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 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 ...
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59 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 $\mathit{m}(X + y) = \mathit{m}(X)$. ...
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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.
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1answer
12 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 ...
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1answer
28 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 $ ...
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2answers
30 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 ...
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0answers
14 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
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1answer
19 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 $f$ ...
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1answer
21 views

Complete set in $L^2(\mathbb{R}^3)$ from 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 ...
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2answers
18 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. ...
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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 ...
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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 ...
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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 ...
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2answers
256 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 ...
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0answers
40 views

Largest subset on which a function is continious

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 ...
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1answer
32 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 ...
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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 ...
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2answers
67 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 ...
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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 ...
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15 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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1answer
33 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} ...
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25 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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23 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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25 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 ...
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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 ...
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2answers
45 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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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 ...
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2answers
45 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
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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 ...
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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 ...
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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 ...
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1answer
27 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 ...
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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
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
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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 ...
0
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1answer
22 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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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.
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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 ...
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1answer
44 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 + ...
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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 ...
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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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17 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
29 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 ...
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2answers
69 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 ...
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0answers
38 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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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 ...
2
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1answer
83 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
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0answers
29 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
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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
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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 ...