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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66 views

$ P( A \mathrel{\triangle} B ) \le P(A \mathrel{\triangle} C) + P(B\mathrel{\triangle} C)$

Show that $$ P( A \mathrel{\triangle} B ) \le P(A \mathrel{\triangle} C) + P(B\mathrel{\triangle}C)$$ where $\mathrel{\triangle}$ indicates the symmetric difference I cannot write my idea, because ...
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1answer
29 views

Definition of Multiple .

Definition of multiple is : In mathematics, a multiple is the product of any quantity and an integer. In other words, for the quantities a and b, we say that b is a multiple of a if b = na for ...
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0answers
74 views

Can anyone check if this correct?

Convert to spherical coordinates and evaluate:$$\iiint_{E}z(x^2+y^2+z^2)^{-3/2}dV$$ where E is the region satisfying the following inequalities:$$x^2+y^2+z^2\le16,z\ge 2$$ This is what i have done so ...
2
votes
1answer
90 views

Second moments from survival function

Let X be a non-negative continuous random variable with probability density function f(x). Let $$G(t) = \int_{t}^{\infty} f(x)dx$$ Show that$$E(X^{2}) = 2\int_{0}^{\infty} tG(t)dt$$ My thoughts: I ...
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1answer
38 views

limt of the function as $\mu\rightarrow\infty$ or $\mu\rightarrow-\infty$ .

$\lim_{\mu\rightarrow\infty}\frac{\exp(\bar x-\mu)^2}{(\bar x-\mu)}=? $ Also, $\lim_{\mu\rightarrow-\infty}\frac{\exp(\bar x-\mu)^2}{(\bar x-\mu)}=? $ I know, ...
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2answers
2k views

Is $g(x)=\log x$ convex function?

The graph of convex function is : In a book it is written that $g(x)=\log x$ is strictly convex function. So i searched for graph of $g(x)=\log x$ and found that Though it has been said that ...
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1answer
486 views

How to Self-Study Mathematical Methods?

Edit: Ok, user Chinny84 made comment that truly helps narrow the focus of my question. Basically, I'm asking for a self-study course of Mathematical Methods. Thanks to his recommendation I ...
2
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0answers
27 views

Uniform probability bound - checking my understanding

Let x and y be two independent random variables. What is the difference between (1) $P_x[\forall y, f(x,y) < \epsilon] >1- \delta$ (uniform bound), and (2) $\forall y, ...
2
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1answer
87 views

To prove set is a group

Given a non empty set together with associative binary operation $*$ on $G$ such that $a*x=b$ and $y*a=b$ have solutions in $G$ for all $a,b$ in $G$ To prove it is a group Hints to get started ...
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1answer
68 views

gluing together continuous functions

HI I was checking and old question here and I have troubles to proof the following: Proposition: Let $X$ be a space with subspaces $Y,A,B$ such that $X \backslash Y= A \sqcup B$ (disjoint union). Let ...
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1answer
463 views

Sample Points Calculation.

A coin is tossed three times. There are three sample points that i can get one head and two tail. I can count the sample points after writing the sample space, as ...
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1answer
36 views

To find right cosets of H in G where G=<a> and H=$<a^{2}>$ ,where o(G)=10

To find right cosets of H in G where G= and H=$<a^{2}>$ ,where o(G)=10 Since order of $G =10$ , so $a^{10}=e$ .We have $G= { a,a^{2},a^{3},a^{4},a^{5},a^{6},a^{7},a^{8},a^{9},e}$ and $H = ...
3
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0answers
159 views

On the importance of the Riesz–Markov–Kakutani representation theorem.

I am following big Rudin and I have arrived at the representation theorem. Before doing the full long proof I would like to know what results are based on this theorem that for completeness I state ...
1
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0answers
38 views

How to show that a function is continuous in the topology of weak convergence

Let $\Omega$ be compact, and let $\omega^* \in \Omega$ be arbitrary. Let $\Delta (\Omega)$ denote the set of all probability measures over $\Omega$, and endow $\Delta ( \Omega)$ with the topology of ...
1
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3answers
112 views

Showing $(\mathbb{Q},+)$ is not isomorphic to $(\mathbb{R},+)$

To show: $(\mathbb{Q},+)$ is not isomorphic to $(\mathbb{R},+)$ Now, the equation $x^{2} =3$ has a solution in $\mathbb{R}$, but not in $\mathbb{Q}$. Hence they are not isomorphic to each other. Is ...
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1answer
37 views

how to get the second equation (related to summation)

$$V(Y) = \sum_{i=1}^N\sum_{j=1}^N [\frac{N^2}{n^2}] (Y_i-Y_j)^2 \frac{n(N-n)}{N(N-1)} $$ for $i< j$ Equation(2.5) $$=(\frac{(N-n)}{n(N-1)})\sum_{i=1}^N \sum_{j=1}^N (Y_i-Y_j)^2 $$ for $i< j$ ...
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3answers
394 views

How do i visualize Cosets of a group

The Lemma asserted in Herstein as given by $[a] = Ha$ seems very non intuitive to me. How do I think in order that this thing makes sense to me? LEMMA 2.4.4 For all $a$ in $G$ , $$Ha = \{ x \in ...
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1answer
46 views

Are the following Stopping Times?

I've been working through the following list of stopping time questions. I am have problems with the final two (e and f). I appreciate any assistance offered. $\textbf{Question:}$ Let $S,T : ...
4
votes
1answer
267 views

Quality of Videos Lectures and Lectures vs Textbooks

I am a student trying to learn different subjects by watching video lectures and reading on my own time. I was wondering if the lectures from ICTP and nptelhrd are a great use of my time. I tried ICTP ...
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4answers
63 views

Prove $R$ is an equivalence relation.

I think I'm on the right track. Set $S = N \times N$, and for any two members $(a,b),(c,d)$ of $S$, define $(a,b) \simeq (c,d)$ provided that $ad = bc$. Prove that $\simeq$ is an equivalence ...
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0answers
59 views

Compactness & Continuity - Looking for feedbacks on a specific setting

I am trying to get the implications of the following general setting concerning compact spaces and continuous maps. Any feedback would be greatly appreciated, because I have some difficulties in ...
2
votes
1answer
87 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)?
1
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1answer
111 views

Determinant of an almost-diagonal matrix

I would like to compute the determinant of the $(k+1)\times (k+1)$ matrix below $$J=\begin{vmatrix} y_{k+1}& 0 & \ldots & 0 & y_1 \\ 0& y_{k+1}& \ldots& 0& y_2 \\ ...
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1answer
73 views

Group of all $2\times2$ matrices where $a$, $b$, $c$, and $d$ are integers modulo $p$, Herstein Q$26$ Page $37$ [duplicate]

Let $G$ be group of all square matrices of order $2$ $$\begin{bmatrix} a & b\\ c & d \end{bmatrix}$$ such that $a$, $b$, $c$, and $d$ are integers modulo a prime number $p$, such that ...
0
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1answer
54 views

Non-abelian group $G$ satisfying $(a \cdot b)^i=a^i \cdot b^i,$ for two consecutive integers.

Given an example of a non-abelian group $G$ satisfying $(a \cdot b)^i=a^i \cdot b^i, \forall a, b \in G$ for two consecutive integers. This is question 5 from Herstein Page 35. I have proved that ...
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3answers
56 views
3
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4answers
207 views

What books should a high school calculus student read to learn more about truly beautiful mathematics? [closed]

I love mathematics! Unfortunately, I don't know as much about it as I would like to. I honestly spend a large portion of my free time reading further in my Calculus textbook, and it's very ...
1
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1answer
887 views

Multivariable/Vector Calculus Textbook Recommendation Please!

S.E friends, I am a college sophomore with a major in mathematics. I am trying to self-study multivariable and vector calculus (they means the same, right?) and prepare for Summer course on ...
1
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1answer
67 views

The mean value theorem in $\mathbb{R}^n$ and its application to show that functions are independent of a variable

I am currently reading through several multivariable calculus books to understand the proofs better (most of which go back to introduce functions in $\mathbb{R}$ for which the results are already ...
2
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1answer
58 views

Looking for an online course

My friend and I are interesting in doing an online math course together. He has the basic high school math up to Calculus AB and will be doing BC while we are doing the course. I, however have done ...
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2answers
59 views

mathematical induction to establish inequality

Studying for a test in discrete mathematics and I cannot seem to grasp the explanations in the textbook regarding these questions. Using mathematical induction, prove that $$2^n > n^2, \text{for ...
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1answer
94 views

What to read after Shreve's “Stochastic calculus for finance 2”?

I am finishing the last pages of Shreve's Stochastic calculus for finance 2, and I was wondering what would be the best book to follow. I would like to go on with a book introducing more technical ...
2
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1answer
34 views

Application of the chain rule for curves

Problem: Let $f: \mathbb{R}^3 \to \mathbb{R}$ be a differentiable function such that $$y \frac{\partial f}{\partial x}(x,y,z) -x \frac{\partial f}{\partial y}(x,y,z) + \frac{\partial f}{\partial ...
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2answers
50 views

Problem in Primitive Pythagorean Triples (PPT)

I'm new to number theory. So now I'm starting my journey of 'number theory' by reading this book. I'm currently in chapter 2 which is Pythagorean Triples. I don't understand. It says there are ...
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3answers
481 views

Initial value problem for a linear system.

Consider the linear system $$ \frac{dY}{dt} = \begin{pmatrix} 1 & -1 \\ 1 & 3 \\ \end{pmatrix} Y $$ (a) Show that the function $$ Y(t) = \begin{pmatrix} te^{2t} \\ -(t + 1)e^{2t}\\ ...
1
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1answer
139 views

In balancing effort and advancement in what concerns learning

What's new besides showing modern advancements in modern mathematics as well as eloquently written notes also contains some good advice to young people like me in this room (career advice:!). in ...
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4answers
843 views

Forgotten old results break my motivation

I'll begin graduate school next year and I am very impatient to learn new things such as theories, ways of thinking and so on (I enjoyed reading about category theory on my own and I find Galois ...
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0answers
64 views

Question about branches of functions (complex power)

I have the following question, I really appreciate if someone can help me to clarify ideas and I apologize if is a stupid question: This is from Conway's complex analysis book: Let $f: G \to ...
2
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0answers
117 views

What is the best way to master my algebra skills without taking an algebra class?

I was in advanced math my entire life. I got through all the math I needed for my original degree. 8 years later here I am changing degrees and I need more math. I just took calculus I and I passed ...
2
votes
2answers
49 views

On integration of a simple random variable in measure theory.

Suppose we have a simple Random variable $X$ defined on a probability space $(\Omega, F, P)$. A random variable is simple if $X(\Omega) = \{ \alpha_1, \ldots , \alpha_n \}$. We define the integral of ...
2
votes
1answer
123 views

Dual Pairs, topology of weak convergence and weak* topology

Edit for Bounty: I decided to put a bounty on this question because I would really like to get it properly. Thus, I would like to get feedbacks on my basic questions, and a detailed answer on my ...
2
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3answers
64 views

Proving that the generator of $U$ is normal if $\forall u \in U, g\in G$ $gug^{-1} \in U$

This is from Herstein. $4.$ $\;a)$ Given a group $G$ and a subset $U$ denote by $\hat U$ the smallest subgroup of $G$ which contains $U$ (the subgroup generated by $U$). Prove there is a ...
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1answer
62 views

how to calculate sum of a series? (me or Wangenmakers is wrong)

Wagenmakers in his critical article about p-values wrote that: $$\sum_{i=12}^{\infty} {{n-1} \choose {2}} \cdot \left(\frac{1}{2}\right)^n \approx .033$$ How could he do his calculations if the ...
3
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1answer
167 views

Non-isomorphic $\mathbb{C}$-algebras

The question is as follows: Show that the $\mathbb{C}$-algebras: $A=\mathbb{C}[x,y]/(x^2y-xy)$, $B=\mathbb{C}[x,y]/(x^2y+xy^2)$, $C=\mathbb{C}[x,y,z]/(xy, yz, zx)$, and ...
0
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1answer
50 views

Definition of continuity in practice

In general I have a problem to recognise if a function is continuous or not. I simply don't know where I should start to actually see it. Here there is an example of my problem that I found in a ...
2
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2answers
227 views

Easy papers on fundamental groups (for beginners)

I'd like to read some papers concerning fundamental groups, for example, papers written to explain some basic facts about homotopy explicitly for undergraduate students. All the papers I have ...
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3answers
67 views

Why does = change to $\leq$ and then to = in this proof of |a+b| = |a|+|b|?

From Spivak's Calculus. This proof is motivated by the observation that |a| = $\sqrt {a^2}$. $\sqrt x$ denotes the positive square root of x; this symbol is defined only when x $\geq 0$. We may ...
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8answers
133 views

Proof that $|\sqrt{x}-\sqrt{y}| \leq \sqrt{|x-y|},\quad x,y \geq 0$

Any hints on how I can prove the inequality: $$|\sqrt{x}-\sqrt{y}| \leq \sqrt{|x-y|},\quad x,y \geq 0$$ Thank you.
4
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0answers
107 views

The canonical height of a point on an elliptic curve

I am struggling with exercise 3.3 in Silverman-Tate Rational Points on Elliptic Curves. Here is the paraphrased problem with necessary background: Let $C:y^2 = x^3 + a x + b$ be a nonsingular cubic ...
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1answer
22 views

Can I substitute $\beta A \alpha^{1-\gamma}$ with $c^\gamma$?

I reach a point where in the book the author substitutes $\beta A \alpha^{1-\gamma}$ with $c^\gamma$ to simplify the rest of notation, where $\beta, \gamma \in (0,1)$ and $\alpha, A$ two other ...