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
16 views

Graph of the same function represented differently?!

I'm taking pre-calculus classes - learning about functions, limits and that stuff right now - and I came to $y = \sin(\frac{1}{x})$. Google represents it like this: Google_graph_of_$y = ...
0
votes
1answer
34 views

Solving a quadratic complex equation

What is the approach to solving this equation? $$ iz^2 + 2(1 − i)z + 2i + 2(\sqrt{3} − 1) = 0 $$ I do not think that I need the complete solution. Just the approach on how to do it. Please only help ...
0
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2answers
30 views

Vector norm and relationship with euclidean distance

If $y\in E_n$ (n dimensional euclidean space) show that $||\textbf{y}||\leq|\textbf{y}|\leq \sqrt{n}||\textbf{y}||$ Where $||\textbf{y}||$ is the euclidean length of the vector $\textbf{y}$ and ...
0
votes
1answer
36 views

Continuity of $F(x)=\int_{(-\infty,x]}fd\lambda$

For a homework assignment I was told to prove that given $f\in L^1(\mathbb R)$, the following function is continuous $$F(x)=\int_{(-\infty,x]}fd\lambda.$$ I thought to use DCT and show sequential ...
4
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4answers
99 views

Evaluation of $\,\displaystyle \lim_{n\rightarrow \infty}\sum_{k=1}^n\sin \left(\frac{n}{n^2+k^2}\right)$

Evaluation of $\displaystyle \lim_{n\rightarrow \infty}\sin \left(\frac{n}{n^2+1}\right)+\sin \left(\frac{n}{n^2+2^2}\right)+\cdots+\sin \left(\frac{n}{n^2+n^2}\right)$ $\bf{My Try::}$ We can ...
4
votes
1answer
95 views
+100

A sequence of continuous functions which is pointwise convergent to zero and not uniformly convergent on any interval.

The exercise is to construct a sequence of continuous functions $f_n:\mathbb{R}\rightarrow \mathbb{R}, n\in \mathbb{N}$ , which is pointwise convergent to $f(x)=0 , x\in \mathbb{R}$ and not uniformly ...
1
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0answers
22 views

Partition of Unity, FEM, show basic estimate

I've been reading this paper and in remark 2.1. it is mentioned that the estimate (9) from Def 2.1. $$\|\nabla \varphi \|_{L^\infty (K)}\leq \frac{C}{diam \Omega}$$ follows from the regularity of the ...
2
votes
2answers
96 views

Is $C[0,1]$ reflexive?

I.e. is the embedding $C[0,1]\hookrightarrow \left( C[0,1] \right)^{**}$ surjective? I am having a hard time answering that question. It would be enough to find a closed subspace of $C[0,1]$ which ...
0
votes
1answer
14 views

Limits of the derangements proportion within the permutations of the set $[1,n]$

Let be $D_n$ the number of derangements of a set of $n$ elements, by convention we have $D_0=1$ Ifound that $D_n=n!\sum\limits_{k=0}^{n}\frac{(-1)^k}{k!}$ For all $n\in \mathbb{N*}$, we write ...
0
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1answer
64 views

A weaker than “zero derivative” condition implies that a function is constant?

Let $f:(a,b)\to R$ be a continuous function such that $\limsup\limits_{n\to \infty}\frac{f(x_n)-f(x_0)}{|x_n-x_0|}\leq 0$ for every $x_0\in(a,b)$ and sequence $x_n$ converging to $x_0$ such that ...
0
votes
1answer
28 views

Is this operator a distribution?

Is this operator: $$T: \mathcal{C}^{\infty}_0 \ni \varphi \to \lim_{x \to \infty} x^2 e^{-x} \varphi'(x) \in \mathbb{R}$$ a distribution (generalized function)? I need to check two things: whether ...
2
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0answers
23 views

Methods for first order PDEs in higher dimensions

What are the possible known methods for solving first order PDEs in higher dimensions? Is there anything else besides the method of characteristic curves? In particular, I have four first order, ...
7
votes
3answers
94 views

$f'(x) = g(f(x)) $ where $g: \mathbb{R} \rightarrow \mathbb{R}$ is smooth. Show $f$ is smooth. [duplicate]

Suppose $f: \mathbb{R} \rightarrow \mathbb{R} $ is differentiable and $g: \mathbb{R} \rightarrow \mathbb{R} $ is infinitely differentiable, i.e. $ g \in C^{\infty}(\mathbb{R})$, where we know ...
0
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0answers
28 views

Baby Rudin theorem 6.16, explanation that a Riemann Stieltjes integral could be expressed as a infinite series.

The theorem says: Suppose $c_n \geq 0$ for $1,2,3 ...$. $\sum c_n$ converges, $\{s_n\}$ is a sequence of a distinct points in $(a,b)$, and $\alpha (x) = \sum^{\infty}_{n=1} c_n I(x-s_n)$. Let $f$ be ...
3
votes
1answer
42 views

Prove a vector in $\ell^2(\mathbb{Z})$ is zero

Suupose we take a vector $\vec{c}\in\ell^2(\mathbb{Z})$ where $$c(i)=\sum_{k=1}^\infty\frac{c(-k+i)+c(k+i)}{k+1}$$ That is, every elements of the vector is a series with the other terms in $\vec{c}$. ...
5
votes
1answer
75 views

If $\sum a_n$and$\sum b_n$diverge, can$\sum \min\{a_n,b_n\}$converge? [duplicate]

Do there exist sequences $\{a_n\}$ and $\{b_n\}$ satisfying all of the following properties? $a_n>0$ and $b_n>0$ $\{a_n\}$ and $\{b_n\}$ are both decreasing $\sum a_n$ and $\sum b_n$ both ...
0
votes
1answer
34 views

Derivative of an integral on a level set

Consider a mapping $\xi:\mathbb{R}^d\rightarrow\mathbb{R}^k$ such that $D\xi \, D\xi^T>\delta\, I_k$. Here $D\xi:\mathbb{R}^d\rightarrow \mathbb{R}^{k\times k}$ is the Jacobian. Consider a ...
1
vote
1answer
13 views

Conditions for invariance under flow.

I am beginning to study dynamical systems. We are given $U \subset \mathbb{R}^n$ open, a vector field $f: U \to \mathbb{R}^n$, and an associated evolution operator for fixed $t \in \mathbb{R}$ ...
2
votes
4answers
49 views

Is $\{\frac{m}{10^n}\mid m,n\in\mathbb Z,\quad n\geq 0\}$ dense in $\mathbb R$?

The set $S$ of real numbers of the form $m/(10^n)$, $m,n$ integers and $n$ greater than equal to $0$, is dense sunset of $\mathbb R$ or not?? I know dense means closure of $S$ in $\mathbb R$ is ...
1
vote
1answer
19 views

Let $n>1$ and $g_1,…,g_{n-1}$ be $C^2$ scalar fields over $\mathbb R^n$ , then for any scalar field $f$ , is $\det J(f,g_1,…,g_{n-1})=0$?

Let $n>1$ and $g_i:\mathbb R^n \to \mathbb R$ be scalar field for each $1\le i\le n-1$ such that all second order partial derivatives of each $g_i$ exist and are continuous ( i.e. each $g_i$ is ...
0
votes
1answer
32 views

Epsilon delta proofs of theorems of continuity

Can anyone suggest a book which contains epsilon delta prooves for properties and theorems of continuity rather than sequential proofs.
2
votes
1answer
37 views

A relation between two properties of sequences of operators

We have $(T_l)_l$ a sequence of bounded linear operators from $\ell^2$ to $\ell^2$. $\bullet$ We say $(T_l)_l$ satisfies the property "A" if ...
1
vote
3answers
58 views

Integration of a real valued function on complex plane

Suppose $f: \mathbb{C}\rightarrow \mathbb{R}$ $f$ is continuous, bounded, $f(z)\geq 0$. Can we claim that the following integration $$\int_{C_R}f(z)dz$$ is equal to zero? ($C_R$ is a circle ...
0
votes
1answer
25 views

Lipschitizianity of the square root of a positive $C^2$ function

I was trying to solve this exercise. Let $f\in C^2(\mathbb{R})$ a strictly positive function such that $f''$ is bounded. Then prove that $\sqrt{f}$ is Lipschitz. A first idea was to prove that it's ...
1
vote
1answer
20 views

Inequality for $u \in L^r(\Omega)$: $\int_{\Omega} (a_1+a_2|u|^{r/s})^s dx \leq a_3 \int_{\Omega} (1+|u|^r)dx$

This question is from one of the steps of the Proof of Proposition B.1 in Appendix B of P. H. Rabinowitz's "Minimax Methods in Critical Point Theory with Applications to Differential Equations." Let ...
1
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0answers
34 views

Fourier coefficients of the Gaussian.

I would need to find the fourier coefficient of this gaussian for a problem. I'm now stuck with this integral, \begin{equation} c_{n}=\int_{-1}^{1}e^{\frac{x^{2}}{2}}\left(\cos\left(\pi ...
4
votes
4answers
50 views

limit of form “$∞ \cdot 0$”

I am trying to formally prove that limit of $2^n\sin(π/2^n)$ as $n$ approaches infinity is $π$. Generally I can tell limit of each term of product of $∞$ and $0$ respectively, but am little confused ...
0
votes
1answer
21 views

question about weierstrass approximation theorem true or false justify [closed]

Is the following assertion true or false? There exists a nonzero function $f \in C([0,1])$ such that $$\int_0^1f(x)x^ndx=0 (\forall n \in \mathbb N)$$ holds. (Hint: use the weierstrass approximation ...
1
vote
1answer
25 views

When is the Stieltjes integral of bounded variations?

I was trying to figure out when a Riemann or Lebsgue Stieltjes integral is of bounded variation. For simplicity let $f$ be a increasing RCLL function; when is that $$\int_0^t g(x) df(x)$$ is of ...
1
vote
1answer
17 views

Application of uniform boundedness principle for sequence of operators

Let $T_n:X:=C([0,1])\rightarrow C([0,1])=:Y$ be a sequence of linear operators. Suppose: $T_n$ is bounded $(Tf)(x):=\lim\limits_{n\to\infty}(T_nf)(x)$ exist for every $f\in C([0,1])$ and $x\in ...
0
votes
1answer
36 views

The derivatives of Riemann xi function

What are the first few values of derivatives of Riemann xi function at zero? Is there any general formula for calculating the nth derivative of the riemann zeta function at zero? What happens to the ...
0
votes
1answer
50 views

Why is continuity needed to substitute value of derivative inside Riemann-Stieltjes Integral?

Given $f$ increasing on $[a,b]$, $g(x)\in R(\alpha)$ on $[a,b]$, $\alpha \in C([a,b])$ and $\alpha \in BV([a,b])$ $$ \beta(x)=\int_a^xg(z)d\alpha(z) \text{ on [a,b]} $$ Why is the additional ...
0
votes
1answer
37 views

If $(X,d_1)$ and $(X,d_2)$ two connected metric spaces if only if $X\times Y$ is connected metric space

$(X,d_1)$ and $(X,d_2)$ are two connected metric spaces if and only if $X\times Y$ is a connected metric space with metric $$ D((x_1,y_2), (x_2,y_2)) = \max(d_1(x_1,x_2),d_2(y_1,y_2)).$$ I know that ...
0
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0answers
15 views

Dense convex set in $*$-weak topology

Let $X$ be a Hausdorff topological vector space over $\mathbb{K}$. Suppose $W$ is a convex subset of its topological dual $X'$. How to prove that if for any $x\in X\setminus\{0\}$ set $\{f(x):f\in ...
0
votes
1answer
19 views

Proving that a set infers a norm given certain conditions

Let $\bar{B}$ be a set in a vector space $E$. Then $\bar{B}$ is the closed unitary ball for a norm iff $\mathbb{N} . \bar{B} = E$ $\lambda x + (1-\lambda)y \in \bar{B}$ for all $x \, ,y \, \in ...
0
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0answers
17 views

Show that the function is identically Zero in certain subset

We are given a open ball D (radius = 1) in $\mathbb R^2$. and let $\{x_n\}$ be the dense sequence in the set D. Around each point $x_n$ we make a hole of radius $r_n$. The sequence $r_n$ satisfy the ...
0
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0answers
14 views

Determining the unit normal field of a paraboloid $P$, and integrating a vector field over $P$

Let $M \subseteq \mathbb{R}^n$ be a $n-1$-dimensional manifold, and $N_x M$ the normal vector space of $M$ at a point $x \mathbb{R}^n$, that is, the (1-dimensional) space of vectors that are ...
1
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2answers
52 views

homogeneous differential equations $y' = f(y/x)$

There is a weird Theorem that comes about when considering whether a function is homogeneous (in the sense of the title definition). I was unable to prove it, or to find a proof to it. Can any one ...
0
votes
1answer
27 views

prove $\sup X \le \sup Y$ if $X$ is a subset of $Y$ both sets are nonempty and $Y$ is bounded above

Working through foundations of mathematical anlysis by johnsonbaugh per suggestion and wondering if the following proof works? (no solutions to book) Problem: Let X and Y be nonempty subsets of real ...
1
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0answers
8 views

Performing and interpreting a Fourier Transform with the Excel Data Analysis Pack

I am confused about how to perform a discreet fourier transform in excel and what the output means. There are many math.stackexchange questions about this topic (see for example ...
0
votes
1answer
21 views

Continuous map from $L^r(\Omega)$ to $L^s(\Omega)$.

The following theorem appears in the appendix of P.H. Rabinowitz monograph on Critical Point Theory: Let $\Omega \subset \mathbb R^n$ be bounded. Let $g$ be such that (i) $g \in C(\overline{\Omega} ...
0
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0answers
16 views

Runge's Theorem Application

Below is a question out of Gamelin's Complex Analysis which I cannot quite figure out. Any tips would help appreciated! "Let $(z_j)$ be a sequence of distinct points in a domain $D$ that accumulates ...
0
votes
1answer
43 views

About the proof of the four colour theorem

The proof of the classification of finite simple groups is thousands of pages which does not seem to be a human readable proof because its too long (more than 10000 pages) to be read by one person ...
1
vote
2answers
50 views

Proof by induction: $(a+b)^n=a^n+na^{n-1}b+\frac{n(n-1)}{2!}a^{n-2}b^2+…+nab^{n-1}+b^n$

I'm studying in preparation for a Mathematical Analysis I examination and I'm solving past exam exercises. If it's any indicator of difficulty, the exercise is Exercise 1 of 4, part $a$ and graded for ...
1
vote
2answers
44 views

ELI5: Riemann-integrable vs Lebesgue-integrable

I am wondering what the difference between riemann-integrable and lebesgue-integrable means. Does it have anything to do with the absolute value of the integrand, something like ...
0
votes
0answers
45 views

Bound the first derivative of the following function: $f(x)=g(x)+h(x)$

Consider a decreasing function $f(x)$, resulting from the sum of two other decreasing functions: $f(x)=g(x)+h(x)$. All these 3 functions are positive. In addition, we have $g(0)=h(0)=1$. Further, we ...
-2
votes
0answers
15 views

Volume Problem in Munkres' analysis on manifolds [closed]

I am having trouble with problem (a) of this question. I figured that the volume of $\triangle_1(R)$ is $|(\alpha(a+h, b)-\alpha(a, b))\times (\alpha(a+h, b+k)-\alpha(a, b))|$ but don't know how I ...
4
votes
1answer
58 views

How to prove that for all $k\in\mathbb N$, $h(kx)=kh(x)$ and $h(x+y)\le h(x)+h(y)$?

Suppose $X$ is a commutative monoid and $f:X\to\mathbb R\cup\{\infty\}$ a function and $$g(x)=\inf\left\{\sum_{i=1}^nf(x_i)~\middle\vert~\sum_{i=1}^nx_i=x,n\in\mathbb N\right\}$$ ...
2
votes
2answers
92 views

Theorem 2.43 in Baby Rudin: How to understand the proof?

Here's Theorem 2.43 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition: Let $P$ be a non-empty perfect set in $\mathbb{R}^k$. Then $P$ is uncountable. Here's the ...
-2
votes
0answers
39 views

Show that $\lim_{z\to z_0} cf(z)=ac$ (where $c$ is a complex number) [on hold]

How do I prove this? Suppose that $a, b$ and $c$ belong to $\mathbb C$ and that $$\lim_{z\to z_0} f(z)=a$$ and $$\lim_{z\to z_0} g(z)=b.$$ a - $\lim_{z\to z_0} cf(z)=ac$ (where $c$ is a ...