Questions about studying mathematics without formal instruction.

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3
votes
1answer
52 views

How to get $\sum_{k=1}^{n-1}n\binom{n-1}{k-1}y^k \bigg(\frac{1-z^k}{k}- \frac{1-z^n}{n}\bigg)$"?

I asked a question here and got answer committing : $$(1+yz^n)(1+y)^{n-1} - (1+yz)^n=\sum_{k=1}^{n-1}n\binom{n-1}{k-1}y^k \bigg(\frac{1-z^k}{k}- \frac{1-z^n}{n}\bigg) ...
2
votes
1answer
78 views

find the general integrals of the given P.D.E

I tried to find the general integrals of the given P.D.E in the yellow box. And I found $c_1$. But I cannot find another one, say $c_2$. Please help me finding $c_2$. Thank you.
1
vote
1answer
276 views

Find the integral curves of the equation

Question: Find the integral curves of the equation: $$\frac{dx}{y^2x-2x^4}=\frac{dy}{2y^4-x^3y}=\frac{dz}{2z(x^3-y^3)}$$ I could not find any similar example to understand this type of questions ...
2
votes
1answer
76 views

Number of lines required to split a plane into N regions

I was wondering if there was any theorem out there that talks about the number of lines required to split a plane into any given number of regions. I don't really have much of a mathematical back ...
5
votes
7answers
361 views

Good Number Theory books to start with?

I'm in Grade 11. I'm interested in elementary number theory and would like properly study it. I'm not intending to enter any competitions.
0
votes
3answers
84 views

Question on Proof of the Contraction Mapping Theorem

Contraction Mapping Theorem If $T\colon X\to X$ is a contraction mapping on a complete metric space $(X,d)$ then there is exactly one solution $x\in X$. Proof: Let $x_0$ be any point in $X$. We ...
2
votes
2answers
323 views

Book that is more accessible than Shoenfield

My logic course is based on my Computer Science education and on some random Internet pages (mostly Wiki). I want to make my knowledge of logic more coherent and fill in missing gaps. Thus I started ...
6
votes
1answer
470 views

What's the best way to teach oneself both Category Theory & Model Theory?

I've done a bit of reading around both Category Theory & Model Theory (CT & MT) as a novice in each field. I'm interested in how they might combine, particularly when applied to Algebra. [So ...
9
votes
5answers
550 views

What is the easy way to calculate the roots of $z^4+4z^3+6z^2+4z$?

What is the easy way to calculate the roots of $z^4+4z^3+6z^2+4z$? I know its answer: 0, -2, -1+i, -1-i. But I dont know how to find? Please show me this. I know this is so trivial, but important ...
0
votes
0answers
38 views

The part I dont understand while calculating contour integral.

Question is the following; $$\int_{0}^{\infty}[x^{m-1}/(1+x^n)]dx$$ for $m,n=1,2,\dots$ and $n>m>0$ Solution: its poles were found as this $$a_k=e^{i(2k+1)\pi/n}$$ for $k=0,1,...(n-1)$ And ...
0
votes
1answer
53 views

Properties of Quasi-Coherent Modules

Let $X=\mathrm{Spec}\,A$ be an affine scheme and $M$ an $A$-module. Show that the following two conditions are equivalent: (a) $\tilde{M}$ is a locally free $\mathcal{O}_{X}$-module of finite type. ...
0
votes
1answer
48 views

Limit of a sequence that is Cauchy with respect to a family of seminorms

Definition: Let $p$ be a seminorm on some space $X$. A sequence $(x_n)$ in $X$ is said to be Cauchy with respect to $p$ if for any $\epsilon > 0$, there is an $N > 0$ such that $p(x_m - x_n) ...
0
votes
2answers
56 views

The calculation of roots of complex numbers.

How to calculate the roots of $x^6+64=0$? Or how to calculate the roots of $1+x^{2n}=0$? Give its easy and understanble solution method. Thank you. In general, the results of "exp" are obtained.
4
votes
1answer
122 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. ...
2
votes
1answer
63 views

Scheme-Theoretic Nakayama's Lemma

Let $X$ be a noetherian scheme and $\mathscr{F}$ a coherent $\mathscr{O}_{X}$-module. For a point $x \in X$, let $k(x)=\mathscr{O}_{X,x}/\mathfrak{m}_{x}$ be the residue field at $x$. (a) Suppose $x ...
1
vote
0answers
39 views

Classify all invertible meromorphic functions

I am studying complex analysis. And I see a topic, namely meremorphic functions But I cannot find any enough information about this function. So I have insufficient knowledge about this topic. ...
1
vote
0answers
46 views

What areas of mathematics are taught in a Computer Engineering course?

I'm planning on taking a Computer Engineering course next year, I study hard when it comes to math so I wanna know what area of mathematics I'm going to tackle during my course so I can study it ...
2
votes
1answer
438 views

Finding a conformal map from semi disk to upper half plane.

Find a conformal mapping $f$ of semi-disk$S=\{z: \vert z\vert \lt 1, Im z\gt 0\}$ onto the upper plane. Again I used composition of conformal map. First of all, let's define a conformal map ...
4
votes
0answers
279 views

Find a conformal map from the disc to the first quadrant.

Find a conformal mapping of the disk $x^2+(y-1)^2\lt 1$ onto the first quadrant $x, y \gt 0$ I did something, which may be false or not, I cannot exactly say anything. I used the composition of ...
2
votes
1answer
580 views

How to find a conformal mapping of the first quadrant.

Find a conformal mapping of the first quadrant onto the unit disc mapping the points $1+i$ and $0$ onto the points $0$ and $i$ respectively. I think that i need to use "the change of variables ...
2
votes
2answers
286 views

Conformal map example $ f(z)=e^z$

I an studying the example-1. I understand $f(z)=e^z$ has a nonzero derivative at all points, hence it is everywhere conformal and locally $1-1$. But I dont understand th part I underlined with ...
3
votes
2answers
151 views

Why not $f(z)=z^2$ conformal at $z=0$?

$$f(z)=z^2$$ is not conformal at $z=0$ Why? Conformal definition: $f$ is conformal at z if f preserves angles there.
1
vote
3answers
172 views

Solutions to $x+y+z=31$ and $x+2y+3z=41$

For the equations $$x+y+z=31$$ $$x+2y+3z=41$$ is there a elegant way or method to find all the positive solutions in integers? Thus far, I have been using trial and error (which is time consuming). ...
1
vote
1answer
59 views

How to construct a term of a particular type

I am reading the article "Introduction to Type Theory" by Herman Geuvers, the chapter explaining the Fitch style of natural inference. I stuck at the exercise 1.3 (first two are simple): ...
0
votes
0answers
66 views

Why this sequence should be terminated as soon as $7$ or $8$ is obtained?

I asked a question here on probability, but can't get why should we terminate as soon as we get $7$ or $8$. What if we extend it? Let me elaborate my doubt: Suppose we have a sequence such that ...
0
votes
1answer
257 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, ...
1
vote
1answer
58 views

Open immersion from a proper scheme to a separated, irreducible scheme.

Fix a scheme $S$ and let $X$ and $Y$ be $S$-schemes. Assume that $X$ is proper over $S$ and $Y$ is separated over $S$. Let $f: X \rightarrow Y$ be an open immersion of $S$-schemes. If $Y$ is ...
2
votes
3answers
973 views

What is the equal sign with 3 lines mean in Wilson's theorem?

I'm reading up on Wilson's Theorem, and see a symbol I don't know... what does an equal sign with three lines mean? I'm looking at the example table and I still can't infer what they are trying to ...
1
vote
1answer
128 views

Resources for exploring math without a teacher

The ability to understand the beauty of math requires rigorous study. However, most people do not have access to the kind of training pure math requires. Many of my friends easily get interested in ...
1
vote
1answer
105 views

Trying to understand an exercise using factorials with induction

Exercise: Prove that (n + 1)! - n! = n(n!) for any n $\ge$ 1 Given Answer: I will skip the basic step since I understand that part. (n + 2)! - (n + 1)! = (n + 1)!(n + 2) - n!(n + 1) I understand ...
1
vote
1answer
309 views

Showing how to find the vertices of the circle.

Find that the circle has four vertices. $$\gamma (t)=\langle R\cos (t/R), R \sin (t/R)\rangle$$ for $t\in [0,2\pi]$ I know the theorem: Every simple closed convex curve has atleast four ...
0
votes
0answers
20 views

Question regarding differentation with respect to functions

I am reading some papers which include differentiation wrt functions rather than real numbers. I follow the proofs, and am able to verify that they hold, but still do not feel comfortable that I would ...
3
votes
1answer
138 views

Definition of $ 1 + \cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{4+\cfrac{1}{\ddots}}}}$

Is there a definition of $ 1 + \cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{4+\cfrac{1}{\ddots}}}}$? I am somewhat familiar with continued fractions; that is, I am aware that their convergence depends on whether ...
3
votes
1answer
130 views

Showing the parametrically representation of hyperbolic paraboloid. And how to find the curves $u$ and $v$ be constant.

Show that the hyperbolic paraboloid can be represented parametrically as $$r(u,v)=\langle a(u+v), b(u-v), uv\rangle$$ Find the curves $u$ is constant and $v$ is constant. I guess I need to use the ...
4
votes
1answer
88 views

The curve has constant torsion.

Question: Show that when the curve $c_1=c_1(t)$ has constant torsion $\tau$, the curve $$c_2=c_2(t)=-\frac{1}{\tau}N+\int_{t_0}^{t}B(u)du$$ has constant curvature $-\tau$ or $+\tau$. What I ...
2
votes
1answer
100 views

21, Not Touched Maths Since GCSE. Want to start learning again. Where to Start?

I am 21 and have got into computer programming. Doing very well in my degree. Would love to get into computer science but feel I am being held back by my basic knowledge of maths. I got an A at GCSE, ...
0
votes
1answer
89 views

$(\frac{1}{\kappa})^2+(\frac{\dot{\kappa}}{\kappa^2\tau})^2=r^2$

Show that for a curve lying on a sphere of radius r with nowhere vanishing torsion, the following equation is satisfied: $$(\frac{1}{\kappa})^2+(\frac{\dot{\kappa}}{\kappa^2\tau})^2=r^2$$ Please ...
3
votes
1answer
55 views

Show that $\dot{n_s}=-\kappa_s t$

I found the question in a differential geometry textbook while studying. This question seems so intesting to me. So please help me solving it. Show that, if $\gamma$ is a unit-speed plane curve, ...
0
votes
1answer
145 views

Partial derivative of piecewise function of two variables

I'm having some difficulty figuring out $\frac{\partial}{\partial x}$ of the following function: $ f(x,y) = \left\{ \begin{array}{lr} x^2+y^2 & : x \not= 0\\ y^4 & : x = 0 ...
3
votes
1answer
79 views

If $\gamma$ is spherical, then the equation $\frac{\tau}{\kappa}=\frac{d}{ds}(\frac{\dot{\kappa}}{\tau \kappa^2})$ holds.

Question: Let $\gamma (t)$ be a unit-speed curve with $\kappa(t)\gt0$ and $\tau(t)\neq0$ for all $t$. Show that, if $\gamma$ is spherical, i.e., if it lies on the surface of a sphere, then ...
1
vote
1answer
87 views

Supremum of sets of extended reals

Hi everyone I'd like to know if following is really correct, looks kinda cumbersome, I think, it is for the great quantity of cases to analyze. To be honest I don't know if this is the better way to ...
1
vote
1answer
61 views

Showing a set of functions $F$ is bounded

I have a set of functions given by; $$F = \{f:[0,1]\rightarrow\mathbb{R}|\int_0^1 f(x)dx = 0, |f(x)-f(y)|\leq|x-y|, x,y\in[0,1]\}.$$ I have a solution for the question so my questions are about the ...
4
votes
4answers
134 views

Calculate $\sum\limits_{n=1}^\infty (n-1)/10^n$ using pen and paper

How can you calculate $\sum\limits_{n=1}^\infty (n-1)/10^n$ using nothing more than a pen and pencil? Simply typing this in any symbolic calculator will give us $1/81$. I could also possibly find this ...
8
votes
2answers
122 views

How to obtain $y$

The question was written with dark-blue pen. And I tried to solve this question. I obtained $x$ as it is below. But I cannot obtain $y$ Please show me how to do this. By the way, $\gamma (t)$ ...
2
votes
1answer
72 views

Verify that an ellipse has four vertices.

Verify that an ellipse has four vertices. The ellipse is given by $$ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$ And I took $$x=a\cos t$$ and $$y=b \sin t$$ for $t\in [0,2\pi]$ Please can someone help ...
0
votes
1answer
67 views

From $ \sum^\infty_{\lfloor \log n \rfloor + 1}n/{2^r} $ to $ \sum^\infty_{r=0}1/2^r $?

$$ E[h] = E[\sum^\infty_{r=1}I_r] = \sum^\infty_{r=1}E[I_r] $$ $$ = \sum^{ \lfloor \log n \rfloor}_{r=1}E[I_r] + \sum^\infty_{\lfloor \log n \rfloor + 1}E[I_r] $$ $$ \leq \sum^{ \lfloor \log n ...
9
votes
2answers
850 views

Learning Abstract Algebra for a graduate degree

I would like to do a graduate degree in mathematics, and I have a full year before I will be able to do so (for personal reasons). I mainly have my weekends available to study. I am interested in ...
0
votes
1answer
104 views

Joint distribution of two marginal normal random variables

Question: Suppose we have: \begin{align*} \begin{bmatrix} X_1 \\ X_2 \end{bmatrix} \sim N\left(\begin{bmatrix} 6 \\ 3 \end{bmatrix}, \begin{bmatrix} 12 & 3 \\ 3 & 2 \end{bmatrix} \right) ...
-1
votes
1answer
45 views

how to find the signed normal

$$\gamma (t)= (R\cos (t/R), R\sin (t/R))$$ $$\dot {\gamma (t)}=(-\sin (t/R), \cos (t/R))$$ $$n_s= (-\cos (t/R), -\sin (t/R))$$ where $n_s$ is the signed normal. the instructor has found the $n_s$. ...
0
votes
0answers
33 views

Convergence of $xe^x - R$

Basing my question on one of the previous questions I have passed before Root of the function $f(x)=xe^x-R$, I was wondering why does $xe^x - R$ always converge? I was told that the function will ...