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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7 views

If two monic polynomials have no common roots, are the coefficients of their product locally diffeomorphic to the product of the coefficients?

Let $P^d (t,\lambda)$ be the "generic" d-th degree monic polynomial $P^d (t,\lambda) = t^d + \sum\limits_{i=1}^d \lambda_i t^{d-i}$ with real coefficients. Let $\lambda(\xi,\eta)$ be given by the ...
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0answers
10 views

Maximal interval of solutions existence: $x'(t)=-x(t)+\sin x(t)+t^3$

$$x'(t)=-x(t)+ \sin x(t)+t^3$$ I consider the function: $$ f(t,x)=-x+\sin x + t^3 $$ $$\frac{\partial f}{\partial x}=\cos x-1$$ I see that: $$\left| \frac{\partial f}{\partial x} \right| < 2$$ ...
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0answers
22 views

Finding a function $g:\mathbb{R}^k\rightarrow \mathbb{R}$ such that $g\geq 0$ when $f_1=\dots=f_{n-1}=0$

I have a function $f:\mathbb{R}^{k+n}\rightarrow \mathbb{R}^n$ with Jacobian nonsingular, and I got some $y$ with $f(x, y(x))=0$ by implicit function theorem. Now I am looking for a function ...
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0answers
19 views

Function from $\mathbb{R}^9$ to $\mathbb{R}^6$ with zero set the orthogonal $3\times 3$ matrices

I am trying to construct a $C^\infty$ function from $\mathbb{R}^9$ to $\mathbb{R}^6$ with zero set the orthogonal $3\times 3$ matrices. I am thinking about mapping $M$ to $MM^T-I$, but am not sure ...
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0answers
19 views

Absolute Convergence of Infinite Weierstrass Product

I am really stuck on something. I need to show the following: Let $U$ be a domain in $\mathbb{C}$. If $f_n: U \to \mathbb{D}$ are analytic functions satisfying that $\sum |f_n - 1|$ converges ...
2
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1answer
18 views

Sequential compactness of smooth functions

Suppose I have a sequence $u_n$ of smooth functions on the $N$-dimensional reals. If $\|D^{\alpha}u_n\|_{\infty} \leq C_{\alpha}$ for all multi-indices $\alpha$, then is it possible to deduce that ...
10
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1answer
132 views

Analytic function $f$ in $\overline{\mathbb{D}}$ satisfying $\left\lvert\,f'(\tfrac{1}{2})\,\right \rvert\leq 8.$

Let $f$ be an analytic function on the closed unit disk $\overline{\mathbb{D}}$. On its boundary $\partial \mathbb{D}$ it holds that $\lvert\,f(z) -z\rvert < \lvert z\rvert$. I now have to show ...
3
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2answers
49 views

Construct $f: X\to Y$ such that $f(p)=p$

Let , $X=[-1,1]\times [-1,1]$ and $Y=\{0\}\times \left[-\frac{1}{2},\frac{1}{2}\right]$. Construct an example of a continuous map $f:X\to Y$ such that $f(p)=p$ for each $p\in Y$. I construct a ...
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1answer
395 views

Bounded functions and infimum/supremum

Question from my homework: ![question] My proof: Let $ S = \{f(x) | x \in \mathbb{R} \} $, since $f$ is bounded it has an infimum & supremum (is this true? I tried to prove it, but have no ...
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2answers
364 views
+150

Laplacian of a radial function

Let $f:\mathbb{R}^n\to\mathbb{R}$ be a radial function, i.e. $f(x)=f(r)$ with $r:=\left\|x\right\|_2$. As stated at Wikipedia $$\Delta f=\frac{1}{r^{n-1}}\frac{d}{dr}(r^{n-1}f')$$ What's the most ...
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2answers
41 views

Show analytically that $te^{-t}$ is not decreasing monotonically.

How does one show analytically that $te^{-t}$ is not decreasing monotonically on $(0, \infty)$? One can consider numbers in the interval $(0, 1]$ and show a counterexample to monotonicity, but ...
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2answers
42 views

Complex analysis using definition of the derivative [on hold]

Question: $f(z) = z + 2iz^2 \operatorname{Im}(z)$ Is the function differentiable at $z = 0$? Where is $f(z)$ analytic? Is there any way to do this using the definition of a ...
0
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1answer
21 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 ...
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1answer
2k views

Convergence and accumulation points proof.

Suppose $\{a_n\}_{n=1}^\infty$, converges to $A$ and $\{a_n: n \in J\}$ is an infinite set. Show that $A$ is an accumulation point of $\{a_n: n \in J\}$. So far I have done the basics of convergence ...
1
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1answer
25 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 ...
0
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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 ...
3
votes
1answer
58 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 ...
0
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3answers
88 views

Norm of vectors inequality [duplicate]

I tried proving this with triangular inequality but i was not right can any one help me with this
2
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2answers
94 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 ...
4
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2answers
47 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 ...
2
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1answer
36 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 ...
5
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4answers
3k views

Computing $ \int_{0}^{2\pi}\frac{\sin(nx)}{\sin(x)} \mathrm dx $

I would like to compute: $$ \int_{0}^{2\pi}\frac{\sin(nx)}{\sin(x)} \mathrm dx $$ We have: $$ \int_{0}^{2\pi}\frac{\sin(nx)}{\sin(x)}\mathrm dx=2\int_{0}^{\pi}\frac{\sin(nx)}{\sin(x)}\mathrm dx$$ ...
1
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1answer
393 views

Continuity of distance function and its generalization

The starting is an easy undergraduate problem. The distance function $d: X \times X \rightarrow \mathbb{R}$ in a metric space $(X,d)$ is continuous. Please check if my proof is correct. If it is wrong ...
0
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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 ...
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0answers
23 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) = ...
1
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1answer
32 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 ...
0
votes
1answer
41 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 + ...
1
vote
1answer
30 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
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2answers
37 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 ...
1
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0answers
33 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 ...
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0answers
32 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.
2
votes
1answer
81 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. ...
2
votes
1answer
34 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
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2answers
42 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 ...
4
votes
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 ...
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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.
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0answers
16 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 ...
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1answer
27 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 ...
0
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2answers
70 views

Criteria to be in weak $L^{p}$ space

Let $X$ be a $\sigma$-finite measure space. Let $f : X \rightarrow \mathbb{C}$ be a measurable function and $1 < p < \infty$. Suppose for $f$ there is a constant $C$ such that ...
1
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2answers
67 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
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0answers
36 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 $f(x)=0$?
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0answers
27 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 = ...
6
votes
1answer
986 views

Check my answer: Prove that every open set in $\Bbb R^n$ is a countable union of open intervals.

I have a question. I have solved this but please can you check my solution? Thank you. If there are any mistakes or something is missing and so on, please tell me. This is important to me. Is this ...
3
votes
1answer
412 views

Selecting the Real Analysis Textbooks

I am a sophomore in US with double majors in mathematics and microbiology. I am interested in self-studying the real analysis starting now since it will help me on my current research on computational ...
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 ...
0
votes
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
28 views

Solutions for the dependency problem

Currently I read about the dependency problem of interval arithmetic. Mainly it's the problem that in the equation $X-X$ for $X$ being an interval the following is calculated: $$X-X=\{x-y:x\in X, y\in ...
0
votes
1answer
63 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 ...
2
votes
3answers
466 views

$E$ is closed $\iff\partial E$ (boundary of set $E$) $\subseteq E$

I am studying topology of euclidean space from William Wade's text book. I saw this question. But I cannot come up with any ideas. Please show me the solution in an instructive an clear way. ...