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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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}} ...
3
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1answer
82 views

How to avoid losing the woods for the trees in daily study/lecture time

When facing to some new material in mathematics, I feel easily to be overwhelmed by lots of details with losing the woods for the trees. So is there some good strategy to study the materials ...
1
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1answer
37 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 ...
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0answers
39 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 ...
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1answer
44 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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3answers
87 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
68 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
52 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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1answer
88 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.
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0answers
61 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 ...
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5answers
5k views

Is memorization a good skill to learn or master mathematics?

I sometimes spend inordinate amounts of time memorizing math articles or theorems/proofs or formulas. My question is "am I wasting time?" and will 'active thinking' or 'working out problems' be faster ...
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1answer
143 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 ...
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1answer
58 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 ...
2
votes
1answer
55 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 ...
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2answers
94 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. ...
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0answers
40 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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2answers
95 views

Self learning game theory and probability

I am teaching myself mathematics, my objective being a thorough understanding of game theory and probability. In particular, I want to be able to go through A Course in Game Theory by Osborne and ...
2
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1answer
92 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. ...
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0answers
12 views

On inverse images of a function from a $\sigma$-algebra into a topological space.

Let $f$ map $X$ into $Y$. Call the $\sigma$-algebra $X$ and let it be the set $\{\{1,2\},\{1\},\{2\},\{\emptyset\}\}$. Call the topological space Y and let it be the set of all open sets of $R$. ...
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0answers
31 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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0answers
36 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 ...
2
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1answer
267 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 ...
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1answer
73 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 ...
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2answers
92 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 $\lambda$ ...
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1answer
2k 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
122 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 ...
2
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2answers
586 views

Getting up to speed enough to tackle mathematics and logic applied in programming?

I'm a games programmer with an interest in the following areas: Calculus Matrices Graph theory Probability theory Combinatorics Statistics More linguistic related fields of logic such as natural ...
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1answer
252 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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votes
2answers
52 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
83 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 ...
5
votes
1answer
191 views

indefinite integral $\int\sin\sqrt[3]{x}~dx$

I want to determinate the integral $\int\sin\sqrt[3]{x}~dx$ . I tried to use integration by partitions and integration by substitution but I came to no result. I know the result which is shown here ...
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1answer
44 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
19 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
61 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 ...
0
votes
1answer
118 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 $s \gt d$. ...
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0answers
44 views

Inequality of gamma distribution

Let $X_{\alpha} \sim \text{Gammma}(\alpha,1),\alpha >0$ with distribution function $G_{\alpha}$ and $X_{\beta} <_{(c)} X_{\alpha}\,\forall ~0<\alpha<\beta<\infty$. Then show ...
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1answer
17 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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1answer
33 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' ...
3
votes
1answer
83 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 ...
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1answer
30 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= ...
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1answer
36 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 ...
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2answers
23 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}}$ ...
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2answers
386 views

Find the equation of the parabola given the tangent to a point and another point.

I have a problem with derivatives, I've been trying to solve but I was not able to do it. A parabola is tangent to the line $3x-y+6 = 0$ in the point $(0,6)$ and goes through the point $(1,0)$. ...
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0answers
61 views

(soft question) Kreyszig's Functional Analysis or Rudin's Real and complex analysis for an introduction into more advanced analysis?

I realise they are quite different in their approach and material covered, but they share the central stuff like normed/Banach/Hilbert spaces, Hahn-Banach theorem etc. Not really understanding what ...
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1answer
39 views

How to show this inequality in the law of iterated logarithm

Let $\psi(t) = \sqrt{2t\log\log t}$ and $q>4$. Then we have for large $k$, $$ \psi\left(q^k - q^{k-1}\right) \geq \psi\left(q^k\right) \left(1-\frac{1}{q}\right). $$ To prove this, I can tell by ...
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1answer
182 views

Issue with proof: Cauchy Completeness of Real Numbers

Having trouble understanding a cardinality-related argument when proving that all Cauchy sequences of reals numbers converge to a real limit. Came across it on CC Pugh's Real Mathematical Analysis ...
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1answer
64 views

Categories and the direct image functor

I've just started learning about sheaves and yesterday I had a look at MacLane/Moerdijk's book "Sheaves in Geometry and Logic". I've discovered something in this book on which I'd like some ...
0
votes
1answer
40 views

A Question on the Scaling Invariance of Brownian Motion

I read the following paragraph. Let $B_t, \ t \in [0, \infty)$ be a standard linear Brownian motion. For each $q > 4$, define the following sequence of sets. $$ \Omega_k := \left\{\omega \in ...
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2answers
77 views

A Question on Bolzano-Weierstrass Theorem

The following are two equivalent statements about the Bolzano-Weierstrass theorem. Every bounded real sequence has a convergent subsequence. A subset of $\mathbb R$ is compact if and only if it is ...
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0answers
55 views

Total Variation of a Signed Measure

In Royden's Real Analysis the total variation $|\mu|$ of a signed measure $\mu$ is defined to be $$|\mu|(E) := \sup\sum_{k=1}^n |\mu(E_k)|,$$ where the supremum is taken over all finite disjoint ...