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 the fact that you're self-studying is what your question is _about_.

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23 views

On functions and their linear independence

How would you access the following problem: Show that the set of functions $$ \phi_n : \mathbb{R}_{>0} \rightarrow \mathbb{R}$$$$\phi_n(x) = \frac{1}{n+x}$$for $n \in \mathbb{Z}^{\ge 0}$ is ...
2
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1answer
96 views

Conservative vector field, potential function and work done

For (i), is that I have to show $curl F = 0$ ? For (ii) and (iii), what should I do in order to find the potential function and work done? Also, is the answer $4$ for (iii)?
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2answers
61 views

Convergence of Expectations

Suppose $\{X_n\}$ is a sequence of non-negative random variables such that $$EX_n<\infty, \text{ }\lim_{n\rightarrow \infty}EX_n =\infty$$ and $\lim_{n\rightarrow \infty}X_n$ exists a.s. May I ...
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1answer
49 views

Multiple integrals: Double integrals

For this question, how to evaluate the integral by changing the order of integration? Also, how to sketch the region of integration? I really get stuck.
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1answer
34 views

Finding partial derivatives from a given function

For i), is the answer $df/du = (a)df/dx + (2cu)df/dy, df/dv = (b)df/dx + (2dv)df/dy$ For ii), is the answer $d^2 f/du^2 = (a^2) d^2 f/dx^2 + (4acu) d^2 f/dxdy + (4c^2 u^2) d^2 f/dy^2$ , $d^2 f/dv^...
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0answers
28 views

On a summation manipulation

I have $R_t= \frac{1}{h} \sum_{j=0}^{h-1} E_tr_{t+j} + \theta_t$ where $E_tr_{t+j} = E[r_{t+j}| I_t]$. By subtracting $r_t$ from both sides and after some manipulations I should get: $$R_t - r_t= \...
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2answers
40 views

How to do this Integration (in Orthonormal Family for Continuous Functions)

Here is a common example in the discussion on orthonormal family. Let $\mathcal L = C[a, b]$. For $k \in \mathbb Z$, let $e_k \in \mathcal L$ be defined by $$e_k(\xi) := \frac{1}{\sqrt{b-a}} e^{\frac{...
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1answer
62 views

Convolution Operator and Integration Operator

I have some questions about the following two operators. A convolution operator $T$. If $k \in \mathcal L^1(\mathbb R)$, then $$f(x) \mapsto \int_{-\infty}^\infty k(x-y)f(y) dy: \mathcal L^2(\mathbb ...
3
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1answer
123 views

Closed Subspaces of Hilbert Spaces

I read the following statements. But I do not know how to show it or any example to support it. Could anyone provide some explanation and examples, please? Thank you! The subspace $C^\infty$ ...
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0answers
49 views

Basis for Finite Dimensional Hilbert Spaces

Verify that a Hilbert space orthonormal basis in a finite dimensional Hilbert space is the same as an orthonormal basis in the sense of linear algebra. Here is what I know. Hilbert space orthonormal ...
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3answers
102 views

$f$ an analytic function on $\mathbb{C}$ which takes values in $\mathbb{C}\backslash(-\infty,0]$ implies constant

Let $f$ be an analytic function on $\mathbb{C}$ which takes values in $\mathbb{C}\backslash(-\infty,0]$, i.e. takes values in the complement of the nonpositive part of the real axis. Show that $f$ is ...
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1answer
72 views

Eigenvalue of Compact Operators

To prove that the set of eigenvectors of a compact linear operator on a normed space $X$ is countable, I read "it suffices to show that for every real $k > 0$ the set of all eigenvalues whose ...
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1answer
74 views

A Question on Compact Operators on Hilbert Space

I read this question which I have no idea how to start. Could anyone provide me with some detailed answer, please? Thanks. Suppose that a linear operator $F$ from a Hilbert space $\mathcal H$ to ...
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2answers
58 views

To discuss differentiability of function at origin and my attempt

P1: $F = |x| + |y| when x,y is not equal to 0, = 0 when x = y = 0 P2 :Discuss the differentiability at origin of $F = y sin(1/x)$ :
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0answers
84 views

Can any one recommend a way to “quickly” learn a subject?

I would love to read a well written book on a subject - provided that I have the time. But sometimes we do not need to become experts on a particular field but still need the basics. For example, a ...
3
votes
3answers
100 views

limits multivariable calculus. where am i wrong with my attempt?

P : $\lim_{(x,y) \to (0,0)} f(x,y)$ where $$f(x,y) = y\sin\frac1x + \frac{xy}{x^{2} + y^{2}}$$ Text book says Limit doesnot exist . So where i am wrong with my proof below ? EDITED ATTEMPT : Or ...
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1answer
536 views

Real Analysis : Self Studying vs Doing a Course

I am an engineering graduate student. Recently I got interested in studying Maths. So, I have started self-studying Real Analysis(let's call it RA) using a few books. I will also be using problem ...
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1answer
59 views

Sum converging a.s.

Let $X_k$ be independent random variable s.t. $\sum_{k=1}^nX_k\rightarrow_{a.s.} X$. So, $$X=\sum_{k=1}^\infty (X_k^+ -X_k^-)$$ Is it true that $X=\sum_{k=1}^\infty (X_k^+)-\sum_{k=1}^\infty (X_k^-)...
2
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2answers
49 views

On subgroups of isometries and their respective fixed-points

I am working on the following problemset: Let $G < \DeclareMathOperator{\Iso}{Iso}\Iso(E)$ be a finite subgroup of the isometries in the euclidean plane. Denote: $$G = \{g_1, \ldots , g_n \} ...
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1answer
185 views

Proving Euler Summation by Parts Without Using Integration by Parts

Assume $f$ has continuous derivative $f'$ on [a,b]. Prove the following summation formula, without using partial integration: \begin{equation} \sum_{a< x \le b}f(n)=\int_{a}^{b}f(x)dx+\int_{a}^{b}...
2
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1answer
69 views

Exercise on measure theory, (verification and suggestion)

Hi everyone I'd like to know if the following is correct and also I'd appreciate any suggestion to improve the argument. Thanks in advance For every positive integer $n$, let $f_n:{\bf{R}}\to [0,\...
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0answers
59 views

Proof check - if a set is a $\sigma$-algebra or not.

From Rudin Real and Complex Analysis Theorem 1.12. Suppose $M$ is a $\sigma$-algebra in $X$, and $Y$ is a topological space. let $f$ map $X$ into $Y$. If $\Omega$ is the collection of all ...
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1answer
146 views

Prove that $\frac{1}{a^3(b+c)}+\frac{1}{b^3(a+c)}+\frac{1}{c^3(a+b)}\ge \frac32$

$a,b,c$ are positive reals with $abc = 1$. Prove that $$\frac{1}{a^3(b+c)}+\frac{1}{b^3(a+c)}+\frac{1}{c^3(a+b)}\ge \frac32$$ I try to use AM $\ge$ HM. $$\frac{\dfrac{1}{a^3(b+c)}+\dfrac{1}{b^3(...
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0answers
48 views

Riemann-Stieltjes Integrals with $n$ discontinuities (Proof Review)

First of all is the proof legitimate? If so, then are there other methods to achieve the same result that are neater than mine? Let $\alpha : [a,b] \to \mathbb{R}$ be a step function with ...
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2answers
121 views

Borel-Cantelli (proof and application)

Hi I was reading the second volume of the Tao's Analysis book and in one exercise he's asking for a proof of Borel-Cantelli If we have a sequence $s_n\in \Omega$ of measurable sets s.t. $\sum_{n\...
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0answers
46 views

Convergence equivalent random sequences

Suppose we have a sequence of independent random vars $X_n$ and consider a sequence of truncated random variables $Y_n=X_n1_{X_n\le n}$ s.t. $E[Y_n]=0$. We know that $X_n$'s and $Y_n$'s are ...
1
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1answer
94 views

Riemann-Stieltjes Integral and the Step Function

Let $a < c< b$ and let $\alpha (x)$ be defined as $\alpha (x) =\begin{cases} 0 & a \le x \le c \\ 1 &c<x \le b \end{cases}$. Show that $f \in \mathcal{R}(\alpha)$ if and only if $f$ ...
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3answers
4k views

James R. Munkres' TOPOLOGY, 2nd edition: How to check my work?

I'm trying to learn, or revise, some topology from James R. Munkres' TOPOLOGY, 2nd edition. I'm working alone; that is, I'm self-learning. It is quite fun. But the problem is how do I check if I've ...
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2answers
469 views

Proof that every open set in the plane is a countable union of rectangles.

I came across this statement in the first chapter or Rudin Real and Complex Analysis. Rudin states that every open set in the plane is a countable union of rectangles. Looking for a proof I ...
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1answer
327 views

Norm of the Resolvent of a Self-Adjoint Operator

Let $\mathcal H$ be a Hilbert space and $\mathcal L$ is a self- adjoint operator with a discrete spectrum $\{\lambda_{j}\}$. I read that it is well known that for, $\lambda \notin \sigma(\mathcal L)$,...
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2answers
125 views

Spectrum of Self-Adjoint Operators

This is an exercise (5-i) from here. It has two parts as follows. For a self-adjoint operator $A$. Show that $A \geq k I, \ k \in \mathbb R$ if and only if $\lambda \geq k$ for all $\lambda$ ...
2
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1answer
700 views

Most efficient mental way to convert Decimal to Hexadecimal

My question is as follows: What is the most efficient mental way to convert Decimal to Hexadecimal? I've heard of many methods. Some people divide the decimal by 16 and find the remainder. Others ...
3
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1answer
103 views

Nash's Axiomatic Bargaining: Source of problems sets and practice questions.

From where can I practice questions related to the following topic: Nash's Axiomatic Bargaining. Any form of book reference or a link to some online problem set would be highly appreciated.
2
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1answer
47 views

On loss of generality proving the Cauchy-Schwarz inequality.

In Rudin's Real and Complex Analysis the Schwarz inequality: $|(x,y)| \le ||x|| \ ||y||$ is proven in the following manner: Put $A = ||x||^2, B=|(x,y)|$ and $C = ||y||^2$. There is a complex number ...
8
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1answer
2k views

Vladimir Zorich vs Rudin/Pugh/Abbott

There have been various comparisons between books on Analysis. I was surprised to find out that Zorich's book on Analysis was not compared anywhere. Can anyone give a comparison between Zorich and ...
0
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1answer
431 views

In what order should I learn linear algebra and multivariable calculus?

I took AP calculus in high school and I really enjoyed it, but when I got to my university I was upset that I couldn't take Calculus II as it didn't fit in my schedule. I feel kind of behind now that ...
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2answers
60 views

How to prove $\dfrac{a}{b+c}+\dfrac{b}{c+a}+\dfrac{c}{a+b} <2$ [duplicate]

Prove the inequality for a triangle with sides $a,b,c$ we have $$\dfrac{a}{b+c}+\dfrac{b}{c+a}+\dfrac{c}{a+b} <2$$ Trial: Since $a,b,c$ are sides of a triangle I know $a+b>c,b+c>a,a+c>b$
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0answers
92 views

How to deal with a lack of ability to apply ideas in math?

I am currently studying Theoretical Computer Science, but as a Computer Science student who does not have a formal background in mathematics, past A Level (High School), I find that the ideas I learn ...
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1answer
64 views

Determine all the generators of $\mathbb{Z}_{25}^{\times}$

Determine all the generators of $\mathbb{Z}_{25}^{\times}$. Is there some way that I can use the fact that $\mathbb{Z}_{25}^{\times}$ is cyclic generated by $3$?
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1answer
24 views

Show that zero sequences satisfy the following equation

I am working on the following problem and got puzzled: $\\$ Show that every zero sequence $(a_n), a_n \neq 0$ satisfies the equation: $$ \lim_{n \rightarrow \infty} \frac{\sqrt {1 + a_n} - 1}{a_n} = \...
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2answers
68 views

Optimizing a box

I'm learning the use of derivatives and I have found a problem: Supposing we want to build a box of $4000\, \textrm{cm}^3$ of volume without top and a square base. Which are the measures so we ...
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1answer
349 views

Existence of a unique maximizer of a strict quasi-concave function defined over a convex set

Set $S \subset \mathbb R^2$ is compact and convex. A typical element of $S$ is $s=(s_1,s_2) \in S$. Also, $d \in \mathbb R^2$ is a fixed element such that there exists $s \in S$ such that $s \gt d$. ...
0
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1answer
19 views

If $s_1 +s_2 \gt 1$ and $(t_1,t_2)$ be a convex combination of this with $(0.5,0.5)$ then show that $t_1t_2 \gt 0.25$

Let $(s_1,s_2)$ be such that $s_1 + s_2 \gt 1$. Let $(t_1,t_2)=((1-\epsilon )(0.5) + \epsilon s_1 , (1- \epsilon)(0.5) +\epsilon s_2)$, where $0< \epsilon \lt1$. I need to show that for $\epsilon $...
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2answers
336 views

A Counter Example about Closed Graph Theorem

This is an example I read around closed graph theorem. Let $Y=C[0, 1]$ and $X$ be its subset $C^\infty[0, 1]$. Equip both with uniform norm. Define $D: X \to Y$ by $f \mapsto f'$. Suppose $(f_n, f_n') ...
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1answer
39 views

A mapping defined by an equivalence relation $\sim$ that is compatible with a semigroup

I am working on the following: Problem: Let $(H, \cdot)$ be a semigroup and $\sim$ an equivalence relation on $H$. We say that $\sim$ is compatible with $(H, \cdot)$ $$: \iff \forall a,a',b,b' \...
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1answer
42 views

question on a stopping time problem.

I borrowed some lecture notes on stochastic calculus, which contained the following exercise: Let $(X_n)_{n>0}$ be a sequence of random variables with $X_n: \Omega \to [0,\infty)$. We set $S_n= ...
3
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1answer
108 views

Self-learning Book recommendation for topics in ring-theory

I failed badly in my Internal examination in ring theory , and at any cost want to improve upon my grades in the final eamination,with a month and a half to go .... I haven't yet covered the below ...
16
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1answer
493 views

IMO programs of different nations?

We in Albania have a good team in the IMO, and this year I will probably be part of it. Since Albania does not have a public training programme, I have to consult the training programmes of other ...
0
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1answer
45 views

Continuity and Differentiability of a series of functions

Consider the function $f(x)=\sum_{n=1}^{\infty} 2^{-n}g(2^{2^{n}}x)$ where \begin{equation} g(x)=\begin{cases} 1+x &-2 \le x \le 0 \\ 1-x &0 \le x \le 2 \end{cases} \end{equation} where $g(x)...
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2answers
26 views

Converges of a sequences defined through a continued fraction

Consider the following sequence $(b_n)_{n \geq 1}$ recursively given through the continued fraction $b_1 = \frac{1}{1}$, $b_2 = \frac{1}{1+ \frac{1}{2}}$, $ \dots , b_n = \frac{1}{1+b_{n-1}}$ ...