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).

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3
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3answers
109 views

Weird calculus limit

How to find the following limit? $$ \lim_{n \to \infty} \dfrac{ 5^\frac{1}{n!} - 4^\frac{1}{n!} }{ 3^\frac{1}{n!} - 2^\frac{1}{n!} } $$ Edit done to the question. Thank you!
1
vote
3answers
195 views

Showing a function is not differentiable at $(0,0)$

Let $\displaystyle f(x,y)=\begin{cases} \frac{x^3+y^4}{x^2+y^2} \text{ if } (x,y) \neq (0,0)\\ 0 \text{ if } (x,y)=(0,0). \end{cases}$ Show this is not differentiable at $(0,0)$. My strategy ...
0
votes
2answers
176 views

Understanding limits at infinity with regard to the definition of a limit

This is sort of a follow up to my previous question Say you have $$ \lim_{x\to +\infty} f(x) $$ where $f : \mathbb{R} \to \mathbb{R} , x \in \mathbb{R}$ What exactly does this mean? From the last ...
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2answers
43 views

Computing a gradient knowing only the Directional Derivative and unit vectors.

Suppose that $f: \mathbb{R}^2 \to \mathbb{R}$ is differentiable at $p$. Also suppose that $D_uf(p)=1$ and $D_vf(p)=1$ where $u=\left( \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)$ and $v=\left( ...
7
votes
0answers
412 views

On the weak and strong convergence of an iterative sequence

I have some difficulties in the following problem. I would like to thank for all kind help and construction. Let $H$ be an infinite dimensional real Hilbert space and $F: H\rightarrow H$ be a ...
0
votes
0answers
34 views

Conditions under which a real function is measurable [duplicate]

Consider a function $f: \mathbb R \to \mathbb R$ such that $|f|$ is measurable and $f^{-1}(F)$ is measurable for every finite set $F \subset \mathbb R$. Under which conditions will $f$ be measurable? ...
1
vote
1answer
211 views

Weak Differentiability of Holder functions

Is it true that every Holder function is weakly differentiable? If not please give counterexample. Thanks
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votes
1answer
28 views

Approximation related to resonance

Can someone help me with this problem. We have $$x(t)=N \sin (w_{0} t)+\frac{w_0}{w_1}e^{\frac{-t}{T}}\sin (w_{1}t)$$ and $w_1=(1+\frac{\delta_1}{N^2})w_0$ for some $|\delta_1|\leq 1$. I need to ...
3
votes
1answer
451 views

Smooth approximation of characteristic function of a bounded open set

Let $U$ be an open bounded set of $\mathbb{R}^n$. Is it possible to approximate $\chi_U$ as almost everywhere limit of increasing sequence of smooth functions?
5
votes
1answer
232 views

Expressing $\sin\pi/n$ in terms of radicals of integers

Are values of $$\sin \frac{\pi}{n}$$ where $n$ is a positive integer all expressible in terms of radicals of integers? If not, what is the first $n$ for which it is not?
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votes
2answers
42 views

Definition of a covering and how it applies to the following example

Let $S= \{(x,y): x,y > 0\}$. The collection $F$ of all circular disks with centers at $(x,x)$ and radius $x$, for $x>0$, is a covering of $S$. Then all disks such that $x$ is rational is ...
5
votes
3answers
167 views

Grasping the definition of open and closed sets

In a metric subspace $S = [0,1]$ of $\mathbb{R}^1$, why is it that every interval of the form $[0,x)$ or $(x,1], x\in (0,1)$, is an open set in $S$? I understand that if you were to remove either, ...
1
vote
2answers
147 views

Existence of smooth function with given compact support

Let $K$ be a compact set in $\mathbb{R}^n$. Does there exist a smooth function $\phi$ such that $0<\phi\leq 1$ on $K$ and 0 outside of $K$
1
vote
1answer
394 views

Difficult and unusual probability problem, how to solve?

Let $n_i$ be the $i$'th randomly chosen element from $\mathbb N$ with replacements. All elements have probability greater than zero of being chosen. After a number of trials $k$, the probability ...
1
vote
1answer
99 views

To show that a partial dertivative (of a piecewise function) is continuous at $0$

$$f(z)=\cases{\frac{x^4-6x^2y^2+y^4}{x^2+y^2} +i\frac{4xy(x^2-y^2)}{x^2+y^2},& $z\ne0$\cr 0, &$z=0$}$$ Let $u=\Re(f)$. I have shown from first principles that $\frac{\partial ...
3
votes
1answer
326 views

(Baby Rudin) To show the set of all condensation points of a set in Euclidean space is perfect

This is from Rudin's Principal of Mathematical Analysis, Chapter 2, Problem 27. Let $E \subseteq \mathbb{R}^k$. Let $P$ be the set of all condensation points of $E$. Let $\{ V_n \}$ be a countable ...
0
votes
1answer
46 views

Topological degree of a map with finite energy

Suppose that $\phi:\mathbb{R}^3 \to S^2$ is of class $\mathscr{C}^1(\mathbb{R}^3\setminus \left\{a\right\}) \cap \mathscr{C}^0(\mathbb{R}^3\setminus \left\{a\right\})$, that is $\phi$ might have a ...
3
votes
3answers
1k views

What sets contain $\infty$ and $-\infty$ and why are the Integers closed?

So I'm currently studying from Rudin's Principles of mathematical analysis or colloquially "Baby Rudin" and have stumbled into the second chapter namely basic topology. He lists some sets and states ...
1
vote
3answers
2k views

How to use the inverse function theorem?

I have a function $F (x, y) = (x^2+y^2, xy)$ and I need to show that it has an inverse. How do I find the inverse of this function using the inverse function theorem? I have not learned this before in ...
13
votes
1answer
530 views

Real analysis textbok that develops the subject in a self-motivated, coherent fashion?

Well, it seems as though I just failed my analysis prelim for the second time... I have one more try in about $5$ months. I'm failing to build up a framework for how to think about analysis problems. ...
2
votes
2answers
136 views

Implicit function theorem - how to approach?

I have a question that I have been working on for a while. I was wondering how I should approach the following question: Are there any points on the graph of the equation $$x^3+3xy^2+2xy^3=1$$ ...
1
vote
1answer
123 views

Using the Implicit Function Theorem, prove that it can be solved.

Show that the equation $x + y - z + \cos(xyz) = 0$ can be solved for $z = f(x,y)$ in the neighborhood of the origin.
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vote
1answer
60 views

Lipschitz Continuity and Slope

Lipschitz-Continuity is defined as follows: A function is Lipschitz-Continuous if there exists a K $\in \mathbb R$ s.t.: $|f(x) - f(y)| \leq K|x-y| \forall x,y \in D$ Now I was wondering if it is ...
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2answers
306 views

How to construct this Laurent series?

How do I construct the following Laurent series (clipped off Wolfram Alpha)? I know that the numerator can be written as $-1+\frac{\pi}2 z-...$ Alternatively (without the Laurent series), how can I ...
2
votes
1answer
44 views

Existence of a certain inner product

Does there exist an inner product, such that the all of the monomials $1,x,x^2,\ldots,x^n,\ldots$ (viewed as real valued functions) form an orthogonal (or even orthonormal) set? And what about a ...
1
vote
1answer
232 views

Rudin Real and complex analysis question[Differentiation]

At the beginning of the chapter on differentiation, the following theorem is stated without proof. Apparently it is so trivial that it does not require justification. I however don't find it so ...
4
votes
0answers
123 views

What are norms used for?

These two questions are quite similar to this one, so I apologise if this irritates anyone. Also, I suspect that a lot can be said in the answer, so I am really just looking for some main points ...
1
vote
2answers
132 views

Convex function which has a limit in $-\infty$ is non decreasing

I have a homework question, looks simple but I can't figure out a way to solve it. Any clue or help will be helpful. Let $f: \mathbb R\rightarrow\mathbb R$ be a convex function such that ...
1
vote
1answer
93 views

If the derivative approaches zero then the limit exists

I have a homework question, looks simple but I can't figure out a way to solve it. Any clue or help will be helpful. Let $f: [0,\infty)\rightarrow\mathbb R$ be a differentiable function such that ...
0
votes
0answers
40 views

Mandelbrot set and normalizing

Hi guys we were given a set of presentations in class, and I didnt go to these two presentations and we needed to find the answer for this. If anyone could show me how to do it i would be really ...
0
votes
0answers
40 views

Limit of the ratio of the logs, knowing the ratio

Consider $f,g: \mathbb R \to \mathbb R^+$ and suppose that $\liminf_{x \to \infty} \frac{f(x)}{g(x)}=+\infty$. Can anything be proved (equality or inequality) in general about $\liminf_{x \to \infty} ...
1
vote
1answer
105 views

$y''(t)=-g+y'(t)^2/y(t)$ unique solution

I am looking for a theorem of proof that tells me, that $$ y(t)=r \left( 1-\cos \left( \sqrt{\frac{g}{r}}t \right) \right) $$ is a unique solution to the differential equation ...
3
votes
0answers
77 views

State of the art of the Implicit Function Theorem

What is the most general form of the Implicit Function Theorem? Quite a general form of this theorem was given by Kumagai (1980): An implicit function theorem. So I am wondering what are the weakest ...
0
votes
1answer
365 views

Show that this piecewise function is differentiable at $0$

I have shown (from first principles) that the Cauchy-Riemann equations for the following function are satisfied at $z=0$. But to properly prove differentiability at $z=0$, what should I do next? Do I ...
5
votes
1answer
557 views

Is a Cauchy sequence - preserving (continuous) function is (uniformly) continuous?

Let $(X,d)$ and $(Y,\rho)$ be metric spaces and $f:X\to Y$ be a function and suppose for any Cauchy sequence $(a_n)$ in $X$, $(f(a_n))$ is a Cauchy sequence in $Y$. Is $f$ continuous? Let $f$ be ...
3
votes
1answer
57 views

conditional convergence

This is an practice question from "Advanced Calculus, Folland" Chapter 6.3, Q.2 (not HW) I am not sure how to go about this question :: suppose $\sum { { a }_{ n } } $ is conditionally convergent. ...
1
vote
1answer
58 views

Consider the mapping $f: \mathbb R^ 3 \to \mathbb R ^3$ defined by $f(x,y,z) = (x, y^3, z^5)$

Consider the mapping $f: \mathbb R^ 3 \to \mathbb R ^3$ defined by $f(x,y,z) = (x, y^3, z^5)$. $f$ has a (global) inverse $g$, despite that the matrix $f '(0,0,0)$ is singular. What does this imply ...
2
votes
1answer
90 views

using the M. Riesz Interpolation Theorem

I'm trying to decipher a particular claim in a paper I'm reading, but I just can't seem to figure it out. The M. Riesz Interpolation Theorem says: Let $T:L^{p_0}\cap L^{p_1} \to L^{q_0} \cap ...
0
votes
2answers
219 views

How to show that $z^4$ is not uniformly continuous?

$f$ maps $z$ in $\mathbb C$ to $z^4$ in $\mathbb C$. How do I show that this function is not uniformly continuous?
1
vote
1answer
146 views

Open or Closed set in sup norm

$ B=\big\{x= (x_n)_{n\in\mathbb{N}}:|x_n|<\epsilon\text{ for all }n\in \mathbb{N}\big\}$ , where $\epsilon>0$ is given in $(\ell^{\infty},||\cdot||_{\infty})$ How do I go about showing ...
0
votes
2answers
70 views

Determine if series are convergent in $(C([0,1]),\| \cdot\|_{\infty})$

Determine whether the following series $\sum_{n=1}^\infty f_n$ are convergent in the space $(C([0,1]),\|\cdot\|_\infty)$, where (i) $f_n(t)= \frac{t^n}{n!}$; (ii) $f_n(t)=\frac{t^n}{2n}$ ...
1
vote
3answers
107 views

Differentiable function and tangent line

a. Prove that differentiable function $f(x)$ on $[a,b]$ and its tangent line $T(x)$ at $a$ satisfies $|f(x) - T(x)|\le C|x-a|$ where $C=\sup\limits_{a\le y \le x}|f'(y) -f'(a)|$. No idea! I am ...
0
votes
3answers
634 views

Intersection of chord with circle knowing the length and a point

Let's take a circle with radius R, and center in O (0, 0). We take on this circle a point A with coordinates xA and yA. We know that point A is one of the endings of a chord with length l. Which is ...
2
votes
1answer
65 views

Small question about a lemma of measurability

Hi ; I have this lemma , and i want to ask tow questions : 1) What is the diffrence between say that $\varphi$ is measurable and to say that $\varphi$ is $(\mathcal{T},\mathcal{B}(U))$measurable . ...
1
vote
1answer
231 views

Strong and weak-* convergence for bounded linear maps

The textbook I am using says that a sequence $(T_n)$ in $\mathcal{B}(X,Y)$ for $X$, $Y$ normed linear spaces converges strongly to $T$ if $\lim_{n\rightarrow\infty}T_nx=Tx$ for every $x\in X$. The ...
1
vote
2answers
415 views

If $f(x) \le g(x) \le h(x)$ for all $x\in D$ and $f$, $h$ are Riemann integrable, then so is $g$. True or False? (Check my work) [duplicate]

If $f(x) \le g(x) \le h(x)$ for all $x\in [a,b]$, and f and h are Riemann integrable on [a,b], then so is g. True or false? Explain. A: True Proof: Since $f\in R[a,b]$, $\bar \int_a^b{f} = ...
3
votes
4answers
94 views

Show that $f '(x_0) =g'(x_0)$.

Assume that $f$ and $g$ are differentiable on interval $(a,b)$ and $f(x) \le g(x)$ for all $x \in (a,b)$. There exists a point $x_0\in (a,b)$ such that $f(x_0) =g(x_0)$. Show that $f '(x_0) ...
1
vote
1answer
159 views

Banach Space continuous function

On the Banach space $(C([-1,1]), ||\cdot||_\infty ) $ consider the operator given by $(Tf)(x)= \dfrac{1}{3} \displaystyle\int^1_0txf(t)\ dt + e^x - \dfrac{\pi}{3} $ 1) prove that the mapping is a ...
1
vote
2answers
100 views

Accumulation points of $A\subseteq C([0,1])$

Consider the subset A of $C([0,1]) $ consisting of continuous functions f with $f(0)=f(1)=0$ In $(C([0,1]), ||\cdot||_1)$ determine whether the follow are accumulation points of the set A 1) ...
0
votes
1answer
549 views

Expected hitting time of one of two barriers

In the webpage "hitting time of one of two barriers", the probability that a non symmetric random walk hits one of two barriers is computed. The walker starts from $x=0$ and the barriers are located ...