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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1answer
31 views

Estimating the number of nodes of a discretization scheme

Working on a problem in partial differential equations, I have come across a function $$f\colon [0, T]\to \mathbb{R}_{\ge 0}$$ which is continuous and non-decreasing, and whose maximum is $M=f(T)$. I ...
0
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1answer
81 views

To show continuity at a point using boundedness of partial derivatives

Suppose $$U = (-1,1) \times (-1,1) \subset \mathbb R^2,\ f: U\to \mathbb R\ .$$ Assume that $\partial f/\partial x$ and $\partial f/\partial y$ exist at each point of U and are bounded on U. Show ...
7
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3answers
220 views

2013th derivative of rational function

I am struggling to find $f^{(2013)}(0)$ for $$f(x) = \frac{1}{1 + x + x^3 + x^4}$$ I know that I should use power series, and following a hint I rewrote the equation as the following: $$1 = (1 + x + ...
1
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0answers
45 views

If $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ exist and are bounded on $U$, then $f$ is continuous at $(0,0)$.

Suppose $U = (-1,1)\times(-1,1) \subset \mathbb{R}^2$, $f: U \to \mathbb{R}$. Assume that $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ exist at each point of $U$ and are bounded ...
3
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1answer
140 views

Prove it doesn't exist any function f:R→R that is continuous only at the rational points.

Prove it doesn't exist any function $f:\mathbb R \to \mathbb R$ that is continuous only at the rational points. Suggestion: For every $n \in \mathbb N$, consider the set $U_n=\{x \in \mathbb R : ...
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0answers
33 views

Is there a 'simple' test one can perform to see if a function maps measurable sets to measurable sets?

I know I'm probably missing a huge key point somewhere in my analysis background, but is there such a thing as a 'simple' test to check if a function does this? (Note: I am specifically considering ...
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1answer
51 views

What is the gradient of this function

Imagine you have a function $f:\mathbb{R}^n \rightarrow \mathbb{R}$ $f(z):=z^TAz$, where $A$ is a symmetric matrix. Now I was wondering what $\nabla f(x_1,...,x_{n-1},\gamma(x_1,...,x_{n-1}))$, where ...
3
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1answer
69 views

Show that $\operatorname{div} X = - \delta X^\flat$

I want to show the equality $\operatorname{div} X = -\delta X^\flat$, where $X \in \Gamma(TM)$ and $M$ is some Riemannian manifold with metric tensor $g_{ij}$. If I'm not mistaken it holds for the ...
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2answers
42 views

A math analysis exercise question.

my teacher gave me this exercise: let $f:I\to\Bbb R$ a non decreasing function, where I is an open set in $\Bbb R$, and let $S=\{x\in \Bbb R : f $ is not continuous in $x\}$; knowing that every $x ...
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1answer
93 views

How is min max f(x,y) defined when solving a dual problem?

I am trying to solve a dual problem. And it is said that min max f() is always smaller or equal to max min f(). For example, $\max_{y \in Y} \min_{x \in X} f(x,y)$ is always smaller or equal to $ ...
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1answer
31 views

$W = \{x\in l_0 : <x,a>=0\}$ where $a=(1,\dfrac12,\dfrac13,…)$ and $l_0$ is sequences with finitely many non-zero terms. Show $W$ is separable

Consider the inner product space $l_0$ consisting of all infinite sequences of complex numbers with only finitely many non-zero terms, with the inner product of $l^2$ (space of square summable ...
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1answer
36 views

Show that $\sum_{n=1}^{\infty}\dfrac{(-1)^{n+1}(x+\sqrt{n})}{n}$ does not converge absolutely for $x\in [-1,1]$

Show that $\sum_{n=1}^{\infty}\dfrac{(-1)^{n+1}(x+\sqrt{n})}{n}$ does not converge absolutely for $x\in [-1,1]$ Consider ...
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1answer
28 views

Inequality with partial integration in one dimension

Is it possible to prove $ \| u \|_{L^2(0,1)} \leq \| u' \|_{L^2(0,1)} $ for $u \in C^1([0,1])$ with $u(0)=0$ by using partial integration?
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1answer
26 views

$(z_{2n})_{n\in \mathbb{N}}$ and $(z_{2n+1})_{n\in \mathbb{N}}$ converges to same limit. Show $(z_n)_{n\in \mathbb{N}}$ converges to the same limit.

$(z_n)_{n\in \mathbb{N}}$ is a complex valued sequence such that subsequences $(z_{2n})_{n\in \mathbb{N}}$ and $(z_{2n+1})_{n\in \mathbb{N}}$ converges to same limit. How could I show that the ...
0
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1answer
53 views

$A = \{e_i:i\in \mathbb{N}\}$ where $e_i = (0,…0,1,0,0,…)$. Show that $\overline{span(A)}=c_0$, set of infinite sequences that converges to 0

Let $A = \{e_i:i\in \mathbb{N}\}$ where $e_i = (0,...0,1,0,0,...)$. It has $1$ in the $j^{th}$ entry. Define $span(A)$ as the set consisting of all finite linear combinations of elements in $A$. ...
3
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1answer
363 views

How to prove that $2\sqrt{a^{ea}b^{eb}}\ge a^{eb}+b^{ea}$ for $a > 0, b > 0$?

Let $a,b\in R^{+}$. Show that $$2a^{\frac{ea}{2}}b^{\frac{eb}{2}}\ge a^{eb}+b^{ea} \>.$$ My attempt I know the following inequality is true: $$a^{ea}+b^{eb}\ge a^{eb}+b^{ea} \>.$$ See this ...
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1answer
67 views

If $\sum_{n=1}^\infty f_n(x)$ and $\sum_{n=1}^\infty g_n(x)$ converges uniformly, then $\sum_{n=1}^\infty [ f_n(x)+g_n(x)]$ converges uniformly

Prove that if $\sum_{n=1}^\infty f_n(x)$ and $\sum_{n=1}^\infty g_n(x)$ converges uniformly on $x\in X$, then $\sum_{n=1}^\infty [ f_n(x)+g_n(x)]$ converges uniformly on $x\in X$. My working so ...
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1answer
68 views

Continuity in Metric Spaces

$8.\,\,\,$Let $d$ and $d'$ denote the usual and discrete metrics respectively on $\Bbb R$. Show that all functions $f$ from $\Bbb R$ with metric $d'$ to $\Bbb R$ with metric $d$ are continuous. ...
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1answer
458 views

Open and Closed Sets Examples [duplicate]

Ok so well Im struggling to find examples for the first two parts and for the last, well I don't think it is open but can't again find an example. Thanks.
0
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1answer
70 views

Convergence of $\sum_{n=1}^\infty\dfrac{\tan^{-1}n}{n+\sqrt{n}}$

How could I determine the convergence or divergence of this series? $$\sum_{n=1}^\infty\dfrac{\tan^{-1}n}{n+\sqrt{n}}$$
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1answer
70 views

Counterexample to: if $1\le p<q<\infty$, then $L^q(X)\subset L^p(X)$ with $\mu(X)=\infty$

We know if $\mu(X)<\infty$, and if $1\le p<q<\infty$, then $L^q(X)\subset L^p(X)$ (can be proved by using Holder's inequality). Is this still true if $\mu(X)=\infty$? Counterexample? ...
0
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1answer
279 views

example of a Riemann integrable function (on a bounded rectangle) that is discontinuous on a dense subset of the rectangle

Construct a nontrivial example of a Riemann integrable function (on a bounded rectangle) that is discontinuous on a dense subset of the rectangle. A (trivial) example would be to rede fine a nice ...
1
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1answer
161 views

Isolated points over specific intervals

I am having some trouble understanding isolated points. I am using the textbook "Real Analysis" by Manfred Stoll. Isolated points are defined as "a point p in set E that is not a limit of E." Limit ...
3
votes
3answers
212 views

Compute $\int_0^{\infty}\frac{\cos(\pi t/2)}{1-t^2}dt$

Compute $$\int_0^{\infty}\frac{\cos(\pi t/2)}{1-t^2}dt$$ The answer is $\pi/2$. The discontinuities at $\pm1$ are removable since the limit exists at those points.
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1answer
143 views

Applying Fatou's Lemma to a sequence of positive functions which converges in measure

I would like to prove the following: If $f_n \geq 0$ and $f_n\rightarrow f$ in measure, then $\int f d\mu \leq \liminf \int f_n d\mu$. So far I have Since $f_n \rightarrow f$ in measure, then there ...
4
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1answer
91 views

Prove that $\lim_{\substack{b\to\infty \\ a\to0+}}\int_a^b\frac{\hat{f}(\xi)}\xi d\xi=-\pi i\int_0^\infty f(x)dx$

Suppose $f\in L^1(\mathbb{R})$ and that $f$ is odd. Prove that $$\lim_{\substack{b\to\infty \\ a\to0+}}\int_a^b\frac{\hat{f}(\xi)}\xi d\xi=-\pi i\int_0^\infty f(x)dx$$ Here $\hat{f}$ denotes the ...
0
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1answer
78 views

Convergent Sequence from Introduction to Analysis

Consider the sequence of real numbers $$\frac 12, \cfrac 1{2+\cfrac 1 2}, \cfrac 1{2+\cfrac 1{2+\cfrac 12}}, \ldots.$$ Show that this sequence is convergent and find its limit by first ...
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1answer
27 views

Extending the path of integration to the boundary of the domain of definition of $g:\Omega \to \mathbb{C}$

Working in Shakarchi and Stein's Complex Analysis, they show an example (p.231) of a conformal map from the $\mathbb{H} = \{z: Im(z)>0 \}$ to $\{z: 0 < arg(z) < \alpha z\}$ using the ...
3
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1answer
159 views

Strange mistake in this proof about normed separable vector spaces.

I am supposed to prove that an infinite-dimensionale separable normed space $X$ constains a countable subset $Y$ that is linearly independent and dense. There are three hints given: First, prove ...
2
votes
1answer
165 views

Problem with notation: Laplacian on a manifold

In the Aubin's book "Nonlinear analysis on manifolds" the Laplacian operator on functions on some smooth manifold is defined by the formula $$ \Delta = -\nabla^\gamma\nabla_\gamma, $$ where ...
2
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1answer
175 views

Techniques for determining whether the weak limit of an absolutely continuous sequence of probability measures is absolutely continuous?

Let $(\mu_n)$ be a sequence of probability measures on $\mathbb{R}$ that converges weakly to a probability measure $\mu$ on $\mathbb{R}$. So $$ \lim \int hd\mu_n = \int h d\mu $$ whenever $h$ is a ...
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0answers
35 views

introductory real analysis - convergence and series Q

Show that $\forall \epsilon > 0 $$ \exists H \in \mathbb{R} $ s.t. P>H & N>H implies $\left|1 - \displaystyle\sum_{p=2}^P\sum_{n=2}^N\left(\dfrac{1}{n}\right)^p \right| < \epsilon $ my ...
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1answer
220 views

differentiability and local Holder continuity

My analysis is really rusty, so apologies if this is a stupid question. If $f\in C^1$ in a compact set $\Omega$, does this mean $f$ is Holder continuous for any $\alpha$ in $\Omega$? I have tried ...
0
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1answer
193 views

$l_0$ is all sequences with finitely many non-zero terms. Show $W^\perp=\{y: <x,y>=0, x\in W\}=\{0\}$ where $W = \{x : <x,a>=0\}$.

Consider the inner product space $l_0$ consisting of all infinite sequences of complex numbers with only finitely many non-zero terms, with the inner product of $l^2$ (space of square summable ...
1
vote
1answer
31 views

does $(\overline{E})^{'}= E^{'} \cup ( E^{'})^{'}$ holds?

My question is as follows: Suppose $E$ is a set in metric space $X$, let $\overline{E}$ denote the closure of E, let $E^{'}$ be the set of all the limit points of $E$. We all know that ...
0
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1answer
74 views

Find the limit of $\lim_{x\to\infty}\frac{\sin[xf(x)]}{x\cdot\sin[f(x)]}$

Let $f(x)$ is a continuous function which satisfies that $\lim_{x\to\infty}f(x)=0$. Then whether $$\lim_{x\to\infty}\frac{\sin[xf(x)]}{x\cdot\sin[f(x)]}=1$$ If not, then can we impose some ...
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1answer
105 views

Spectrum of operator in infinite dimensional hilbert space

We know that if a complex hilbert space $H$ is separable, then for every compact set $K$, there exists a bounded linear operator $T : H \to H$ s.t $\sigma (T) = K$. My question is if this still holds ...
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2answers
123 views

How take example such this three conditions.is continuous at all irrationals, discontinuous at all rationals

Example: The function $f(x)$ such this follow three conditions: (1): $x\in [0,1]$ (2): such $f(x)$ is continuous at all irrationals, discontinuous at all rationals; (3):and $f$ have many ...
3
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2answers
207 views

Why isn't the parallel between the Fourier transform and the Laplace transform complete?

I mean the question in the following sense. For Fourier, we can do it on compact intervals and then we get a sequence of coefficients. We can do it continuum-style, and then we get a superposition ...
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6answers
912 views

Good PDE books for a graduate student?

I am now a graduate student in mathematics, and I really want to learn more about PDE. I would say I have a very solid foundation in soft analysis, including functional analysis and harmonic analysis, ...
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2answers
100 views

Example of a sequence of functions

Construct an example of a sequence of functions $(f_n)$ defined on $[0,1]$ such that $f_n$ converges pointwise to $0$ and for every sequence of numbers $(a_n)$ that tends to $\infty$, sequence ...
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0answers
51 views

A question about curl

Is there a new proof? Or it is just trivial?
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1answer
51 views

Determine the tangent plane on this paraboloide

given a paraboloide $\frac{x^2}{\alpha^2}+\frac{y^2}{\beta^2}=\frac{z}{\gamma}$ The tangent on a point $(x_0,y_0)$ along the paraboloide is given by $(x,y,f(x_0,y_0) + \langle \nabla f (x_0,y_0), ...
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4answers
129 views

Show $\lim\limits_{x \to 0}\frac{e^x-1}{x} = 1$ [duplicate]

I have a homework problem which asks me to show $$ \lim\limits_{x \to 0}\frac{e^x-1}{x} = 1 $$ Any help is appreciated.
4
votes
1answer
112 views

I can't understand why Ahlfors' statement is true (isolation of singular points)

In Ahlfors' complex analysis text, page 264, he writes: When $\Omega$ is the whole plane $F(\zeta)$ has isolated singularities at $\zeta = 0$ and $\zeta=\infty $. Reading the previous sections, ...
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0answers
41 views

Checking if a function is in the Schwartz space of rapidly decreasing functions.

Is there any neat bi-implication other than the definition that I can use to check this? This question was motivated by a question that asked if $ f(x) = e^{-|x|^3}$ was in S. It isn't infinitely ...
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0answers
174 views

Classifying Singularities

This is my first post here. I'm working on a complex analysis questions, where I am asked to find and classify all of the the singularities of a given function. My function is as follows: $$f(z) = ...
0
votes
1answer
240 views

Discontinuity of second kind with some properties

My friend ask me to construct a function with infinite discontinuity of second kind(i.e. one of $\lim_{x\to x_0^-}f(x)$ and $\lim_{x\to x_0^+}f(x)$ doesn't exists) defined on $[0,1]$, such that the ...
1
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0answers
63 views

Computation of the series: $\sum_{n=1}^{\infty} \frac{\sin (n)}{n}$ [duplicate]

How to compute the series above? Thanks in advance.
0
votes
1answer
56 views

Convergence of iterative method

Assume that iterative method: $x_{k+1}=F(x_{k})$ where $(k=0,\,1,\,2,\,...)$ converges to $\alpha$ which is root of $f(x)=0$ equation. Prove that if $F(\alpha)=\alpha$, $ ...