Questions about the process of studying mathematics without formal instruction.

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Can the given transformation possible for given determinant?

In forth step $(x-1)(x-2)$ is obtained by applying transformation R$1 \frac{1}{(x+1)}$ and R$2 \frac{1}{(x+2)}$. But we get value of $x = -1$ or $ x = -2$ so $\frac{1}{(x+1)}$ and ...
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25 views

Good starting point for learning noncommutative geometry?

Currently, I am attempting to learn noncommutative geometry. My interests mostly lie on the boundaries of pure mathematics and theoretical physics, so I am not only interested in the mathematical ...
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0answers
21 views

Double Integral of an Exponential Function with an Absolute Value in the Numerator of the Exponent

This is a question related to statistics, but my major concern relates to the setup and evaluation of integrals. So I decided this question was better suited for Mathematics Exchange than CV. I know ...
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1answer
20 views

A proof related to diameter of a simplex S

Question: Prove that the diameter $\mathcal p(S)$ of a simplex $\mathcal S$ equals the greatest Eucledian distance between two vectors in the simplex. My opinion: We all know what every vector in the ...
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49 views

Which version of MIT Single Variable Calculus should one take? [on hold]

MIT offers nine versions of the single variable calculus course. five with course number 18.01..., and four versions under the course number heading "Supplemental." 1) Which do you recommend to ...
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1answer
22 views

Show that a positive definite (not necessarily symmetric) matrix induces a hyperellipse

Consider $A\in M_n(\mathbb{R})$ a positive definite matrix and a matrix $B\in M_{n \times p}(\mathbb{R})$, with $n\geq p$ and $rank(B)=p$. i) Show that $C=B^TAB$ is positive definite. ii) Show that ...
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1answer
44 views

How to get this inequality

Let $c>0$, $n \in \mathbb N$ and $q>1$. How to get the following approximating inequality when $n$ is large, please? To be more specific, I cannot see how to get rid of the square root. $$ ...
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2 views

Variance of Inhomogenous Poisson process in a given window

Consider some variable $X\sim \operatorname{Poi}(\lambda(t))$ to be Poisson-distributed with some parameter $\lambda$ dependent on time, where we know how the random variable $\lambda$ is distributed. ...
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76 views

Are “Transition Books” (Spivak/Apostol/Courant) really necessary?

Why do so many people recommend Spivak, Apostol, and Courant calculus textbooks, especially as a preparation toward the advanced courses like analysis and abstract algebra? Are they really necessary? ...
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2answers
34 views

An orthogonal projection matrix in $ \Bbb{R}^{3} $.

Consider the vector space $\mathbb{R^3}$ with usual inner product. Find the orthogonal projection matrix on the xy plane. I've found sometimes the orthogonal projection of a vector in a given ...
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3answers
29 views

Linear algebra, inner product and matrix

Let $A\in M_{m \times n}(\mathbb{R})$, $x\in \mathbb{R}^n$ and $b,y\in \mathbb{R}^m$. Show that if $Ax=b$ and $A^ty=0_{\mathbb{R}^m}$, then $\langle b,y\rangle=0$. Also make a geometric ...
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1answer
468 views

Linear algebra and Multivariable calculus prerequisites for Stochastic Calculus

Which topics are considered "graduate-level" for the following subjects: Linear algebra Multivariable calculus On Internet, it is said that you need "graduate level" Linear algebra and ...
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15answers
7k views

A good way to retain mathematical understanding?

What is a good way to remember math concepts/definitions and commit them to long term memory? Background: In my current situation, I'm at an undergraduate institution where I have to take a lot of ...
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0answers
33 views

Textbook with full solutions: To self-learn logic for the first time [on hold]

I never studied logic before; so I'm seeking an intelligible textbook (written in simple English) with practice problems that MUST be accompanied with full detailed solutions. I read ...
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1answer
26 views

Understanding two sided t-test

Assume we have two search engines, A and B. I get a list of scores for 10 different queries. Now, I model this with a t-test in order to test significance. These are my hypothesis: $H_0: B-A=0$ ...
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1answer
438 views

Equivalent systems of Linear equation

I've just begun to re-learn linear algebra because is so important, the book that I chose is naturally the Hoffman's for a lot of reason. Well, In the first chapter I'm stuck with the following, ...
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0answers
11 views

Reference request for conditional and unconditional covariance of n-times integrated Brownian motion

I'm working through an old Diaconis paper on Bayesian numerical analysis, and am currently calculating the details behind his brief comments on using $n$-times integrated Brownian motion as a function ...
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2answers
88 views

Are basic trigonometry functions ( sine, cosine, tangent ) intuitive or memorized?

First, I'm really sorry for this somewhat vague and possibly just silly question. I also apologize if the following context runs a bit long. But please trust me that I'm asking with total sincerity ...
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2answers
36 views

The maximum and minimum of five independent uniform random variables

Let $U_1,\dots,U_5$ be independent, each with uniform distribution on $(0, 1).$ Let $R$ be the distance between the minimum and maximum of the $U_i^{'}$s. Find the joint density of the max and the ...
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1answer
304 views

Book on discrete mathematics for self study

I am searching for book on discrete mathematics which is suitable for self study. This mean I want it to have exercises with answers (It would be ideal if it had solutions). I have already read ...
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0answers
24 views

How to partially differentiate an integral with a density function?

I am given this result: $$\frac{\partial}{\partial x(t)} \left[\lambda \int u(x(t)) f(t) \mathrm{d}t\right] = \lambda u^\prime(x(t)) f(t)$$ Where $\lambda$ is a constant, and we have the probability ...
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29 views

What should I study, if I want to learn about higher dimensional spaces and objects? Also, what resources should I obtain?

I am becoming interesting in learning about higher dimensions. What are subjects I could study, and what are some good resources for those subjects?
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2answers
56 views

Companion Books for Rudin's PMA

S.E friends, I am a college sophomore with double majors in mathematics and microbiology. I wrote this email to seek your recommendation on selecting the introductory analysis textbook, particularly ...
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2answers
340 views

Self-Contained Books / Series / Lectures for Comprehensive Introduction to College-Level Math for Someone with VERY Poor Math Foundation?

I've long been interested in various math related subjects (technology, philosophy, sciences, computer science, languages, etc.) without really invested time to actually any learn any of them. I ...
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3answers
612 views

Algebra book(s): Beginner through to advanced

I admit, I am not great at Maths; I'm situated in one off the lowest Maths class for my year, partly due to myself losing focus last year. Though, I like maths, I really do. Especially since I do a ...
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2answers
24 views

Find the distribution function of bivariate distribution

Find the distribution function of $$f_{X,Y}(x,y)=\begin{cases} e^{-y}, & \text{if $0< x<y < \infty$} \\ 0, & \text{ otherwise} \end{cases}$$ Trial : According to my calculation ...
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Doubt on asymptotics of continous functions (little-o notation and taylor expansion).

Suppose I have $e^{(\frac{1}{n}b + o(\frac{1}{n}))}$ then $\lim_{n \rightarrow \infty} = e^0 = 1$ so $$e^{(\frac{1}{n}b + o(\frac{1}{n}))} = o(1) +1$$ But if I take the Taylor expansion of ...
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1answer
18 views

Characteristic Function and Density Function

Consider a random variable $X$ with density function $f(x)$, moment generating function $M(t):= \int e^{tx}f(x) dx$ (existing in an interval containing $0$), cumulant generating function $K(t):=\log ...
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1answer
51 views

How to solve this integral in moment generating function

The moment generating function of generalised Pareto distribution eventually comes down to the following integral (here). $$ M_X(\theta) = \mathbb Ee^{X\theta} = \int_\mu^\infty e^{\theta ...
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1answer
43 views

Matrix multiplication memorisation

So I'm writing an exam about matrices in a few weeks time, and I'd like to know if anybody has any tips about multiplying matrices.
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1answer
30 views

Requirements for learning and understanding trigonometry?

Im reaching a point in programing where I need to create basic shapes which I simply cant since my math skills are very bad. After finding out that the skills required are trigonometry I read a few ...
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2answers
69 views

Study a math course on my own, suggestions? [closed]

I would like to study some math on my own. I am currently studying my second semester at a university and I have too much freetime so I would like to study something on my own, but I can't decide what ...
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1answer
24 views

Concave and convex functions [closed]

Does the function $ \frac{1}{x} $ in (0 ,10] , represent a convex function ? And the function $ log(x^{2}) $ , in the same interval, is concave function ? Thanks
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1answer
32 views

What textbook(s) do I need to self-study grade 9 - 11 math? I failed Gr 11 Math twice.

I need a book that will help me to master all the concepts in math up until at least toronto's grade 11 math level by June. It's not a long way away, and I'm a slow learner. So what can I do to speed ...
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0answers
18 views

Prove that minimum of the matrix norm is achieved at certain parametres

Given matrix $A\in R^{n\times m}$ prove that minimum of the $||A-xy^T||$, $||B||=tr(B^TB)$, is achieved when $x$ is an eigenvector of $AA^T$, corresponding to its greatest eigenvalue, and $y$ is an ...
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1answer
18 views

Inverse Laplace Transformation of a heaviside function.

I'm working through an example of an inverse laplace transformation: $$\mathscr{L}^{-1}[\frac{e^{-3s}}{s+1}] = u_3(t)e^{-(t-3)}$$ I am having trouble seeing how this works. I know that: ...
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2answers
207 views

Prerequisits for Gauss-Green theorem

Consider the following theorem from the appendix C from Evans PDE book: I know about integration in $\mathbb{R}^n$ but not about how to make sense of the integrals on the right-hand side. As my ...
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23 views

Matrixproduct of A'A expressed as a sum

I have difficulties in proving (understanding, seeing) the following identity: $ \mathbf{A'A} = \mathbf{(a_1, a_2, ...,a_n)} \begin{pmatrix}\mathbf{a_1'\\a_2'\\ \vdots \\ a_n'}\end{pmatrix} = \sum ...
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26 views

Does Thompson's Calculus hit all the essentials needed?

I found out in my preliminary research for my undergraduate degree that I need a bit of undergraduate-level mathematics for my topic at the very minimum to start with, up to ODEs. So I found a ...
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6answers
8k views

Best Sets of Lecture Notes and Articles

Let me start by apologizing if there is another thread on math.se that subsumes this. I was updating my answer to the question here during which I made the claim that "I spend a lot of time sifting ...
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1answer
50 views

Infinite horizon cost function

The following quote is from Bertsekas's Dynamic Programming and Optimal Control. I'm only looking for a nudge in the right direction as to how to interpret the following equations, particularly ...
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1answer
42 views

Seeking Recommendation on Theoretical Multivariable Calculus textbooks

I am a college sophomore with double majors in mathematics and microbiology. I wrote this email to seek your advice on selecting a theoretical, proof-based textbook on the multivariable calculus. I ...
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1answer
77 views

Importance of Exercises in Mathematics for Self-Studying

I am a high school student wanting to major in Mathematics in the future. I started to like Mathematics recently, starting a year ago and I watched some interesting math videos on YouTube for fun (ex: ...
3
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2answers
146 views

Zero divisors in ring of real valued functions.

I'm working though Pinter's A book of Abstract Algebra and would like a quick verification on a simple problem. Exercise 17.B2 asks Describe the divisors of zero in $\mathcal{F}(\mathbb{R})$. ...
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1answer
156 views

Mathematical introduction to machine learning

At first glance, this is once again a reference request for "How to start machine learning". However, my mathematical background is relatively strong and I am looking for an introduction to machine ...
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1answer
18 views

Dual Vectors and Dual Metric

In the book of Nadir Jeevanjee „An Introduction to Tensors and Group Theory for Physicists“ it is stated as an exercise that: 2.17 Given a basis $\{e_i\}_{i=1,...,n}$ , under what circumstances do ...
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1answer
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The definition of the $false$ truth value

In "Topoi: The Categorial Analysis of Logic" by R. Goldblatt the $false: 1 \to \Omega$ truth value is defined as the characteristic arrow of the arrow $0_1: 0 \to 1$. This definition requires that ...
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1answer
93 views

Complete example of haar measure on compact groups like $GL(n,R)$

I am currently reading the proof of existence of haar measure, but I learn better mostly by examples so I would like examples of explicit computation of haar measure mainly on any $Gl(n,R)$ or any lie ...
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1answer
42 views

How to find conditional expectation $\mathbb E(X|X<M)$

Consider a random variable $X$ following the so-called folded normal distribution. That is, $X$ has density function $$ f_X(x) = \sqrt{\frac{2}{\pi\tau}}e^{-\frac{x^2}{2\tau}}, x>0. $$ ...
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1answer
43 views

What's in a name? (Sum of Squares)

I have always believed that in order to fully understand and appreciate the mathematical subtleties and ideas behind a certain concept is to understand the name given to it. For now, can someone ...