Mathematical Analysis generally, including Real Analysis, Harmonic Analysis, Complex Variable Theory, the Calculus of Variations, Measure Theory, and Non-Standard Analysis.

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4
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1answer
84 views

Is any of the sets a subset of a union of other sets?

I have eleven sets, all of them are subsets of $X:=\{(a,b,c,d)\in[-1,1]^4: a\le b,\text{ and } c\le d\}$: $$\begin{align*} A_1&:=\{(a,b,c,d)\in X: b\ge 0,\ c\le a+b+d\}\\ ...
1
vote
0answers
128 views

Equi-convergence of a function family at a point?

The definition of uniform integrability of a family of $L^1$ functions is: If μ is a finite measure, a subset $K \subset L^1(\mu) $ is said to be uniformly integrable if $\lim_{c \to \infty} ...
2
votes
1answer
57 views

A mapping dominates another

I was wondering what the definitions of one mapping dominating another in some general settings are? A special case I inferred from Dominated Convergence Theorem is that: for mappings $f$ and $g$ ...
4
votes
1answer
175 views

how to construct a polynomial

I have a question here: A finite sequence of real numbers $c_1, c_2, \dots, c_{n−1}$ is called saw– like if we have $(−1)^k(c_k − c_{k+1}) \leq 0$ for all $k = 1, \dots , n − 2$ or if we have ...
1
vote
1answer
108 views

partitions of unity

I'm trying to prove that the function $f$ is of class $C^\infty$ as follows. Given any integer $n\geq 0$, define $f_n : \mathbb R \rightarrow \mathbb R$ by $$f_n(x) = \frac{e^{-1/x}}{x^n}, \quad ...
1
vote
2answers
129 views

Calculating an ugly integral

Let $A$ be the set in $\mathbb{R}^2$ defined by $$A = \left\{(x,y)\left| x \gt 1\text{ and }0\lt y\lt\frac{1}{x}\right.\right\}.$$ Calculate ...
1
vote
2answers
195 views

Suprema and Infima of subsets of $\mathbb R$.

Real Analysis Question (Suprema, Infima and $\mathbb R$) Question: Suppose $U$ is a non-empty subset of $\mathbb R$, bounded above, with supremum $s$. If $a$ is any number satisfying $a < s$, ...
2
votes
1answer
268 views

On the Hardy-Littlewood Maximal function in $L^{2}(\mathbb{R}^n)$

Let $f$ be in $L^{2}(\mathbb{R}^n)$ then $f^{*}\in L^{2}(\mathbb{R}^n)$? Is there any weaker result like $f^* \in L^2_{loc}(\mathbb{R}^n)$ or $ff^*\in L^{1}((\mathbb{R}^n)$? Notation: ...
1
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2answers
129 views

A minimum of three maximums

Let $a,b,c,d$ be constants in the interval $[-1,1]$. Let $A$ be the minimum of the following three numbers: $$\max\{2-a-c, |b-d|\},$$ $$\max\{2-b-c, 2-a-d, 2-b-d\},$$ $$\max\{2+b-c, 2-a+d, 2+b+d\}.$$ ...
1
vote
0answers
47 views

How to show existence of two functions satisfying certain conditions? [duplicate]

Possible Duplicate: Finding two functions (density) $g,f$ satisfying some conditions I've asked this board before if they knew of a clever way to construction two functions $f$ and $g$ ...
0
votes
1answer
74 views

Relation between $\lim_{n\to\infty}\int_{0}^{1}\left|f_{n}(x)-f(x)\right|dx=0$ and $\lim_{n\to\infty}\int_{0}^{1}\left|f_{n}(x)-f(x)\right|^2dx=0$

Let $f$ and $f_n$ for $n=1,2,\ldots,n$ be Riemann integrable real-valued functions defined on $[0,1]$. For each of the following statements, determine whether the statement is true or not and prove ...
3
votes
1answer
176 views

Construct a function that is differentiable only on the rationals.

In the spirit of having handy counter examples, is it possible to construct a function that is differentiable on all of the rational numbers and nowhere else? Similarly, if a function is holomorphic ...
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0answers
77 views

Trying to create an algorithm that takes into account multiple variables

I've hit a road block on how I can put what I have into a calculation that works. Simply I am trying to work out the "value" of each company that we have on my companies books. The information I want ...
2
votes
1answer
113 views

Regularity of the distance function

Let $\Omega \subseteq \mathbb{R}^N$ be open and bounded, and set: $$d(x):=\text{dist} (x,\partial \Omega) =\inf_{y\in \partial \Omega} |x-y|\;.$$ I would appreciate if somebody could verify my proof. ...
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0answers
210 views

How can I find the Laurent series of $f(z) = \ln(1+\exp(z))$ about its singularities?

I think my problem below can be solved by finding the Laurent series of $f(z) = \ln(1+\exp(z))$ about its points of singularities. (Any better suggestion is more than welcome!) How might I find such a ...
2
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0answers
72 views

Estimation of Lebesgue measure of some set.

Assume that a function $f: \mathbb{R^n} \rightarrow \mathbb{R^n}$ is of class $C^2$, convex if necessary, $x \in \mathbb{R^n}$, $r>0$ and $z \in B(x, \frac{r}{2})$ (where $B$-open ball in ...
1
vote
1answer
161 views

How to calculate $\int_{|z|=r}\ln(1-z)\,dz$ in dependence of $r\neq1$?

With the integration I mean one counter-clockwise turn around the origin, i.e. $$\int_{\phi=0}^{2\pi}\ln(1-re^{i\phi})ire^{i\phi}d\phi$$ For $r<1$, this is simply a contour integration on a ...
2
votes
2answers
120 views

Showing that a sequence converges (in metric space)

In $(\ell ^\infty,{\Vert .\Vert_\infty)}$, how would I show that $x_n=\left(\frac{n+1}{n},\frac{n+2}{2n},\frac{n+3}{3n}, ...\right)$ converges and how would I find the limit? I tried using the fact ...
2
votes
1answer
173 views

Radius of convergence composite function

If the Taylor series of 2 complex functions $f,g$ have radii of convergence $r_f, r_g$ respectively, does it follow that the radius of convergence of their composition has radius of convergence equal ...
2
votes
3answers
357 views

Combining Taylor expansions

How do you taylor expand the function $F(x)={x\over \ln(x+1)}$ using standard results? (I know that WA offers the answer, but I want to know how to get it myself.) I know that $\ln(x+1)=x-{x^2\over ...
5
votes
1answer
1k views

Show a $\sigma$-algebra contains the Borel sets : with $(a,\infty)$ or $(-\infty,b)$?

For a certain $\sigma$-algebra $A$ on the real line, I would like to show that it contains the Borel sets. I can show that $A$ contains the left and right half-line $(a,\infty)$ and $(-\infty,b)$ for ...
3
votes
1answer
153 views

How do I show that this metric space is not convex?

Denote $X$, the space of all sequences $\in$ $\mathbb R$. I have a metric $$d(x,y):=\sum_{n=1}^\infty 2^{-n}\frac{| x_n-y_n|}{1+| x_n-y_n|}$$ and $(X,d)$ is a metric space. How would I show that the ...
1
vote
1answer
105 views

Showing boundedness of metric space

Denote $X$, the space of all sequences $\in$ $\mathbb R$. I have a metric $$d(x,y):=\sum_{n=1}^\infty 2^{-n}\frac{| x_n-y_n|}{1+| x_n-y_n|}$$ If $(X,d)$ is a metric space and if $A$ is a subset of ...
4
votes
1answer
203 views

Polynomial interpolation

Let $P=[a,b]\times (c,d)$. Assume that we have given $n$ points $(x_1,y_1),...,(x_n,y_n)\in P$, such that $x_i\neq x_j$ for $i\neq j$; $i,j=1,...,n$. Does there exist a polynomial $f$ such that ...
7
votes
4answers
262 views

Modified Intermediate Theorem Implies Continuity… Counterexample to Question from Spivak

Question 13b of chapter 7 in Spivak's Calculus (which I've been slowly working through over the last few months) says this: 'Suppose that f satisfies the conclusion of the Intermediate Value Theorem, ...
2
votes
1answer
853 views

Show a function is continuous if and only if it is both upper and lower semi-continuous

Let $f$ be a real-valued function on $\mathbb{R}$. Show $f$ is continuous if and only if $f$ is both upper and lower semi-continuous, using the definition of continuity and semi-continuity based on ...
1
vote
1answer
73 views

Completeness of $(C[0,1],\|\cdot\|_p)$

For $p=2$, $(C[0,1],\|\cdot\|_{p})$ is not a complete metric space and its closure is $L^{p}[0,1]$? I am curious as to whether this is true for all $p<\infty$?
6
votes
1answer
314 views

Finding two functions (density) $g,f$ satisfying some conditions

Is there a clever way to find two density functions, $f$ and $g$, that satisfy the following conditions? $$\begin{align*} ...
1
vote
0answers
299 views

An exercise in measure theory

In the treatise of analysis volume 2 second edition of Dieudonné, there is an exercise which I found not easy because of a wrong & misleading hint. Can somebody tell me if I missed something ? ...
2
votes
1answer
205 views

How to evaluate $\int \limits_{-\infty}^{\infty}\frac{e^{-|x|}}{|1-\sin x|^{\frac{1}{4}}} \,dx$?

$$\int \limits_{-\infty}^{\infty}\frac{e^{-|x|}}{|1-\sin x|^{\frac{1}{4}}} \,dx$$ Any advice and comments will be appreciated
4
votes
1answer
315 views

Integration over a bounded set

Let $f,g: S\to \mathbb{R}$. If we assume $f$ and $g$ are integrable over $S$, then I'm trying to show: If $f$ and $g$ agree except on a set of measure zero, then $\int_S f=\int_S g$. Also, how ...
2
votes
1answer
155 views

An approach to Borel-Cantelli for the $l^p$

Let $\mu$ be a non-negative measure and $\{E_{k}\}$ a sequence such that $\sum \mu(E_k)^p<\infty $ then show that $F=\lim \ \sup E_{k}=\cap_{k=1}^{\infty}\cup_{n\geq k}E_n$ has $\mu$ measure zero.
2
votes
0answers
171 views

When can I decompose a random variable $Y=X'-X''$?

I am wondering if I can find a decomposition of $Y$ that is absolutely continuous nto two i.i.d. random variables $X'$ and $X''$ such that $Y=X'-X''$, where $X'$ is also ...
3
votes
1answer
692 views

speed of convergence to infinity

Lets take for example $\lim_{x\rightarrow\infty} \log(x)$, from a mathematical point of view this is $+\infty$, but from a logical point of view it's clear that $x$ converges to $+\infty$ much more ...
1
vote
1answer
464 views

set dense in $\mathbb{R}$ and its containing set; also its intersection

I'm using the definition (without much topology) that a set $S$ of real numbers is dense in $\mathbb{R}$ if $S \cap (a,b) \neq \varnothing$ for all $a,b \in \mathbb{R}$ and $a<b$. My questions: ...
0
votes
2answers
104 views

Sequence of Functions

I'm having difficulty determining where this sequence of functions $\displaystyle f(n,x)=\frac{x^n}{(1+x^n)}$ converges, and whether it converges uniformly. Thanks.
2
votes
1answer
106 views

Where is the mistake? Two different methods to compute directional derivative lead to do two different results!

Problem: Let $$f(x,y) = \begin{cases} \frac{xy\sin x}{x^{2}+y^{2}}, & (x,y) \neq (0,0) \cr 0, & (x,y) = (0,0) \end{cases}$$ Let $u=\left ( u_{1},u_{2} \right )$ be a unit vector. ...
13
votes
1answer
510 views

Is there a 'far' irrational number?

I recently learned of so-called 'far' numbers at a talk. In the talk, it was proven that there is a dense subset of the interval $[0,1]$ of far numbers (however, far numbers were only a minor point of ...
6
votes
2answers
412 views

How much of Stirling is in Stirling's formula?

This is a naive question about history. My understanding is that Stirling's formula or something trivially equivalent to it first appeared in an early edition of Abraham de Moivre's book The Doctrine ...
1
vote
1answer
720 views

What is the total derivative of a partial derivative of a function, whos total derivative depends on that function?

I have a function $c(t,\lambda)$ with $$\frac{\text dc}{\text dt}=f(c,\lambda),$$ or $$\left(\frac{\text dc}{\text dt}\right)(t,\lambda)=f(c(t,\lambda),\lambda),$$ How to compute ...
0
votes
1answer
309 views

Taylor's theorem-multivariable

We are given a function: $f:\mathbb{R}^{n}\rightarrow \mathbb{R}$ be a $C^{2}$ function such that: $D^{2}f\left ( x \right )\left ( t,t \right )> 0$, $\forall x\in \mathbb{R}^{n}, \forall t\in ...
1
vote
1answer
220 views

differentiation-chain rule- function of two variables

Problem Let $f\left ( x,y \right )$ be a differentiable function on $\mathbb{R}^{2}$. Find a formula for $\frac{d}{dt}f\left ( t,t^{2} \right )\mid _{t=1}$ in terms of the partial derivatives: ...
2
votes
2answers
359 views

Why solve polynomial equations?

Most people learn in linear algebra that its possible to calculate the eigenvalues of a matrix by finding the roots of its characteristic polynomial. However, this method is actually very slow, and ...
3
votes
0answers
287 views

Shrinking Map and Fixed Point via Iteration Method

Let $T$ be a map from a compact metric space $X$ into itself satisfying $ d(Tx,Ty) < d(x,y)$ for all distinct $x,y$ in $X$. It is true that $T$ has a unique fixed point. Fix $x_0 \in X$ any point, ...
2
votes
2answers
179 views

Looking for a proof that the number of non-zero derivatives of a polynomial $f(x)$ is equal to the number of its roots

I can see why this works for a root $p$ with multiplicity $k\geq 1$, when $f(x)=(x-p)^k$. But, why is that true if $f(x)=(x-x_1)(x-x_2)\cdots(x-x_n)$ has distinct roots $x_1\neq x_2\neq \cdots \neq ...
2
votes
2answers
325 views

Möbius transform which completely preserves circles (how to map a circle?)

(remmert theory of complex function) I am trying to solve this exercise, however it seems impossible because I don't know how to map a circle, and I will be very thankful if somebody points out to ...
3
votes
1answer
130 views

Handling Cross ratios ( Fractional linear transformations )

According to remmert there is a relationship between the crossratios: $$C(z,u,v,w) = \frac{(z-v)(u-w)}{(z-w)(u-v)} \text{ and } C(z,v,u,w)= \frac{(z-u)(v-w)}{(z-w)(v-u)}$$ where $z,u,v,w \in ...
2
votes
2answers
171 views

differentiability/partial derivatives

This is a problem from a previous graduate preliminary exam in multivariable analysis/calculus that I am trying to solve for my own practice: Problem: Let $f:\mathbb{R}^{2}\rightarrow \mathbb{R}$ be ...
2
votes
2answers
682 views

Is the graph $G_f=\{(x,f(x)) \in X \times Y\ : x \in X \}$ a closed subset of $X \times Y$?

I'm thinking about Hausdorff spaces, and how mappings to Hausdorff spaces behave. Suppose I have an arbitrary (continuous) function $f:X \longrightarrow Y$, where $Y$ is a Hausdorff space (I think it ...
4
votes
2answers
282 views

Continuity of the Characteristic Function of a RV

Defining the Characteristic Function $ \quad \phi(t) := \mathbb{E} \left[ e^{itx} \right] $ for a random variable with distribution function $F(x)$ in order to show it is uniformly continuous I say ...