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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27 views

Help in derivative with summation

I have forgotten how to handle the derivative $\frac{\partial}{\partial x}[(n-1) \sum_{y_i} \log x({y_i})]$ where $x$ is a function of a vector $\mathbf{y_i}$. How do I evaluate this? Thank you
2
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1answer
52 views

Open neighbourhoods, Polish spaces, and basis for the Baire space

This is a follow-up of a question I asked yesterday answered by GEdgar. I think I see now GEdgar’s answer, but I am not sure about an issue related to it. Thus, I will write my general understanding ...
3
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0answers
69 views

Totient Function: if $\gcd(m,n)=2$, then $\varphi{(mn)}=2\varphi{(m)}\varphi{(n)}$

Suppose we know that $(m,n)=2$. Show that this implies that $\phi{(mn)}=2\phi{(m)}\phi{(n)}$. My attempt: So let $m=p_1^{r_1}p_2^{r_2}...p_k^{r_k}, n=p_1^{s_1}p_2^{s_2}...p_k^{s_k}$. Then ...
3
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2answers
53 views

Open neighbourhoods & polish spaces - typo in Marker's notes?

A very (very!) easy question that merges together the very basic concepts of two fields that I find always problematic for my understanding, namely topology and descriptive set theory. Everything ...
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1answer
37 views

Suggestion or help to choose a book for theory of ODE

I have recently have a course in theory of ordinary differential equations , I have learned main ideas and now I want to study theory of ode from a book for filling gaps. I have choosed three book byt ...
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1answer
51 views

How to solve this implicit equation involving integral

Consider the following equation with respect to $\alpha$ defined through a probability density function $f(x) = \exp[x-\exp(x)]$. $$ \int_{-\infty}^\infty (x-2) e^{\alpha(x-2)}e^{x-e^x}dx = 0. $$ ...
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4answers
59 views

On the intuition behind a conditional probability problem.

This is a very similar question to this one. But notice the subtle difference that the event that I define $B$ is that I am dealt at least an ace. Suppose I get dealt 2 random cards from a standard ...
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2answers
140 views

Continuous, bijective - yet not a homeomorphism

I'm going through the earlier chapters in books and making sure I can do everything (and addressed many short-comings, like compactness) but I've come across something I can't do. In "Introduction to ...
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2answers
77 views

Which rational functions $\mathbb{P}^1\rightarrow k$ are regular at the point at infinity?

I am trying to solve a problem in § 1.7 of Shafarevich's "Basic Algebraic Geometry 1": "Let $k$ be an algebraically closed field. Which rational functions $\mathbb{P}^1\rightarrow k$ are regular at ...
3
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2answers
89 views

What happens if we consider an algebra instead of $\sigma$-algebra in probability theory?

I understand the difference between algebra (of sets) and $\sigma$-algebra. But which are the implications if we use algebra instead of $\sigma$-algebra in probability theory? If it exists, could you ...
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2answers
81 views

What mathematics should I study to understand Neural Nets / Machine Learning?

I am strongly fascinated by neural nets, and perhaps other forms of machine learning. There are so many (potential) applications: teaching a robot with shaft encoders to drive along different ...
0
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1answer
153 views

How to re-learn math: books or websites?

To re-learn math, both websites and books provide visual content (text and some of them shows illustrations). So are websites an alternative to books (content quality-wise)? My goal is to re-learn ...
4
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1answer
183 views

Construct a conformal mapping from $\Bbb C$ Onto $R$ if such a map exists. And explain why if does not exist.

Let $R$ be the domain obtained by removing the non negative real numbers from $\Bbb C$. Construct a conformal mapping from $\Bbb C$ Onto $R$ if such a map exists. And explain why if does not exist. ...
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0answers
28 views

Projection matrix?

If $\mathbf{X}_{n \times K}= \begin{bmatrix} \mathbf{x}'_1 \\ \cdots \\ \mathbf{x}'_n \end{bmatrix}$, then can we say that ...
1
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1answer
100 views

A little help for a 14 year old.

I am 14 years old and i am really into maths. I can say that i am really good at it(at what we do at school) and i really want to make a step ahead and start exploring a more complicated sector in ...
2
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1answer
55 views

Arc length of curve (regular condition)

I have a question regarding the defintion of arc length of a curve in $\mathbb{R}^n$. If $\gamma$ is a regular curve, the define arc length as $S(t)=\int_{t_0}^t|\gamma'(t)|dt$. Since $\gamma$ is ...
1
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1answer
40 views

Trying to prove a sequence of functions is increasing.

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 $k\delta_n \le t < (k+1)\delta_n$. Define $$\psi_n(t)= ...
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1answer
40 views

If $f$ is differentiable and $f'\geq m\geq0$, $|\int_a^b\cos{f(x)}dx|\leq2/m$

Suppose $f:[a,b]\to\mathbb R$ is a differentiable function such that its derivative is monotonically decreasing and $f'(x)\geq m>0$ for all $x\in[a,b]$. Prove that $$|\int_a^b\cos ...
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2answers
53 views

Corollary of theorem 1.14 Rudin real and complex, supremum of a sequence of measurable functions.

Theorem 1.14 states: If $f_n:X\rightarrow [-\infty, \infty]$ is measurable, for $n = 1,2,3, ...,$ and $$g = \sup_{n \ge 1} f_n, \ h = \lim_{n \rightarrow \infty} \sup f_n$$ then g and h are ...
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1answer
649 views

Looking for an easy lightning introduction to Hilbert spaces and Banach spaces

I'm co-organizing a reading seminar on Higson and Roe's Analytic K-homology. Most participants are graduate students and faculty, but there are a number of undergraduates who might like to ...
2
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1answer
26 views

Transformation of Extreme Value Distribution

Let $X$ be a random variable following distribution function (i.e., generalized Pareto distribution) $$ F_{\gamma, \sigma}(x) = 1-\left( 1+\frac{\gamma x}{\sigma} \right)^{-\frac{1}{\gamma}}, $$ ...
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5answers
4k views

Where can I download Discrete Mathematics lecture videos?

Good morning, I'm doing a course in Discrete Mathematics (so far: Four Colour Theorem, Intro Graph Theory, Intro Logic Theory, Intro Set Theory and Intro Proofs) at University, but unfortunately they ...
2
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2answers
46 views

Does $\int_0^1 \frac{x \ln x}{1+ x^2}dx$ converge?

$\int_0^1 \frac{x \ln x}{1+ x^2}dx$ converges? Kinda stuck doing this problem. I just need a hint on what to start with. I know that it is an improper integral and I have to use limits but I need to ...
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1answer
39 views

Borel Sigma-Algebra on $\mathbb{R}$

Show that the Borel sigma-algebra on $\mathbb{R}$, denoted $B_R$ is generated the open intervals in $\mathbb{R}$. My attempt: Let $I$ be the collection of all open intervals, let $\sigma I$ be the ...
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0answers
241 views

Every Riemann Integrable Function can be approximated by a Continuous Function

Prove that given any Riemann Integrable function $f$ on $[a,b]$, and given any $\varepsilon>0$ one can find a continuous function $g$ on $[a,b]$ such that $$\int_a^b|f(x)-g(x)|dx<\varepsilon ...
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2answers
66 views

Awodey's first UMP example

I am reading Awodey's "Category Theory" by myself and got stuck in a simple passage. He writes: If $g:A^\ast\rightarrow N$ satisfies $g(a)=f(a)$ for all $a\in A$ then, for all ...
2
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1answer
64 views

Learning from Alternative Sources

I have a very general question about people's experiences with learning math. I can think of a couple of times where I had the following situation. I was seeking to learning about topic A. However, ...
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0answers
63 views

Understanding a measure theory statement from Wikipedia

I was reading the Measure (mathematics) page on wikipedia: http://en.wikipedia.org/wiki/Measure_%28mathematics%29 I was confused with one of their sections at the end, "Additivity". They were ...
0
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1answer
28 views

Find $\sup$ and $\inf$ of $A$ and justify

$$A=\left \{x\in \Bbb R :x<\dfrac{2}{x} \right \}.$$ a) $\sup A = -1$ since $\max A=-1;$ b) $\inf A$ does not exist since $A$ is not bounded below. Is this the only justifications? Can anyone ...
2
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1answer
77 views

What are the steps to this functional derivative problem?

I'm trying to derive equations from Matthew Beal's Thesis, Variational Algorithms for Approximate Bayesian Inference pg.55, but I'm stuck on one of the equations (well I'm stuck on a lot of equations ...
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0answers
37 views

Homework : Anti log expression

I have this expression $x(r) = y(a)r^a$ where $r$ is a random variable and I want to express the expression in terms of $r$. The objective is to substitute the variable $r$ into the pdf of $r$, ...
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1answer
44 views

Find the extremals of a functional of the form $\int^{x_1}_{x_0}F(y',z')dx$

I was working on Problem 3 in Ch. 2 of Gelfand & Fomin's Calculus of Variations, which reads: Find the extremals of a functional of the form $$\int^{x_1}_{x_0}F(y',z')dx$$ given that ...
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0answers
26 views

Integration of unknown derivative

I am unable to solve this integral, have forgotten basics and so need help. Shall be very thankful If a way out is provided: $\int_0^R \ln[p'(t)]dN(t) - \int_0^R p'(t) dt$ If $p(t)$ was known then I ...
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2answers
339 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 ...
2
votes
1answer
28 views

Points defining plane - starting step? [closed]

If the points $P, Q, R$, not all lying on the same straight line, have position vectors $a, b, c$ respectively, show that $(a \times b) + (b \times c) + (c \times a)$ is a vector perpendicular to the ...
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1answer
31 views

An Example of Taylor Series

Consider a positive function $f(x)$ and suppose that we would like to approximate its value around some point $x_0$. One way to do so is to use two-term Taylor series expansion as follows. $$ f(x) ...
0
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0answers
57 views

Proving Frobenius Theorem for Eigen Values

In my mulitivariable calculus class to justify second derivative test my professor used a theorem he called the frobenius theorem. But when I searched on wiki all I could find was Perron Frobenius ...
0
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0answers
55 views

Calculating Power of a Paired T Test

$ 239$ subjects had their cholesterol measured, and then were put on high-fiber diets. After a month on the high-fiber diet, the cholesterol was measured again. The mean LDL cholesterol level before ...
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2answers
97 views

Strange behaviors of finitely additive probabilities

Watching a lecture on youtube I heard the lecturer stating that in general finitely additive probabilities behaves strangely. For example, it is possible that every open interval around a point $x$ ...
0
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1answer
27 views

How to determine a function whose minima falls on a specified curve?

I have a family of curves given by $g(x,y)=C_0 yx^{-n}$. How can I determine the function $f(x,y)$ for the family of curves that satisfies the condition that the local minima $\frac{\partial ...
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2answers
378 views

How can a beginner researcher or Ph.D. student efficiently and effectively learn new concepts while staying motivated?

I hope this question is appropriate for MSE. The situation is that someone is reading a book (e.g. a monograph), possibly helpful in his/her research, and the content is sufficiently extensive or ...
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1answer
24 views

Independence of two multivariate normals.

Suppose we have two multivariate normals $X_1 \sim N(u_1, \Sigma_{11}\Sigma_{22}$) and $X_2 \sim N(u_2, \Sigma_{21} \Sigma_{22})$ . Why are $X_2 $ and $X_1-\Sigma_{12} \Sigma_{22}^{-1}X_2$ ...
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0answers
75 views

What are some interesting, atypical mathematical topics that a student who has taken an introductory calculus sequence can learn about?

I understand that usually the next step after $3$ semesters of calculus and $1$ semester of ordinary differential equations (plus one semester of linear algebra, for some) is something like an ...
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1answer
57 views

Conditional expectation of $X$ given $Z$, where $Z = 1$ if $X > Y$ and $-1$, otherwise

Let $X\sim\operatorname{Exp}(1)$ and $Y\sim\operatorname{Exp}(2)$ be independent random variables. Define $Z$ by $$ Z = \begin{cases} 1,& X>Y\\ -1,& X\leqslant Y. \end{cases} $$ I want to ...
0
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1answer
214 views

sum of two dependent random variables

Let $X$ be a cotinuous random variable uniformly distributed over $[-10,10]$. Let $Y$ be a random variable with pdf $f_Y(y) = \frac{1}{40}\ln \frac{20}{|y|}, -20 \leq y \leq 20$. $X$ and $Y$ ARE NOT ...
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0answers
18 views

transformation and functions of random variables

Let $X,Y$ be independent random variables. I already have the distribution of $XY$ over a certain subinterval of $\mathbb{R}$, by convolution. My question is, is it possible to get the distribution of ...
2
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1answer
58 views

The inverse of the sum of two matrices in *Applied statistical decision theory *.

I am following Applied statistical decision theory [by] Raiffa, Howard. Which can be consulted online here. A theorem at the page linked states that if two matrices $A,B$ are non-singular and of ...
3
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2answers
67 views

Logic behind a proof in Topological Vector Spaces

I found the following result at the beginning of some notes on topological vector spaces (TVS). This is a quite fundamental result, that apparently is considered the corresponding version of the ...
2
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1answer
137 views

Doubts: Proof of Deduction Theorem

I am reading Robert Wolf's A Tour Through Mathematical Logic and am enjoying it. But the author omits proofs for the Deduction and Generalization Theorems. I looked through Intermediate Logic by ...
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1answer
449 views

Real Analysis : Self Studying vs Doing a Course

I am an engineering graduate student. Recently I got interested in studying Maths. So, I have started self-studying Real Analysis(let's call it RA) using a few books. I will also be using problem ...