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)

1
vote
1answer
76 views

Is a closed n-dimensional disk compact necessarily compact?

As the title asks, is a closed n-dimensional disk compact necessarily compact? I'm thinking the answer would be no. If you consider the case in $\mathbb{R}^1$ then can you define the radius to be ...
3
votes
1answer
55 views

Evaluating limits by subsituting special sequences, justification for that

Sometimes I saw people using transformations like $$ \lim_{x\to 0} f(x) = \lim_{n\to \infty} f(\frac{1}{n}) $$ or $$ \lim_{x \to p} f(x) = \lim_{n \to \infty} f(x + \frac{1}{n}). \quad (*) $$ I know ...
0
votes
2answers
78 views

Taylor expansion of $\log(1+ix)$

How do I obtain the Taylor expansion of $\log(1+ix)$ around $x=0$? I know how to do it for $\log(1+z)$ if $z$ is a real number. But how do I do it (formally correct) in the case of the complex ...
0
votes
1answer
64 views

$a + b = a$ in machine precision [closed]

I have the following statement: "If $a + b = a$, then $b = 0$" may not true with the floating point operations. Actually, if $|y| ‎< (\varepsilon / B) |x|$, then $fl(x+y) = x$, where ...
1
vote
2answers
266 views

convergence in $L^1$ for product of functions

If $f_n$ converges to $f$ in $L^1$ and $g_n$ converges to $g$ in $L^1$. Does it necessarily mean that $f_ng_n$ converges to $fg$ in $L^1$ for finite measure spaces.
3
votes
0answers
47 views

inequality of some integrals of continuously differential function.

Let $f:[a,b]$→$\mathbb{R}$ be a continuously differential fuction satisfying f(a)=0. My goal is to show that $$\int_{a}^{b} |f(x)|^2 dx \le \frac{(b-a)^2}{2} \int_{a}^{b} |f'(x)|^2 dx $$ My ...
3
votes
1answer
132 views

Solution to a tricky inequality (math analysis)

Let $p>1$ and put $q=\frac{p}{p-1}$, so $1/p+1/q=1$. Show that for any $x>0$ and $y>0$, we have $$ xy \le \frac{x^p}{p}+\frac{y^q}{q}$$ And find where the equality holds. So far, I have ...
2
votes
1answer
31 views

how to show that $A(x)\nabla u\in L_\mathrm{loc}^{2}(\Omega) $ for $u\in H_\mathrm{loc}^{1}(\Omega)$

Let $\Omega\subset \mathbb{R}^n$ be a connected open set containing $0$, $u\in H_\mathrm{loc}^{1}(\Omega)$, $A(x)\leq C|x|^{-1+\epsilon}$, where $\epsilon$ is small, and we also have $$ \|\nabla ...
0
votes
1answer
33 views

Why would $f_n(x) = (\lfloor 2^nf(x)\rfloor/2^n)\wedge n$ converge to $f(x)$?

Why would $$f_n(x)=\frac{\lfloor 2^nf(x)\rfloor}{2^n}\land n$$ converge to $f(x)$? I saw this step in the proof of change of variable formula in Rick Durrett's Probability Theory and Examples.
0
votes
1answer
168 views

Integrals Regulated functions

stuck on an example for this question, Give an example of a regulated function $f \colon [a,b] \to \mathbb{R}$ with the properties that $\forall x \in [a,b] f(x) \ge 0 , \int_a^b f = 0$ and there is ...
0
votes
1answer
18 views

Using Taylor's series

Using Bayes Theorem I have solved a problem to the equation P(Dc|-) = (0.98-0.98p)/(0.98-0.93p) between the interval [0,0.1]. Show (e.g., by means of Taylor series) that in this interval the P(Dc|−) ...
2
votes
3answers
77 views

Minimum of set $\{\frac{m}{n} + \frac{4n}{m}\}$

We have the following set: $\mathcal{A} = \{ \frac{m}{n} + \frac{4n}{m};\ \ m, n \in \mathbb{N} \} $ Attempting to prove that the set's minimum is 4 yields: $$\frac{m}{n}+\frac{4n}{m} = \frac{m^2 + 4 ...
0
votes
1answer
20 views

Addition of distributions in statistics

Is it possible to add distributions? I've worked out "Say that you are given ten identical coins for which you assume Beta(4,4) prior distribution on the unknown probability θ of any of the coins ...
2
votes
0answers
99 views

A problem concerning measures on locally compact spaces

I am stuck on a question for quite sometime now, although in the text (http://www.latp.univ-mrs.fr/~chaabi/ARTICLES%20IMPORTANTS/Autres%20articles%20interessant/Jewett.pdf , Pg. 10, 2.3E ) it is said ...
2
votes
1answer
121 views

Maximal unique solution to an IVP.

In class we learned the existence and uniqueness theorems for differential equations. The weaker Picard-Lindelof states that for any IVP, $$ \begin{cases} x'(t) = f(t, x(t))\\ x(t_0) = x_0 \end{cases} ...
1
vote
2answers
155 views

Prove that $\lim_{n\to\infty}\frac1{n}\int_0^{n}xf(x)dx=0$.

Let $f$ be a continuous, nonnegative, real-valued function and $$\int_0^{\infty}f(x)dx<\infty.$$ Prove that $$\lim_{n\to\infty}\frac1{n}\int_0^{n}xf(x)dx=0.$$ A start: If ...
10
votes
2answers
284 views

Find all functions $f:\mathbb{R}^+\to \mathbb{R}^+$ such that for all $x,y\in\mathbb{R}^+$, $f(x)f(yf(x))=f(x+y)$

Find all functions $f:\mathbb{R}^+\to \mathbb{R}^+$ such that for all $x,y\in\mathbb{R}^+$$$f(x)f(yf(x))=f(x+y)$$ A start: set y=0 to get $f(x)f(0)=f(x)$. So $f(0)=1$ unless $f$ is identically zero.
1
vote
2answers
86 views

Determine the value of $p>0$ for which $\sum_{n=1}^{\infty}(-1)^{\lfloor{\sqrt{n}}\rfloor}/n^p$ converges.

Determine the value of $p>0$ for which $$\sum_{n=1}^{\infty}\frac{(-1)^{\lfloor{\sqrt{n}}\rfloor}}{n^p}$$ converges. By considering $\lfloor{\sqrt{n}}\rfloor$, we see the series is $$\sum_{k\ge1} ...
2
votes
1answer
165 views

Could we write Fourier transform as a matrix?

I have heard that Fourier transform is a linear transformation. I have also heard that any linear transformation can be written as a matrix multiplication. (probably I'm missing some details in the ...
0
votes
2answers
58 views

Why is this set closed?

Let $(X,d)$ be a metric space. Let $a \in X$ and $r \ge 0$. Define: $E_r(a) = \{b \in X : d(a,b) \le r\}$ I want to show that $E_r(a)$ is closed. Here's what I know: $E_r(a)$ is closed if every ...
3
votes
2answers
594 views

Prove open set is not closed

The question might sound ridiculous, but I am not able to prove it with rigor. I tried proving it by the following definitions ONLY. Open set: A set $U$ is open if for every $a$ belonging to $U$, ...
1
vote
0answers
49 views

The Fourier Stieltjes transform is uniformly continuous

Let $G$ be a locally compact Abelian group and $\hat{G}$ be its dual group, that is the group of all complex functions $\gamma:G\to\mathbb C$ such that ...
4
votes
2answers
94 views

$\{(x,y):x\in{U},y>f(x)\}$ is an open subset of $\mathbb{R}^{n+1}$

Let $f$ be a continuous real-valued function defined on an open subset $U$ of $\mathbb{R}^n$. Show that $\{(x,y):x\in{U},y>f(x)\}$ is an open subset of $\mathbb{R}^{n+1}$ I known $f(x)$ is open ...
1
vote
1answer
75 views

Natural Analysis: a possible new field?

I have studied real analysis for two years now, yet I often find that when applying it, (say to finding a path of shortest time) I often have to get solutions without closed forms (i.e. defined point ...
2
votes
1answer
180 views

Uniform closure of an algebra $\Rightarrow$ uniformly closed algebra

In PMA, Rudin's book, there is the following theorem (7.29): Let $B$ be the uniform closure of an algebra $A$ of bounded functions. (Here, an algebra means a family of function satisfying that it is ...
0
votes
1answer
62 views

Differentiability of scalar function

Let $f:\mathbb R→\mathbb R$ be a continuous function, with $f(0)=0$. Let $F(x,y)=xf(y)+yf(x)$. Analize if $F$ is differentiable at the origin.$$$$ I've proved that ...
2
votes
1answer
45 views

If $A$ is null set, then $\int\limits_A f dm = 0 $

Define $ \int_E f dm = \sup Y(E, f) $ where $ Y(E,f) = \{ \int_E \phi : 0 \leq \phi \leq f \} $ $\phi$ is simple Suppose $A$ is a null set. We show $Y(A, f) = \{ 0 \}$. Pick $x \in Y(A, f)$. So, we ...
0
votes
0answers
68 views

How prove this series $\sum_{n=1}^{\infty}a_{n}$ converges

Question: let $E$ is a point set on $(-\infty.+\infty)$,and let $x_{0}$ is a limit point of $E$(maybe $x_{0}=\pm \infty$ possible),if the series $\sum_{n=1}^{\infty}U_{n}(x)$ converges uniformly ...
2
votes
1answer
521 views

Generalization of absolute continuity with $f(x) = x^a \sin(1/x^b)$

As a generalization of Prove that $x^\alpha \cdot\sin(1/x)$ is absolutely continuous on $(0,1)$ : Let $f : (0, 1] \to \mathbb{R}$ be the function denoted by $f(x) = x^a \sin(1/x^b)$. Determine for ...
1
vote
1answer
221 views

Show that $f$ is constant?!

Problem: Suppose $f$ is a non-vanishing continuous function on $\bar{\mathbb{D}}$(closure of unit disk) that is holomorphic in $\mathbb{D}$. Prove that if $$|f(z)|=1, \mbox{whenever} ...
0
votes
3answers
150 views

Composite bounded functions

Prove $f(x)$ is bounded $\rightarrow$ that $f(g(x))$ is bounded. For all x in $f(x)$ ang $g(x)$. To my understanding, suppose $f(x)$ is bounded, then do we need to show that the composition function ...
0
votes
1answer
38 views

conversion of discrete to continuous

Given $N_{j+1}-N_j=kN_j$ How can I substitute some time variable in to make $delta(t)$ small? Meaning change in time. I need to show $N_j=e^{(j\ln(1+k))}$ How can I rewrite the given in terms of ...
0
votes
1answer
43 views

Reparametrization of an absolutely continuous curve

If $\alpha : [0,1] \rightarrow \mathbb{R^n} $ is $C^1$ and $\alpha'(t) \neq 0$ for all $t\in[0,1]$ then there always exists a reparametrization in which $\| \alpha'(s) \| = 1$. Is there an equivalent ...
1
vote
1answer
32 views

Asymptotic behaviour of a function of a bivariate normal vector

Let $(Z_1,Z_2)$ be a bivariate standard normal vector and $x\in\mathbb{R}$. We consider $$f(\sigma_l):=\left| \operatorname{E}[1\{Z_1\leq x/\sigma_l\}1\{Z_2\leq ...
0
votes
1answer
51 views

Is such a function of bounded variation?

Let $f:[a,b] \rightarrow \mathbb R$ and let $(x_n) $ be a given sequence of points such that: $$ a<x_{n+1} <x_n<b \textrm{ for } n\in \mathbb R, \atop x_n \rightarrow a. $$ Let's assume that ...
2
votes
0answers
44 views

Is $||f||_p$ continuous in $p$ [duplicate]

I just started learning about $L^p$ spaces today and I have this question: Let $(X,\scr{M},\mu)$ be a measure space. Let $f:X\rightarrow \mathbb{C}$ be measurable. Consider ...
0
votes
1answer
90 views

Integration and uniform norm

Suppose that $f$ is twice differentiable on $\mathbb{R}$ and $\|f\|_\infty = A$ and $\|f''\|_\infty = C$. Prove that $\|f'\|_\infty\leq \sqrt{2AC}$. Hint: $f'(x_0) = b > 0$. Show that $b-C|t| ...
0
votes
0answers
129 views

Are there bump functions which have infinitely many smooth integrals?

While it is well-known that bump functions are smooth yet non-analytic $C^\infty$ functions, I was wondering if like the latter bump functions do also possess infinitely many smooth integrals, i.e. ...
0
votes
1answer
15 views

Asymptotics of a real sequence

Let $(a_n)_{n\in\mathbb{N}}$ be a real sequence with $a_n\in O(n^d)$ $(d\in (-1,0))$. Now we consider the expression $$ b_n:=(1-\sqrt{1-a_n}).$$ Is $b_n\in O(\sqrt{n^d})$? Thanks!
2
votes
1answer
39 views

Is the set where $\mathrm{dist}(x,\{1,1/2,1/3,\ldots\})$ is not differentiable a closed set?

Suppose that $ A=\{1,1/2,1/3,...\}$ and $f: \mathbb{R}\to\mathbb{R}$ such that $f(x)=\inf \{|y-x|;y \in A\}$. Let $K$ be the set of points where $f$ is not differentiable. Is $K$ closed? Can $K$ be ...
3
votes
0answers
81 views

$|f(t) - f(s) |\leq \int_s^t g $ then $f(t) - f(s) = \int_s^t h.$

Let $f : [0,1] \rightarrow [0, + \infty)$. If there exists $g \in L^1([0,1]) $ s.t. for every $t,s \in [0,1]$ holds $$ |f(t) - f(s)| \leq \int_s^t g(u) \, du \quad (t>s),$$ then there ...
0
votes
1answer
70 views

Network throughput and message delay

I'm trying to figure out how to calculate the throughput. Throughput is defined as the rate (bits/sec) bits are transferred between a sender and receiver; also, . I have a source node and a ...
0
votes
1answer
203 views

Is $\mathbb R$ a normal topological space?

As in the title, in euclidean space is it always possible two find for two disjoint closed sets $A,B$ two open sets $U,V$ disjoint such that $A \subseteq U$ and $B \subseteq V$ (T4-property, normal)?
0
votes
1answer
34 views

Need help clarifying a proof ( limSn=SupS)

Let $S$ be a bounded nonempty subset of $R$ such that $Sup(S)$ is not in $S$. Prove $\exists$ a sequence $(S_n)$ of points that belong to $S$ such that $ limS_n=Sup(S)$. Let $t=Sup(S)$.then for ...
0
votes
1answer
101 views

differentiability of jump functions

In Stein's real analysis book, we consider a bounded increasing function $F$ on $[a,b].$ Consequently, we know that the set of discontinuities of $F$ on this interval is countable. Because $F$ is ...
3
votes
0answers
106 views

On the existence of $\sqrt{2}$ (guided)

Introduction: This is a homework assignment of mine, first I want to mention that I am aware of that there are many proofs all over the internet (including this site) about the existence of ...
2
votes
3answers
108 views

Prove $\forall r \in \mathbb{R}. \exists k \in \mathbb{Z}. r < k$

I would like to prove that for every real number there exists an integer that is greater than it. My problem lies in that I am not sure how to construct the real numbers and provide their theory with ...
0
votes
1answer
123 views

prove Cauchy sequence

I have a problem in this exercise Suppose that ${(a_n)}$ is a sequence such that ${a_{2n}}$ ${}\le{}{}$ ${a_{2n+2}}$ ${}\le{}{}$ ${a_{2n+3}}$ ${}\le{}{}$ ${a_{2n+1}}$ for all n ${}\geq{}{}$ 0. Show ...
6
votes
2answers
104 views

Zeros of $C^\infty$ functions

If $f(x) \in C^\infty(\Bbb{R})$,and $f(a)=0$, do we have $$f(x)=(x-a)g(x)$$? where $g(x) \in C^\infty(\Bbb{R})$ and $g(a)=f'(a)$
2
votes
2answers
118 views

Natural growth conditions and weak solutions for inhomogenous systems.

Let $\Omega\subset \mathbb{R}^n, \ n\geq 2$ be a bounded Lipschitz domain. Consider the following inhomogenous system (in divergence form) subject to zero Dirichlet boundary conditions: ...