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)

0
votes
0answers
17 views

Constrct a smooth curve whose range is dense in $R^2$

How to construct a smooth curve $f: R \to R^2$ such that whose range is dense in $R^2$?
-3
votes
0answers
7 views

MATLAB -Banded LU Factorization No pivoting [on hold]

Please show MATLAB code and explain all steps. I know that I need to make q and p arguements, but I am not sure how to make sure there's no pivoting and the bandwidths exist. I'm not sure how to ...
0
votes
1answer
18 views

Suppose a set A is infinite and B is countable. Show that A∪B ∼ A.

Suppose a set A is infinite and B is countable. Show that A∪B ∼ A. What I am thinking: Choose S, a countable subset of A. Construct a bijection from S to B. Proof S∪B~S Proof A∪B~S Question: ...
0
votes
2answers
23 views

Representation of a linear functional

Let $f, f_{1}, . . . , f_{n}$ be linear functionals on a linear space $L$ such that $f_{1}(x) = . . . =f_{n}(x) = 0$ implies $f (x) = 0$. Prove that there exist constants $a_{1},, . . , a_{n}$ such ...
0
votes
1answer
21 views

Power series and zeros

When is a power series equal to zero? Example: Take $\sum_{n=0}^\infty a_n(z-z_0)^n$. Is this power series equal to zero only at $z=z_0$ if we assume that we have infinitely many nonzero $a_n$? ...
1
vote
1answer
19 views

Convergence and metric - Proof?

Let $(x_n)$, $(y_n)$ be two sequences in a metric space $(P,d)$. Suppose $(x_n)$ converges to $x$ and $(y_n)$ converges to $y$. Prove that $\displaystyle\lim_{n \to \infty} d(x_n,y_n) = d(x,y)$ My ...
1
vote
1answer
12 views

Horizontal Cylinder Gas Problem

We have a perfect cylinder with a diameter of 3 ft that lies horizontal. The gas gauge is broken so we are forced to use a dipstick to determine how much gas in our tank. In this problem we are ...
0
votes
1answer
8 views

Subsets of a metric space in which Hausdorff semi-distance is symmetric

These are the definition of Hausdorff distance and Hausdorff semi-distance for subsets of a metric space $X$. ‎‎Hausdorff semi-distance of two subsets ‎$‎A‎, B‎ \subset X$ is defined as below: ‎$‎d(A ...
0
votes
0answers
7 views

bounding supremum with Sobolev embedding theorem

I consider $d \geq 2$ and the domain $ D = [0,1]^d$. I consider a function $f : D \to \mathbb{R}$ that is $k$ times continuously differentiable. [$k$ can be specified as large as possible.] I am ...
1
vote
3answers
21 views

Infimum of a set with two variables

I have encountered a problem concerning the infimum of a set: Prove that. $$\mathrm {inf} \left\lbrace\sqrt{a^2+{1\over b^2}}:a,b\in(0,1) \right\rbrace=1$$ What I've been able to do is to prove ...
0
votes
3answers
55 views

A problem on simple analysis [on hold]

If at least one of two natural numbers isn't a perfect square, proof that $\sqrt{m} + \sqrt{n}$ isn't rational.
1
vote
1answer
65 views

How prove this $\int_{a}^{b}f(x)dx=\frac{1}{2}(b-a)[f(a)+f(b)]-\frac{1}{12}(b-a)^3f''(\xi)$

Let $f(x)$ be a twice-differentiable function on $(a,b)$,show that there exsit $\xi\in(a,b)$ ,such $$\int_{a}^{b}f(x)dx=\dfrac{1}{2}(b-a)[f(a)+f(b)]-\dfrac{1}{12}(b-a)^3f''(\xi)$$ if this problem ...
-4
votes
0answers
43 views

Identifying Symbols [on hold]

When you see $x$ written on a piece of paper you automatically identify it. When you yourself write $x^2 + 2x = 0$ The $x$ you write in $x^2$ differs from the $x$ you write in $2x$ just by a ...
1
vote
0answers
45 views

What is a everyday example of a non measurable set?

I'm working on my understanding of measurable sets and my immediate intuition wants to know what's not a measurable set? Initially I think of some space where divisions go to infinity, like a ...
0
votes
2answers
24 views

Prove that Archimedean Property implies that $lim_{n->\infty} 1/n$ =0

I am very curious how to prove this. To start off, we assume the Archimedean Property, or there exists an $\epsilon>0$ s.t. for a natural number n, 1/n < $\epsilon$. But From there I am simply ...
0
votes
2answers
76 views

$f(x)$ is everywhere differentiable on $[a,b]$ then give examples

$f(x)$ is everywhere differentiable on $[a,b]$ then give examples for each (they are independent) (1) $f'(x)$ is not Riemann integrable (2) $f''(x)$ does not exist (3) $f'(x)$ is not continuous
4
votes
0answers
46 views

Writing the roots of a polynomial with varying coefficients as continuous functions?

Consider the monic polynomial $$p_{\zeta}(z) = z^n + a_{n-1}(\zeta)z^{n-1} + \dots + a_0(\zeta), $$ where the $a_{i}$'s are continuous functions defined over $\mathbb{C}$. As is well known, the ...
1
vote
0answers
15 views

What assumptions should be made?

take a problem like A trough is 12 feet long and 3 feet across. Its ends are isosceles triangles with altitudes of 3 feet. Water is being pumped into the trough at 2 cubic feet per minute. How fast ...
0
votes
1answer
25 views

Finding the set of analytic functions whose image is a subset of a given set

Let $A=${$z\in\mathbb{C}||z|=1$} and $B=${$z\in\mathbb{C}||z|<2$}. I want to find the the set of analytic functions such that $f(B)\subset A$. Is there a way to solve this? Hope someone could help ...
0
votes
0answers
16 views

Typo in Caffarelli-Silvestre?

I am reading two papers by Caffarelli and Silvestre, namely Regularity results for fully nonlinear equations by approximation and The Evans-Krylov theorem for nonlocal fully nonlinear equations. From ...
2
votes
2answers
42 views

Select a subsequence to obtain a convergent series.

Does there exists strictly increasing sequence $\{a_k\}_{k\in\mathbb N}\subset\mathbb N$, such that $$ \sum_{k=1}^{\infty}\frac{1}{(\log a_k)^{1+\delta}}\lt \infty, $$ where $\delta>0$ given and ...
3
votes
2answers
37 views

Convergence of $\sum{a_kb_k}$ if $\sum{a_k}$ converges and $\sum{b_k}$ absolutely converges.

Convergence of $\sum{a_kb_k}$ if $\sum{a_k}$ converges and $\sum{b_k}$ absolutely converges. I tried to think that Since $\sum |b_k|$ is bounded I thought that $\sum a_k b_k$ $<$ $S\sum a_k$. ...
1
vote
1answer
29 views

Question about the answer to Kac's problem: 'Can one hear the shape of a drum?'.

I'm looking at the article of Gordon, Webb and Wolpert http://www.math.upenn.edu/~kazdan/425S11/Drum-Gordon-Webb.pdf, having only basic notions of group theory. In this article the authors describe ...
0
votes
0answers
26 views

The differentiability of the complex valued function $(Rez)(Imz)z\over|z|^2$

$$ f(z) = \left\{ \begin{array}{ll} \Re(z)\Im(z)z\over|z|^2 & \quad z \neq 0 \\ 0 & \quad z = 0 \end{array} \right. $$ I want to prove that this ...
0
votes
1answer
17 views

Square integrability

Given a function $g(y)=\int_y^{\infty}f(x) dx$ and given that I know that for $y\rightarrow-\infty$ the function $g(y)\rightarrow C$, where C is a constant, why is the last condition implying that the ...
0
votes
0answers
33 views

Algebra of limits- Is this proof correct?

If you go to http://math.wikia.com/wiki/Algebra_of_Limits Shouldn't the line before the last line read $$\lim_{n \to \infty} \frac {1}{y_{n}} = \frac {1}{y}$$ Instead of $$\lim_{n \to \infty} \frac ...
0
votes
1answer
10 views

Inequality in Evans PDE section 5.7

I'm stuck in the proof of the Compactness Theorem in Evans PDE 2nd edition book. On page 287, last line, how do you get the inequality $$ \epsilon ...
0
votes
2answers
27 views

Is the following true for $f: \Bbb R^3 \rightarrow \Bbb R$ continuous?

For $f: \Bbb R^3 \rightarrow \Bbb R$ continuous I am asked to prove that if there is an $x$ such that $f(x)=0$ but $f(o,o,o)$ is not zero. Then there exists another $y$ closer to the origin, such ...
1
vote
2answers
42 views

Conditions for convergence of $\sum\limits_{n=1}^\infty{a^nf(n)}$

assume $a>0$, and for all $n$ we have $0 \leq f(n) \leq 1$. Is there a necessary and sufficient condition on the series $f(n)$ for which $\sum\limits_{n=1}^\infty{a^nf(n)}<\infty$ ? Thanks!
0
votes
0answers
55 views

Prove that $\{\sqrt[n]{e^{n+1}}\}$ is convergent.

I need to prove that $\{\sqrt[n]{e^{n+1}}\}$ is convergent, and find its limit using this theorem: Let $f:E \to R$ with $x_{0} \in E$ an accumulation point of $E$. Then these are equivalent: 1)$f$ ...
0
votes
1answer
56 views

Correctness of proof that $\lim_{n\to \infty}\sqrt n*c^n=0$

My proof is as follows: Assume $|c|\lt 1$ and $c$ can be written as 1/1+d for d>0 The definition of the mentioned limit is: For all $\epsilon>0$ there exists a natural number N s.t. for all n ...
0
votes
1answer
23 views

Proof that the set of irrational numbers is dense in the reals

The hint I was given was to simply prove that y=xz is irrational given that x is nonzero, x is rational and z is irrational. Here's how I did it: Claim: y=xz is irrational Proof: Assume $x\neq0$, x ...
2
votes
1answer
23 views

Correctness of proof that an ordered field S that has the supremum property also has the infimum property

First question I have is how would you describe the relationship between an ordered field and an ordered set and continue the proof by treating the field as a set? I want to say that right in the ...
1
vote
0answers
13 views

Why are weak-mixing systems considered “random” and compact systems considered “ordered”?

As I understand it, weak-mixing systems sort of tend to become "orthogonal" to themselves on the long run, and compact systems tend to become almost periodic. How is this related to them being called ...
2
votes
1answer
35 views

Does $\lim_{x \to 0}({z^2\over \overline z})$ exist? $(z\in \mathbb{C})$

I am trying to figure out if $\lim_{x \to 0}({z^2\over \overline z})$ exists or not. This is a way I though to show that this does not exist but I am not entirely sure. Let $a_n={1\over n}$ and ...
2
votes
0answers
25 views

In which metric spaces other than the discrete spaces are the closures of open balls different from closed balls?

Let $(X,d)$ be a metric space such that $d$ is not the discrete metric. Let $x_0 \in X$, let $r>0$, and let $$B(x_0;r) \colon= \{ x \in X \colon d(x,x_0) < r \}$$ be the open ball with center ...
1
vote
0answers
40 views

Norm equivalence of p-forms

Let $U$ be a bounded domain in the Euclid space $(\mathbb{R}^d,g)$. $g_{\wedge^p}$ denotes the fiber metric of $\wedge^pT^{\ast}\mathbb{R}^d$ derived from $g$. $A^p$ denotes the set of p- forms on ...
2
votes
1answer
31 views

Distance between a point and a closed set in metric space

Here is what I am thinking. Let (X,d) be a metric space and let C be a closed subset of X. Fix any poin p in X. Then, there exists a point q in C such that d(p,q) = distance(p,C). I think this ...
1
vote
0answers
21 views

How to compute the derivative of this functional on a manifold?

I'm a little puzzled by the following computation. Suppose $\Sigma$ is a compact two-dimensional Riemannian manifold and $M$ is any Riemannian manifold. Let $\Omega$ be some space of functions from ...
0
votes
1answer
56 views

Continuous iff Oscillation is zero

For a bounded function $f:D\subset \Bbb R^n \rightarrow \Bbb R$, $b$ in $\Bbb R^n$, and a real number $\delta>0$. Define the following: $M(f,b,\delta)$=sup{f(x)$: x$ in $D$ and ...
0
votes
2answers
78 views

Prove that $X$ is complete but not inner product, and vice versa

Let $X$ be the space $C[0, 1]$ under the norm $||·||_{p}$ for $1 \leq p \leq \infty$. (a) Show that $X$ is complete for $p = \infty$, but it is then not an inner product space. (b) Show that $X$ is ...
1
vote
1answer
27 views

Prove that for $0<p<1$, $|x-y|^p$ is a metric space on $R^{n}$

Define the function $f_p : R^{n} → R^{n}$ for $n ≥ 2$ by $f_p(x) = \sum_{k=1}^{n} |x|^{p}$. Show that for $0<p<1$, we get $d_b(x,y) = f_p(x-y)$ is a metric on $R^{n}$. I tried to use ...
0
votes
0answers
17 views

Mean value theorem and Multi-valued functions

Let a point $x$ map to a set of points $\{y | y \in U \}$ where $U \subseteq \mathbb{R}$ or $U \subseteq \mathbb{C}$. Can MVT be generalized to multi-valued functions ?
0
votes
0answers
18 views

Generator of smooth complex-valued functions vanishing at infinity

Let $C_0(\mathbb{R})$ be the $C^{\ast}$-algebra of complex-valued functions vanishing at infinity, with involution given by $f^{\ast}(x) = \overline{f(x)}$. How can I prove that this commutative ...
2
votes
1answer
38 views

Prove that any derivative of a given function is bounded

Let the function $f\left( x \right) = \left( {\frac{{1 - \cos x}} {{{x^2}}}} \right)\cos (3x)$ if $x\ne 0$ and $f(0)=\frac{1}{2}$. Prove that any derivative of $f$ is bounded on $\mathbb{R}$. Thank so ...
0
votes
0answers
56 views
+50

Example about the difference between Morse and degree theory

i found this example but i don't understand how we applyed Morse theory and why we can't applyed degree theory. if the functional $f$ behaves like $<lu,u>$ at infinity where the symmetric ...
2
votes
2answers
75 views

Does the series $\sum_{n=1}^\infty (-1)^n \frac{\cos(\ln(n))}{n^{\epsilon}},\,\epsilon>0$ converge?

Numerical results showed that the series $$Q=\sum_{n=1}^{\infty} (-1)^n \frac{\cos(\ln(n))}{n^{\epsilon}},\epsilon>0\tag{1}$$ with $\epsilon=10^{-6}$ converged to $-0.53259554096828...$ We can ...
0
votes
0answers
10 views

Extension of norm from integral domain to fraction field

Let's say I have an integral domain $D$ with norm $\|\cdot\|:D\to\mathbb{R}.$ I aim to show that $\|\cdot\|$ can be uniquely extended to $\text{Frac}(D).$ I think the simple extension ...
0
votes
1answer
19 views

How do induction on a recursive defined function?

I made a mistake on some homework, because I didn't prove by induction, but I am lost upon how to prove this my induction. How I understand in induction: Show base case (n=1) is true Assume n ...
1
vote
1answer
29 views

Why $f (x):= \frac{1}{\sqrt{x}\left(1+\left|\ln x\right|\right)}$ only belongs to $L^2(0, \infty)$

This is a result given in Royden and Fitzpatrick (p. 143). Show that $$ \int_0^\infty \left[ \frac{1}{\sqrt{x}\left(1+\left|\ln x\right|\right)} \right]^p < \infty $$ if and only if $p=2$. That ...