Questions about studying mathematics without formal instruction.

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0
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0answers
9 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 ...
1
vote
1answer
12 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 ...
-3
votes
0answers
48 views

Equivalence Relation between sets $X \times Y$

If $X$ and $Y$ are sets a relation between $X$ and $Y,$ is a subset $R\subset X \times Y$ . For a relation $R\subset X \times Y$ we have $\{(a,b):a\in X, b\in Y\}.$ if $(a,b)\in R$ or $(a,b)\notin R.$ ...
0
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0answers
17 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
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1answer
14 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
10 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
99 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
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1answer
30 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 ...
0
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1answer
30 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 ...
-1
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0answers
28 views

Lagrange multiplier method

I am doing some data mining algorithm self learning tutorial. I came up with a problem which I need your help to solve. In order to minimize the resource consumption, a car manufacturer considers how ...
0
votes
1answer
34 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 ...
0
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0answers
45 views

What is mathematic manipulative skills? [closed]

Hye.Anyone know what is actually manipulative skills in mathematics? especially geometric manipulative skills. I am currently doing my assignment on that
17
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0answers
194 views
+50

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 ...
2
votes
4answers
78 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] ...
6
votes
2answers
331 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, ...
3
votes
2answers
115 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
45 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 ...
2
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0answers
45 views

Proving that $\lim_{n \to\infty} \frac{X_n}{n} = 0$

If ${X_n}$ are nonnegative random variables such that $\sup_{n\ge1} E(X^a_n) \lt \infty$ where a $\gt$ 1 is a constant. Prove that $$\lim_{n \to\infty} \frac{X_n}{n} = 0$$ Now my question is, what ...
0
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1answer
23 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.
0
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1answer
27 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.) ...
3
votes
1answer
36 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
28 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
21 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
28 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
16 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
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1answer
30 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
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1answer
35 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
41 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
101 views

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

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. ...
0
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3answers
44 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 ...
0
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0answers
22 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
53 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 ...
1
vote
2answers
75 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
35 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 $k$ ...
0
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0answers
30 views

Videolectures and Spivak's Calculus

I'm reading Spivak's Calculus but I have problems understanding some topics, so I would be glad if someone share with me some Videolectures that will make my self-learning more efficient. Sorry if I ...
2
votes
1answer
60 views

How to master Calculus for physics [closed]

I would like to know how to master calculus for use in a discipline such as physics. I have an excellent textbook and a good working knowledge of calculus, but I have by no means mastered it. Is the ...
0
votes
0answers
31 views

Where can I get detailed and comprehensive notes of a functional analysis course taught using the book by Erwine Kryszeg?

Where on the Internet can I find detailed and comprehensive lecture notes of an elementary functional analysis course taught using the book Introductory Functional Analysis with Applications by Erwine ...
1
vote
1answer
37 views

How to show the given expression is geometric mean

Let $a_1,a_2,\dots,a_n$ be any $n$ positive real numbers. Show that $$\lim_{t \to 0^+}\left[\frac1n \sum_{i=1}^{n}a_i^t\right]^{1/t}$$ is the geometric mean of $a_1,a_2,\dots,a_n$. I know ...
0
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0answers
41 views

How to substitute for a nonlinear function

I have a nonlinear function which is an infinite geometric series $P(X_t) = \inf{\sum_{k=1}^\infty U_k^d}$. $\mathbf{x_t} = (x_t,x_{t+1},\ldots,x_{t+d-1})$, $t=1,2,\ldots, N$ $|U_k| = ...
4
votes
2answers
60 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 $\sup\limits_{i\in\Bbb{N}}|{a_i-b_i}|$. ...
2
votes
0answers
48 views

From newbie to professional, the path.

I left school long time ago because of family stuffs and with this I left many subjects behind and since then I try to find something to do and when I discovered the computer and what I can do with it ...
0
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0answers
15 views

Help with understanding step in Optimisation Book

I am reading an Optimisation book. My knowledge on multi-variable calculus is minimal. Hence I do not understand the block-quoted step. We take $ \underline w = \underline x^* + t(\underline x - ...
1
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1answer
81 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
80 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
49 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
26 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
24 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
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1answer
15 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
50 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 = ...
0
votes
0answers
35 views

Upper bound and Chernoff bound

I would be happy to get help for the following question: Given n students and n courses, is also well known that every student has successfully passed k ($ 1< k <n $) courses (in randomly). In ...