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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2
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2answers
80 views

What is the definition of closed subspace?

I am trying to understand what is intended with closed subspace, I took the following guess: A closed subspace $M$ of a Hilbert space $H$ is a subspace of $H$ s.t. any sequence $\{x_n\}$ of elements ...
3
votes
3answers
79 views

On the intuition behind the projection theorem.

I have recently proved the projection theorem in a Hilbert space setting. The statements were: If $M$ is a closed subspace of a Hilbert space $H$ and $x \in H$, then: There is a unique element ...
7
votes
2answers
531 views

For someone who is self-studying topology: what are the main topics to focus on?

I will have to teach myself topology for the Math GRE Subject Test because, although I graduated with a math major, I never took topology. I have Munkres and Kelley, along with the Schaum's Outlines ...
7
votes
3answers
114 views

What does one study to increase understanding of the $P \stackrel{?}{=} NP$ problem?

If one were to learn more about the $P \stackrel{?}{=} NP$ problem, where would one start? I understand what the problem is—but not enough to be able to read anything technical about it. ...
2
votes
1answer
97 views

Length of A Diagonal Line of Square

After watching this video to calculate the length of diagonal of square , a question arises to me is : Why is length of diagonal of square $$\frac{\text{side}}{cos45^0}$$? Why $cos45^0$ ?If i ...
10
votes
2answers
360 views

Importance of the zero free region of Riemann zeta function

I have heard that for improving the error term in the Prime Number Theorem, we need better and better estimates on the zero free region. I have also heard that the best possible error term comes from ...
1
vote
1answer
72 views

Let $X$ and $Y$ have the joint pdf $f(x,y)=8x(1-y),0<y<1,0<x<1-y$. Compute $P(Y<X\mid X \leqq \frac{1}{4})$

Let $X$ and $Y$ have the joint pdf $f(x,y)=8x(1-y)$, $0<y<1$, $0<x<1-y$. Compute $P(Y<X\mid X \leqq \frac{1}{4})$. I know: $$P(A\mid B) = \frac{P(A \cap B)}{P(B)}$$ Therefore I ...
1
vote
2answers
193 views

What is the probability that $X$ and $Y$ are within 0.1 of each other given a uniformly distributed joint pdf.

Two construction companies make bids of $X$ and $Y$ (in \$$100,000$'s) on a remodeling project. The joint pdf of $X$ and $Y$ is uniform on the space $2<x<2.5,2<y<2.3$. If $X$ and $Y$ ...
1
vote
1answer
231 views

Probability and Measure Theory by Ash

Has anyone used the textbook above? If so how does it compare with billingsley, Chung and similar such books in terms of rigor, coverage, and ease if use for self study?
0
votes
0answers
49 views

Find the marginal pdf of a joint distribution

Let $X$ and $Y$ have a uniform distribution on the set of points with integer coordinates in $S = \{(x,y):0\leqq x\leqq7, x \leqq y \leqq x+2\}$. That is, $f(x,y) = \frac{1}{24}\in S$, and both x and ...
2
votes
1answer
47 views

How to find all Dirichlet characters

I want to know all the Dirichlet characters modulo $m$. I know that the number of such characters are $\phi(m)$. But how do find each and every character. for small moduli I could do it using some ...
0
votes
0answers
51 views

Calculus of Variations-First and Second Order Deviations

I'm new to Calculus of Variations and the Method of Least Action (L=T-V) What I'm unsure about is how first and second order deviations are used in finding the least action. I know it's used to find ...
0
votes
1answer
18 views

On the polar representation of an inner product.

Take $H$ an inner product space. $x,y \in H$. Take $b = |<x,y>|$ . Then the polar representation of $<x,y>$ is: $$<x,y> = be^{i\theta}$$ for some $\theta \in (-\pi, \pi]$. Why is ...
0
votes
2answers
183 views

Difference among the same distribution , identical distribution and similar distribution.

$X\sim N(\mu_1,\sigma)$ and $Y\sim N(\mu_2,\sigma)$ are similar but not identical. $X\sim N(\mu,\sigma)$ and $Y\sim N(\mu,\sigma)$ are identical. But what is same distribution? Do same and ...
1
vote
3answers
221 views

Is natural numbers set $\mathbb N$ infinite set?

A set with uncountable number of elements is called an infinite set. Is that the set of all natural numbers, $\Bbb N=\text{{$1,2,3,\ldots$}}$ infinite set? As far i know $\Bbb N$ is "countably" ...
2
votes
1answer
76 views

Proof of the starting part of theorem 1.17 Rudin ( Complex and Reals)

The proof I would like is of the following fact: Put $\delta_n = 2^{-n}$. To each positive integer n and each real number t corresponds a unique integer $ k = k_n(t)$ that satisfies $k \delta_n \le t ...
1
vote
1answer
47 views

$\mathbb E[\bar X_n]=0$

A conditional normal rv sequence, does the mean converges in probability, in this question how can i get $\mathbb E[\bar X_n]=0$? Here is my attempt; $$\mathbb E[\bar ...
0
votes
1answer
74 views

Partial derivative is bounded

Let $f(t,z)$ be a bounded (say by a constant $M$) continuous function on $\mathbb{R}_t \times \mathcal{U}$ where $\mathcal{U}$ is an open neighborhood of $0 \in \mathbb{C}_z$. Moreover, for each fixed ...
44
votes
2answers
864 views

Effective Research Notes

Note-taking for research is vital to your success as a mathematician. As I look back at some of my handwritten notes, I realized how poor they were. I had thought to myself, "What happened?" I was ...
12
votes
6answers
296 views

Sources for mathematics outside the mathematics world

In this question I would like to ask you about material showing the uses (or occurrences) of mathematics in the everyday world. The aim is to encourage with it a group of young undergraduate ...
3
votes
4answers
139 views

prove that $\sqrt{2} \sin10^\circ+ \sqrt{3} \cos35^\circ= \sin55^\circ+ 2\cos65^\circ$

Question: Prove that: $\sqrt{2} \sin10^\circ + \sqrt{3} \cos35^\circ = \sin55^\circ + 2\cos65^\circ$ My Efforts: $$2[\frac{1}{\sqrt{2}}\sin10] + 2[\frac{\sqrt{3}}{2}\cos35]$$ $$= 2[\cos45 \sin10] ...
11
votes
2answers
686 views

How can I pick up analysis quickly?

I have a 2-3 week recess from university for winter break. In this time, I would like to learn analysis, starting with Walter Rudin's Principles of Mathematical Analysis, and then, if at all possible, ...
4
votes
2answers
196 views

Is “mixed math” a useful way to learn math?

I was reading a book about how mathematics was taught in Cambridge in the 19th century, and it struck me how much physics was included in the syllabus, and it wasn't optional but everyone had to learn ...
2
votes
0answers
141 views

Directional derivative (Vector)

Given $f:\mathbb{R}^2 \to \mathbb{R}^2$ is a map $f(x,y)=(u(x,y),v(x,y))$ and $\alpha=(\alpha_1,\alpha_2)$ is a point, then how does one show that $f$ is differentiable (or not) in the direction ...
0
votes
1answer
82 views

Solve laplace equation inside a rectangular

My answer is $U = Acos(nπx/L)e^-nπy/L$ I really have no idea how to solve the particular solution. Please advise me.
1
vote
1answer
37 views

On the isomorphisms $(\mathcal{O}_{Z,X})_\mathfrak{p}\cong\mathcal{O}_{Y,X}\text{ and }\mathcal{O}_{Z,X}/\mathfrak{p}\cong\mathcal{O}_{Z,Y}$.

Suppose you have two closed, irreducible subvarieties $Z\subseteq Y$ in some variety $X$. (I'm not sure if it matters, but for ease I'll just assume everything is over an algebraically closed field.) ...
4
votes
1answer
282 views

Learning functional analysis and measure theory

I have taken a first course in real analysis and I'm currently studying analysis in $\mathbb{R}^N$ on my own. I want to start functional analysis after this, and I also want to learn measure theory ...
1
vote
1answer
31 views

Bound for Outlyingness

Given a sample of $n$ data, $x_1, \dots, x_n$. Define the sample mean $$\bar x := \frac{1}{n}(x_1+\cdots+x_n),$$ and sample variance $$s^2 := \frac{1}{n-1} \sum_{i=1}^n (x_i-\bar x)^2.$$ To measure ...
1
vote
0answers
38 views

Continuity of set function on field and relation with continuity in topological space

I am trying to understand how continuity of measures relates to the definition of continuity in topological sets : Every open set in range corresponds to an open set in domain. A real valued set ...
2
votes
0answers
46 views

Continued fraction approximation to a function and its derivative

I am recently working on fitting a model with incomplete beta function. In order to put it into my optimization algorithm, I must find out the derivatives of the incomplete beta function $B_p(x,y)$ ...
0
votes
1answer
49 views

Convergence of Remainder from Taylor Expansion

For a distribution function $F$ and its variance functional $T(F)$, it can be shown that the Taylor expansion of $T(F)$ at $F$ in the direction of the empirical distribution function $F_n$ gives the ...
1
vote
1answer
46 views

Problem of Partial Differential Equations

For this question, I get stuck when I apply the second initial equation. My answer is $θ= Ae^-(kλ^2 t)\cos λx$, where $A$ is a constant. Would anyone mind telling me how to solve it?
0
votes
1answer
59 views

Lebesgue-Stieltjes Integral (Several Variables)

Let $\mathcal F$ be a convex set of probability measures or distribution functions and $F, G$ be two elements in $\mathcal F$. Let $T$ be a functional on $\mathcal F$ defined as follows. Note that $h$ ...
2
votes
1answer
103 views

Showing that the indicator/characteristic function is not a regulated function

I want to show that the indicator function (aka. the characteristic function) is not a regulated function. \begin{align} \chi : \begin{cases}[a,b] & \longrightarrow \mathbb{R} \\ x & ...
5
votes
2answers
205 views

Still forget even if theorem-proof “self-discovered”; Importance of intuition/proficiency of concepts in research work… [closed]

It is widely said if we go through concepts/theorems/proof on our own by actively doing instead of passively reading, the idea will be ingrained in mind. I agree with that, it really often helps. ...
1
vote
3answers
150 views

Confused about transfinite induction

QUESTION: I seem to be confused about how transfinite induction is carried out. I have looked at several examples and they seem to follow a procedure consisting of grounding the induction, proving the ...
1
vote
0answers
54 views

Observed and expected Fisher information of a Bernoulli Random Variable

If $X$ is a Bernoulli random variable with parameter $p$, the probability mass function is given by $$ f(k) = p^k(1-p)^{1-k} $$ and the loglikelihood, $\ell(p)$, is given by $$ \ell(p) = ...
3
votes
1answer
91 views

Why is a Hyperplane called a “Hyper”plane?

I just had this curious question. In other fields, the word "hyper" is actually used to refer to something which is "over; beyond; above" as defined by Google. An example of such terms would be ...
2
votes
2answers
248 views

How to learn math? [closed]

I am 19 years old and I'm computer programmer and Software Engendering college Student, And I am smart (mean: I am not stupid) and know programing better than other, I think math is like programming. ...
5
votes
1answer
55 views

The map $f\colon\mathbb{A}^2_k\to\mathbb{A}^2_k$ given by $f(x,y)=(x,xy)$ is birational?

I'm reading a bit about rational maps, and I'm still trying to get get my head around birational maps. Consider the map $f\colon\mathbb{A}^2_k\to\mathbb{A}^2_k$ on the affine $2$-space over $k$ ...
3
votes
2answers
94 views

A metric that makes $l^\infty$ separable

I know that "The metric space $l^\infty$ is not separable with the metric defined between two sequences $\{a_1,a_2,a_3\dots\}$ and $\{b_1,b_2,b_3,\dots\}$ as $\sup\limits_{i\in\Bbb{N}}|{a_i-b_i}|$. ...
1
vote
1answer
97 views

In war with exercise, any future for me?

I love theory with theorems, definitions & proofs, but i don't like exercise, I need more context around it. Is there a different way of practicing theory except given exercises, maybe some ...
5
votes
6answers
93 views

What is the value of $a+b+c$?

What is the value of $a+b+c$? if $$a^4+b^4+c^4=32$$ $$a^5+b^5+c^5=186$$ $$a^6+b^6+c^6=803$$ How to approach this kind of problem. Any help. UPDATE: Thank you all for answers. Now I ...
1
vote
1answer
94 views

Properties of a differentiable and strictly convex $f:(a,b) \to \mathbb{R}$

Let $f:(a,b) \to \mathbb{R}$ be a differentiable and strictly convex function I tried to explore some of the properties of such a function. For all $x,y \in (a,b)$ with $x \neq y$ I could apply ...
1
vote
1answer
107 views

Expectation of multinomial distribution

Three fair dice are cast. In 10 independent casts, let X be the number of times all three faces are alike and let Y be the number of times only two faces are alike. Find the joint pdf of X and Y and ...
0
votes
1answer
139 views

Probability of unbiased die

One of the numbers 1,2,...,6 is to be chosen by casting an unbiased die.Let this random experiment be repeated five independent times.Let this random variable $X_1$ be the number of termination in the ...
1
vote
1answer
73 views

How do I compute the normalisation of $A=k[X,Y]/(Y^3 - X^5)$?

I'm trying to solve exercise 4.7 in Reid's UCA: "Find the normalisation of $A=k[X,Y]/(Y^3 - X^5)$." I can easily show $A$ is not normal: let $x$ and $y$ denote the images of $X$ and $Y$ in $A$. Thus ...
0
votes
2answers
80 views

Prove that $A \subset B$ if and only if $A \setminus B = \emptyset$

Prove that $A \subset B$ if and only if $A \setminus B = \emptyset$. What is the correct and mathematically strict way to prove the above? (slightly different than Prove that if $A \setminus B = ...
2
votes
1answer
61 views

How to show that $\lim_{x\to \infty}f'(x)=0$

Let $f$ be a real-valued, bounded, twice differentiable function defined on $(0,\infty)$ with $f'(x)\ge 0$ and $f''(x)\le 0$. Show that $$\lim_{x\to \infty}f'(x)=0$$ I understand $f: (0,\infty) ...
46
votes
14answers
4k views

Nobody told me that self teaching could be so damaging…

Even though I've been teaching myself math for a couple of years now I only just started (a month ago) at the university. My experience is rather mixed. For starters, I'd like to mention that I'm 21 ...