Questions about studying mathematics without formal instruction.

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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
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1answer
252 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
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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
949 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 ...
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1answer
126 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 ...
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1answer
103 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
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1answer
307 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 ...
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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
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1answer
136 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
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1answer
122 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
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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
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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, ...
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1answer
85 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
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1answer
53 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
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1answer
143 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
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1answer
77 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
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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
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1answer
59 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
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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 ...
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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
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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 ...
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1answer
65 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
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2answers
820 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
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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) ...
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1answer
44 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
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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 ...
2
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2answers
56 views

How to calculate Frenet-Serret equations

How to calculate Frenet-Serret equations of the helix $$\gamma : \Bbb R \to \ \Bbb R^3$$ $$\gamma (s) =\left(\cos \left(\frac{s}{\sqrt 2}\right), \sin \left(\frac{s}{\sqrt 2}\right), ...
0
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0answers
166 views

Proof verification of some properties of the exponent.

I'd like to know if the proof of the following laws of exponent is correct. First of all thanks in advance and I apologize for the extension of the following proof but it's one exercise. I hope there ...
0
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1answer
52 views

calculation of vector product $\ddot{\gamma} (t) \times \dot{\gamma} (t)$

$$\ddot{\gamma} (t) \times \dot{\gamma} (t)=(-a\cos t, -a\sin t, 0)\times (-a\sin t, a\cos t, b)$$ The writer will get the following result $$(-ab\sin t, ab\cos t, -a^2)$$ but I don't know how to ...
0
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1answer
734 views

how to calculate the curvature of an ellipse

how can I compute the curvative of an ellipse given by $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$ do i need to take $x=acos(t)$ and $y=bsin(t)$? please show me a way how to solve this? thank you for ...
0
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1answer
90 views

Abelian Group elements and inverses

Let G be a finite abelian group, say, $G={e,a_1,a_2...a_n}$ Prove the following: a)$(a_1a_2...a_n)^2=e$ b)If there is no element x $\neq$ e, x=x^(-1), then $a_1a_2...a_n=e$ c)If there is exactly ...
2
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1answer
65 views

solve this equation $z(z+y)dx+z(z+x)dy=0$

I need to solve this following equation $$z(z+y)dx+z(z+x)dy=0$$ I get this from above equation $$\frac{dx}{z(z+x)}+\frac{dy}{z(z+y)}=0$$ After there, I dont know what I need to do.
0
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1answer
99 views

$\liminf$ and $\limsup$ question

I am trying to learn about $\liminf$ and $\limsup$, as I have struggled with the definition of these, and mostly just avoided questions about this in the past. I have had a go at answering a question ...
0
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1answer
61 views

Find solution(primitive) of the equation

I want to find its solution of the following equation $$ydx+xdy+2zdz=0$$ answer: Keeping $z$ constant; I obtain that $$ydx+xdy=0$$ or $$\frac{dx}{x}+\frac{dy}{y}=0$$ Then I get $$U(x,y,z)=xy$$ ...
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1answer
55 views

How to obtain the last ratio $\frac{d(x+z)}{(x+z)}$

I am studying example-2.3 In the first line, it says that "the numerators and denominators in the first and last ratio" And the following is obtained $$\frac{d(x+z)}{x+z}=\frac{dy}{y}$$ But I ...
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1answer
476 views

Prove the least upper bound property using $\mathbb{Q}$-Cauchy sequences.

Hi everyone I'd like to know if the next proof is correct. I'd appreciate any suggestion mainly in the points marks with (1) and (2). Theorem: Let $E$ be a nonempty subset of real numbers which has ...
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2answers
275 views

The Decision of three methods of the solutions $dx/P=dy/Q=dz/R$

Question: (A) $$\frac{adx}{(b-c)yz}= \frac{bdy}{(c-a)xz}=\frac{cdz}{(a-b)xy}$$ (B) $$\frac{dx}{xz-y}=\frac{dy}{yz-x}=\frac{dz}{1-z^2}$$ These are simultaneous diff eq. of the first order and the ...
0
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3answers
146 views

Natural Deduction (FeedBack)

I am looking for feedback to three proofs (alternatively derivations) that I have constructed. The first is: Theorem. Injectivity does not imply surjectivity. Proof: Suppose $\{\phi\} \vdash ...
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1answer
73 views

How to show that the limaçon has only two vertices.

Question: Show that the limaçon has only two vertices. I researched what is limaçon. And I reached the following result; Note that I only know that The limaçon is the parametrized curve ...
4
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1answer
255 views

Proof verification that Q is dense in R

Hi everybody I'd like to know if the next argument is sound. Definitions: A real number x is said to be positive if can be written as a formal limit of a Cauchy sequence of rational numbers $(x_n)$ ...
2
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1answer
92 views

Writing a parametrization of the cissoid by using $\theta$

The cissoid of Diocles is the curve whose equation in terms of polar coordinates $(r,\theta)$ is $$r = \sin\theta \tan\theta, −\pi/2 < \theta < \theta/2$$ Write down a parametrization of the ...
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1answer
96 views

interactive training in mathematics olympiad competition for 8th Grade: Ages 13–14.

I'll enjoy your kindness to ask this question, despite that it seems it'snt the right destination. Please show me an url, for training in mathematics olympiad competition for 6th Grade: Ages 11–12. ...
3
votes
1answer
118 views

linear algebra from beginner to intermediate level

I was wondering if there is a way to learn linear algebra from beginner to advanced level by studying it myself. I want to collect a number of books, video lectures, tutorials and other resources to ...
3
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2answers
110 views

Proof of the infinite descent principle

Hi everyone I wonder to myself if the next proof is correct. I would appreciate any suggestion. Proposition: There is not a sequence of natural numbers which is infinite descent. Proof: Suppose for ...
2
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2answers
195 views

Natural Deduction and Identity

In Logic and Structure, p. 99, Van Dalen gives a brief summary of the rules for predicate logic involving identity. Now, I would like to think of myself as quite proficient in terms of constructing ...
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3answers
74 views

Evaluation of the integral.

$$I\left(n,\epsilon\right)=\int_{-{\rm i}\infty}^{+{\rm i}\infty} \frac{{\rm e}^{\epsilon z}}{\left(z+\epsilon\right)^n}\,{\rm d}z$$ The integration is taken along the imaginary axis, an integer ...
1
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0answers
33 views

Confirm my working for the conditional posterior of $\beta$

So I have the following question from my textbook, the answer I get is slightly different from the book's answer, which I think may be wrong, could someone please confirm? Question: Suppose $y_{1:T} ...
5
votes
2answers
271 views

Which books to study category theory?

I am an amateur math researcher in the field of general topology. I've set the purpose to learn enough category theory for my research. After reading Steve Awodey, "Category Theory", 2010, is it ...
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4answers
279 views

Advanced Mathematics

I am a high school student and would like to pursue a career in mathematics and I am hoping to find a serious explanatory book on math (geometry, algebra, calculus, functions and trigonometry) for ...
3
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3answers
370 views

Proving linear independence of vectors which are functions of other independent vectors

If the $n$-component vectors $a,b,c$ are linearly independent, show that $a+b, b+c, a+c$ are also linearly independent, Is this true of $a-b,b+c,a+c$? What I did was write the new vectors as sums ...