Mathematical analysis. Consider a more specific tag instead: (real-analysis), (complex-analysis), (functional-analysis), (fourier-analysis), (measure-theory), (calculus-of-variations), etc. For data analysis, use (data-analysis).

learn more… | top users | synonyms (1)

1
vote
0answers
58 views

Question in ergodic theory

Again the source is http://www.math.ucla.edu/~biskup/275c.1.13s/PDFs/HW1.pdf this time I'm looking at #6 the part that is left as an open-ended question. If $f \in L^1$ and $\phi$ is a measure ...
1
vote
1answer
141 views

Find local maxima of this quadratic function

How can I find local maxima of this quadratic function? $$f(x) = \sum _{i=1}^n -\frac{(z_i - x)_+^2}{2} - \left\{((\frac{(z_i - x)_+^2}{2})-(\frac{(y_i - x)_+^2}{2}) ) * c_i\right\} $$ which ...
1
vote
0answers
62 views

Homework questions in ergodic theory

Let $X_1, X_2, ...$ be iid. If $f: \mathbb{R}^\mathbb{N} \rightarrow \mathbb{R}$ is measurable wrt the product structure it's $L^1$ under the distribution measure induced by the $X_i$ then why is it ...
2
votes
2answers
74 views

Showing that $ \sum \limits_{m=1}^{n} b_m x_{m-n}~\to~ ab$ as $n~\to~\infty$

If $x_n ~\to ~a$ as $n~ \to~ \infty$ Does: $ \sum \limits_{m=1}^{n} b_m x_{n-m}~\to~ ab$ as $n~\to~\infty$? $b_m ~\geq~0$ and $ b~\equiv~ \sum \limits_{m=1}^{\infty} b_m < \infty$ My attempt: ...
0
votes
0answers
74 views

Green's function

Please can someone told me how to find the Green's function $G(t,x)$ of BVP : $$u'''(t)=0 , \quad t\in (0,1)$$ and BC : $$u(0)=u'(p)=\int_q^1 w(s)u''(s) ds =0 $$ where $\frac12 < p<q<1$ are ...
4
votes
1answer
100 views

The geometric interpretation [duplicate]

In the course of mathematical analysis, there was one problem that i excited to know more about it: What is the geometric interpretation of $$ \int_a^b f(x)\,d(\alpha(x)) $$ and $\alpha(x)$ is ...
2
votes
0answers
271 views

Proof on showing F(x,y) is continuous by $\epsilon - \delta$ definition

The task is as follows: Given: $$F(x,y) = \frac{xy(x^2 - y^2)}{x^2 + y^2}$$ Goal: Prove that $F(x,y)$ is continuous everywhere on the plane Here is my attempt so far: (1) By the ...
1
vote
0answers
146 views

Is zero a cluster point of $n\sin n$?

I have seen somewhere that if $0\le \alpha<1$, then zero is a cluster point of the sequence $n^\alpha \sin n, n=1,2,\cdots$. My question is what if $\alpha=1$? Or $\alpha>1$?
3
votes
1answer
976 views

Possible ways to do stability analysis of non-linear, three-dimensional Differential Equations

For example Lorenz system, $$ \frac{d}{dt}\begin{pmatrix} x\\ y\\ z \end{pmatrix}=\begin{pmatrix} -\sigma & \sigma & 0\\ \rho & -1 & -x\\ y & 0 & -\beta ...
1
vote
1answer
79 views

Composition of $\mathrm H^p$ function with Möbius transform

Let $f:\mathbb D\rightarrow \mathbb C$ be a function in $\mathrm{H}^p$, i.e. $$\exists M>0,\text{ such that }\int_0^{2\pi}|f(re^{it})|^pdt\leq M<\infty,\forall r\in[o,1)$$ Consider a Möbius ...
0
votes
3answers
75 views

What wrong with pointwise convergent

It is true that the uniform limit of continuous functions is continuous as it has proved as a theorem, but what is wrong with pointwise limit? I mean why this theorem it doesn't work for if the ...
9
votes
4answers
279 views

What will be a circle look like considering this distance function?

I am working on some exercises in the book Geometry: A Metric Approach with Models by R.S. Millman. He defines the following map: $$d_S(P,Q):\mathbb R^2\times\mathbb R^2\to\mathbb R\\\ ...
1
vote
1answer
119 views

Finding Partial Derivative ($n$-dimensional) using implicit differentiation vs explicitly solving

This is a book example (not a homework question) about implicit differentiation on a composite of functions in $n$-dimensional space. But my book explains this example in a very unclear manner. So I ...
2
votes
1answer
161 views

About measure theoretic interior and boundary

Reference:- Evans-Gariepy, Federer, other books and notes of geometric measure thoery. I just want to clarify whether these definitions of measure theoretic interior and boundary are correct. Given ...
3
votes
1answer
145 views

Show that sequence approaches fixed point of a function

Problem Let $f(x)$ be a differentiable function on $\Bbb R$ with $\left|\,f ' (x)\right| \leq r < 1$, where $r$ is constant. Then consider the sequence $\{x_n\}$ such that $x_1 = 0$, $x_{n+1} = ...
30
votes
4answers
1k views

When $f(x+1)-f(x)=f'(x)$, what are the solutions for $f(x)$?

The question is: When $f(x+1)-f(x)=f'(x)$, what are the solutions for $f(x)$? The most obvious solution is a linear function of the form $f(x)=ax+b$. Is this the only solution? Edit I should ...
3
votes
2answers
74 views

How to prove that there exists a $z_0 \in U_{1} [0]$ such $ \prod_{k=1}^{n} |z_0 - a_k | \geq 1 $ for $a_1, \dots , a_n \in U_{1} [0] $?

Let $a_1 , \dots , a_n $ be points in the unit circle/ball in $\mathbb{C}$ around $(0,0)$ (also known as $U_{1} [0]$), which do not necessarily differ from one another. How to prove that there exists ...
5
votes
1answer
205 views

A problematic integral: $\int_0^{2\pi} e^{-2\pi i\lambda\cos(t)}\,dt$

Is there a special trick to calculate this integral? $$\int_0^{2\pi} e^{-2\pi i\lambda\cos(t)}\,dt$$ for $\lambda>0$.
6
votes
1answer
71 views

Exercise on convergent series

I am stumped by the following exercise (3.24 in Biler--Witkowski's book "Problems in mathematical analysis"): Let $f$ be a continuous, increasing function from $[0,+\infty]$ to itself. Show that ...
5
votes
0answers
260 views

Open Problem in Fixed Point Theory [Prize]

This open problem appeared on the bulletins of Evans Hall at Berkeley this week. I hope this doesn't violate StackExchange policy (the solution carries a $500 prize), but I thought why not re-post ...
1
vote
0answers
67 views

How to prove $C^1$ class is a proper subset of Lipschitz class?

Let $Lip(A)$ be the set of vector-valued functions $f$ on the closed set $A\in\mathbb R^n$ such that $$f(0)=0,$$ $$||f|| \text{ is finite, where by definition: } ||f||=\sup ...
0
votes
2answers
696 views

A few problems on sup and nested intervals

I've been doing these 3 problems for a `proof´ oriented class, one i have found a solution (in fact has been asked here before but the threads are all closed), and checked a correct solution in the ...
8
votes
1answer
411 views

Regular open set whose boundary has nonzero volume.

I found this question quite interesting, but its answers were disappointingly non-geometric. I'd be interested to know whether there exists a geometric example. To be precise about what I mean by a ...
4
votes
2answers
141 views

Domain whose boundry has non zero volume.

Can There be a domain in $\mathbb{R^n}$, for any $n$ such that some domain has non zero boundry volume? I.E. volume of boundry is non zero? Motivation: In some theorems, it is specified that volume ...
4
votes
0answers
108 views

Differential calculus on Banach space

I'm revising for my upcoming test, and this problem dated back some years ago. I've been working on this problem for almost a day, but I don't even know how to start it correctly. Problem Given the ...
1
vote
1answer
100 views

The infinity version of Blumenthal's 0-1 law

Blumenthal's 0-1 law states that on the space of continuous maps with domain $[0, \infty)$ with the appropriate (Wiener) measure making the coordinate maps Brownian motions starting at $x$, any event ...
0
votes
1answer
42 views

Question on the demonstration of Morse theorem

We have theorem of Morse and this is the proof i dont understand this : "$(c_i)$ has no cluster point since each $M^a=f^{-1}]-\infty,a]$ is compact " Thank you.
1
vote
0answers
79 views

Show derivative of integral equals integral of partial derivative if M[0,1]-measurable

I am trying to determine a method of approaching the following: Suppose that $f:[0,1] \times (0,1)$ $\rightarrow$ $\mathbb{R}$ is such that, for each $y \in (0,1)$, the function $f^{[y]}(x) = f(x,y)$ ...
1
vote
1answer
70 views

Inverse Function Thorem

Let $f,g:\mathbb R\to\mathbb R$ be smooth functions with $f(0)=0$ and f'$(0)\neq 0$. Consider the equation $f(x)=tg(x), t\in \mathbb R$. Show that in a suitable small interval $|t|\leq \delta$, there ...
4
votes
5answers
879 views

Checking whether a polynomial of high degree is bijective or not.

Let $P(x)$ be a polynomial of degree $101$. Then $x\mapsto P(x)$ cannot be a one-one onto mapping, i.e., bijective function from $\Bbb{R}$ to $\Bbb{R}$. True or false? I think is when we take ...
3
votes
3answers
241 views

Topological boundary vs geometric boundary

Let $M_1=B((0,0),1)=\{(x,y) \mid x^2+y^2<1\}$ $M_2=\{(x,y) \mid x^2+y^2\le1\}$ What are the interior of $M_1$ and $M_2$ ? And What are the boundary of $M_1$ and $M_2$ ? How to find them? ...
2
votes
2answers
375 views

Parametrization of $n$-spheres

This comes from Guillemin and Pollack's book Differential Topology. The book claims that one cannot parametrize a unit circle by a single map. I thought we could (by a single angle $\theta$). I ...
2
votes
1answer
81 views

What is the difference between the spaces $L^1$($\mu$) and $L^1$(d$\mu$)? And is one a subset of the other?

What is the difference between the spaces $L^1$($\mu$) and $L^1$(d$\mu$) ? And is one a subset of the other? $\mu$ is the Lebesgue measure.
16
votes
1answer
512 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 ...
0
votes
0answers
54 views

Big $O$ notation and little $o$ notation [duplicate]

I am a bit confused about the big $O$ and little $o$ notations. In other words, can any one show me with examples how these notations works? Explain in examples please?
82
votes
9answers
4k views

How far can one get in analysis without leaving $\mathbb{Q}$?

Suppose you're trying to teach analysis to a stubborn algebraist who refuses to acknowledge the existence of any characteristic $0$ field other than $\mathbb{Q}$. How ugly are things going to get for ...
1
vote
1answer
57 views

Suppose E is an infinite subset of a metric space X.

Prove that x is a limit point of $E$ if and only if there is a sequence $\left \{ x_n \right \}^\infty_{n=1} \subset E $ that converges to x. This was part of our practice final and I have no idea ...
0
votes
1answer
99 views

Newton-Raphson in $\mathbb{R}^n$

Let $U\in\mathbb{R}^n$ be open and $f:U\to\mathbb{R}^n$ be a $\mathcal{C}^1$ map. $\exists p\in U$ such that $\;f(p) = 0$ and $Df_{|p}$ is invertible. Define $\phi(x)= x-(Df_{|x})^{-1}f(x)$, show ...
5
votes
2answers
105 views

Determine the character of $\sum_{n=1}^{+\infty}{\frac{e^{i\theta n}}{n}}$

Determine the character of the following series: $$\sum_{n=1}^{+\infty}{\frac{e^{i\theta n}}{n}}$$ where $\theta$ is a real parameter. I try to divide the series with De Moivre' s formula: ...
1
vote
1answer
180 views

Determining the Lipschitz constant

Determine the corresponding Lipschitz constant of $f(t,y(t))=e^{(t-y)/2}$, where $D=\{(t,y) : 0\leq t \leq 1,-\infty<y<+\infty\}$.
5
votes
1answer
231 views

Algebraic transformations to continuously extend functions

Lately I was browsing through my analysis lecture notes (since right know I'm somewhat rusty in analysis) and the proof that $x \mapsto \frac{1}{x}$ is differentiable at every $x'\neq 0$ captured my ...
0
votes
1answer
37 views

Calculating fairness in a competition across offices

We want to run a fitness competition in our company using pedometers to determine which office location is the healthiest. The problem is, we can't use total number of steps, because each office has a ...
0
votes
0answers
173 views

Show $f$ is in $L^1$ (d$\mu$) space and $_X\int f $ d$\mu=\lim_{n\to \infty}\int_X f_n d\mu$

Suppose $f$ is in $L^1$($\mu$). Prove that for each $\epsilon > 0$ , there exists a $\delta > 0$ so that the $\int |f|\mathrm d\mu$ < $\epsilon$ over the set $E$ whenever $\mu(E) < ...
1
vote
0answers
128 views

Simpson's rule characteristics

I just wanted to ask a quick question in regards to simpson's rule for integration. I have been reading up on the trapezoidal rule, and have found the notations and have an understanding such that: ...
1
vote
3answers
145 views

Is a function always a monotonically increasing function

For $\lim_{x \rightarrow a}f(x) = L$, if $f(x)$ is not a constant, is $\delta(\epsilon)$ always a monotonically increasing function? Alternatively, how does the definition of a limit guarantee that ...
1
vote
1answer
128 views

Counter Example to Darboux's theorem.

I am looking for a function that does not satisfy Darboux's theorem (the Intermediate Value Theorem). Maybe it will not be continuous [a,b] or not differentiable on (a,b).
1
vote
1answer
66 views

What are some non-constant functions in the following sets? Are these sets a subspace of $Y = C^1[a,b]$?

Let $Y = C^1[a,b]$. What are some non-constant functions in the following sets? Are these sets a subspace of $Y = C^1[a,b]$. a) $D = \{ y \in Y: y'(a) = 0,\, y(b) = 1\} $ b) $D = \{y \in Y : ...
1
vote
0answers
101 views

Pull Back (change of variables)

Let be $h:\mathbb{R^2}\rightarrow\mathbb{R^2}$ a change of variables (diffeomorphism). Let be $X$ a vector fields in $\mathbb{R^2}$ and $f:\mathbb{R^2}\rightarrow\mathbb{R}$ a continuous application. ...
3
votes
2answers
63 views

Differentiability of function $f(x,y) = |x|^a + |x-y|$.

I am trying to figure out the points at which the function $f(x,y) = |x|^a + |x-y|$ is differentiable. Could you please help me out. I have considered the cases x>0, y>0 etc, but am having difficulty ...
5
votes
1answer
1k views

Prove that to each $\epsilon >0$, there exists a $\delta >0$ so that the Lebesgue integral…

Suppose $f$ is in $L^1$ space of $\mu$, where $\mu$ is the Lebesgue measure. Prove that to each $\epsilon >0$, there exists a $\delta >0$ so that the Lebesgue integral of the absolute value of ...