All Questions

1
vote
2answers
250 views

A doubt in Hatcher's Algebraic Topology.

I refer to pg. 27 of Hatcher's Algebraic Topology. I refer to the part where Hatcher proves that $f.(g.h)\cong (f.g).h$ For the life of me, I cannot figure out how the diagram on the right proves ...
0
votes
1answer
73 views

Proving some properties about the expected first order statistic (maximum) with respect to sample size.

Question: Consider $n$ random variables $x_1, x_2,\cdots x_n\sim \mathcal{N}(0,1)$. The expected value of the $i$th order statistic (the maximum) can be written as $E(\mathcal{O}^n_1)=\int_{-\infty}^{...
3
votes
1answer
58 views

not following a step in ash and novinger example of analytic but does not have primitive

I'm trying to self-study complex analysis and am currently reading the book "complex analysis" book by ash and novick. 0n the top of page 14, they write that , if $f(z) = \frac{1}{z}$ and the path ...
0
votes
0answers
194 views

residue theorem in complex analysis(Rudin)

In P.224, Rudin's real and complex analsysis, I doubt an equation (3). The full statement containing equation (3) is following : If $\Gamma$ is a cycle and a $\notin$ $\Gamma$*, then $Q(z)=\...
1
vote
5answers
7k views

In a triangle ABC, prove that cot(A/2)+cot(B/2)+cot(C/2) =cot(A/2)cot(B/2)cot(C/2)

In a triangle ABC, prove that $\cot \left ( \frac{A}{2} \right )+\cot \left ( \frac{B}{2} \right )+\cot \left ( \frac{C}{2} \right )=\cot \left ( \frac{A}{2} \right )\times \cot \left ( \frac{B}{2} \...
-3
votes
1answer
99 views

axiom of regularity and power sets [closed]

So axiom of regularity works for any non-empty set. Is that means that a set like this {aab, +, 50, ), (, **} and all of it's subsets are actually made of empty sets? And not just sets, but power sets?...
1
vote
4answers
169 views

Show that $\langle f_n \rangle$ is a Cauchy sequence, where $f_n=1-\frac12+\frac13-\frac14+\dots+\frac{(-1)^{n-1}}{n}$

Show that $\langle f_n \rangle$, where $$f_n=1-1/2+1/3-1/4+\dots+\frac{(-1)^{n-1}}{n}$$ is a Cauchy sequence. My attempt: Consider $$|f_{2m}-f_m| = \left| \frac{(-1)^m}{m+1}+\frac{(-1)^{m+1}}{m+2}+\...
1
vote
1answer
174 views

How to characterize elements in the Bruhat open cell?

This might be an elementary question. For simplicity, let's assume $G=GL(n,F)$, where $F$ is a local field. Let $U$ be the subgroup of upper triangular unipotents, $A$ the subgroup of diagonal ...
2
votes
4answers
118 views

Convergence of $a_n=1+1/5+1/9+\ldots+\frac{1}{4n-3}$

Show that the sequence $$a_n=1+1/5+1/9+\ldots+\frac{1}{4n-3}$$ does not converge but the sequence $b_n=\frac{a_n}{n}$ converges. I can show the first part. For the second part, will it be sufficient ...
1
vote
1answer
594 views

Power series and their inverses (radius of convergence of each)

Suppose I have a power series approximation $y$ to an invertible function $f(x)$, and I know that $y$ convergences around $x$ on an interval $(-R,R)$, $R$ being the radius of convergence. How are the ...
0
votes
1answer
713 views

Prove that $\small\sin x\sin y\sin(x-y) + \sin y \sin z \sin(y-z) + \sin z \sin x \sin(z-x) + \sin(x-y) \sin(y-z) \sin(z-x) = 0$.

Prove that $$\sin(x) \sin(y)\sin(x-y) + \sin(y) \sin(z) \sin(y-z) + \sin(z) \sin(x) \sin(z-x) + \sin(x-y) \sin(y-z) \sin(z-x) = 0 \; .$$ I tried all identities I know but I have no idea how to ...
0
votes
3answers
3k views

number of ordered pairs of integers $(x,y)$ satisfying the equation

I need to find the number of ordered pairs of integers $(x,y)$ satisfying this equation: $$x^2 + 6x + y^2 = 4.$$ I have tried, and I think $x<0 . $ Is there a specific way to solve such ...
0
votes
0answers
39 views

Finding dynamic range of rotation matrices

How do I theoretically calculate the maximum value of the transformed output can reach after transforming a vector? If it is an eigen vector then the eigen value will tell the max scaling possible, ...
0
votes
1answer
21 views

contraints on equation of a cylinder

(x-a)^2+(y-b)^2 = r^2 any way to adjust this formula to add constraints to the z-axis? recently introduced to the idea this goes forever in z-axis and I want to see if theres way's to adjust formula ...
2
votes
1answer
505 views

Blow-up in Projective Space: Choosing the Appropriate Chart

Consider $x^2+y+\alpha=0$ (made-up example), where $(x,y)\in\mathbb{C}^2$ and $\alpha\in\mathbb C$ is a free parameter. The equation defines a family of curves. The curves have no common points in ...
1
vote
0answers
91 views

can we expect a bounded approximate identity in $L^{1}(\mathbb R)$, whose Fourier transform at particular point is zero?

Let $\phi:\mathbb R \to \mathbb C$ be a function; and define $$\phi_{t}(x):=\frac{1}{t}\phi (x/t), (t>0).$$ We let, $\phi\in L^{1}(\mathbb R),$ then $\phi_{t}\in L^{1}(\mathbb R);$ in fact, we ...
3
votes
2answers
82 views

Finding $f(x)$.

If $$f(x)=1+x+x^2+\displaystyle\int_{0}^{x}e^k f(x-k) dk$$ then how do we find the function $f(x)$? Is there a way to solve it, with or without arriving at a differential equation? This a homework ...
1
vote
1answer
56 views

Linear Transforms & Matrices

$T:R^4 -> R^3$ Linear Transform This matrix is $[T]_{B2}^{B1}$ = A =\begin{pmatrix}1&2&3&4\\1&4&0&2\\2&2&9&10\end{pmatrix} After elimination we get: \begin{...
1
vote
0answers
122 views

Maximize profit

My book (George F. Simmons - Calculus with analitic geometri) hasthe following question: An library could buy from the book publisher the book "Rituals" with a cost of $40.0$ each. The manager from ...
0
votes
1answer
55 views

how to find uniform continuity

I have some questions on continuity. What is the difference between continuous and uniformly continuous function? Please explain with this question. Find $f(x)=x^2$ is uniformly continous on [0,$\...
0
votes
1answer
555 views

Sketching the unit ball centered at the origin of the metric $d(x,y)=\vert x_1 -y_1 \vert + \vert x_2-y_2 \vert$ in $\mathbb{R}^2$

I am having some diffucilty sketching the unit ball centered at $(0,0)$ for the metric given by $$d(x,y)=\sum_{i=1}^n \vert x_i -y_i \vert$$ in $\mathbb{R}^n$ for $n=2$. If $n=2,$ the unit ball is the ...
4
votes
0answers
151 views

What is the solution to the system $\frac{df_n}{dt} = kf_{n-1}-(k+l)f_n+lf_{n+1}$?

I'm trying to solve the system $$ \begin{matrix} & \frac{df_1}{dt} = kf_1+lf_2 \\ & \vdots \\ & \frac{df_n}{dt} = kf_{n-1}-(k+l)f_n+lf_{n+1} \\ & \vdots \\ & \frac{df_N}{dt} = kf_{...
3
votes
1answer
209 views

Why is there no $[X,[X,[X,Y]]]$ and $[Y,[Y,[X,Y]]]$ in the fourth order term of BCH formula?

While trying to deal with a problem involving BCH (Baker-Campbell-Hausdorff) formula, I've noticed something strange. Everywhere in the literature I've managed to fetch (for example: this and this ...
2
votes
2answers
2k views

Prove that $\sin(12^\circ)\sin(48^\circ)\sin(54^\circ)=\frac18$ [closed]

Prove that $$\sin(12^\circ)\sin(48^\circ)\sin(54^\circ)=\frac18.$$ Without using a calculator. I tried all identities I know but I have no idea how to proceed. I always get stuck on finding $\sin36^\...
0
votes
1answer
282 views

Rotate $xyz$ by use of pitch and yaw around origin

I have a project for a game which uses pitch/yaw for the direction of a players head. The pitch ranges from $0$ to $180$ and the yaw is $0$ to $360$. Yaw modifies $X$ and $Z$, pitch modifies the $Y$, ...
6
votes
5answers
312 views

Solutions to functional equation $f(f(x))=x$

Is there any more solutions to this functional equation $f(f(x))=x$? I have found: $f(x)=C-x$ and $f(x)=\frac{C}{x}$.
4
votes
2answers
3k views

The meaning of Inverse Matrix

I am studying Linear Algebra, I have 3 questions in my mind What does an inverse matrix mean. I am trying to have a meaning of it, but I don't really understand. When a matrix does not have an ...
1
vote
3answers
6k views

Linearize a first order differential equation

The system described by $x'=2x^2-8$ is linearized about the equilibrium point -2. What is the resulting linearized equation? Answer is $x'=-8x-16$. How? I have no idea how it went from the first ...
1
vote
1answer
1k views

Is it possible to have a inflection on a vertical asymptote?

I found the derivative of a function to be f'(x)=8/x^3 and thus its second derivative as f''(x)=0/3x^2. After setting the second derivative to zero and doing the substitution into the parent function, ...
0
votes
1answer
125 views

Minkowski Distance Metric

Given compact sets $A$, $B$, define the Minkowski distance between the two sets as: $$ \delta(A,B):= \inf \{ r: B \subseteq \mathscr{N}_r (A) \, \, \text{and} \, \, A \subseteq \mathscr{N}_r (B) \}$$ ...
-1
votes
1answer
227 views

Weak nullstellansatz in Atiyah-Macdonald 5.17

$\newcommand{\fm}{\mathfrak{m}}$ Problem 17 in the exercises after the 5th chapter of Atiyah-Macdonald is the following (with some references and hints omitted): Let $X$ be an affine algebraic ...
6
votes
1answer
135 views

Radius of convergence continuous?

Let $ f: [0,1] \rightarrow \mathbb{R} $ be analytic. Let $ r_f(x) $ be the radius of convergence of $ f $ at $ x $. Is $ r_x(f) $ continuous? Alternatively, is there an $ r_{min} $ I can choose so ...
10
votes
1answer
1k views

Flat connection with non-trivial holonomy? I cannot get it

maybe this is a dumb question, but I cannot understand how a principal $G$-bundle can have non-trivial holonomy with a flat connection. Maybe I'm missing something, but doesn't Ambrose-Singer theorem ...
3
votes
1answer
60 views

Distribution of random variable

I need help with this problem: Let $(X_n)_{n\in \mathbb{N}}$ a sequence of i.i.d$\sim $Uniform$(\{0,\dots,9\})$ random variables. What is the distribution of $$X= \sum_{n=1}^{\infty} X_n 10^{-n}$$ ...
2
votes
3answers
2k views

Can an inflection exist if there's no max/min?

Very quick question: if a function doesn't have a maximum nor minimum, can it still have a point of inflection? I believe that these two go hand in hand and without one you can't have the other but ...
1
vote
3answers
60 views

number of solution to the given equation.

a,b,c, are all non-negative integers such that a + b + c=100 and 1000a + 300b + 50c = 10000 How many such triplets are possible? i have tried to reduce ...
33
votes
4answers
2k views

A closed form of $\sum_{k=1}^\infty\frac{(-1)^{k+1}}{k!}\Gamma^2\left(\frac{k}{2}\right)$

I am looking for a closed form of the following series \begin{equation} \mathcal{I}=\sum_{k=1}^\infty\frac{(-1)^{k+1}}{k!}\Gamma^2\left(\frac{k}{2}\right) \end{equation} I have no idea how to ...
3
votes
2answers
2k views

Show that two spaces are not homeomorphic

Let $H=[-1,1]\times \{0\}$ and $V=\{0\}\times [-1,0)$ in the plane. Let $T=H \cup V$. Show that $T$ is not homeomorphic to the unit interval $I=[0,1]$. My idea for this problem is that , if we remove ...
2
votes
0answers
61 views

Finding whether a sum of numbers in a set generate another number

I have a set of numbers $\{a_1,\dots,a_n\}$ and another number $k$. I need to find whether sum of any combination of numbers in the set produces $k$. It can be $a_1 + a_2$ or $a_1 + a_2 + a_3 + a_7$. ...
0
votes
1answer
643 views

Proof of convergence of Kaprekar's Constant

I've tried googling this one a bit but nothing seems to come up, even though its considered to be a well known fact. Why does the kaprekar process of taking a 4 digit number: L, generating L' and L'' ...
1
vote
1answer
95 views

Alternative function definitions

If you go to the wikipedia page on the sine function or the log function you'll find a number of different definitions of these functions. I know that what defines a function are it's values, for ...
2
votes
4answers
273 views

Books that integrate physical reasoning with mathematical reasoning? mathematicians?

As the title says, can anyone help me to find any book that shows how physical reasoning using concepts from classical/quantum mechanics and physics in general can enlighten us about mathematical ...
2
votes
3answers
246 views

Lebesgue integral of $\chi_{\mathbb{Q}}: \mathbb{R} \rightarrow \mathbb{R}$

Suppose $(X, \mathfrak{A}, \mu)$ is a measure space. Let $\phi$ be a simple function with canonical representation $\sum^{k}_{n=1} a_{n} \chi_{E_{n}}$. I know we define the Lebesgue integral of $\phi$ ...
18
votes
5answers
1k views

Is there a way to calculate the area of this intersection of four disks without using an integral?

Is there anyway to calculate this area without using integral ?
0
votes
1answer
572 views

Finding the area of a triangle from vertices? Linear Algebra

I pretty much did this problem, but I failed to get the few last blanks where they ask the area. Its confusing, they say its half the volume of matrix (u v w) in the start of the question. which means ...
9
votes
2answers
925 views

Is there a characteristic property of quotient maps for smooth maps?

If $\pi\colon X\to Y$ is a quotient map, and $f\colon Y\to Z$ is a continuous map between topological spaces, then the characteristic property of the quotient map says $f$ is continuous iff $f\circ \...
0
votes
2answers
1k views

Finite dimensional subspaces of inner product spaces are orthogonally complemented

Can someone please explain the proof of the theorem below? I've been looking at it for hours and couldn't figure out how to prove it. Thanks! Suppose $U$ is a finite-dimensional subspace of $V$. ...
2
votes
2answers
141 views

Is there any subset of Complex numbers that is algebraically closed?

That any polynomial that is allowed to have coefficients from that subset has also a root in that subset
2
votes
5answers
167 views

How to show $n(n+1)(2n+1) \equiv 0 \pmod 6$?

I've been asked to show that: $n(n+1)(2n+1) \equiv 0 \pmod 6$ I found in a previous question that: $n(n+1)$ was divisible by $2$ and resulted in an even number e.g $n(n+1) \equiv 0 \pmod 2$ so I ...
0
votes
2answers
99 views

product of comprime numbers and UFD

It is well-known that if a product of coprime numbers is a perfect square, so are the numbers. The proof depends on fundamental theorem of arithmetic, and this implies that in a UFD, if ab is a ...

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