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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2answers
122 views

Is this true about the open intervals on the real line?

Let $a<b$ and let $m$ be a positive integer such that $$3^{-m} < \frac{b-a}{6}.$$ Then can we find a positive integer $k$ such that the open interval $$\left(\frac{3k+1}{3^m}, ...
1
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4answers
53 views

How to give a rigorous proof of this fact about closures of open balls in the euclidean spaces?

Let $n$ be a positive integer, $\vec{a} \in \mathbb{R}^n$, and $r > 0$. Then it is intuitively clear that the closuer of the open ball $$B(\vec{a} ; r) \colon= \{ \vec{x} \in \mathbb{R}^n \colon ...
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0answers
9 views

Looking for a motivating example for backward error analysis

For many people, it seems self-evident that backward error is a powerful tool in numerical analysis. But for me, it is hard to imagine a situation in which backward error analysis provides any useful ...
1
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2answers
37 views

Inequality for all real numbers

I am trying to prove a hw problem from Taos Analysis 1 book. I would like some help proving the following statements if they are true which I do not necessarily believe. Let $x,y \in \Bbb R$. Show ...
0
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1answer
43 views

(absolute) Convergence of a series

I want to prove that the following series is convergent for $x>0$: $$ \sum_{n=1}^\infty \left( \prod_{p\mid n} \frac{1}{p-1}\right) n^{-x} $$ I tried to estimate the product but I didn't get so ...
1
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2answers
28 views

Branch of $n$th root of $f$ is holomorphic

The problem states to prove that if $h$ is a branch of $f^{1/n}$ for integer $n > 0$ (i.e. $h(z)^n = f(z)$ for $z \in G$, $h$ continuous), then $h$ is holomorphic, where $f$ is a holomorphic ...
1
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1answer
40 views

Subalgebra generated by selfadjoint operator $A_0\in\mathscr{L}(H,H)$

Let $\mathscr{L}(H,H)$ be the Banach algebra of bounded operators defined on a complex Hilbert space $H$ and let $B(A_0)$ be the subalgebra generated by the selfadjoint operator $A_0$, i.e. ...
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2answers
28 views

$C^n[a,b]$ as a normed algebra

I would like to prove that the space of complex valued functions, differentiable $n$ times with continuous derivative, $C^n[a,b]$, with the metric defined by the norm ...
0
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1answer
20 views

An ODE with boundary conditions at infinity

I have a problem where: $\ddddot{x} - 2 \ddot{x} + x = 0$ With boundary conditions $x(0) = 1, \dot{x}(0) = 2, x(\infty) = 0, \dot{x}(\infty) = 0$ So I get my characteristic equation: $s^4 -2s^2 + ...
0
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1answer
21 views

show existence of subsequence $\{a_{i_b}\}_b^{n+1}$

Suppose $\{a_n\}_{n=1}^{m^2+1}$ is a strictly increasing sequence of $n^2+1$ positive integers, show that there exist a subsequence $\{a_{i_b}\}_b^{n+1}$ of length $n+1$ such that $a_{i_k}$ is ...
3
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1answer
23 views

If $f$ is equal to an affine function up to $1$-th order at $a$, then $f$ is differentiable at $a$, proof more subtle then it appears?

I came across the following exercise: Two functions $f, g : \mathbb R \to \mathbb R$ are equal up to $n$th order at $a$ if $$ \lim_{h \to 0} \frac{f(a + h) - g(a + h)}{h^n} = 0. $$ Show that $f$ ...
0
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0answers
13 views

What is a generalized inverse to generalized nondecreasing left-continuous function?

Assume that $f:(a,b)\rightarrow \mathbb R$ is a nondecreasing function. Let $c=\inf f$, $d=\sup f$. We define $$ f^-(y)=\sup_{\{x: f(x)<y \}} x= \inf_{\{x: f(x)\geq y\}}x \textrm{ for } y\in ...
0
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1answer
20 views

Proof of inequality containing a function and its integral

I came across a proof that used following equalty, but for me it didn't look that obvious and I was not able to prove it, can you give me a hint (is it even true)? The statement of the inequality was ...
0
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1answer
27 views

What does monotonically convergent mean in this example.

Suppose that $$f(x)=\sum_{n=1}^{\infty}f_n(x)\,\,\,\,\,\,\,\,\,\,(x\in X)$$Where $f_n:X\rightarrow [0,\infty]$ for $n=1,2,3...$ Let $g_N=f_1+...+f_N$. Then the sequence $\{g_N\}$ converges ...
0
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2answers
33 views

How to prove that a function is left continuous

I cannot work out this problem even though it seems not that difficult. Could anyone kindly give me any hint? Thanks! If $f(x)$ is measurable on $E \subset \mathbb R$, then $$ \varphi (t)=m\big(\{x ...
1
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1answer
40 views

fastest and most efficent way to count all combinations in many sets and sum them together

I am a Java programmer who has reached the limits of brute computer power. My relational database (and non relational databases) is not producing results quick enough and I have hit a bottleneck in ...
1
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2answers
49 views

Derivatives vanishing at infinity

Take a subinterval of Euclidean space, for instance, which has infinite length. WLOG let it be $\mathbb{R}$. Is there an example of a function $f$ such that $f$ and $f''$ vanish at infinity, but not ...
1
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1answer
35 views

Supremum and Infimum with proof

Let $x\in (0,1)$ Compute with careful proof: The greatest lower bound of $(x^n : n \in N)$ and the least upper bound of $(x^n : n \in N)$ Hint: For the infimum (greatest lower bound), first prove ...
1
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0answers
42 views

isometric isomorphism between normed spaces and its dual

Let $E$ and $F$ be normed spaces. If $E \equiv F$ (isometry isomorphic), Does $E^* \equiv F^*$ (isometry isomorphic)? Where $E^*$ and $F^*$ are continuous dual spaces.
0
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1answer
50 views

Convergence of $\sum_{n=1}^\infty\left(1+\frac{2}{n}\right)^{n^3+n^2+1} \mathrm{e}^{-2n^2}$

Does the series $$ \sum_{n=1}^\infty\left(1+\frac{2}{n}\right)^{n^3+n^2+1} \mathrm{e}^{-2n^2} $$ converge? The ratio test is inconclusive, so I think I must use the comparison test. But I couldn't ...
1
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3answers
67 views

Prove $\frac{n^2+2}{(2 \cdot n^2)-1} \to \frac{1}{2}$

Prove $\frac{n^2+2}{(2 \cdot n^2)-1} \to \frac{1}{2}$ for $n \to \infty$. I've been looking at this for hours! Also, sorry I don't have the proper notation. This is where I'm at: $$ \left| ...
2
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1answer
27 views

Closure and subbasis

Let $X$ be a topological space and $A \subset X$ with a subbasis $S$. Does it then hold that $x \in \overline{A}: \Leftrightarrow \forall s \in S: (x \in s \Rightarrow s \cap A \neq \emptyset).$ This ...
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2answers
52 views

Question analysis so amazing [duplicate]

How this process has been calculated in a manner added.
2
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1answer
18 views

Give example where an outer measure is strictly less than the set function from which it is defined.

Let $K $ be a class of subsets of $X $ where for every subset $A $ of $X $ there is a sequence $\{E _n \} $ of sets in $K $ such that $A \subset \bigcup _{n=1 }^{\infty } E _n $. Let $\lambda$ be a ...
0
votes
1answer
25 views

Reparametrization of a curve which is not regular

Let $\alpha : [a,b] \rightarrow \mathbb R^3$ be a $C^1$ mapping (curve). Then $\alpha$ has a length. If $\alpha'(t)\neq 0$ for all $t\in [a,b]$ then, denoting $$ \sigma(t)=\int_a^t |\alpha'(u)|du, $$ ...
4
votes
5answers
71 views

If $f=u+iv$ is an entire function such that $u^2\geq v^2,$ then $f$ is constant

Let $f=u+iv$ be an entire function such that $u^2(z) \geq v^2(z), \forall z \in \mathbb{C}.$ Could anyone advise me how to prove $f \equiv$ constant $?$ Hints will suffice. Thank you.
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1answer
25 views

Proving that an analytic function that maps on to {$z\in \mathbb{C}| |z-2|=1$} from some connected open set is constant

This is the approach I took to solve this but I got stuck. Suppose$f=u+iv\in $ {$z\in \mathbb{C}| |z-2|=1$} and that $f$ is analytic on an open connected set. Then we have that $(u-2)^2+v^2=1$. ...
1
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0answers
27 views

Analytic continuation Dirichlet series

I have a Dirichlet series $A(s)$ with an absolutely convergent Euler product for $\sigma >0$. The zeros of the factors converge to $0+2\pi k$. I now have to proof that there can't be an analytic ...
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votes
1answer
27 views

On Pseudometric

How a pseudometrics induces topology? Can anyone discuss on this topic or give any good reference?
1
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1answer
46 views

If $f$ is differentiable everywhere, is $f'$ the weak derivative?

Let $f \in C^0([0,T])$ be such that $f'$ exists in the classical sense everywhere, but $f'$ may not be continuous. Is it true that $f'$ is the weak derivative of $f$ too, if it exists? I know this is ...
0
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1answer
45 views

Exam Question from multivariable calculus.

This question from a previous multivariable calculus exam.I don't know how to start with this question: Let $f$ be differentiable at every point of line segment joining $x_0$ and $x_0+h$.Show that ...
0
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1answer
13 views

What conditions have to be met for a subset $A$ of a measurable set $X$ to be also measurable?

What conditions have to be met for a subset $A$ of a measurable set $X$ to be also measurable? I understand that the union of measurable of sets is also measurable. But I am wondering if there is ...
0
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1answer
21 views

I do not understand why $E_n$ is measurable in this proof

Suppose we have a sequence of measurable functions $\{f_n\}$ on $X$. Also, suppose that (a) $0\leq f_1(x)\leq f_2(x)\leq ...\leq\infty$ for every $x\in X$ (b) $f_n(x)\rightarrow f(x)$ as ...
0
votes
1answer
29 views

Hölder norm bounded by $L^p-$norm?

Let $C_0^{\alpha}(\mathbb{R})$, $0<\alpha<1$ denote the space of Hölder-continuous functions on $\mathbb{R}$ with compact support. Is it true that for any $f\in C_b^{\alpha}(\mathbb{R})$ one ...
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votes
3answers
24 views

sum of two closed sets

Sum of two closed sets in $\mathbb{R}^2$ need not be closed. If one of them is compact, then the sum is closed. But if I'm given any two closed sets in $\mathbb{R}^2$, then how do I check whether it ...
3
votes
4answers
294 views

Is it true that every bounded sequence with the following property converges?

Is it true that every bounded sequence $\{a_n\}$ of real numbers such that $|{a_n - a_{n-1}}|<1/n$ for all $\ge2$ is convergent?
0
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4answers
30 views

Let $E \subset \mathbb{Z}$ be non-empty, bounded below. Prove that $\inf{E} \in E$

I know that since $E$ is non-empty and bounded below that $\inf{E}$ exists, but I'm not sure how to show that it is in $E$.
0
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1answer
30 views

Constructing a countable dense subset of a totally bounded set

Given a metric space $(X,d)$, and (non-empty) totally bounded set $E$ in $X$, is it possible to construct $D \subseteq E$ which is countable and dense? I feel that this should definitely be possible. ...
1
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1answer
28 views

Is $\Omega$ not open?

Suppose $\Omega\subset\mathbb{R}^d$ is connected. Let $z:[0,1]\to\Omega$ be a continuous path. Suppose $\underset{t\in[0,1]}{\inf}\text{dist}(\partial\Omega,z(t))=0$. Is $\Omega$ necessarily not ...
0
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2answers
50 views

Subject GRE question - set of points of discontinuity

I was just working on a Math Subject GRE practice test, and I got the following problem wrong: Let $f$ be the function defined on the real line by $\displaystyle f(x) = \begin{cases} \displaystyle ...
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3answers
142 views

Let $a$ be a real number, such that $a^7,\,a^{10}\in\mathbb Q$. [on hold]

Let $a$ be a real number so $a^7,a^{10}\in$ $\mathbb Q$. Can we prove that $a\in\mathbb Q$. Could you please provide a hint?
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2answers
35 views

Is this correct the rational numbers?

Determine the rational numbers $a,b$ , if , $($$2a-b$$)$ - $2b\sqrt3$ $=$ $3$ + $2\sqrt3$ I'm thinking that $-2b\sqrt3$ = $2\sqrt3$ $=>$ $ b = -1 $ $2a-b$ resembles with $3$ , and i solved ...
1
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0answers
39 views

Bounded variation in the context of Feller's paper on Muntz' Theorem

The paper I have posted a picture of is a paper of Feller. He shows that the functions $f_k$ are Laplace transforms of $C^\infty$ functions $u_k$. In order to execute his suggested proof, I ...
1
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1answer
28 views

Is the number of subsequential limits of a sequence always countable

I know that a sequence can have many different subsequential limits but is the number of subsequential limits always countable? How do we know?
2
votes
1answer
24 views

Is every Lipschitz continuous function is holder continuous with exponent $\in (0,1)$?

Is every Lipschitz continuous function is holder continuous with exponent $\in (0,1)$? This seems to be true,but I haven't found such a conclusion in any textbook.
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2answers
23 views

Prove the existence of a greatest lower bound of $X$ if $X \subset \mathbb{R}$ is a non-empty set that is bounded below

Attempt: Let $C \subset \mathbb{R}$ be the set of all lower bounds of $X$. Since $C$ is not empty and bounded above, every $x \in X$ is an upper bound of every element $c \in C$. Thus, there exists ...
1
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1answer
34 views

$\partial(S') \subset \partial S$ iff $S' \cap S^o \subset (S')^o$

Usually I can come up with some ideas but this time I don't. It would be great if you can tell me how I would make use of the first part of the question to prove the equivalent relation. Question: ...
1
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1answer
49 views

partial derivative of $f(x,y)$ who satisfies $f_{xx}-f_{yy}=0$

Suppose that $z=f(x,y)$ and its second-order partial derivative is continuous. It also satisfies $\displaystyle\frac{\partial^{2}f}{\partial x^{2}}-\frac{\partial^{2}f}{\partial y^{2}}=0$,$f(x,2x)=x$ ...
0
votes
1answer
22 views

A simple question about the laplacian

Suppose that $f:X\subseteq R^n\to R$ depends only on the distance $(x_1,x_2,...,x_n)$ is from the origin in $R^n$ (i.e. $f(\vec x)=g(r)$ where $r=\left | \vec x \right |$) Show that for all $\vec x\ne ...
-1
votes
3answers
69 views

Is $f(x)=x$ the solution of an integral equation? [closed]

Suppose that $f:[0, \infty)\longrightarrow \mathbb{R}$ is continuous and $f(x) \neq 0 $ for all $x>0$. If $$ \big(\,f(x)\big)^2=2 \int_0^x f(t)\,dt, $$ for all $x>0$, is it then true that ...