# Tagged Questions

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

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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### 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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### 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 ...
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### 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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### 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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### 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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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### 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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### 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 ...
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### 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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### If G is a group such that $(a.b)^{2}=a^{2}.b^{2}$ for all a and b,Then show that G is abelian

This is problem from I.N Herstein Page 35 Q3 .How should i start doing this ?Hints ? Thanks
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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 ...
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### 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 ...
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### 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 ...
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### 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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### 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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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}\\ ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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$$How could he do his calculations if the ... 1answer 167 views ### Non-isomorphic \mathbb{C}-algebras The question is as follows: Show that the \mathbb{C}-algebras: A=\mathbb{C}[x,y]/(x^2y-xy), B=\mathbb{C}[x,y]/(x^2y+xy^2), C=\mathbb{C}[x,y,z]/(xy, yz, zx), and ... 1answer 50 views ### Definition of continuity in practice In general I have a problem to recognise if a function is continuous or not. I simply don't know where I should start to actually see it. Here there is an example of my problem that I found in a ... 2answers 227 views ### Easy papers on fundamental groups (for beginners) I'd like to read some papers concerning fundamental groups, for example, papers written to explain some basic facts about homotopy explicitly for undergraduate students. All the papers I have ... 3answers 67 views ### Why does = change to \leq and then to = in this proof of |a+b| = |a|+|b|? From Spivak's Calculus. This proof is motivated by the observation that |a| = \sqrt {a^2}. \sqrt x denotes the positive square root of x; this symbol is defined only when x \geq 0. We may ... 8answers 133 views ### Proof that |\sqrt{x}-\sqrt{y}| \leq \sqrt{|x-y|},\quad x,y \geq 0 Any hints on how I can prove the inequality:$$|\sqrt{x}-\sqrt{y}| \leq \sqrt{|x-y|},\quad x,y \geq 0 Thank you.
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 ...
### 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 ...