The process of studying mathematics without formal instruction. _Don't_ use this tag just because you were self-studying when you came across the mathematical question you're asking; it is only for when fact that you're self-studying is what your question is _about_.

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Proving this binomial identity

I'm required to prove the following binomial identity: $$\sum\limits_{k=0}^l {n \choose k} {m \choose l-k} = {n+m \choose l}$$ I tried various arrangements but reached nowhere. Finally I turned to ...
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1answer
69 views

Every metrizable space with a countable dense subset has a countable basis

I'm working on this problem from Munkres: Show that every metrizable space with a countable dense subset has a countable basis. Here's my attempt at a proof. Let $X$ be a metrizable space with ...
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3answers
265 views

Laplace transforms: Convolution

Find $$1*1*1*\cdots*1\quad n\,\,\text{ factors}$$ that is, a function $f(t)=1$ convolution with itself for a total of $n$ factors. Would anyone mind helping me? I have no idea what I should do. ...
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2answers
67 views

Two sequences are equivalent. Prove that one is Cauchy iff the other is Cauchy.

This question has already been asked and answered here Let $ϵ>0$ be given. With loss of generality, we may assume $ϵ$ is rational. Suppose $a_n$ is a Cauchy sequence and $b_n, a_n$ are ...
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0answers
103 views

Continuity of the right-hand derivative of a Convex function (help with the proof)

Hi everyone I have some trouble with one point in the following proof. Let $f$ be a convex function (strict convex function) on a real interval. If $f'_-(a)=f'_+(a)$ where $f'_-$ and $f'_+$ are ...
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1answer
2k views

Laplace transformation: second shifting theorem

I know the answer is $1/(s^2) +e^-6s (2/s^3 -14/s -1/s^2 )$, but can anyone tell me how to evaluate the solution? I really get stuck.
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282 views

Isolation and self-study

A little background: I am currently a sophomore (studying mathematics) at an unknown university in the Middle East. My mother is European so it does not make sense to study mathematics in the Middle ...
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1answer
33 views

Why is this a boolean algebra

Let $A = \{a,b\}$. The $\mathcal P(A) = \{\emptyset,\{a\},\{b\},A\}$. Let $+$ be $\cup$, $\cdot$ be $\cap$, complement be set complement, $1$ be $A$, and $0$ be $\emptyset$. I need to explain why ...
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2answers
66 views

strict midpoint convex $\Rightarrow$ strict convex (help with a proof)

Hi everyone I have trouble with the following I think is something very simple, but I cannot figure out yet the correct approach for the strict inequality If $f$ is continuous and $f$ is strict ...
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3answers
70 views

CDF of $max(X,X^2)$

$X$ is uniformly distributed on $[-1,1]$. And $Y=max(X,X^2)$. What is $F_{Y}(t)$ , the CDF of $Y$? My attempt: I tried to graph it, but I think I found wrong. I found the joint pdf $5/6$. Is this ...
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3answers
331 views

Any good introductory book/tutorial on Fourier Transform (up to FFT) with plenty of exercises and solutions?

I wonder what could be a good book to start learning in depth all aspects of the Fourier transform up to the FFT algorithm, and beyond. I am going to dedicate quite some time on the subject, so I ...
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1answer
21 views

Confused about a proposition about rational polynomial

I'm studing complex analysis.I saw that $R(z)$ is a rational function.Consider the function $R(1/z)$ which we can rewrite as a rational function $R_1(z)$,and set $R(\infty)=R_1(0).$ with the ...
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2answers
78 views

why $\infty$ +$\infty$, $\infty$ - $\infty$ and 0⋅$\infty$ are left undefined.

I'm reading http://en.wikipedia.org/wiki/Riemann_sphere, and having the following question. 1. What's the mean of symbol $\infty$?Is it a surreal number? 2. they write note that ∞ + ∞, ∞ - ...
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0answers
69 views

Create a finite-state machine

I need to create a finite-state machine which accepts strings whose characters are in {a,b,c} and produce output strings of T's and F's. The machine outputs a T once the characters ab is encountered ...
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1answer
32 views

A doubt regarding splitting field.

$\Bbb Q(\omega)= \Bbb Q(\sqrt3,\iota)$ This is written in my text book that i am following. But I think this is a typo. Since $\Bbb Q(\sqrt3,\iota)$ is a larger field. in which $\Bbb Q(\omega)$ is ...
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1answer
45 views

Deriving Stochastic Euler Equation

If a consumer has utility function \begin{equation*} u(c_t) = ac_t - \cfrac{b}{2}c_t^2 \end{equation*} and present value budget constraint \begin{equation*} \sum_{j=0}^\infty E_t[\beta^jc_{t+j}] = ...
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1answer
84 views

symmetric normalized Graph Laplacian and symmetric normalized Adjacency matrix eigenvalues

I am trying to show that the symmetric normalized Graph Laplacian and symmetric normalized Adjacency matrix have corresponding eigenvalues $\lambda_i$ and $1 - \lambda_i$ for i=1 to n. $\lambda$ is ...
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2answers
219 views

Application of Picard-Lindelöf to determine uniqueness of a solution to an IVP

I am still struggling quite a lot with the Picard-Lindelöf-Theorem (also known as the Cauchy-Lipschitz-Theorem). Problem: Consider the following IVP with $\alpha \neq 1$ $$\begin{cases} y'&= ...
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1answer
135 views

Given a second countable space $X$, show $A \subset X$ has uncountable limit points

Here's my attempt at a solution and I'm wondering if it's correct. Let $X$ have a countable basis with $A \subset X$ an uncountable set. Show $A$ has uncountably many limit points. Let $A'$ be the ...
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1answer
55 views

Why is the set of positive definite matrices in $\mathbb R^{n\times n}$ a positive cone

The set of positive definite matrices in $\mathbb R^{n\times n}$ is geometrically a positive cone. This statement appears in almost every article on real positive definite matrices I read but without ...
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2answers
33 views

Finding a cycle with a specific property

I am reading the book Dummit and Foote - Abstract Algebra . One of the exercises is to find an $n$-cycle $(n \ge 5)$, $\sigma$ such that $\sigma^k = \tau$ for some positive integer $k$, where ...
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1answer
165 views

Question about $G_\delta$ set

I'm on summer break but I want to keep my math skills sharp so I'm self-studying a bit from Munkres. This question is from pg 194, chapter 4 about the Countability and Separation Axioms. I've ...
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1answer
54 views

Show that for $f,g: [0, + \infty) \to \mathbb{R}$ convex functions of Class $C^2$ their product $fg$ is convex

Problem: Let $f,g: [0, + \infty[ \to \mathbb{R}$ be two convex functions of Class $C^2$. Assume that $$ f(0) \geq 0, g(0) \geq 0 \text{ and } f'(0) \geq 0, g'(0) \geq 0 \tag{!}$$ Show that $fg$ ...
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4answers
193 views

Some aspects of inner products in $\mathbb R^3$

I read several descriptions about inner products recently due to an exercise (here it is no longer active so I like to ask again). I am still very confused about this concept. Any clarification will ...
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1answer
35 views

Differentiability of Norm $N: U \subset \mathbb{R}^n \to \mathbb{R}, \ x \mapsto \sum_{i=1}^n i|x_i|$

Problem: Let $U:= \lbrace x \in \mathbb{R}^n \mid x_i \neq 0 \text{ for } 1 \leq i \leq n \rbrace $ and show that the Norm given by $$ N: \begin{cases} U & \longrightarrow \mathbb{R} \\ x ...
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1answer
32 views

Question on closed sets using a convergent sequence

Intro: The following two questions are from my exam preparation sheet, it is not mandatory and will not be accredited (or improve marks and the like). There won't be a correction, merely an online ...
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1answer
47 views

Show a simple strategy.

Imagine that we have 49 cards with the values written on their faces, (they are all visible ) as follows; $$25, 24, 23, 22, ........3, 2, 1, 2, 3, .........23, 24, 25$$ suppose Paola and Victor are ...
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1answer
45 views

Help me writing Payoff matrix.

I guess, in order to answer this question, I need to write Payoff matrix. But I cannot write it. And then, I Will able to answer this question by myself. Thank you for helping. (These are just ...
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1answer
71 views

Check my answers: Dominant strategy.

I saw another question on Game theory. My answer for part a the nash equlibria (T, L) and (B,R). for part-b, Player-1's action T is strictly diominated. So Player1 never choose T. For part ...
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0answers
50 views

Is every curve measurable? [duplicate]

Does there exist a function $f:\mathbb R \rightarrow \mathbb R$ such that the set $E=\{(x,f(x)\mid x\in\mathbb R\}$ is non-measurable in $\mathbb R^2$?
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1answer
89 views

A question on Game theory

I'm studying Game theory, I saw the question: Consider two players; player A and player B playing the following estimation game. Each player chooses a number from {1, 2, 3}. If the difference ...
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0answers
84 views

Continuous transition kernel for Markov Chains

I am trying to show that the stationary distribution for a Markov Chain on a continuous state space can be obtained by building a transition density kernel, which obeys the detailed balance rule where ...
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1answer
25 views

Minumum value of integral and of partition are eventually the same?

I am trying to compare the minimum value of the integral of some function $f$ over the interval $[a,b]$ to the minimum value of the sum of the rectangles up to a certain point in a partition of ...
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2answers
169 views

Easy exercise (hint) Real Analysis

I've been stuck for a while with this problem. I suppose is something very easy, but I cannot figure out yet the correct approach. I'd really appreciated not a complete solution just some hints ...
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2answers
193 views

Measure Theory or Set Theory?

Having taken Real Analysis I before (the seven first chapters of baby Rudin) I have the option to take Measure Theory now. However I am torn between that and Set Theory. Which course would you go for ...
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1answer
50 views

Find $a,b,c\in \mathbb{R}$ that minimize the integral $\int_0^\infty |t^3 -at^2 -bt-c|^2e^{-t}$dt

Find $a,b,c\in \mathbb{R}$ that minimize the integral $\int_0^\infty |t^3 -at^2 -bt-c|^2e^{-t}$dt Hint:Use orthogonality of $(P_n)_{n=0}^\infty$ in $H=L_{2,\rho}(\mathbb{R}_+)$ with $\rho(t)=e^{-t}$ ...
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2answers
101 views

Show that $f(x,y)= \|x-y\|_2^2$ is differentiable

Problem: Show that $f: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ with $f(x,y)=\|x-y\|_2^2$ is differentiable and compute its differential at every point in the domain of $f$Note: $\| \cdot ...
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1answer
85 views

Good introduction to number theory that develops and/or makes heavy use of commutative ring theory and lattice theory?

I'd like to learn some number theory, since it provides a lot of motivation for commutative ring theory and even some motivation for lattice theory (at least, that's the impression I'm under). ...
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10answers
3k views

Becoming Better at Math

How can I become excellent at math? It really interests me but when I fail I become demotivated and begin to give up. EDIT: Could anyone suggest books for someone with a math education that just ...
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1answer
50 views

Let $X=L_2(0,1)$ and $S=\lbrace x\in X: \int_0^{1/2}x^2 dt\le1\rbrace$

Let $X=L_2(0,1)$ and $S=\lbrace x\in X: \int_0^{1/2}x^2 dt\le1\rbrace$. I'm trying to show wheter S can be written as $\cup_{n=1}^\infty S_n$ where $S_n\in X$ and $Int\bar{S_n}=\emptyset$ I tried a ...
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3answers
153 views

Manifolds and their dimension

Why are the following two sets manifolds and what are their dimensions? The set of all 2 by 2 matrices with determinant 1. The set of all inner products on $\mathbb R^3$. I have difficulty in ...
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1answer
407 views

Prerequisites for understanding G.H. Hardy's 'Divergent Series'

I picked up a copy of G.H. Hardy's 'Divergent Series' a few days ago. So far I love it, as I love the ideas associated with sequences and series, but I am finding it a bit difficult to understand. I ...
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2answers
131 views

On the canonical isomorphism between $V$ and $V^{**}$

I am trying to understand more about the Bidualspace (or double dual space). The whole idea is that $V$ and $V^{**}$ are canonically isomorphic to one another, which means that they are isomorphic ...
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1answer
43 views

Abstract Algebra: prove it is cyclic

I have question in referring to below link. Question. Suppose if I have [M:K]=2 and I know that K is subset of M. M:=$\mathbb{Z}_2[x]/(f(x))$ where f(x)=x$^4$+x+1. Then how this will be cyclic? I ...
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1answer
63 views

One and Two Tailed Independent T Test Questions

The city council for a small town has been receiving complaints from local law enforcement that citizens have been extremely uncooperative when pulled over for minor traffic violations. To remedy ...
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3answers
135 views

Is there a better way to read proofs?

I'm finishing my undergraduate degree in 6 weeks and I'm pretty happy with how my education is coming along so far. I can write proofs, solve many different problems, and I even have some idea as to ...
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1answer
54 views

Determine the languages for the given alphabet

For the alphabet $\sum = \{0,1\}, let A,B,C \subseteq \sum^*$ be the languages below. $i. A = \{1, 0, 00, 11, 000, 111, 0000, 1111\}$ $ii. B = \{w \in \sum^*|||w|| \ge 2 \}$ $ii. C = \{w \in ...
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2answers
50 views

Evaluating expression at infinity

How do I evaluate something like: $$xe^{-(x-\theta)}\text{ from }x = \theta\text{ to }x=\infty?$$ This came up in an integration I tried to do, and I realize it's a very basic question. But I am ...
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3answers
79 views

Normalizer of a subgroup of $GL_2(\mathbb{R})$

I have following subgroup of $GL_2(\mathbb{R}):$ $$A=\Bigg\{\left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right),\left( \begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array} ...
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9answers
2k views

Good Textbooks for Real Analysis and Topology.

I'm currently in my 3rd year of my undergrad in Mathematics and moving onto my 4th year next year. I took a course in Real Analysis I, but the professor was very confusing and we didn't use a textbook ...