Mathematical Analysis generally, including Real Analysis, Harmonic Analysis, Complex Variable Theory, the Calculus of Variations, Measure Theory, and Non-Standard Analysis.
2
votes
0answers
40 views
Checking my question: the function is continuously differentiable.
I solved a question related to first order partial derivatives. Please check my solution. Is it correct and ehough to get sufficient grade from an exam?
I am not sure espacially part-b. Please check ...
0
votes
0answers
7 views
Applying continuous operators to functions defined by parameter integrals.
Let $T\in \mathcal{L}(\mathcal{S}(\mathbb{R}^{n}))$.
For fixed $f\in\mathcal{S}(\mathbb{R}^{2n})$, define $g(x):=\int_{\mathbb{R}^n} f(x,y)\, dy$. Note that this implies that $g$ is also Schwartz.
...
1
vote
0answers
31 views
On continuity of measure
Let $m$ be a probability measure on $\mathbb{R}^n$.
Consider a function $\ f: \mathbb{R}^n \rightarrow \mathbb{R}$.
Say under what conditions the following inequality holds.
$$ m\left(\left\{ x \in ...
3
votes
1answer
59 views
When $f^{(n)}\to g$ uniformly?
Let $f\in C^\infty([0,1])$ and consider the sequence $$f_n=f^{(n)}$$
where $f^{(n)}$ denote the derivative of order $n$ of $f$. My question is: What is a necessary condition to impose on $f$, such ...
0
votes
2answers
50 views
A discrete space of cardinality $\aleph_0$.
How does a discrete space of cardinality $\aleph_0$ looks like? On finite sets I always get finite discrete spaces, countable sets (i.e. sets of cardinality $\aleph_0$) yields spaces of cardinality ...
-1
votes
0answers
32 views
subdivision or classification of pure math [on hold]
urgently need a sound and systematic subdivision of math.
google says you can go 4 main subdivisions:
1. algrebra
2. geometry
3. analysis
4. combinationtrics
well, ok, but what about arithmetic, ...
1
vote
2answers
31 views
prove the limit inferior of $(x_n)$ where $n \in\mathbb{N}$
The problem states let $(x_n)$ be a bounded sequence for each $n \in\mathbb{N}$. Let $t_n=inf\{x_k: k\geq n\}$. Prove that $(t_n)$ is monotone and convergent.
After a little research because I was ...
-4
votes
0answers
53 views
Asking a calculus book's solution key by fitzpatrick [on hold]
I will take real analysis course next semester. so, I am preparing on summer holiday. for that, I Try to solve exercises from fitzpatrick's advanced calculus book. But I dont have answers. I want to ...
0
votes
3answers
60 views
Proving the function f , which has zero first order parital derivatives, is constant
Let the function $f: \Bbb R^{2} \to \Bbb R$
The first order derivatives of f are zero.
i.e $f_x(x,y)$ = $f_y(x,y)$ = $0$
How can I prove that $f(x,y)$ is constant for all $(x,y)$
4
votes
3answers
111 views
Why such function does not exist?
I could not prove the following:
A function $f \in \mathscr{C}^2([0, \pi])$, such that $$f(0) = f(\pi) = 0,\\
\int_0^{\pi} (f'(x))^2dx = 2,\\
\text{and }\int_0^{\pi} (f(x))^2dx = 1.$$
Then such ...
0
votes
0answers
41 views
approximate Fourier transform
Let $\mathcal{F}$ stand for the Fourier transform. Suppose $f : [-\delta/2,\delta/2] \to \mathbb{C}$ is a "nice" function. Is it true that $$\left|\mathcal{F} \left(e^{imx} \left(e^{ix^2}-1 ...
5
votes
1answer
104 views
Check my answer: Prove that every open set in $\Bbb R^n$ is countable union of open interval
I have a question. I solved this. But please can you check my question? Thank you.
If there are any mistake or lack or and so on, please say me. This is important for me. And is this proof enough to ...
4
votes
2answers
66 views
right continuous continuous function is measurable
Let $f: S \times [0, \infty)\rightarrow \mathbb{R}$ satisfy $f(x, t)$ is continuous in $x$ for each $t$ and right continuous in $t$ for each $x \in S$. Here $S$ is a metric space. Why is $f$ Borel ...
2
votes
1answer
63 views
Inequality involving definite integral
Just wondering, what may be the best way to show that $$\int_0^1 xf(x)dx \leq \frac{1}{2}\int_0^1 f(x)dx,$$ provided that $f(x) \geq 0$ over the interval $[0,1]$ and that $f(x)$ is monotonically ...
4
votes
0answers
43 views
Dual and completion of metric spaces
Say we have a metric space $(M,d)$, and we want to complete it in the following sense:
Definition: A completion of $(M,d)$ is a complete metric space $(\widetilde{M},d')$ together with a Lipschitz ...
1
vote
0answers
53 views
Try the exercise using green
Let $\phi$ and $\psi$ be functions $\left [a,b \right ] \to \mathbb{R}$ of class $C^1$, such that $\phi(a) = \psi(a)$, $\phi(b)= \psi(b)$, $\phi(x) \leq \psi(x)$, $\phi$ is convex, and $\psi$ is ...
0
votes
1answer
32 views
Books to learn about predictive analytics [on hold]
I have a personal data tracking website - to quantify your life basically. I'd like to add some sort of predictions to the systems. Just a few examples:
if you're tracking your mood you possible ...
3
votes
2answers
36 views
Confusion on Derived Sets and the $n$-the Derived Set
I want to solve the following exercise from R. Engelking: General Topology.
For every positive integer $n$ the $n$-th derived set $A^{(n)}$ of a subset $A$ of a topological space $X$ is defined ...
0
votes
1answer
19 views
Linear approximation definition of differentiability
A function $f:\Bbb R^n \rightarrow\Bbb R$ is differentiable at $a$ iff there exists a linear map $L$ and a function $g$ tending to $0$ as its argument tends to $0$ such that:
$$f(a + h) - f(a) = L(h) ...
1
vote
1answer
37 views
Is just continuity enough to prove this?
Sorry if that´s an idiot question. Let $f: D \longrightarrow \Omega$, such that $D$ is the unitary open disc centered at the origin and $\Omega = \{z \in \mathbb{C}; \mathscr{Re}(z) \geq 0 \}$. If ...
0
votes
1answer
17 views
Question on different definitions of upper (hemi)semicontinuity for set-valued maps
In this thesis(page $8-10$), it is asserted, two definitions are equivalent, if the set-valued map $f$ maps to a compact space.
Definition $1$:$f : X \to 2^Y$ is upper semicontinuous if:
$f(x)$ ...
0
votes
1answer
30 views
Lebesgue Measure of a k-cell
Working through Rudin's RCA construction (Theorem 2.20, p. 53) of the Lebesgue measure using the Riesz Representation Theorem. Rudin constructs a linear functional $\Lambda$ on ...
0
votes
0answers
48 views
How can I prove that if $x_n\to s$ as $n\to\infty$ and also if $f'(s)\neq0$, $s$ is a zero of $f$ [on hold]
How can I show that in the secant method, if $x_n\to s$ as $n\to\infty$ and also if $f'(s)\neq0$, then $s$ is a zero of $f$?
I am supposed to use the secant method as well as the mean value theorem ...
4
votes
3answers
46 views
Closure Operator and Set Operations
In Engelking, General Topology stand the following exercise:
Show that for any sequence $A_1, A_2, \ldots$ of subsets of a topological space we have
$$
\overline{\bigcup_{i=1}^{\infty} A_i} = ...
3
votes
1answer
42 views
Distinguishing between the different eigenvalues
Consider the symmetric matrix
$$A=\begin{pmatrix} 2 & t & \cos t-1 \\ t & 2 & 0 \\ \cos t-1 & 0 & 2 \end{pmatrix}. $$
The (real) eigenvalues of $A$ can be found easily using ...
1
vote
1answer
31 views
Show that $x^{\alpha}$ is uniformly continuous on $[1, \infty)$
Fix an $0 < \alpha < 1$ and consider $f(x) = x^{\alpha}$. Show that $f$ is uniformly continuous on $[1, \infty)$.
Work so far: If $\alpha = 1/2$, then we can prove $f$ is uniformly continuous ...
1
vote
2answers
31 views
Inequality for finite harmonic sum and logarithm
How do you prove the inequality:
$|\sum_{k=1}^n 1/k - \log n | \leq 1$
?
3
votes
0answers
41 views
Using geometric arguments to solve an analysis problem
This question is beyond my skills, I think that is because Im not good in geometric interpretations... any help is very welcome.
Consider the unitary disc $$D=\{(x,y,0)\in\mathbb{R}^3, ...
0
votes
3answers
94 views
Upper Bound of Logarithm
For $1\leq x < \infty$, we know $\ln x$ can be bounded as following:
$\ln x \leq \frac{x-1}{\sqrt{x}}$.
Then what is the upper bound of $\ln x$ for following condition?
$2\leq x <\infty$
0
votes
0answers
23 views
Orthogonal decomposition
Can someone explain me what is : the ortogonal decomposition according to $d^2 f(z_i)$ ?
Please help me
Thank you
2
votes
0answers
26 views
Symbol for functions that vanish on boundary?
If I have a domain $ M \subset \mathbb{R}^n $, is there a standard symbol for the set of functions $ f \in C^\infty(M) $ that vanish on $ \partial M$ ? I feel like I have seen this before, but I'm ...
2
votes
1answer
25 views
Showing that this Coercivity condition implies uniform boundedness of a minimising sequence.
The following problem is in Dacorogna's book "Introduction to the Calculus of Variations":
Let $\Omega\subset\mathbb{R}^n$ be open and bounded with a Lipschitz boundary. Let $f\in C(\mathbb{R}^n\times ...
0
votes
0answers
25 views
In set $E$, If $\{y_j\} \to y \in E$ where $y_j$ is an upper bound of $E$ …
I am trying to prove the following.
In set $E$, If $\{y_j\} \to y \in E$ where $y_j$ is an upper bound of $E$ and if $\{x_j\} \to y$ where $\forall j, x_j \in E $, show that $y$ is the supremum ...
2
votes
2answers
44 views
Considering a sum of a monotonically increasing and decreasing sequence.
The following is the problem that I am working on.
Let $\{z_n\} = \{x_n\}+\{y_n\}$ be a sequence where $\{x_n\}$ is monotonically increasing, $\{y_n\}$ monotonically decreasing, and $\{z_n\}$ is ...
1
vote
2answers
39 views
Trying to understand $\sup$ and $\limsup$ of a sequence.
The following is the sequence and a problem that I am working on.
$\{x_n\} = (-1)^n + \frac{1}{n} + 2\sin(\frac{n\pi}{2})$
Find the $\sup$, $\inf$, $\limsup$ and $\liminf$ of this sequence.
...
5
votes
1answer
99 views
Better Proofs Than Rudin's For The Inverse And Implicit Function Theorems
I am finding Rudin's proofs of these theorems very non-intuitive and difficult to recall. I can understand and follow both as I work through them, but if you were to ask me a week later to prove one ...
1
vote
0answers
37 views
Unique continuity property
Can someone told me what is :"the unique continuity property" in the following paragraph ?
and what is the meaning of : .... and either $v\in E(k)$ or $v\in E(k+1)$
Please help me
Thank you .
0
votes
0answers
25 views
Critical groups
I have a small question : What is the relation between the critical groups and the change in topology near a critical point ? ,how we can see this changement ?
Please help me
Thank you .
1
vote
0answers
74 views
Circle in a simplex
Let $T$ be a $2$-dimensional simplex in $\mathbb{R}^2$. A circle $C(x,y,r) \subset \mathbb{R}^2$ is given by its center $(x,y) \in \mathbb{R}^2$ and radius $r\ge 0$.
Show that the set of circles in ...
1
vote
0answers
13 views
question about the conditions on the weak derivative of periodic function
Let $f: \mathbb R \rightarrow \mathbb R$ be a periodic, say with the period 1, and locally integrable function.
Assume that $g: \mathbb R \rightarrow \mathbb R$ is a weak derivative of $f$ of order ...
3
votes
2answers
25 views
question about continuity: using polar coordinates
Given a function $f\colon\mathbb R^2\rightarrow \mathbb R$ I want to study continuity. So I know the $\varepsilon-\delta$ and sequence criterion.
Now we had polar coordinates in lectures: set ...
2
votes
0answers
39 views
Relationship of two generalizations of the real/complex calculus
On the one hand, one has the various functional calculi from Operator Algebras. The continuous functional calculus for C* algebras, the bounded borel functional calculus for Von Neumann Algebras, the ...
0
votes
1answer
60 views
Question on Linearised system
I have this question :
study the nature of the critical point for the linearized system of : $x''+x'^3+x=0$
please how we find the linearised system of $x''+x'^3+x=0$.
Please help me ,
Thank you
1
vote
1answer
38 views
adjoint of an operator
I have a very simple question
Let $V$ be the real (finite-dimensional) inner product space of polynomials of degree at most $2$, with inner product $(p,q):= \int_0^1 p(x)q(x) \, dx$.
Let $T$ be the ...
2
votes
2answers
70 views
$\sum _{p\leq n}\frac{1}{p}=C+\ln \ln n+O\left(\frac{1}{\ln n}\right)$
$\sum _{p\leq n}\frac{\ln p}{p}=\ln n+O(1),n\geq 2,$ where $p$ is a prime number, prove:
$$\sum _{p\leq n}\frac{1}{p}=C+\ln \ln n+O\left(\frac{1}{\ln n}\right)~~~(1)$$
one examination ...
3
votes
3answers
87 views
Are $\dfrac {a_{n+1}} {a_n}$ and $\sqrt {|a_n|}$ divergent?
Suppose $a_n$ is a convergent sequence. Which of the following may diverge?
$$\dfrac {a_{n+1}} {a_n}$$
$$\sqrt {|a_n|}$$
$$\sqrt[2n+1]{|a_n|}$$
I think that the last two should share the same answer ...
1
vote
2answers
37 views
Stability of solution
I have this differential equation $x'=-x^3$ ,
How to study the stability and the asymtotic stability of $x=0$ ?
Please help me
Thank you .
1
vote
2answers
64 views
Prove $\displaystyle\lim_{n\to \infty}{nb^n}=0$
Prove $\displaystyle\lim_{n\to \infty}{nb^n}=0, 0<b<1$
One last thing I used a certain theorem in my proof which I will give.
Theorem 3.1.10 Let $(x_n)$ be a sequence of real numbers and ...
1
vote
1answer
22 views
Compact $C \subset$ Jordan-measurable $A$ such that $\int_{A-C} 1 < \epsilon$.
This question is (3-22) in M. Spivak's Calculus on Manifolds.
If $A$ is a Jordan-measurable set and $\epsilon > 0$ show that there is a compact Jordan-measurable set $C \subset A$ such that ...
2
votes
2answers
26 views
Non Identical Closure
I'm working on counterexample here. Can we construct two bounded non empty open sets $A,B$ with $A \subset B$ that are $\lambda(A)=\lambda(B)$ but $\overline{A}\ne\overline{B}$? Here $\lambda$ is the ...
