Nonresidues of modulo 15

In an effort to determine these, I crafted the following table mod 15. In turn the answer is given to be: Since a quadratic residue is said to be an integer q such that $x^2 \equiv q\mod 15 $, if ...
3
votes
1answer
185 views

Solvable by radicals or not?

Let $f=2x^5-5x^4+5\in \Bbb Q[x]$. Then, how to prove that it's not solvable by radicals? Since $f$ is solvable by radicals iff $Gal(f)$ is solvable and $Gal(f)\subset S_5$ (up to isomorphism), I ...
3
votes
1answer
61 views

How prove this $f(x)=\sum_{n=1}^{\infty}\left(\frac{\sin \frac{1}{x-r_n} }{2}\right)^n$ such follow condition?

Question: let $r_{1},r_{2},\cdots.$ is the interval $[0, 1]$ rational sequence in a line, and define: $$f(x)=\begin{cases} ...
7
votes
1answer
90 views

Prove this is an equivalence relation.

Define a relation of $\mathbb{Q} -$ {$0$} as follows: $x$ ~ $y$ $\Leftrightarrow$ $\dfrac {x} {y} = 2^k $ for some $k \in \mathbb{Z}$ Prove this is an equivalence relation. ATTEMPT: Reflexive: ...
1
vote
3answers
631 views

Prove a set is closed

Suppose $f\colon \mathbb R \to \mathbb R$ is a continuous function and $K$ is a closed subset of $\mathbb R$. Prove that the set $A = \{x \in \mathbb R : f(x) \in k\}$ is also closed. ...
5
votes
3answers
267 views

An Example of Abelian Group with exactly one maximal subgroup.

Let $G$ be a finitely generated group and $G$ has exactly one maximal subgroup. Then I can conclude that $G\cong\mathbb Z_{p^k}$. Now, I am looking for an example of infinite abelian group $G$ such ...
0
votes
0answers
38 views

Proving a DTFT relation

I'm trying to prove that the inverse DTFT of: $$\frac{1}{1-ae^{-j\Omega }} $$where |a|<1 is: $$a^{n}u[n]$$ The way to prove it is by the integral below but I'm not sure how to proceed: $$ ...
2
votes
0answers
63 views

Given a sequence and find the smallest natural number s.t.

The question itself comes from a daily life problem which I think could take a large wall of text to explain. Unfortunately, I don't have any clue about this question and I'm also not sure how to ...
1
vote
1answer
74 views

Chebychev's formula with upper bound

Suppose $X$ & $Y$ are independent with respective variances $9$ & $16$. $E(X)=E(Y)$ Use Chebychev's inequality to put an upper bound on $P(|X-Y|\ge 10)$ I have been looking around online ...
-2
votes
2answers
76 views

Find the probability of $ x_2/x_3 \leq a $ where $x_2,x_3$ are uniform i.i.d.

Let $x_1,x_2,...,x_n $ be independent and identically distributed, uniformly on $(0,1)$. How to find $P(x_2/x_3 \leq a)$?
6
votes
1answer
63 views

Defining natural numbers without $0$ or $1$.

Let's define Peano's axioms having $2$ as the first number: $\newcommand\Nt{\mathbb N''}2\in\Nt$. $\newcommand\next{\mathop{\mathrm{next}}}\forall n\in\Nt:\next n\in\Nt$ (or $\next:\Nt\to\Nt$). ...
4
votes
1answer
47 views

Does every graph arise as the commutativity graph of some group?

By graph let us mean a set $G$ together with a relation $\bot$ that is reflexive and symmetric. Now every group gives rise to a commutativity graph by defining $x \,\bot\, y \iff xy=yx.$ Does every ...
2
votes
1answer
65 views

Can someone show me step by step how to handle this convergence problem?

I just took my final and one of the questions read: Use the integral test to determine the convergence or divergence of the series $$\sum_{n=1}^{\infty} \frac{e^n}{1+e^{2n}}.$$ I struggled ...
2
votes
1answer
284 views

equivalence relation composition problem

Let $R_1$, $R_2$ be two equivalence relations on $X$, prove that $R_1\circ R_2$ is an equivalence relation if and only if $R_1\circ R_2= R_2\circ R_1$ First I´m trying to prove that $R_1\circ R_2= ...
1
vote
1answer
66 views

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.
3
votes
1answer
53 views

Does every orthocomplemented lattice satisfy the shuffle laws?

Let $L=(X,\wedge,\vee,^\bot)$ denote a non-empty lattice having a unary operation $x \mapsto x^\bot$ that satisfies both "shuffle" laws: $$x \wedge y \leq z \iff x \leq y^\bot \vee z.$$ $$x \leq y ...
0
votes
1answer
44 views

Bivariate normal PDF

Suppose that X & Y are bivariate normally distributed with E(X)=2, Var(X)=9, E(Y)=1, Var(Y)=16, & p=.25 Determine P(X<3.5) & P(X<3.5 | Y =2) The only mention of bivariate normal ...
1
vote
1answer
44 views

Number theory question on finite simple continued fractions

I have to show that for any positive integer n, the following holds, some help: a) √(n^2+1) = [n,(2n) ̅] b) √(n^2+2) = [n,(n,2n) ̅] c) √(n^2+2n) = [n,(1,2n) ̅]
1
vote
1answer
113 views

complete subset of a metric space

Let $f:X\to Y$ be a continuous map between metric spaces. Then $f(X)$ is a complete subset of $Y$ if A. the space $X$ is compact B. the space $Y$ is compact C. the space $X$ is complete D. the ...
4
votes
4answers
132 views

negative exponent problem

$$\sqrt{\frac{1}{3^0 + 3^{-1} + 3^{-2} + 3^{-3} + 3^{-4}}}$$ Does this equal = $$ \begin{align*} & \sqrt{3^0 + 3^1 + 3^2 + 3^3 + 3^4} \\ =&\sqrt{1 + 3 + 9 + 27 + 81} \\ =&\sqrt{121} \\ ...
1
vote
1answer
98 views

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 ...
0
votes
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 ...
2
votes
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 ...
-1
votes
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 ...
1
vote
1answer
56 views

Using the step and shift theorems to find the Laplace transform

The Problem $$f(t):=\begin{cases}e^{-t}&t\lt4\\e^{-2t}&4\le t\le10\\0&t\gt10\end{cases}$$ What I know I can write this as one function: $$f(t)= e^{-t} + (e^{-2t} - e^{-t})u(t-4) + ...
5
votes
1answer
374 views

Minkowski Content

Could someone provide some intuition behind the $n$-dimensional Minkowski Contentthe $n$-dimensional upper Minkowski Content of $\mathcal{A}$ as $$\mathfrak{M}^{*m} (\mathcal{A}) : = \lim_{\epsilon ...
1
vote
2answers
90 views

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?
3
votes
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 ...
2
votes
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 ...
2
votes
1answer
24 views

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 $$\int_{-\infty}^{\infty}d_1(x)dx=1.$$ ...
0
votes
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 ...
1
vote
1answer
73 views

A diffusion partial differential equation, or Sturm-Liouville eigenvalue ODE

What is the analytical solution for the following diffusion partial differential equation (initial value problem)? $$\frac{\partial f}{\partial t} = (ax^2+b)\frac{\partial f}{\partial ...
0
votes
1answer
205 views

Prove that the three statements are equivalent

I need to show that the following statements are equivalent. A $\subset$ B, A $\cap$ B$^c$ = $\emptyset$, and A$^c \cup$ B = U (U is the universal set) So to show that A $\subset$ B is true I said ...
1
vote
3answers
263 views

Examples of root, parabolic, and borel subgroups corresponding to roots

I'm interested in seeing a few examples of root, parabolic, and Borel subgroups given a specific reductive group $G$. Here is what I know. Let $G$ be a reductive algebraic group over an ...
0
votes
1answer
68 views

Unsuccesful Google search for “euclidean geometry proof of collinearity”

I am looking for a clear explanation of techniques one might use for proving collinearity. It's easy in coordinate geometry and with vectors. My question is: in Euclidean Geometry what precisely is ...
0
votes
1answer
49 views

Limit of max of a vector/series?

Do you have any clue on how this can be solved? $$ \lim_{n \to \infty} \max \left \{ \cos (\alpha -2\pi\left ( \frac{i-1}{n} \right ),\ldots,\cos (\alpha -2\pi\left ( \frac{i-1}{n} \right ) )\right ...
0
votes
1answer
495 views

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, ...
2
votes
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 ...
2
votes
2answers
299 views

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 ...
2
votes
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 ...
1
vote
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.
1
vote
2answers
505 views

General solutions of Trigonometric functions

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 ...
1
vote
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 ...
3
votes
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 $A$. ...
1
vote
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.
0
votes
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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