Questions about studying mathematics without formal instruction.

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0answers
69 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 ...
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1answer
168 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 ...
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0answers
45 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 ...
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1answer
296 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 ...
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2answers
44 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 ...
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2answers
83 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 ...
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2answers
19 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 ...
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0answers
178 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 ...
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1answer
154 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 ...
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1answer
73 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
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1answer
51 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 ...
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1answer
93 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
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1answer
105 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. ...
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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 $$ ...
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1answer
120 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 ...
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1answer
111 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 ...
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0answers
22 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 ...
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1answer
61 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
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1answer
200 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
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1answer
72 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} ...
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4answers
140 views

Linear Algebra and Set Theory book recommendations.

I would like to studying linear algebra and set theory. Does anyone have a a good recommendation of books/resources/etc.?
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0answers
26 views

Error of the norm of solution in linear least-squares

How can we estimate the solution norm ($\Vert x \Vert$) error, separate from the solution ($x$) error in solving $Ax=y$ (linear least-squares problem)? Is the error of $\Vert x \Vert$ higher or lower ...
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0answers
51 views

Between any rational and $\sqrt{2}$ is another rational [duplicate]

I'm self-studying analysis from Rudin, and in there is this proof. I understand the proof after reading it, but how would a mathematician come up with this or approach doing a proof like this? It ...
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1answer
174 views

Machine learning: beginner study material.

Can anyone suggest to me some beginner study material for Machine learning applications in fields of 1) Financial forecasting and 2) Online advertisement? Thanks in advance!
3
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1answer
161 views

What are the benefits or losses of learning real analysis through a constructivist approach instead of a standard apporach?

Recently I've found some courses on real analysis that use the constructivist approach and I got curious on some aspects: What are the benefits of learning through this approach? Is it ok to learn ...
3
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0answers
89 views

Interchange of finite sum with convergence sequences.

Hi everyone I'm wondering if the following proof is correct (to be honest at the beginning I have some troubles to understand what the sequences of the sums of convergent sequences has to be, but I ...
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1answer
120 views

countably compact subset of a metric space is sequentially compact?

I'm currently trying to get my head around a proof in Lipschutz [1]. Solved exercise 21 of Chapter 11 (page 164) asks for a proof of the following statement: Let $A$ be a countably compact subset of ...
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1answer
72 views

Let $f$ be Lebesgue integrable on $\mathbb{R}$. Show that $ \sum_{n=1}^{\infty} f(x+n) $ converges almost everywhere. [duplicate]

Let $f$ be Lebesgue integrable on $\mathbb{R}$. Show that $ \sum_{n=1}^{\infty} f(x+n) $ converges almost everywhere. I was thinking that maybe the finite sum could be compared to a simple function ...
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1answer
70 views

Calculus: computation of $\sum \frac{2^i}{i!}$

$$\sum_{i=0}^\infty \frac{2^i}{i!}$$ Would anyone mind telling me what is the answer? I know this may be a silly question but I would like to know.
2
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1answer
110 views

An eigen-decomposition/diagonalization question

I'm data analyst without any good math background. I'm struggling to understand and to code myself eigen-decompositions. So far, I know QR algorithm of eigen-decomposition. My problem. Let $\bf A$ be ...
2
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1answer
83 views

Help understanding a counting and probability exercise

I need help in trying to understand the answer to this exercise. [Question] A club is considering changing its bylaws. In an initial straw vote on the issue, 24 of the 40 members of the club favored ...
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0answers
51 views

Finding an unbiased estimator for function of Poisson

Let $X_1,...,X_n \sim Poi(\lambda)$ then unbiased estimator for $\lambda$ is obviously $\bar{X}$. What about $\tau(\lambda)=\sqrt{\lambda}$? Also how would one derive UMVUE for this lambda?
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1answer
387 views

Prerequisites for studying Homological Algebra

I have read the answers here and here and need to ask something more. I wish to study the book on Homological Algebra by Weibel but am not sure of the prerequisites. In particular how much ...
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1answer
50 views

counting and probability question - help needed

I am stuck on how to start this exercise. Any help is welcome. An instructor gives an exam with 14 questions. Students are allowed to choose any 10 to answer. Suppose the exam instructions specify ...
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3answers
76 views

Recursion, multiplication and exponential

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} ...
0
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1answer
29 views

Field extension $F\subseteq L_1$ and $F\subseteq L_2$ and $[L_1L_2:F]<[L_1:F][L_2:F]$.

I'm searching for an example of field extensions $L1$, $L2$ of $F$ for which $[L_1L_2:F]<[L_1:F][L_2:F]$. Infact I'm trying prove the problem below. So any hint can be helpful. Let $K$ be a ...
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1answer
67 views

How to show that the triangle is equilateral triangle?

If $\cot A+\cot B+\cot C=\sqrt3$ then prove that the triangle is equilateral triangle. Trial: I can counter check that this is true as $\cot 60+\cot 60+\cot 60=3 \frac{1}{\sqrt3}=\sqrt3$.Here I ...
6
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1answer
336 views

Learning Complex Geometry - Textbook Recommendation Request

I wish to learn Complex Geometry and am aware of the following books : Huybretchs, Voisin, Griffths-Harris, R O Wells, Demailly. But I am not sure which one or two to choose. I am interested in ...
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2answers
284 views

How much Category theory one must learn?

I have learnt very basic category theory (up to Yoneda lemma from Hungerford's Algebra text). My question is how much category theory should every Mathematics student who is not planning to specialize ...
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0answers
89 views

Gamma distribution and generalized gamma distribution

The generalized gamma distribution is described as follows $$f(x)=\frac{\gamma \cdot \left(\frac{x-\mu}{\beta} \right)^{\alpha\cdot \gamma-1}}{\beta\cdot\Gamma(\alpha)}\cdot e^{- ...
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0answers
40 views

Why can the group of isomorphism classes of line bundles be identified with $H^1(C,\mathbb O_C^*)$?

This is a reference request to the fact in the title. Is there a book at most as advanced as Hartshorn's which explains this result?
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1answer
152 views

Elementary/Intermediate Algebra book with proofs

I am looking for an elementary or Intermediate Algebra book which has proofs. I would like book to present proofs for statements like If $P(x)$ is a polynomial of degree n then will have exactly ...
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1answer
92 views

Bernoulli Related Probability Distribution

Find the probability of having 4 or more girls in a family of 6 children. Find also the probability that among 5 families, each with 6 children, at least 3 of the families have 4 or more girls. An ...
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0answers
50 views

$f(x)=\sum^\infty_{n=1}\frac{1}{n(1+nx^2)}$ for $x\in\mathbb{R}$ for what interval (a,b) does it CV - I can make it con uni for |a|>0

$f(x)=\sum^\infty_{n=1}\frac{1}{n(1+nx^2)}$ for $x\in\mathbb{R}$ for what interval (a,b) does it converge uniformly - I can make it converge uniformly for a given |a|>0 For a non-zero x the denom ...
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0answers
27 views

How is $\lvert f\rvert=\int^1_0\lvert f\rvert dx$ not a norm (f is a regulated function, or funtion with bounded variation 0)

How is $\lvert f\rvert=\int^1_0\lvert f\rvert dx$ for $f\in R[0,1]$ where R are the regulated functions (bounded variation zero, for an epsilion I can give you a step function where the suprememum of ...
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12answers
3k views

I almost quit self-studying mathematics, but should I continue?

Before I move on to the main idea of this post, I need to tell you some background information about myself. Hopefully, it proves useful for you in giving me advice. I'm a 16 year old high school ...
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1answer
71 views

Disagreement with textbook (maybe); something about “theorem about the derivative of the limit of convergence sequences to $C^1$ functions”

I've been told to note that this is "self-learning" by someone with 29k rep, this is self learning! "If we assume $f_n:[a,b]\rightarrow\mathbb{R}$ and $f_n(x)=(1-x/n)^n$ converges uniformly we can ...
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1answer
52 views

A plane is a surface.

I am trying to show that a plane is a surface. I posted what I did. I find $σ$ But I cannot find $σ^{-1}$. Also as I said, I need to verify $σ$ is 1-1 continuous and continuous inverse. How can I do ...
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1answer
61 views

Geometric distribution question for $p=0.5$

Wiki says that geometric distribution $p(k)=p \cdot (1-p)^k$ describes a probability of Bernoulli trials required to get success. My question is: Let us assume that we're tossing a coin. A success is ...
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1answer
57 views

How do I model a real life observation with mathematical expression or equation(s)?

I am looking for a guide to learn how to model real life situations into mathematical equations and able to simulate them. My target is to able to understand and interpret something I observe into a ...