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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3
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2answers
383 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
156 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
52 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
177 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
140 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 ...
5
votes
3answers
198 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
53 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
108 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
41 views

Inequality for finite harmonic sum and logarithm

How do you prove the inequality: $|\sum_{k=1}^n 1/k - \log n | \leq 1$ ?
4
votes
1answer
490 views

Using geometric arguments to solve an analysis problem

Im not good in geometric interpretations... any help is very welcome. Consider the unitary disc $$D=\{(x,y,0)\in\mathbb{R}^3, x^2+y^2\leq1\},$$ parameterized by ...
0
votes
3answers
1k 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? $2\leq x <\infty$
2
votes
0answers
119 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
2answers
76 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
33 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 ...
3
votes
2answers
372 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
88 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. ...
12
votes
1answer
1k 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
58 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 .
1
vote
0answers
89 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 ...
2
votes
0answers
46 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 ...
4
votes
2answers
364 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 ...
1
vote
0answers
54 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
76 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
119 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
227 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
129 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
79 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
91 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 ...
2
votes
1answer
227 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
27 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 ...
0
votes
1answer
58 views

Proving that $x\in E^{o} \iff B_{r}(x)∩ E^{c}\not= \varnothing$

I know it is so easy proof. But I am confused. Remark: $x\in E^{o} \iff B_{r}(x)∩ E^{c}\not= \varnothing$ Proof (İf) suppose $x\in E^c$ and $B_{r}(x)∩ E^{c}=\varnothing$ Then we have $B_{r}(x)⊆ ...
1
vote
1answer
106 views

How can I know the time difference $(\Delta t)$ between two cities aren't in the same latitude?

I'm trying to measure the time difference $(\Delta t)$ between two cities (London, Moscow) (they aren't at the same latitude) but I'm facing problem because the speed of earth rotation ($\nu$) depends ...
2
votes
1answer
101 views

compute an integral with residue

I have to find the value of $$\int_{-\infty}^{\infty}e^{-x^2}\cos({\lambda x})\,dx$$ using residue theorem. What is a suitable contour? Any help would be appreciate! Thanks...
0
votes
1answer
217 views

Slightly confused about the definition of upper limits and lower limits.

I'm reading "The way of Analysis" by Strichartz, and the following is the definition of an upperlimit. The upper limit (limsup) of a sequence $\{x_j\}$ is the extended real number $$\limsup _{k ...
2
votes
1answer
84 views

Completeness for bimetric spaces

Let $(X,d)$ be a complete metric space. Is it possible to find a second metric $d'$ such that $d(x,y) \le d'(x,y)$, $\forall x,y\in X$ for which $(X, d')$ is not complete?
0
votes
1answer
114 views

prove that if $V$ is open in $\Bbb R^n$ then there are open balls such that $V=\bigcup_{j\in\Bbb N} B_j$

Prove that if $V$ is open in $\Bbb R^n$ then there are open balls such that $V=\bigcup_{j\in\Bbb N}B_j$. I have the solution, but it is too short and it is not enough to prove it, also it's too ...
3
votes
1answer
135 views

Differentiate a hypergeometric function expression

I have the following function $$f_\epsilon (p)=\frac{1}{2}(1-p)^\epsilon 2^\epsilon {_2}F_1(1-\epsilon,\epsilon;1+\epsilon;\frac{1-p}{2}),\qquad p\in(-1,1).$$ Here $F$ is the hypergeometric ...
2
votes
1answer
66 views

Making a function in $W^{1,2}$ continuous

Let $\Omega$ be an open domain in $\mathbb{R}^n$, $u\in W^{1,2}(\Omega)$ and assume that for any $y$ in $\Omega$ $$\lim_{\varrho \to 0} \operatorname{osc}(u,B(y,\varrho)) \rightarrow 0 , \varrho ...
1
vote
0answers
93 views

Finding a complex function $f$ and the residue of $f '(z)$ at $z=0$

Let $U,a$ real positive constants, $\varphi_1, \varphi_2$ $C^1$ functions on $[0,a]$ with $\varphi_1(0) = \varphi_2(0)$ and $\varphi_1(a) = \varphi_2(a)$. The problem is to find an analytic function ...
2
votes
2answers
232 views

How can I know the time difference between two cities almost at the same latitude?

Well I know that's the earth rotation speed is: $v=1669.756481\frac{km}{h}$ I have two cities New York, Madrid almost at the same latitude and the distance between them is: $d=5774.39$ $km$ ...
1
vote
1answer
192 views

Unexpected Probability Theory Uses

I am a french student in mathematical engineering. I had to go trough an intensive 3 year "preparation" to pass a "concours" to go to High School. In mathematics, I have been taught a lot of algebra, ...
1
vote
2answers
170 views

How can I know the time difference between two cities by knowing the distance between them and earth speed?

Well I know that's the earth speed is: $v=1669.756481\frac{km}{h}$ and I have two cities Moscow and NewYork the distance between them is: $d=7518.92$ $km$ Actually I know that's : ...
1
vote
0answers
31 views

Distributional convergence question for Feller processes

To briefly go over the setup, $S$ the state space is a separable locally compact metric space, and $C_0$ is the space of continuous functions on $S$ that vanish at infinity. $D$ is the space of ...
0
votes
1answer
95 views

Extension of Cauchy sequentially regular function

To prove: If $A$ is a subset of a metric space $(X,d)$ and there is a function $f$ from $A$ to a complete metric space $(Y,e)$ which maps Cauchy sequences to Cauchy. Then there exists a unique ...
8
votes
1answer
197 views

Algebraic Proof of Stone-Weierstrass

I have seen a few proofs of the Stone-Weierstrass theorem today for the first time. While some were completely analytic, the proof given in Rudin's Principles of Mathematical Analysis hinted at an ...
6
votes
1answer
172 views

Caccioppoli-Leray Inequality for De Giorgi's regularity theorem

I am studying De Giorgi's proof of Holder continuity of solutions of elliptic equations with bounded measurable coefficients. This is the translation of the original paper De Giorgi paper At page ...
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vote
0answers
31 views

Multiplication $ H\times P(1/x) $ in sense of distributions

If $ P(1/x) $ means the principal value of $1/x$ and $ H(x) $ is the Heaviside step function is this then correct (regularization) ...
1
vote
2answers
88 views

To express a maximum in terms of the original function

Suppose that you have a continuous function $$ S \colon \mathbb{R}\times [0, 1]\to [0, \infty).$$ Define an auxiliary function $$S^\star(x)=\max_{t\in[0,1]}S(x, t).$$ Does there exist a continuous ...
0
votes
1answer
288 views

prove that $f$ is continuous on $A$ if and only if $f^{-1}(V)$is open in $\Bbb R^n$ for every open subset $V$ of $\Bbb R^m$

Suppose that $A$ is open in $\Bbb R^n$ and $f$ is a function from $A$ to $\Bbb R^m$. Prove that $f$ is continuous on $A$ if and only if $f^{-1}(V)$is open in $\Bbb R^n$ for every open subset $V$ of ...
7
votes
1answer
155 views

Prove the density of this SDE is not smooth in a parameter

Consider the following, 1-dimensional, equation $$X_t^x = x + \int_0^t \mathbb{E} |X_s^x| \, ds + B_t , $$ where $B$ is a Brownian motion. This a McKean-Vlasov equation, sometimes called a nonlinear ...