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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4
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0answers
9 views

Brezis Exercise 6.10, why does it follow that $Q(S) \in \mathcal{K}(E)$?

Here is the third part of Brezis, Exercise 6.10. Let $Q(t) = \sum_{k = 1}^p a_kt^k$ be a polynomial such that $Q(1) \neq 0$. Let $E$ be a Banach space, and let $T \in \mathcal{L}(E)$. Assume that ...
0
votes
0answers
17 views

Which textbook to use with Problems and Theorems in Analysis by Pólya and Szegő?

I'm currently studying Pólya and Szegő's classic problem collection Problems and Theorems in Analysis. Does anyone know whether they originally wrote it with a specific analysis textbook (or series of ...
1
vote
2answers
25 views

For all $x$ close to $c$

What is the precise mathematical meaning of the phrase "for all $x$ close to $c$"? Does it mean that for all $x$ that satisfy the inequality $|x-c|<\epsilon$, where $\epsilon$ is a small positive ...
1
vote
1answer
38 views

Limit of $\frac{\tan(x)-x}{x^3}$ as $x$ approaches $0$ without L'Hospital's Rule [duplicate]

I am trying to find the limit of $\frac{\tan(x)-x}{x^3}$ as $x$ approaches $0$. I know that this can be found by using L'Hospital's Rule 3 times. Is there a way to solve this problem without using ...
-3
votes
0answers
14 views

find the derivative of sq(A) [on hold]

Consider the function sq : Rn×n → Rn×n , A 􏰀→ A2. Calculate Dsq(A) (i.e., give a formula for Dsq(A)B and prove it). Same question for cu : Rn×n → Rn×n , A 􏰀→ A3. i want to be sure for my solution ...
5
votes
4answers
212 views

(Non)Existence of limits

When we say that a limit of a function does not exist in $\mathbb{R}$ (or some metric space) does it make sense to say that it might exist somewhere else? [I am trying to think along lines of ...
0
votes
0answers
29 views

Measure of the set of convergence of a series

I need some help to solve this exercise. Let $a_n$ a sequence of real numbers such that $a_n\geq 0$ for all $n\in \mathbb{N}$. Let $A$ the set $$A:=\{x\in[0,2\pi]:\sum_{n=1}^\infty ...
1
vote
1answer
25 views

Show sums of complex $\sin$ and $\cos$ series

By considering the series $\sum_{n=0}^\infty r^ne^{in\theta}$ for $0<r<1$ show that $$\sum_{n=1^\infty}r^{n}\cos(n\theta)=\frac{1-r\cos(\theta)}{1-2r\cos(\theta)+r^2} \text{ and } ...
1
vote
3answers
30 views

Analyticity of $\tan(z)$ and radius of convergence

Define $\tan(z)=\dfrac{\sin(z)}{\cos(z)}$ Where is this function defined and analytic? My answer: Our function is analytic wherever it has a convergent power series. Since (I am assuming) $\sin(z)$ ...
0
votes
0answers
14 views

sequential characterisation of limits

There is a proposition left on a 2nd year vector calculus notes provided with no proof. I always having trouble writing these kind of proof and I hope someone could provide an answer for future ...
1
vote
2answers
63 views

Is Lebsegue Measure Translation Invariant?

I am trying to prove that the Lebsegue measure is translation-invariant. Namely, given a set $X\subseteq\mathbb{R}$, I'd like to show $X + y$ is measurable and $\mathit{m}(X + y) = \mathit{m}(X)$. ...
1
vote
0answers
24 views

Does the Cauchy Schwarz inequality hold on the L1 and L infinity norm?

So i am wondering if the Cauchy Schwarz inequality holds for all p-norms, not just when p=2, which is the euclidean space. Thank you.
0
votes
1answer
15 views

2-Norm of Non-Square Matrices

So, the 2-norm of an m x n matrix for m=>n is defined by the max singular value/square of the max eigenvalue. But, if it's not square, and you're only given a matrix A (no x-vector), what do you do if ...
0
votes
1answer
31 views

Show that $ f(x)=\sum_{n=1}^{\infty} 2^{-n} f_n(x)$ defines a continuous function on $(0,\infty)$

Let $f_n$ be a sequence of continuous functions on $(0,\infty)$ with $|f_n(x)|\le n$ for every $ x>0$ and $n\ge1$, and such that $\lim_{x\to\infty} f_n(x) =0$ for each $n$.Show that $ ...
2
votes
2answers
34 views

Differentiability implies continuity

This is a but of a more mathematically juvenile question but I'm trying to get all my intuition in order. When taking a limit we can cancel things that might be zero because in taking a limit, we ...
0
votes
0answers
16 views

how to show the equivalence of density

How to show $E$ is dense if and only if $int(\mathbb{R}- E) = \emptyset$ suggestions please. I do not see how to a direct proff
1
vote
1answer
20 views

Complex power series centered at w

For any $w \in \mathbb{C} \setminus \{1\}$, find a power series for $$f(z)=\frac{1}{1-z}$$ centred at $w$ and give the radius of convergence. Further, find a power series for $f$ ...
1
vote
1answer
31 views

How to construct a complete set in $L^2(\mathbb{R}^3)$ starting with the Spherical Harmonics?

The Spherical Harmonics form a complete set of functions on the sphere $S^2$, so that any function of $f: S^2\to \mathbb{R}$ can be written uniquely as $$f(\theta,\phi)=\sum_{l=0}^\infty ...
0
votes
2answers
23 views

Prove that $C^1[0,1]$ is space of continuously differentaible function with $C_1$ norm is separable.

$C^1[0,1]$ is space of continuously differentiable function with $C_1$ norm.Then the space $ (C^1[0, 1],)$ is a separable space. I am thinking of c^1[0,1] is subset of c[0,1], and c[0,1] is separable. ...
0
votes
1answer
8 views

1D diffusion equation with boundary condition

Suppose we have the diffusion equation defined by $$u_t(x,t) = \Delta_x u(x,t) \ \ \ \ \text{in}\ \ (0,L)\times (0,T)$$ with the boundary condition $$u(0,t)=f(t)$$ $$u(L,t)=g(t)$$ and the initial ...
0
votes
2answers
20 views

Does a closed set not discrete have a limit point?

My Question: Let $U\subseteq \mathbb{C}$ open and $A\subset U$ be a close set not discrete in $U$, then $A$ must have a limit point in $U$. Remark: I do not know if the statement is true. I know that ...
1
vote
1answer
28 views

Stuck on Applying Cauchy Convergence Criterion - Limit Theory

I get stuck on the following problem for a rather long time that I finally decide to ask for help. The problem is as below: Determine whether the sequence converges by applying Cauchy Convergence ...
5
votes
2answers
260 views

The set of integers is not open or is open

Baby Rudin gives the example of the set of all integers being not open if it is a subset of $\mathbb{R}^2$. If we consider the set of integers in $\mathbb{R}$, is this set also not open? I can find a ...
0
votes
1answer
59 views

Largest subset on which a function is continuous

Let $f: \mathbb{C} \to \mathbb{C}$ a function with $$f(x) =0, ~~~ \text{if} ~~ x = 0 $$ and $$f(x) = (e^x - 1)/x, ~~~\text{if} ~~x \neq 0$$ I want to determine the largest subset $A \subset ...
0
votes
1answer
33 views

$ \sum_{n=1}^\infty \frac{x^4}{(1+x^4)^n} $ converges pointwise on $ R$, but not uniformly on $R$.

Prove Or Disprove, The series summation $$ \sum_{n=1}^\infty \frac{x^4}{(1+x^4)^n} $$ converges pointwise on $ R$, but not uniformly on $R$. I am struggling on this problem in real analysis ...
0
votes
0answers
15 views

Small roots of multivariate polynomials

I'm looking for a program/algorithm (or even "theory"?) which checks if a given multivariate polynomial, say $P(x_1, \dots, x_n)$ has a real root in some given region, say a closed $\varepsilon$-Ball ...
0
votes
1answer
35 views

Prove that $\sum_{n=1}^{\infty} \frac{[nx]}{n^2} $ is discontinuous at $x \in \mathbb Q$

$[x] := x - \lfloor x \rfloor$. I can prove that it is continuous at all irrational points using uniform convergence, but I don't know how to prove discontinuity in this case. I looked at this similar ...
2
votes
2answers
70 views

Show that $\partial A$ is always a closed set

First, I believe there are at least two ways to prove this result. One, constructively, by showing that $\partial A$ contains all limit points. The other, by contradiction, is to suppose that ...
0
votes
1answer
17 views

Bounding the Roots of a Complex-Valued Function

Roots: $Z_1$= $\frac{v(1+ \alpha)+ \sqrt{v^2(1+\alpha)^2 -4 \alpha}}{2}$ $Z_2$= $\frac{v(1+ \alpha)- \sqrt{v^2(1+\alpha)^2 -4 \alpha}}{2}$ It is clear that $|Z_2| \leq|Z_1|$ However I'm stuck on ...
0
votes
1answer
17 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 ...
2
votes
1answer
35 views

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

$x'(t)=-x(t)+ \sin x(t)+t^3$ in $\mathbb{R}$ 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} ...
0
votes
0answers
26 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 ...
1
vote
0answers
24 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 ...
0
votes
0answers
26 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 ...
3
votes
2answers
64 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 ...
0
votes
2answers
46 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 ...
2
votes
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 ...
0
votes
2answers
46 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 ...
2
votes
1answer
39 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
48 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
36 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
1answer
28 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
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
vote
1answer
34 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
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 ...
0
votes
1answer
23 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
34 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
36 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
45 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
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 ...