Questions about studying mathematics without formal instruction.

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2
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0answers
187 views

Fubini's Theorem for Infinite series

In the book what I've read, there is one point where the author suggest to begin the proof of the Fubini's Theorem for infinite sum in the case when is non-negative after this try to generalize. But ...
13
votes
3answers
343 views

Being mathematically critical: how should a student approach statements that appear to be obvious?

Very occasionally, I will read or hear a theorem, and think: isn't that obvious? Not in a contemptuous "I can immediately see how to prove this" way, but rather in a "I would have thought this was ...
1
vote
1answer
150 views

$f(z)$ has infinitely many zeros and that each zero is simple.

Let $f(z)=e^z-z$ I want to check $f(z)$ is finite order. And how to show that $f(z)$ has infinitely many zeros and that each zero is simple. Dfn: an entire function f is finite order if ...
1
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0answers
40 views

Find a function $f$ such that $f$ is harmonic on $D$ and $f|_{\partial D}$.

I understand its solutions in general. But my question is how to decide whether I sould take $Im z^4$ or $Re z^4$? I have two similar examples. And in one example, the real part is taken, but in ...
0
votes
1answer
33 views

Extreme Value Distribution

I would appreciate a lead\tip on the next one: Z is a standard random variable from the extreme value distribution. I need to show that $Y=\sigma \cdot Z+\mu$ is an extreme value variable with ...
1
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0answers
40 views

“Fixed Domains” of a Linear Transformation

Given a linear transformation $T$, I need to find the set of all domains $D$ such that $T:D\mapsto D$. Equivalently, I need to find the set of all domains $D$ that are symmetric under $T$. Aside from ...
4
votes
1answer
154 views

Is it possible to learn differential topology before analysis?

Currently I'm self studying for my own enjoyment topology and algebra (munkres and herstein). Since I start at the university next year everything I'm learning now is for my own enjoyment and I will ...
11
votes
2answers
136 views

Is the maximal path through a math book necessarily linear?

I'm studying with two main math books (Munkres and D&F) these couple of months. My method so far is just going through the book page by page constructing everything in it (independently if I can) ...
3
votes
3answers
104 views

Is the concept of a uniform space elementary?

I'm self studying with Munkres's topology and he uses the uniform metric several times throughout the text. When I looked in Wikipedia I found that there's this concept of a uniform space. I'd like ...
1
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1answer
58 views

Testing Boundary Points Of $\sum_{n = 1}^{\infty} \frac{n!z^n}{n^n}$

I'm having some trouble testing the series indicated in the title at its boundary points. I'll sketch the preliminary work, then arrive at the problem. It is clear that the series converges absolutely ...
0
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1answer
42 views

Associativity of Finite Partial Sums in a Convergent Series

So I saw some topics on this, but they didn't seem to answer exactly what I was looking for. Self-learning through Understanding Analysis, a exercise is the following: So my misunderstanding is ...
1
vote
1answer
49 views

Statistics primer for Social Sciences?

I'm a Psychology Major intending to take University level Statistics starting in the Fall. As I'm a mature student, I've been out of school for about 7 years, and it's been at least that long since ...
1
vote
1answer
46 views

An application of LDCT

Consider the sequence of functions: $$F_n (t)=\int_{-\infty}^t \underbrace{ \frac{\Gamma \left[ \left( n+1 \right)/2 \right] }{\sqrt{\pi n} \Gamma \left(n/2 \right)} \frac{1}{\left(1+y^2 /n ...
8
votes
0answers
279 views

How much time is reasonable to complete baby Rudin?

I've been teaching myself math for more than a year. My current aim is towards algebraic topology and differential geometry. Apart from a messy (by which i mean some rigorous and some not) ...
0
votes
2answers
23 views

Is the answer true? $\sup \frac{(\theta-\theta')^2}{\exp(n(\theta'-\theta))-1}=\frac{1}{n^2}$

I am trying to calculate the supremum ($sup$) for some equation, Is the answer true? $$\sup \frac{(\theta-\theta')^2}{\exp(n(\theta'-\theta))-1}=\frac{1}{n^2}$$
0
votes
1answer
45 views

To alternatively prove the theorem(*) by proving that $g^{(n+1)}(z_0)=0$ $\forall z_0\in \Bbb C$

Assume that $g=x+iy$ be an entire function. By a theorem(*), $\vert x(z)\vert \le N \vert z\vert ^n \ \ \forall z$ large enough and for constant $N\gt 0$ and for non-negative $n\in \Bbb Z$ ...
0
votes
2answers
81 views

Cauchy integral formula in complex analysis

Assume $g$ be an entire function. And $\exists \ n\gt 0 \:and\ n\in \Bbb Z $ and also $\exists N \: and\ M \in \Bbb R$ s.t. $\forall z \in \Bbb C , \ \ \vert z\vert \ge M\ \ \: and\ \ \ \vert ...
0
votes
2answers
82 views

Intuition behind log plotting

The two plots are the same, except the 2nd one has been log transformed on the y axis. Could I please draw your attention to what happens when Theta >0.65? After taking the log-lin plot it seems ...
8
votes
3answers
130 views

What to look for in a proof?

I am a physics undergrad, wishing to pursue a PhD in Math. I am mostly self taught in the typical math undergrad curriculum. I am looking for more input, in ways I can improve my mathematical ...
1
vote
0answers
48 views

Learning Resources: Mathematica Notebooks

A while ago, I stumbled upon someone's list of Mathematica notebooks for learning various topics. I've since lost the link, but I'm wondering now if I could find notebooks again. So my question is: ...
0
votes
1answer
100 views

Prove that the three statements are equivalent

I need to show that the following statements are equivalent. A $\subset$ B, A $\cap$ B$^c$ = $\emptyset$, and A$^c \cup$ B = U (U is the universal set) So to show that A $\subset$ B is true I said ...
0
votes
1answer
70 views

Joint MGF of double exponential distribution.

A random variable x has density function; Fx(X) = 1/2λexp(-λ|x|) for -∞ < x < ∞ , λ > 0 1) If we let you U=P+Q and V=P-Q how do we get the mgfs of Mu(S) and Mv(t) and also how would be we ...
1
vote
2answers
175 views

why does double rounding 9.46 give 10 but “regular” rounding gives 9?

What's the correct way to round, or estimate, a number to a specified precision? Starting with wikipedia: Rounding a number twice in succession to different precisions, with the latter ...
2
votes
2answers
77 views

The conditions for Harmonics functions in complex analysis

Let $\sigma(u, v)$ and $\gamma(u, v)$ be harmonic functions on a region $D$ in $\Bbb C$. What are the conditions on $\sigma$ and $\gamma$ such that $\sigma \gamma$ is harmonic on $D$. And I want to ...
5
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0answers
146 views

Study group for working through Spivak/Wilson [closed]

I'm planning on working through Spivak's Calculus or Wilson's "Introduction to Graph Theory" and was wondering if anyone here might be interested in joining a study group for it. There's no ...
5
votes
1answer
107 views

Advice regarding best-practice mathematics / categorial logic.

A good heuristic is: If it doesn't cost anything, generalize. In particular, if we have a theorem, and a proof thereof, then we ought to look for a maximal generalization of this theorem, ...
0
votes
1answer
75 views

How to decide whether PDE is Homogeneous or non-homogeneous.

I am studying second order PDE. And I have seen homogeneous and non-homogeneous PDE. But I cannot decide which one is homogeneous or non-homogeneous. For examples; (1) $(D^3-3D^2D'+4D'^3)u=0$ ...
6
votes
0answers
132 views

What is the best way to go about learning math?

I know this is a very subjective question, but after struggling on my own for a while I figured I might as well ask it. I did all the normal math classes in college (LinAlg, MultiVariable Calc, ...
0
votes
1answer
54 views

Doob's inequality application

I'm working through an example of the application of Doob's inequality in Durrett: Let $X_m$ be a submartingale, and define $\bar{X}_n = \max\limits_{0 \leq m \leq n} X_m^+$. Let $\lambda ...
1
vote
1answer
53 views

Please explain. Really I dont understand and I need to learn. Pde: : example of finding particular integral

When we look at the solution part, there is a statement The PI of the given PDE is obtained as follows After the statement, I dont really understand all of the calculation. Espacially, After the ...
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0answers
66 views

Cut-off Subtraction in Coq

I am new to the world of computer assistant proof programs in general, and Coq in particular. As a result, I have sought to prove some elementary results about integers as a way to … At the moment, I ...
-1
votes
1answer
134 views

Compute the geodesic curvature of any sphere on a sphere.

Compute the geodesic curvature of any sphere on a sphere. Again there exists its answer, but not understandable for me. Please explain it explicitly. Thank you so much. (If required, i can post ...
0
votes
0answers
44 views

Mathematics Necessary for General Relativity and Quantum Physics.

I am a self-learner in Mathematics. I was wondering, given some background in calculus and a tiny bit of topology and group theory, what series of documents I would have to learn to be able to ...
3
votes
1answer
268 views

Good book for logic self-study

I know a similar question has already been asked, but can anyone suggest a good book on mathematical logic that includes answers to exercises? I am looking for something that is conducive to ...
1
vote
2answers
43 views

Matrix and eigenvectors

$\quad$The matrix $\mathbf A=\frac19\begin{bmatrix} 7 & -2 & 0 \\ -2 & 6 & 3 \\ 0 & 2 & 5 \\ \end{bmatrix}$ has eigenvalues $1$, $\frac23$ and $\frac13$n ...
0
votes
2answers
74 views

Is there any formula for summation?

$$0.01\sum_{x=1}^{30}(0.99)^{x-1} = 1-0.99^{30}$$ I wonder if there is a formula for summation and I want to know. Would anyone mind telling me? It would be better for me to solve problems, like the ...
0
votes
2answers
18 views

sum calculation

i have the follwoing values $\sum_{i=1}^{n} x_{i} = 34$ $\sum_{i=1}^{n} x_{i}^{2} = 262.22$ $a = 3.78$ $n = 9$ and i want to calculate $$ \sum_{i=1}^{n} (x_{i}-a)^2 $$ i thought this may work ...
0
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0answers
160 views

Orthonormal matrix and diagonal matrix.

Consider the matrix $$A=\begin{pmatrix}3&-1&1\\-1&3&-1\\1&-1&3\end{pmatrix}.$$ (a) Verify that $x=[3\,\,4\,\,1]^T$ is an eigenvector of the matrix $A$ and determine the ...
1
vote
1answer
129 views

Does De Moivre's Theorem hold for all real n?

I have seen the proof by induction for all integers, and I have also seen in a textbook that we can use Euler's formula to prove it true for all rational n, but nowhere in the book does it say its ...
1
vote
1answer
70 views

Second fundamental form question.

Honestly, I dont have any idea for that question I posted. Please can someone help me solving the question. Thank you.
2
votes
1answer
50 views

Vectors and orthonormal matrix

For 2(a)(i), are the length of a =Sqrt(14) and b = sqrt(38)? For (ii), is the angle = 4.31? For (iii), is the answer 3.73? For (iv), is the answer -i+j+k? For (b) to (e), I have no idea what I ...
0
votes
1answer
89 views

Linear Algebra: Matrix and determinant

For 1(a), is $p =12$ and $q = 6$? For b(i), is the answer $a=b$ where $a$ and $b$ do not equal to 0? for b(ii), is the answer $a\ne b$? for b(iii), is the answer $a=b=0$ and the solution is ...
2
votes
1answer
93 views

The second fundamental form and isometry.

What is the effect on the second fundametal form of asurface of applying an isometry of $\Bbb R^3$ ? Or a dilation? I posted its answer. This answer is not understandable for me in general. ...
0
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1answer
32 views

Surface patch are taken different for sphere, but their second fund. forms are not completely different.

Shpere First of all, I take the surface patch for sphere $$\sigma(u,v)=(\sin u\cos v, \sin u\sin v, \cos u)$$ And then I calculated its second fundametal form. And I got the following result $$ ...
0
votes
1answer
109 views

Why first fundamental form and second fundamental form are the same?

Surface of Revolution $\gamma(u)=(f(u),0,g(u))$ and $\sigma(u,v)=(f(u)\cos v, f(u)\sin v, g(u))$ Fist of all, I calculated the first fundametal form for surface of revolution. And I obtained ...
0
votes
1answer
99 views

Transition map for Möbius band in differential geometry.

Calculate the transition map $\phi$ between the two surface patches for the möbius band. These two surface patches are the following $U=\{(t,\theta) \ | -1/2\lt t\lt 1/2,\ \ 0\lt \theta \lt ...
0
votes
0answers
21 views

Kaczmarz's Projection Algorithm

I am trying to understand the derivation of the following formulas given in my lecture (sadly without any further explanation). It says the key idea is that each new prediction error is of the form ...
0
votes
1answer
55 views

Torus in differential geometry.

I want to write separately parametrizations (surface patches) $\sigma$ for torus when (1) x-axis rotation in the first part of the picture and (2) y-axis rotation in the second part of the picture. ...
3
votes
1answer
199 views

Lawvere theories: an equivalence.

I'm having trouble understanding Lawvere theories (as defined below). Definition: A Lawvere Theory is a category $\mathcal{L}$ with finite products and with a distinguished object $A$ such that ...
2
votes
1answer
70 views

Primitive Recursion Functions (Programs)

The set $F_{n}$ of primitive recursive function symbols of arty $n$ can be defined inductively as \begin{array}[lr] & Z, \text{Succ} \in F_{1} & \\ \pi_{j}^{n} \in F_{n} \quad \text{for each} ...