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0
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1answer
31 views

Approximate root of $\alpha x - \beta y$ over $\mathbb Z$ except origin

Consider the polynomial $ f(x,y) = \alpha x^2 - \beta y^2 $ Prove or disprove: For any choice of $\alpha, \beta \in \mathbb R_0^+$, the polynomial $f$ gets arbitrarily close to $0$ over $\mathbb Z ...
1
vote
1answer
38 views

Regularizing conditionally convergent integrals

For functions $f:\mathbb{R}\to \mathbb{R}$ the definition of a convergent improper integral is straightforward: the integral $\int_\mathbb{R} f(x)dx$ converges iff $$\lim_{(a,b)\to (-\infty, ...
1
vote
0answers
38 views

Lebesgue integrable of composite functions

Let $a, b\in R$ be such that $a<b$ and $u\in L([a, +\infty);R)$. Prove that $$ v(t)=\frac{b-a}{(b-t)^2}u\left(\frac{b-a}{b-t}+a-1\right)\in L([a, b];R). $$ Thank you for all helping and guidance. ...
0
votes
3answers
71 views

How to Show that $S=\{(x,y)\in R^2:x>y^2\}$ is open

Show that $S=\{(x,y)\in R^2:x>y^2\}$ is open quite simple one. We need to choose $\epsilon$ for open balls: $D((x_0,y_o),\epsilon)\subset S$ ,$\forall x_o,y_o\in S$. we can take $\epsilon$ as the ...
0
votes
1answer
34 views

How to show $A=\{(x,y)\in R^2:4x^2+9y^2=36\}$ is path connected and compact?

let $A=\{(x,y)\in R^2:4x^2+9y^2=36\}$ . Show that A is path connected and compact. my attempt: since $\frac {x^2}{9}+\frac{y^2}{4}=1$ is elips. A is bounded and closed. so is compact. (by heine ...
0
votes
1answer
73 views

How to show “ If A and B connected, is $A\cup B$ connected”?

If A and B connected, is $A\cup B$ connected? or give a contrary example. I'd say no because when we take A=[1,2], B=[3,4]. this closed intervals are connected. but when we take $U=]\frac 12,\frac ...
15
votes
4answers
653 views

Is there an elementary proof for Euler's product for Sine?

I've been looking at proofs of this equation: $$\displaystyle \frac {\sin \pi x} {\pi x} = \displaystyle \prod_{k \mathop = 1}^\infty \left({1 - \frac{x^2}{k^2} }\right)$$ All the proofs seem to ...
0
votes
1answer
36 views

Polar of revolution cone

Let $s\in\mathbb{R}^n, \|s\|=1$ and $\theta\in[0,\frac{\pi}{2}]$. Consider the cone $$ K=\{x\in\mathbb{R}^n: \langle s, x\rangle\geq \|x\|\cos\theta\}. $$ and $$ K^*=\{x\in\mathbb{R}^n:\langle x, ...
1
vote
1answer
189 views

Is there a bijective map from the open interval $(0,1)$ to $\mathbb{R}^2$?

I couldn't find a bijective map from the open interval $(0,1)$ to $\mathbb{R}^2$. Is there any example?
3
votes
2answers
214 views

Show that the derivatives of a $C^1$ function vanish a.e. on the inverse image of a null set

Let $A \subset \mathbb{R}$ be such that $\lambda(A) = 0$, where $\lambda$ is the Lebesgue measure on the real line. Let $\Omega \subset \mathbb{R}^N$ be an open set and let $u \colon \Omega \to ...
1
vote
0answers
32 views

Interpolating $G(1)=\sum_{a=1}^{\infty} \frac{1}{a^{a}}$, $G(2) = \sum_{a=1}^{\infty} \sum_{b=1}^{\infty} \frac{1}{(ab)^{ab}}$ on $\mathbb{C}$

Given that: $$ G(1) =\sum_{a=1}^{\infty} \frac{1}{a^{a}} $$ (this is just the Sophomore's dream series, but the rest are not) $$ G(2) = \sum_{a=1}^{\infty} \sum_{b=1}^{\infty} \frac{1}{(ab)^{ab}} ...
2
votes
2answers
183 views

Vector of reduction

Let $f:\mathbb{R}^2\rightarrow \mathbb{R}$ be a locally Lipschitz continuous and $\bar{x}\in\mathbb{R}^2$. Put $$ S(x):=\left\{x^*\in\mathbb{R}^2: \liminf_{u\rightarrow x}\frac{f(u)-f(x)-\langle ...
5
votes
0answers
84 views

Weierstrass product expression for Klein's j-invariant

The first sentence of @ccorn's answer to a previous question of mine was: “Because of the modular symmetries of $j(\tau)$, the zeros of $j(\tau)$ are precisely the ...
2
votes
0answers
50 views

Problem reduced to analyzing solutions of a family of nonlinear systems of equations

This was posted on mathoverflow about two weeks ago and I got no response so I'm asking here in case anyone has any ideas. Original post is here. I was able to reduce a research problem relating to ...
0
votes
1answer
230 views

Continuous functions as regulated functions: a property.

In Differential and Integral by Paul Lorenzen (1971) pag. 148, I read ... every continuous function is trivially approximable by step functions that have no jump at a given arbitrary point .... All ...
2
votes
0answers
82 views

analytic forms of $\frac{\sinh(2\pi/7)}{\sinh^{2}(3\pi/7)} - \frac{\sinh(\pi/7)}{\sinh^{2}(2\pi/7)} + \frac{\sinh(3\pi/7)}{\sinh^{2}(\pi/7)}$

On page 183 of Berndt's Ramanujan's Notebooks Vol. 4, eq. 32.34 reads: $$ \frac{\sin(2\pi/7)} {\sin^{2}(3\pi/7)} - \frac{\sin(\pi/7)}{\sin^{2}(2\pi/7)} + \frac{\sin(3\pi/7)} {\sin^{2}(\pi/7)} = ...
2
votes
3answers
187 views

Limit on a spiral

I was thinking about limits of functions along various spirals and this one stumped me a bit. The limit that needs to be found is ultimately: $$\lim_{\varphi\to\infty} \coth\varphi\csc\varphi$$ ...
3
votes
1answer
43 views

finding rational complex numbers in a disk with least denominators

Suppose that I have a disk of radius $r$ around some complex $\alpha\in\mathbb{C}$: How would one find a complex number $g$ in that disk besides $\alpha$ such that $\mathrm{Re}(g)\in\mathbb{Q}$ ...
1
vote
2answers
125 views

Integral defined as a limit using regular partitions

Definition. Given a function $f$ defined on $[a,b]$, let $$\xi_k \in [x_{k-1},x_k],\quad k=1,\ldots,n$$ where $$ x_k=a+k\frac{b-a}n, \quad k=0,\ldots,n \; .$$ One says that $f$ is integrable on ...
8
votes
1answer
194 views

series for $\pi$ which correspond to apollonian gaskets or hyperbolic tilings of the unit disk

Consider the two partitions of the unit disk in $\mathbb{R}^{2}$, the first an Apollonian gasket and the second is the $\{7,3\}$ hyperbolic tiling: Since the unit disk has radius $1$, both of these ...
1
vote
2answers
421 views

A less known definition of the definite integral of a continuous function

The definite integral of a continuous function can be defined using the bounded monotone sequence property: see Osgood's Functions of Real Variables, p.110. (link to full book) (screenshots: page ...
16
votes
1answer
505 views

A curious theorem by Peano

Let $f$ be defined on $[a,b]$ and there differentiable. Show that for every $ \epsilon>0 $ there exists a partition $\, a=a_0<a_1<...<a_n=b \,$ of $ \,[a,b] \,$ so that $$\left|\frac ...
2
votes
1answer
281 views

The range of the derivative of a differentiable function

I read somewhere that, given a function $f$ differentiable on $[a,b]$, the range of $f'$ can be (1) a closed interval or (2) an open interval or (3) a half-open interval or (4) an unbounded interval ...
5
votes
2answers
504 views

The Constant Function Theorem first of all $\,$?

I quote Thomas W.Tucker $\,$ "... By the way, I view the Constant Function Theorem as even more basic than the IFT. It would be nice to use it as our theoretical cornerstone, but I know of no way to ...
2
votes
0answers
310 views

These unknown uniformly differentiable functions

Let $f$ be defined on $[a,b]$ and there uniformly differentiable ($\,$the $\delta$ in the definition of derivative is independent of the point). Given $\epsilon>0$, choose a partition $P \, : \, ...
3
votes
1answer
68 views

The limit of the integral

How to prove that if $f \in C[0,2 \pi]$, then $$ 2\pi \lim_{n \to \infty} \int_0^{2 \pi} |\sin(nx)-f(x)| \, dx = \int_0^{2 \pi} \int_0^{2 \pi} |\sin(y)-f(x)| \,dx \, dy\ ? $$
0
votes
1answer
89 views

continuity with contraction

In ($\mathbb R, d(x,y)=|x-y|$), $f:\mathbb R \to \mathbb R$ is a contraction,that's for all $x,y \in \mathbb R$, there exists a constant $A$ between 0 and 1 such that $|f(x)-f(y)| \leq A|x-y|$. Then ...
2
votes
0answers
71 views

convergence in metric space

Let $C[-1, 1]$ be the space of continuous functions equipped with the metric $(f, g) = \displaystyle\max_{x \in [-1, 1]} |f(x)-g(x)|$. Consider the sequence $(f_n)$ of functions $f_n : [-1, 1] \to ...
1
vote
1answer
85 views

Cauchy in metric space

Let $C[-1,1]$ be the space of continuous functions with metric $$\rho(f,g)=\max\{|f(x)-g(x)|: x\in [-1,1]\}\;.$$ Then the sequence of functions $(f_n) :[-1,1] \to \Bbb R$ defined as ...
3
votes
2answers
63 views

contraction metric space

"Let $0 < a < 1$ and $f(x) = (x^2 + a)/2$. Show that $f : [0, a] \to [0, a]$ and that f is a contraction. Find the fixed point of $f$." For this problem, since there isn't any metric $d(x,y)$ ...
1
vote
1answer
185 views

determine whether a metric space is complete or not

How to decide if the metric spaces $((0,1)$, $d(x,y)=|x^2-y^2|)$ and $((-\frac{\pi}{2},\frac{\pi}{2})$, $d(x,y)=|\tan x-\tan y|)$ are complete or not. For the first metric, I let any cauchy sequence ...
4
votes
1answer
111 views

A dense subset in $\mathbb{R}^n$

I have some difficulties in the following question. I would like to thank for all comments and kind help. Let $S$ be a dense subset in $\mathbb{R}^n$. In this case, whether we can find a straight ...
3
votes
3answers
353 views

How to show that continuous functions between metric spaces agree on a closed set

Let $(X,d)$ and $(Y,d')$ be metric spaces, and let $D$ be a dense subset of $X$. Show that: If $f:X\to Y$ and $g:X\to Y$ be continuous, then the set $\{x\in X\mid f(x)=g(x)\}$ is closed.
1
vote
1answer
155 views

Property of strictly convex polynomial

I have some difficulties in the following problem. Thank you for all comments and helping. Let $f:\mathbb{R}^n\rightarrow \mathbb{R} (n\in \mathbb{N})$ be a polynomial. Suppose that $f$ is strictly ...
4
votes
2answers
103 views

Are the non-standard models of classical analysis the standard models of non-standard analysis?

I've never quite understood how the non-standard models of classical analysis related to the standard models of non-standard analysis. I know that non-standard analysis got its start with Robinson ...
2
votes
1answer
89 views

equivalent metric

Let $(X; d)$ and $(Y; d')$ be metric spaces, and let $f : X \to Y$ be continuous. Define $df (x; y) = d(x; y) + d'(f(x); f(y))$ for $x, y \in X$. Show that $df$ is a metric on $X$ that is equivalent ...
0
votes
0answers
34 views

Question on relations [duplicate]

-1 down vote favorite Define the relation on Q by [m,n]<[j,k] if and only if jn−mk belongs to N, j and m belong to Z, n and k belong to N. (a) Show that < is well defined, that is if ...
0
votes
1answer
94 views

Question about relation on rational numbers

Define the relation on $\mathbb{Q}$ by $$[m,n]<[j,k]$$ if and only if $jn-mk$ belongs to $\mathbb{N}$, $j$ and $m$ belong to $\mathbb{Z}$, $n$ and $k$ belong to $\mathbb{N}$. (a) Show that ...
0
votes
1answer
66 views

Expansion of $x^{-1/2}$ at $0$

Regard the function $f(x) = x^{-1/2}$ on the non-negative real line. The point $z=0$ is 'distinguished' because it is the boundary of the domain of $f$, and because $f$ has a pole there. It seems ...
2
votes
1answer
77 views

Directional differentiable property

I am stuck in constructing a function that is locally Lipschitz continuous at $x_0$ but it does not have directional differentiation at $x_0 $in any direction. Thank you for all help and comments.
10
votes
1answer
285 views

Is there a nice way to multiply power series the wrong way?

This is a question that comes out of a combinatorics question that is using generating functions. Let me define what I mean by multiplying power series the wrong way (although you may be able to ...
4
votes
1answer
78 views

A reference for some fact in analysis

I am looking for a reference for the following fact. Any hints would be appreciated. Suppose $(x_n), (y_n)\subset [0,1]$ are some sequences, $(a_n)$ is absolutely summable and for each $f\in C[0,1]$ ...
0
votes
1answer
192 views

Proof for Cluster Point properties

I have a proposition on the book Elementary Classical Analysis which states the following: Let $x_n$ be a sequence in $\Bbb R$ and let x $\in \Bbb R$. Then $x_n \to x$ iff every subsequence of $x_n$ ...
0
votes
3answers
94 views

Nonnegative function satisfying integral constraints

Find a real function $w(t)\in L_2[0,1]$ such that: $w(t)\geq 0 \quad \forall t\in [0,1];$ $\displaystyle\int_0^{s}w(t)dt\leq s \quad \forall s\in [0,1];$ $\displaystyle \int_0^1 w(t)dt\leq 2;$ ...
8
votes
3answers
292 views

Function $f$ which isn't smooth but $f^3$ is smooth

In Pugh's Real Mathematical Analysis there is an exercise, marked with three stars (which denotes that the author doesn't know the answer), whether there exist a nonsmooth function $f : \mathbb{R} \to ...
1
vote
4answers
164 views

Finding a differentiable function satisfying some given conditions

Finding a differentiable function $g:\mathbb{R}\rightarrow \mathbb{R}$ satisfying the following condiotions: $\displaystyle g(0)=0, g(1)=1, g(-1)=-1;$ $\displaystyle ...
26
votes
10answers
7k views

Is there a bijective map from $(0,1)$ to $\mathbb{R}$?

I couldn't find a bijective map from $(0,1)$ to $\mathbb{R}$. Is there any example?
3
votes
0answers
169 views

Fourier dimension of a measure restricted to an open set

Suppose that the measure $\mu$ on $\mathbb{R}^n$ has Fourier dimension $\beta$, which is to say that $\beta= \sup\left\{\gamma \leq n : |\hat{\mu}(x)| \leq C(1+|x|)^{-\gamma/2}\right\}$. The Fourier ...
1
vote
1answer
288 views

Two Concepts of Monotonicity

Let $K$ be a closed convex subset in $\mathbb{R}^n$ and $F: K\rightarrow \mathbb{R}^n$. We say that $F$ is strongly monotone on $K$ if there exists $\gamma>0$ such that $$ \left<F(y)-F(x), ...
2
votes
0answers
84 views

Constructing a mapping satisfying given conditions

Let $K=[0,1]\times [0,1]$. Find a continuous mapping $F:K\rightarrow \mathbb{R}^2$ satisfying: $\|F(x)-F(y)\|\leq \|x-y\| \quad\forall x,y\in K,$ There exists $\gamma>0$ such that for all $x,y\in ...