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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2
votes
1answer
33 views

Proving $a^b$ is well defined

How do I prove that $$\lim_{(m,n) \to \infty} a_m^{b_n} = a^b$$ where $a,b \in \mathbb R$, $a_i,b_i \in \mathbb Q$, $a_m \to a$, $b_n \to b$ and $a$ and $b$ are not both zero, and $a_m >0$ I can ...
4
votes
2answers
39 views

Is there an identity that says $|\sqrt {a^2+x^2} - \sqrt {a^2+y^2}| \leq |\sqrt {x^2} - \sqrt {y^2}|$?

Is there an identity that says $|\sqrt {a^2+x^2} - \sqrt {a^2+y^2}| \leq |\sqrt {x^2} - \sqrt {y^2}|$? Because of the nature of the square root function, its derivative monotonically decreases. so ...
1
vote
1answer
28 views

$\exp\left({\frac{-1}{(x-a)(b-x)}}\right) $ is infinitely differentiable on $(a,b)$

Let $a<b$. I'm trying to prove that $$\exp\left({\frac{-1}{(x-a)(b-x)}}\right) $$ is infinitely differentiable in the open interval $(a,b)$. Induction seems like a good way to proceed, and I know ...
1
vote
0answers
19 views

Composition is infinitely differentiable

The funcitons below all map real numbers to real numbers. Suppose that $f(x) = h(g(x)) \ \forall x \in \mathbb{R}$. Suppose that $g(x) \neq 0 \ \forall x \in \mathbb{R}$ and that all derivatives of ...
1
vote
0answers
20 views

Upper Bound of Fisher Equation

Could anyone please give me directions on how to establish a non trivial and as good as possible upper bound ($u(x,t) \le u_0$) of the Fisher equation? \begin{cases} u_t = u_{xx} + u(1-u) \\ u(x,0) = ...
0
votes
1answer
17 views

$λ = f dµ$ and $ρ = g dµ$ for Lebesgue measure $µ$, give necessary and sufficient conditions on $f,g$ for $λ ⊥ ρ$ and $λ << ρ$

Le $f,g : \mathbb{R} → \mathbb{R}$ be extended integrable functions. Let $λ = f dµ$ and $ρ = g dµ$ for Lebesgue measure $µ$. Give necessary and sufficient conditions on $f,g$ for $λ ⊥ ρ$ and necessary ...
2
votes
2answers
35 views

Advanced calculus, Riemann integral.

If $f$ is (Riemann) integrable on $[a,b]$ and if $\int_{a}^{b} fh=0$ for all continuous function $h$, then $f(x)=0$ for all points of continuity of $f$. I know if we have $f$ being continuous on ...
0
votes
0answers
9 views

Find $c, M > 0$ such that $\lvert e^{tA}x_0\lvert \le Me^{ct}\lvert x_0\lvert$

In a system of differential equations $x'=Ax$, where $A$ is a constant matrix, and the equation is a sink (all eigenvalues of $A$ have negative real parts), I need to find constants $c,M>0$ such ...
-2
votes
0answers
31 views

Find the polynomial $P$ of lowest possible degree satisfying the given conditions: $P(-1)= 0, P(0)= 2, P(2)= 7$. [on hold]

Find the polynomial $P$ of lowest possible degree satisfying the given conditions: $P(-1)= 0, P(0)= 2, P(2)= 7$. I'm not sure how to construct the polynomial.
1
vote
0answers
30 views

Best Approximate Solution of Heat Equation (Diffusive Logistic Equation)

If $u(x,t) \ge 0$ in the domain $ (0 \times 1) \times (0,\infty)$, find a function that caps the value of $u(x,t)$ in the region $(0 \times 1) \times (0,T)$. $u(x,t)$ is a solution of the following ...
0
votes
1answer
36 views

How to prove complexity of algorithms

I have three different algorithms which I want to prove if they are solvable in polynomial/subexponential/exponential time. The algorithms are $f(k) = e^{\sqrt{\log{k}}}$, $f(k) = k^2 + ...
2
votes
1answer
30 views

Eigenvalue equation and separable solutions

Recently, studying Quantum Mechanics I found a doubt regarding separable solutions and eigenvalue equations for differential operators. Suppose we are considering some space $\mathcal{H}$ of functions ...
1
vote
0answers
15 views

Prove that if $0\le p_n \lt 1$ and $S:=\sum p_n \lt 1$, then $\Pi (1-p_n) \ge 1-S$. [duplicate]

Prove that if $0\le p_n \lt 1$ and $S:=\sum p_n \lt 1$, then $\Pi (1-p_n) \ge 1-S$. I'm having real trouble proving this inequality. I'd greatly appreciate any help.
0
votes
0answers
14 views

Solving $f(x) \leq 10 f(kx) + 10kg(x)$ for $f, g$ nonnegative on $(0, 1]$

Suppose we are given two nonnegative functions $f$ and $g$ on $(0,1]$ that satisfy $f(x) \leq x^{-1/2}$ and $$f(x) \leq 10 f(kx) + 10kg(x)$$ for all $k$ sufficiently large. Is it possible to reduce ...
1
vote
1answer
26 views

A convergent series of irrational numbers only which is not absolutely convergent

While solving another problem I stumbled upon this. I wonder if such a series exists: "a convergent series of irrational numbers only which is not absolutely convergent". I am thinking but I cannot ...
1
vote
2answers
61 views

how to show that $\{x\in \mathbb R^n: f(x)=b\}$ is closed

(1) Let $f: \mathbb R^n \to \mathbb R^m$ be a continuous mapping. Let $b\in \mathbb R^m$. Show $$\{x\in \mathbb R^n: f(x)=b\}$$ is a closed set. My thought: I want to show that the set ...
0
votes
0answers
34 views

$f(x) = 2x \mod 1$ not equal to zero for all $x$?

If any number $\mod 1$ is zero, then how can $f(x) = 2x \mod 1$ be a Baker's map? For any $x\in \mathbb{R}$, shouldn't $f(x)=0$?
1
vote
2answers
35 views

Analytic function $f,$ such that $f(0) = 1$ and $f'(z) = zf(z),$ for all $z \in \mathbb{C}$

I'm trying to find an example of an analytic function $f$ satisfying the IVP $$ f'(z) = z\,f(z), \quad f(0) = 1, $$ and for all $z \in \mathbb{C}$, but I'm somewhat at a loss of the best way to ...
2
votes
1answer
78 views

Differential equation: $\ddot{y}(x) + \alpha\dot{y}^2(x) + \beta y(x) = 0$

I am interested in finding an approximate solution for this differential equation, since the exact analytic solution seems to not exist. I tried with Mathematica and it spits out nothing. ...
0
votes
0answers
23 views

property of complex polynomials

I can't solve the following problem: Let $p(z) = z^n + a_{n-1}z^{n-1} + ... + a_0$ be a complex polynomial of degree $n \ge 1$. Assume that there exist $j \in \{0, 1, ... n-1\}$ such that $a_j \neq ...
0
votes
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
29 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
votes
2answers
29 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
votes
4answers
96 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 ...
2
votes
0answers
34 views

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
vote
0answers
14 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
84 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
12 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
votes
2answers
61 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
27 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
votes
0answers
22 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
91 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
votes
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
40 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
72 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
32 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
48 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
52 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
vote
0answers
32 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
49 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
20 views

question about weierstrass approximation theorem true or false justify [on hold]

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
23 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 ...