Sunday, October 3, 2021

Recent Questions - Mathematics Stack Exchange

Recent Questions - Mathematics Stack Exchange


please give answer for this question of complex number

Posted: 03 Oct 2021 07:45 PM PDT

3} Find a and b if

a+2b+2ai=4+6i

My Fibonacci Formula (with combinatorics)

Posted: 03 Oct 2021 07:36 PM PDT

Hi I originally posted this on MathOverflow, and was told to post it here. I'm a high school student, and was playing around with pascals triangle. and ended up taking (weird) diagonals. And I saw Fibonacci numbers, from the sum of the diagonals.

Pascall's triangle is just combinatorics. So I made a formula. I said that the nth Fibonacci number starts at 0. So F(0) = 1, F(1) = 1, F(2) = 2, F(3) = 3, F(4) = 5,...

My sum is this: $F(n) = \sum_{k=0}^{floor(n/2)} {n-k\choose k}$

This means each diagonal wil lhave floor(n/2)+1 terms.

So $F(0) = {0\choose 0} = 1$ $F(1) = {1\choose 0} = 1$ $F(2) = {2\choose 0} + {1\choose 1} = 2$.....

I put together a python program that does this, and it seems to work.

Have I done something wrong?

What steps should I go through to prove it?

I don't know much maths stuff. Is this already a commonly known formula?

Thanks.

How to find the Fourier trigonometric series of $f(t)$?

Posted: 03 Oct 2021 07:35 PM PDT

I'm new to Fourier series, and in a section of Laplace transform I found this problem:

Determine the Fourier's trigonometric series of odd extension of the function $$f(t)=\left\lbrace -1\ \ if\ \ -\pi<t<0 \atop 1\ \ if \ \ 0<t<\pi \right.$$

Could anyone provide me some hints on what to calculate? and its relation to Laplace if any?

Convex function obtaining maximum at an interior point of a non-convex set

Posted: 03 Oct 2021 07:34 PM PDT

If a convex function obtains a maximum over a closed, bounded non-convex set, can we say for sure that the function is constant? I sort of "feel like" it must be the case but cannot come up with a formal reasoning. I know that if the set is convex, closed and bounded then the function obtains a maximum over the set on the boundary, and not sure if this fact will help here.

Understanding the claim: "...then for a sufficiently small $h\gt 0$, $f$ will be continuous on $[a,a+h]$ and differentiable on $(a,a+h)$"

Posted: 03 Oct 2021 07:32 PM PDT

There is a reoccurring claim that is made in several of my book's proofs. I would like to know if I am correctly identifying the derivation of this claim. It reads as follows:

Suppose that $f$ is continuous at $a$, and that $f'(x)$ exists for all $x$ in some interval containing $a$, except perhaps for $x=a$. Then, for a sufficiently small $h\gt 0$, $f$ will be continuous on $[a,a+h]$ and differentiable on $(a,a+h)$ (a similar assertion holds for sufficiently small $h \lt 0$).

The purpose of this conclusion, where continuity is defined with a closed interval and differentiability is defined with an open interval (involving the same end points), is to invoke this book's Mean Value Theorem, which is derived from this book's Rolle's Theorem.

Rolle's Theorem is written as:

If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, and $f(a)=f(b)$, then there is a number $x$ in $(a,b)$ such that $f'(x)=0$.


As to the initial claim that I am interested in, is this the correct justification for the claim? Or am I missing something:

Consider some interval $I$ that contains $a$. This interval could be opened, closed, or mixed where $c \lt a \lt d$.

  1. $(c,d)$

  2. $[c,d]$

  3. $[c,d)$

  4. $(c, d]$

For all $x\neq a$ in this interval, $f'(x)$ exists. Now, consider the closed interval $J \subset I$, defined as: $\left[a-\frac{|c-a|}{2},a+\frac{|d-a|}{2}\right]$. Because $J \subset I$, every element $x \neq a$ in $J$ is differentiable. This means that every element $x\neq a$ in $J$ is continuous. Further, if $J$, a closed interval, is differentiable, then every element $x \neq a$ in $J'$ , the open interval version of $J$...i.e. $J'=\left( a-\frac{|c-a|}{2},a+\frac{|d-a|}{2}\right)$...is also differentiable. Remembering that $f$ is continuous at $a$, we can then make the following two claims:

A: Every element in $\left[a,a+\frac{|d-a|}{2}\right]$ is continuous, and every element in $\left(a, a+\frac {|d-a|}{2}\right)$ is differentiable.

B: Every element in $\left[a-\frac{|c-a|}{2},a\right]$ is continuous, and every element in $\left(a-\frac {|c-a|}{2},a\right)$ is differentiable.

If we wanted to make these intervals symmetric, we could use $\min\left(\frac{|d-a|}{2},\frac{|c-a|}{2}\right)$ in our definition of these sub-intervals.

At any rate, A: will dictate the "sufficiently small $h \gt 0$" and B: will dictate the "sufficiently small $h\lt 0$".

Is this the basic idea?

Doubts trying to find the Jordan canonical form of a matrix?

Posted: 03 Oct 2021 07:31 PM PDT

I am reading Gelfand's Lectures in Linear Algebra. I have some questions here:

enter image description here

enter image description here

I am trying to understand how to construct the given vectors with an example:

$$A=\left( \begin{array}{ccc} -1 & -18 & -7 \\ 1 & -13 & -4 \\ -1 & 25 & 8 \\ \end{array} \right)$$

This matrix has a single eigenvector $(-5, -3, 7)$ and three repeated eigenvalues $\lambda_1,\lambda_2,\lambda_3=-2$.

  1. Questions: We have $e_1=f_1=g_1=(-5, -3, 7)$, right? Do we have $e_2,f_2,g_2$? I think the answer is "no".

Now I am trying to use this information with the following section of the book:

  1. Question: It's not very clear to me what I should do here. How do I obtain this Jordan block? I think we need to do some change of basis, but it's not clear what change of basis is that.

This question has more then one correct option. [ 2021 JEE QUES)

Posted: 03 Oct 2021 07:23 PM PDT

Let 𝑆1 = {(𝑖,𝑗,𝑘) ∶ 𝑖,𝑗,𝑘 ∈ {1,2,...,10}}, 𝑆2 ={(𝑖,𝑗)∶ 1≤𝑖<𝑗+2≤10, 𝑖,𝑗∈{1,2,...,10}}, 𝑆3 ={(𝑖,𝑗,𝑘,𝑙)∶ 1≤𝑖<𝑗<𝑘<𝑙, 𝑖,𝑗,𝑘,𝑙∈{1,2,...,10}}
and 𝑆4 = {(𝑖, 𝑗, 𝑘, 𝑙) ∶ 𝑖, 𝑗, 𝑘 and 𝑙 are distinct elements in {1,2, ... ,10}}.

If the total number of elements in the set 𝑆𝑟 is 𝑛𝑟, 𝑟 = 1,2,3,4, then which of the following statements is (are) TRUE ?

(A)𝑛1 =1000 (B)𝑛2 =44 (C) 𝑛3 =220 (D) 𝑛4 =420 12

Suppose $f$ is continuous on $(0,+\infty)$ and for every $x_0>0$, $f(nx_0)$ tends to $0$. Show $f(x)$ tends to $0$ as $x$ tends to $+\infty$.

Posted: 03 Oct 2021 07:43 PM PDT

I meet this problem when doing exercises in mathematical analysis. $f\in C(0,+\infty)$ and $\forall x_0>0,\lim\limits_{n\rightarrow +\infty}f(nx_0)=0,$ then it says that $\lim\limits_{x\rightarrow +\infty}f(x)=0$.

I try to show that $\forall \epsilon>0,\exists[a,b]\subseteq (0,+\infty) and\ N>0,$ $\forall x\in [a,b],k>N,|f(kx_0)|<\epsilon.$ If this was true, I will find $ny,y\in[a,b]$ that is sufficiently close to any large $x$, and $|f(x)|<|f(x)-f(ny)|+|f(ny)|<2\epsilon$ by continuity of $f$ and known condition, which means $\lim\limits_{x\rightarrow +\infty}f(x)=0$.

But I fail to prove the wanted condition, I am grateful to any idea on this problem.

Is the following inequality about the pullback of the Gauss map true?

Posted: 03 Oct 2021 07:21 PM PDT

Let $\omega$ be the area two form on the upper hemisphere and Poincaré's lemma tells that we can find a 1-form $\alpha$ such that $d\alpha=\omega$. Let $N$ be the Gauss map from a minimal surface $\Sigma$ with the second fundamental form $A$ and Gaussian curvature $K$. Then

(i) Why $|A|^2dArea=-2KdArea=2N^*\omega=2dN^*\alpha$? (I don't understand the second equality.)

(ii) Why there is a constant $C_\alpha$ so that $|N^*\alpha|\leq C_\alpha|dN|=C_\alpha|A|$?

e and Pi relation

Posted: 03 Oct 2021 07:41 PM PDT

We notice the number 0.99999... the closer to the number 1. Then the more accurate the math. So are we closer to the truth (the relationship between e and pi)?

$$\big(1+\sin(0.99999\pi)\big)^{\frac{1}{\sin(0.99999\pi)}}\approx0.99999e$$

enter image description here

Graph in Below

enter image description here

Methods for solving the functional equation $f(f(x))=2f^{-1}(x)$

Posted: 03 Oct 2021 07:19 PM PDT

My friend and I have been attempting to find possible solutions that satisfy the functional equation

\begin{equation*} f(f(x))=2f^{-1}(x) \end{equation*}

where $f^{-1}(x)$ denotes the inverse function of $f(x)$. Neither of us are overly familiar with functional equations and so would like some input as to whether (a.) this is actually a valid functional equation in the first place and, if so, (b.) what are some paths we might go down in search of said solutions?

When can we apply Lebesgue Dominated theorem on increasing sequence of compact sets?

Posted: 03 Oct 2021 07:32 PM PDT

Let $A$ be a open and bounded set in $\mathbb{R}^2$ and suppose we have an increasing sequence of compact sets $\{B_n\}_{n\in \mathbb{N}}$ in $A$ such that $A = \cup_{n\in \mathbb{N}}B_n$. My question when can we say that

$\int_A f(x)dx = \lim_{n\rightarrow \infty }\int_{B_n}f(x)dx$ ? where $f\in L^2(A)$ (i.e. $\int_A{|f(x)|^2dx}< \infty)$ and $f$ is differentiable in $A$.

I tried to apply Lebesgue Dominated Convergence Theorem and Monotone Convergence Theorem to prove this but these two theorems work on the same domain and dominated sequence of functions or monotone sequence of functions.

Stochastic Product Rule Example

Posted: 03 Oct 2021 07:21 PM PDT

When given a function I wanted to know the how the application relates to: $Y_t = X_tdY_t + Y_tdX_t + (dX_t)(dY_t)$

(I am okay with using the 2 variable 2 equation function form).

For example consider: $Y_t = X_t e^{-rt}$

The answer supplied in the text is $dYt=d(Xt e^{-rt}) = e^{-rt}(dX_t) - re^{-rt}(X_tdt)$.

I'm assuming you break the initial $Yt$ down into '$Y_t$' and $X_t$ components. ie $X_t = X_t$ and $Yt=e^{-rt}$ to fit the equation.

This would match the first part $e^{-rt}(dX_t)$ as being $(Y_t)(dX_t)$. But I'm not sure were the $X_tdt$ came from in the second part.

(Im assuming since the process is deterministic $\sigma x =0$ to get rid of the $(dX_t)(dY_t)$

Any help would be really appreciated

Evaluating on exponenetial functions

Posted: 03 Oct 2021 07:19 PM PDT

I m stuck on a question on evaluation on exponential functions.

The question is as follows:

Define $x(a) = (e^{ma})(\sin a)$ and $y(a) = (e^{ma})(\cos a)$. Where $m = 0.005$.

Evaluate $(x(a))^2 + (y(a))^2$.

My attempt:

After substituting $m = 0.005$ into $x(a)$ and $y(a)$, I got $(x(a))^2 + (y(a))^2 = e^{(a/100)}$.

I can't go further. But the question is asking for evaluate, I am not sure if I need to find some exact numerical answers to the question or not. But without further information, i.e. the value of a, I am not sure how to find the exact numerical answers.

Any help will be appreciated! Thank you!

Solving an implicit equation to find the exponential generating function of a specific sequence

Posted: 03 Oct 2021 07:40 PM PDT

Suppose we have a set $A$ with $n$ elements. First we partition $A$ in at least two blocks. Then, we partition each block in the previous partition that is not of size 1 in at least two blocks. We continue like this until we have a partition of $A$ with only blocks of size 1. We call this procedure a total partition of $A$.

For example, the following is a total partition of the set $A=[7]$.

$$ \{\{1,2\},\{3,4,5\},\{6,7\}\}\implies\{\{1\},\{2\},\{3\},\{4,5\},\{6\},\{7\}\} \\ \implies\{\{1\},\{2\},\{3\},\{4\},\{5\},\{6\},\{7\}\} $$

Let $a_n$ be the number of total partitions of a set with $n$ elements. I have to find the exponential generating function of the sequence $\{a_n\}$, which I will denote by $G(x)$. To do this, I first had to find an implicit equation which is satisfied by $G(x)$. I found the following equation:

$$ G(x)=x+e^{G(x)}-G(x)-1 $$

which details I will skip since it is a quite long explanation. Futhermore I am certain that this equation is right, since my professor told me it was right, and because later I have to prove that the inverse function to $G(x)$ is $2x-e^x+1$, which is equivalent to the implicit equation I already found.

From that implicit equation, I'm guessing I have to solve for $G(x)$ and then write somethings that would appear on the equation as power series in order to obtain the explicit expresion for $G(x)$. I'm thinking this since all of the examples I have seen where generating functions are used to solve recurrence relations use this method, or a similar one. However, I don't know how to solve for $G(x)$ here, any help you could give me is appreciated.

Asymptotic Normality of Linear Form of Asymptotic Normal Components

Posted: 03 Oct 2021 07:35 PM PDT

Suppose $A_{n}$ and $B_{n}$ are such that $A_{n}\stackrel{d}{\rightarrow}A$ and $B_{n}\stackrel{d}{\rightarrow}B$ where $A\sim N(\mu_{A},\sigma^2_{A})$ and $B\sim N(\mu_{B},\sigma^2_{B})$ respectively, where $\sigma^2_{A}>0$ and $\sigma^2_{B}>0$. Suppose further $$ Cov(A_{n},B_{n})\rightarrow c$$where $c> -(\sigma^2_{A}+\sigma^2_{B})/2$. Is it sufficient to conclude $$C_{n}=A_{n}+B_{n}\stackrel{d}{\rightarrow} C$$ where $C\sim N(\mu_{C},\sigma^2_{C})$ where $\mu_{C}=\mu_{A}+\mu_{B}$ and $\sigma^2_{C}=\sigma^2_{A}+2c+\sigma^2_{B}$ ?

Dirichlet problem and its gradient norm

Posted: 03 Oct 2021 07:41 PM PDT

Let $\Omega \subset \mathbb{R}^2$ de a domain bounded and $v \in C^2(\Omega) \cap C^1(\overline{\Omega})$ solution of $\Delta v = -2$ in $\Omega$ and $v = 0$ on $\partial \Omega$. Show that $u = |\nabla v|^2$ reaches its maximum in $\partial \Omega$.

Motivation: I'm curious to know how to solve it, because if we think about the line case, when taking a function $f$ in an open interval such that $f''< 0$, we have that this function is concave and, consequently, $f'$ is non-increasing. Hence, the maximum of $f'$ is at the boundary of this interval.

how many points inside rectangle so distance is $0<d\leq \sqrt{2}$.

Posted: 03 Oct 2021 07:22 PM PDT

Given a rectangle with dimensions $m$ inches and $n$ inches. How many points inside that rectangle have to be chosen to make sure that two of them have distance $d$ with $0 < d \leq \sqrt{2}$?

Here is my attempt: We want to divide the rectangle into "whole" unit squares each of length 1 and the remaining will be rectangles. We know we have $a = \lfloor m \rfloor \cdot \lfloor n \rfloor$ many unit squares and $b = \lceil m -\lfloor m\rfloor\rceil \cdot \lceil n -\lfloor n\rfloor\rceil$ many rectangles. (EDIT) Then choose $a + b + 1$ many points.

Feedback would be greatly appreciated.

Anti commutativity of multicovectors

Posted: 03 Oct 2021 07:17 PM PDT

Let $V$ be a finite dimensional vector space. Let $ \omega \in \Lambda^k(V^*) $ and $\eta \in \Lambda^l(V^*)$. Then

$\omega \wedge \eta = (-1)^{kl} \eta \wedge \omega$

I am trying to understand the reason for $ (-1) ^{kl} $ and i have understood upto the following

If $I$ and $J$ are increasing multi indices then ,

$ \epsilon^I \wedge \epsilon^J = sign \tau \epsilon^J \wedge \epsilon^I $

where $ \tau $ is the permutation that sends $IJ$ to $JI$

Then $ sign \tau= (-1)^{kl} $ is what I cannot understand.

I tried working out in the simple case of $k=2$ and $l= 3$

Then the number of transpositions required for (1,2,3,4,5) to (3,4,5,1,2) is only 4 and the result of each transposition is given below.

(4,2,3,1,5) (4,5,3,1,2) (3,5,4,1,2) (3,4,5,1,2)

There were only 4 transpositions required, while according to the text, it must require $kl=6$ transpositions. Where am i going wrong?

The $H_{n-1}$ of the connected sum of two closed, non-orientable manifolds

Posted: 03 Oct 2021 07:41 PM PDT

I am attempting to show that the connected sum $M_1 \# M_2$ of two closed, non-orientable, connected $n$-manifolds has $H_{n-1} \cong H_{n-1}(M_1)\oplus H_{n-1}(M_2)$, IF one swaps a $\mathbb{Z}_2$ summand for a $\mathbb{Z}$. This is the last part of exercise 3.3.6 of Hatcher:

enter image description here

His hint is to use Euler characteristics. But, I do not see how they enter into this problem. I know that $M_1 \# M_2$ is non-orientable, thus its $H_{n-1}$ has torsion part $\mathbb{Z}_2$. I also know that the Euler characteristic of an odd-dimensional, non-orientable manifold is zero. But how can I use the Euler characteristic in this case to say something about the homology?

Interpolating power series?

Posted: 03 Oct 2021 07:22 PM PDT

Is there a general algorithm for finding a power series $$f(x) = \sum_{n=0}^{\infty} a_nx^n$$

with certain properties (i.e. given values of $f(x)$) without knowing $f(x)$ in terms of $x$? (This rules out the possibility of using recursive order derivatives.)

Given a function $y=f(x)$ and the $(x,y)$ values $(1,p), (2,q), (3,r),...$ (without knowing any other values of $f(x)$), we need to find a sequence $a_n$ such that:

$\sum_{n=0}^{\infty} a_n=p$

$\sum_{n=0}^{\infty} a_n2^n=q$

$\sum_{n=0}^{\infty} a_n3^n=r$

$...$

Essentially, this would be similar to finding an interpolating polynomial, but rather for an infinite series instead, and the sequence $a_n$ must be non-zero for infinitely $n$.

Edit:

For a single series $\sum_{n=0}^{\infty} a_n=p$, we can construct a sequence of partial sums

$s_k = \sum_{n=0}^{k} a_n$

such that

$\lim_{n\rightarrow\infty} s_k = p$

The problem, of course is to generalize this for multiple values $f(x)=p,q,r,...$

$s_k(x) = \sum_{n=0}^{k} a_nx^n$

such that

$\lim_{n\rightarrow\infty} s_k(x) = f(x)$

For example, given $x=2$ and a series that converges to $-1$, one solution is to use a geometric series:

$$-2 + \sum_{n=1}^{\infty} \frac{x^n}{4^n} = -2 + \frac{x}{4} + \frac{x^2}{16} + \frac{x^3}{64} + ...$$

However, add in another random point, say $(3,7)$ and the above series doesn't work anymore. In fact, I don't think there is even a geometric series solution for those two points.


An approach towards a solution:

So here, we want to solve the example with points $(2,-1)$, and $(3,7)$.

Basically, we need some high degree polynomials whose values are "close enough" to the convergence values. Provided the series above for only $(2,-1)$, I was able to find a degree $8$ polynomial (with rational coefficients) which has a close approximation to the desired convergence values:

$$f(x) = -2 + \frac{x}{4} + \frac{x^2}{16} + \frac{x^4}{4} - \frac{141x^5}{538} + \frac{700x^6}{10018} + \frac{x^7}{636} - \frac{x^8}{1911}$$

$f(2) ≈ -1.097370 ≈ -1$

$f(3) ≈ 6.820336 ≈ 7$

However, as this polynomial is finite degree, it cannot be a power series solution for the given set of points.

Are the following definitions of tuples and Tuples adequate?

Posted: 03 Oct 2021 07:37 PM PDT

Define, for $n\in\mathbf{N}$:

$n^{th}$-Tuples: $\emptyset $ is the $0^{th}$-Tuple $\langle \rangle$. x is an $\langle n+1\rangle^{th}$-Tuple just if there is a $y$ and a $z$ so that y is an $n^{th}$-Tuple & $x=\{\{y\},\{y,z\}\} $. $n^{th}$-Tuples are written as $\langle a_0,a_1, \ldots ,a_{n-1}\rangle$.

$n$-tuples: $\{\emptyset\}$ is the $0$-tuple (). $x$ is an $(n+1)$-tuple just if there is an $n$-tuple $y$ and a $z$ so that $x=y\cup\{\langle n,z\rangle\}$. $n$-tuples are written as $( a_0,a_1, \ldots ,a_{n-1})$.

Does the definition of n-tuples adequately satisfy the intended criteria that $(a_0,\ldots,a_{n-1})$ is $(b_0,\ldots,b_{n-1})$ just if $a_0=b_0 \wedge\ldots \wedge a_{n-1}=b_{n-1}$, and that the empty function $\emptyset$ is a member of all $n$-tuples?

Measure-theoretic probability, push-forward measure

Posted: 03 Oct 2021 07:33 PM PDT

I cannot make sense of the definition of the push-forward measure. According to my script, the domain of $\mathbb{P}_{X}$ is $\left\{ A\subset\mathbb{R}:X^{-1}\left(A\right)\in\mathcal{F}\right\} $ where $\mathcal{F}$ is the $\sigma$-algebra.

Here is what I don't get:

Since $\mathcal{F}$ is a $\sigma$-algebra, it must contain the outcome space. Since it must be closed under complementation, it must contain the empty set.

Let's say I have a set $A$ with values that are not in the range of $X$. Then, $X^{-1}(A)$ is the empty set. Since the empty set is in $\mathcal{F}$, that $A$ is in the domain of $\mathbb{P}_{X}$. So basically anything is in the domain. Where am I wrong?

Question about multiplying a rank $1$ density matrix with a positive semidefinite matrix

Posted: 03 Oct 2021 07:44 PM PDT

Let $\rho$ be a density matrix (positive semidefinite and trace $1$), with its rank being $1$ such that \begin{equation} \rho = v v^{*}, \end{equation} where $v$ is a $n \times 1$ unit vector.

Let $M$ be a positive semidefinite matrix such that all its eigenvalues are between $0$ and $1$.

I am trying to see whether the following two inequalities are correct: \begin{equation} \text{Tr}\left(M^{2} \rho\right) \leq \text{Tr}\left(M \rho\right). \end{equation}

\begin{equation} || M v|| \leq \text{Tr}\left(M \rho\right), \end{equation} where $||\cdot||$ is the $2$-norm of a vector.


The statement is obviously true when $M$ is a projector. But what about more general $M$? Note that if the first inequality is true, it implies the second one.

Area of a Trapezoid with Perpendicular Diagonals and Altitude [closed]

Posted: 03 Oct 2021 07:28 PM PDT

In a trapezoid with perpendicular diagonals, the length of a diagonal is 5 and the length of the altitude of the trapezoid is 4. Find the area of the trapezoid.

enter image

I have tried using similar triangle properties and using pythagorean theorem but have found no solution. I tried using triangle BPC ~ triangle APD but didn't know where to go after that.

All Solutions of a Specific Matrix Pencil Problem

Posted: 03 Oct 2021 07:27 PM PDT

I am trying to obtain all solutions $\mu, \lambda, x$ of following problem:

$(\mu A+\lambda B)x=0,$

where:

$A= \begin{bmatrix}1&0\\0&1\\0&0\end{bmatrix}$,

$B= \begin{bmatrix}0&0\\1&0\\0&1\end{bmatrix}$.

I don't seem to think of any except the trivial solution $\mu=0, \lambda=0, x=$ arbitrary.

Please let me know if you could guide.

Thanks.

Radial Function Problem [closed]

Posted: 03 Oct 2021 07:32 PM PDT

Let $n \geq 3$ and suppose that $u$ is a positive, non-trivial, radial, harmonic function on $\mathbb{R}^n \setminus \{0\}$. Prove that there are $a,b \geq 0$ such that $$u(x) = \dfrac{a}{|x|^{n-2}} + b.$$

Proving that a divisibility map is an isomorphism

Posted: 03 Oct 2021 07:38 PM PDT

$n=p_1^{k_1}p_2^{k_2}\dots p_m^{k_m}$ is the prime factorisation on a natural number. With this I want to prove that the map $f:C_{k_1} \bullet\dots\bullet C_{k_m} \to \mathrm{div}(n)$ is an isomorphism. I figured that this map is defined by $f(x_1,x_2,...,x_m)=p_1^{k_1}p_2^{k_2}\dots p_m^{k_m}$ I assume that $p_i$ are such that $p_1<p_2<...<p_m$ and from the fundamental theorem of arithmetic I know that every $n$ can be written uniquely as the product of prime numbers. The way I approach this proof is by wanting to prove that f is bijective which I think I can do by proving f is surjective and injective, however I find myself stuck at this point. May I please ask some assistance with this?

Second-countable, analytic, completely Baire spaces

Posted: 03 Oct 2021 07:39 PM PDT

First a few definitions. A topological space $(X,\tau)$ is said to be:

  • a Polish space if it is separable and completely metrizable.
  • Analytic, if there exists a surjective continuous map from a (Borel subset of) Polish space to the space $X$.
  • Completely Baire (some say "hereditarily Baire"), if each closed subspace is Baire

Now my question is: do we have a counterexample of a non-Polish space that is analytic, second-countable, and completely Baire? Can we say at least that the existence of such counterexample is consistent with ZFC?

Thanks

Prove that $\sum_{r=1}^n \frac 1{r}\binom{n}{r} = \sum_{r=1}^n \frac 1{r}(2^r - 1)$

Posted: 03 Oct 2021 07:27 PM PDT

Let $n$ be a nonnegative integer. Prove that $\sum\limits_{r=1}^n \dfrac 1{r}\dbinom{n}{r} = \sum\limits_{r=1}^n \dfrac 1{r}(2^r - 1)$.

One thing I have tried is to represent both $\binom{n}{r}$ and $2^r$ as sums of binomial coefficients, i.e. $\sum \binom{i}{r-1}$ and $\sum \binom{r}{i}$ respectively, but it does not seem to be helpful. I have also tried to use binomial identities but I do not see how they can be applied to the problem.

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