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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32 views

Are all solutions to the ODE $ay''(t) + by'(t) + cy(t) = 0$ of the form $y(t)= \alpha e^{(\beta + i\gamma)t}$?

Let $a$ $b$ and $c$ be complex numbers. Consider the complex solution of the ODE $$ay''(t) + by'(t) + cy(t) = 0.$$ If there exist solutions to this, are they necessarily of the form $$y(t)= \alpha ...
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0answers
12 views

Relation of rate of decay of a function with width of peaks of its Fourier transform

Consider a function $f(t)=\theta(t)e^{-\sigma_0 t}\sin(\omega_0 t)$, where $\theta(t)$ is $1$ for positive $t$ and $0$ for negative $t$. Its Fourier transform can be easily computed. It has the ...
0
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0answers
11 views

Every cycle in a domain $D$ is null homolog

Let $D$ be a domain, where every cycle is null homolog and $f$ be a biholomorphism. Proof that every cycle $c$ in $f(D)$ is nullhomolog. Let $c$ be a cylce in D, it is null homolog, if ...
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0answers
11 views

Absolute maximum

I´m trying to find the absolute maximum of $(2N-1)$ partial sum of the Fourier´s series of signum function on $[0,\pi]$, I have: ...
0
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2answers
21 views

Ultrametric example

Can anybody give an example for ultrametric space? i.e., in the metric space definition, instead of triangle inequality, we have strong triangle inequality namely $d(x,y) \leq \max \{{d(x,z),d(z,y)}$} ...
1
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1answer
35 views

Differentiability of a function $f:\mathbb R^2\rightarrow \mathbb R$

I want to prove that the funktion $f:\mathbb R^2\rightarrow \mathbb R,\hspace{0.5cm} f(x,y)=\begin{cases} (x^2+y^2)\sin\big(\frac{1}{\sqrt{x^2+y^2}}\big),& \text{if } (x,y)\neq (0,0)\\ ...
1
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1answer
44 views

Use Power Series to solve system of differential equations

Problem: Hello, I wonder how you would use a Power Series to solve a system of differential equations. Lets say I have the system $$\begin{cases}(1)\text{ }\text{ }x_1'=2x_1+4x_2 \\ (2)\text{ ...
3
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2answers
91 views

A continuous onto function from $[0,1)$ to $(-1,1)$

How I can construct a continuous onto function from $[0,1)$ to $(-1,1)$ ? I know that such a function exists and also I have a function $\displaystyle f(x)=x^2\sin\frac{1}{1-x}$ which is ...
3
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0answers
9 views

Imposing boundary conditions AND self-similarity on a PDE

I have a PDE in the form $$u_t=F(u,u_x)$$ where the unknown is $u(x,t)$ on say $\mathbb R\times[0,\infty)$. $F$ is very nonlinear so I was told to assume self-similarity in the form $$u(x,t)=t\tilde ...
0
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1answer
31 views

Simply Connected sets

In my textbook it states, that the Union of two open docs is simply connected but not connected Why is this. I know simply connected means any closed path or loop can be shrunk to a point ...
0
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1answer
24 views

Linear function: relation between linearity and continuity

Given a linear function $A$ between two normed Vectorspaces i have to show euquality of the follwing statements: $A$ is continuous There exists a point where $A$ is continuous $A$ is ...
1
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2answers
41 views

Prove or disprove continuity of two maps

Yet another time I need help to prove continuity of a certain map and don't know how to do it: Look at the vector space $$C_b^1(\mathbb R; \mathbb C) := \{f \in C^1(\mathbb R;\mathbb ...
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0answers
17 views

Equivalence of statements about a linear map

I need someone to help me solve the following exercise: Let $(X, ||\cdot||_X)$ and $(Y, ||\cdot||_Y)$ be normed vector spaces over a common field $\mathbb K$ $(\mathbb R$ or $\mathbb C)$. For a ...
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0answers
10 views

Proving that for every $2$-form $\omega$, a basis of $V^*$ exists, so that $\omega$ can be written in a certain way

Let $\omega$ be an alternating $2$-form on an $n$-dimensional $\mathbb{R}$ vector space $V$, where $\omega$ isn't $0$ everywhere. I want to prove: There exists a basis $(\alpha_i)_{1 ≤ i ≤ n}$ of ...
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0answers
14 views

economics homework production and consumption

I am studying economics and i have got an home work which i am not grown. So please help me. The question is in the picture below. Thank you very much Best regards DK
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0answers
24 views

Proof that in any set A such that A contains a circumference centered at zero you can't find a continuous square root function.

The following is an exercise in my textbook on complex analysis: Proof that in any set A such that A contains a circumference centered at zero you can't find a continuous square root function. ...
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0answers
32 views

I want to draw one expression of fraction which have sin and cos. [on hold]

I want to know what shape of next expression. $a$ is a real number and $0<a<1$. $t$ moves $0<t<\frac{2\pi}{a}$. $x=\frac{\cos(a+1)t}{\cos(at)}$ $y=\frac{\cos(a+1)t}{\sin(at)}$
0
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0answers
17 views

$e^{\varphi -1}$ characteristic function

So I am trying to figure out whether $e^{\varphi-1}$ is a characteristic function given that $\varphi$ is. I know that linear combinations of characteristic functions and the real part of a ...
-2
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1answer
21 views

Let {xn} be a sequence that has two subsequences converging to different limits. Prove {xn} is not convergent. [on hold]

I can't use that if {xn} is a convergent sequence, then any subsequence {xni} is also convergent and their limits are equal.
2
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0answers
30 views

The rectifiable curves that connect two points of a compact set in $\mathbb R^n$.

This is a question from my homework: Let $a,b \in K \subset \mathbb{R}^n$ where $K$ is compact. Define $$C = \{\gamma : [0,1] \to K \mid \gamma(0)=a, \gamma(1)=b, \gamma \textrm{ is continuous and ...
0
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0answers
16 views

Linear stability of an ODE $\frac{dN}{dt}=-\mu N(t)+\mu N(t-T)\left(1+q\left(1-\left[\frac{N(t-T)}{K}[\right]^z\right)\right)$

This is a part of exercise: Consider the following equation: $$ \frac{dN}{dt}=-\mu N(t)+\mu N(t-T)\left(1+q\left(1-\left[\frac{N(t-T)}{K}[\right]^z\right)\right) $$ where all involved constants are ...
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3answers
76 views

Use $\epsilon$, $\delta$ to prove that $\lim\limits_{x\to\ b} \frac{1}{a+x}$ = $\frac{1}{a+b}$.

I've been working on this epsilon delta proof for the longest time now, and I can't quite get it. Let $a>0$ and $b>0$. Use $\epsilon$, $\delta$ to prove that $\lim\limits_{x\to\ b} ...
2
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1answer
4 views

Find function $h$ so that $h(U,V)$ equals density of $f(a) da$ for $f(a)=\frac{1}{2}e^{-\small|a|}$, $a \in \mathbb R$

Let $f(a)=\frac{1}{2}e^{-\small|a|}$, $a \in \mathbb R$ and let $U,V$ be independant and uniform distributed on [0,1]. Now I want to find a function $h$ so that $h(U,V)$ is equal to the density ...
1
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1answer
25 views

$((n-K)s^2)/\sigma^2$ what is this in terms of matrix linear regression?

$$ \frac{(n-K)s^2}{\sigma^2} $$ what is this in terms of matrix linear regression? Has Chi Squared Distribution with (n-K) df
0
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1answer
20 views

The lenght of rectifiable curve in $\mathbb{R}^n \setminus B[0,r]$ that connects antipodes points.

This question is from my homework, here it goes: Let $\gamma \colon [a,b] \to \mathbb{R}^n \setminus B[0,r]$ be a rectifiable curve such that $\gamma(a)=-\gamma(b)$. Using euclidean norm prove that ...
3
votes
1answer
33 views

Infimum of distance between point and (closed) set

I'm having a little trouble with the following exercise: Let $V \subset\mathbb{R}^p$ be a non-empty, closed set and $a \in \mathbb{R}^p$. For $x, y \in \mathbb{R}^p$ we note $d(x,y) = \|x-y\|$. ...
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0answers
33 views

Trigonometric integral $\int_{[-\pi,\pi]^2}{\frac{1-e^{-in\cdot\theta_1}}{1-\cos(\theta_1)\cos(\theta_2)}\,d\theta_1\,d\theta_2}$

I am trying to compute the following integral (see here). Since it seems to be the wrong approach, I am trying to calculate another one which I hope it will give me what I am looking for. My point is ...
0
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0answers
29 views

Show that the ball $\{x:|x|\le 1\}$ is admissible in $\Bbb{R}^n$ without showing its boundary is measure zero

Show that the ball $\{x:|x|\le 1\}$ is admissible in $\Bbb{R}^n$ without showing its boundary is measure zero. I am taking a course in Analysis 3 again and the notations are quite new and confusing ...
0
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1answer
32 views

a continuous function on $\mathbb{Q}$

Is there a continuous bijective function from $[0,1] \cap \mathbb{Q}$ to $\mathbb{R}$? I think that there is no such function. The set $|[0,1] \cap \mathbb{Q}|$ is countable and $|\mathbb{R}|$ is ...
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1answer
47 views

Number of roots $f(x)(\log(f(x)))^\prime$

EDIT: I am interested in derivative of a function $f(x)$. However, since the function is logarithmically convex/concave it is easier to analyze $\log f(x)$. Therefore, I rewrote the derivative and ...
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0answers
18 views

Properties of first integral in ode

Let $\frac{dx}{dt}=Z(x) $ be an ode in an open subset $U\subset\mathbb{R}^n$, I'm strugling to show the follow statements: i) For every point $p\in U$, such that $Z(p)\neq0$, there is a neighbor ...
1
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1answer
46 views
+100

Harmonic function — Application of Divergence Theorem

Suppose $f$ is a harmonic function on $D=\{(x,y)\in\mathbb{R}^2: x^2+y^2<1\}$. Assume $f$ is twice continuously differentiable on $cl(D)=\{(x,y)\in\mathbb{R}^2: x^2+y^2\leq 1\}$. How do we express ...
0
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0answers
17 views

how to know whether a subset of euclidean space is path connected or not?(2)

I am asked whether $X=\{(x,y,z)|x^2+y^2−z^2=1\}\subset \mathbb{R^3}$,is path connected or not. i just know that $X$ is a closed subset.how can i answer this question? is there any hint? i asked a ...
0
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1answer
29 views

how to know whether a subset of euclidean space is path connected or not?(1)

I am asked whether $X=\{(x,y,z,w)|x^2 + y^2 + z^2 + w^2 = 1 \}\subset \mathbb{R}^4$,is path connected or not. i just know that $X$ is a closed subset.how can i answer this question? is there any ...
0
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1answer
44 views

Are eigenvalues (resp. unit eigenvectors) dependent continuously on elements $a_{ij}$ of a symmetric matrix $A$? [on hold]

Let $A(t)=(a_{ij}(t)),~(t\in \mathbb R)$ is a symmetric matrix such that $a_{ij}(t)=a_{ji}(t)$ is a real-valued continuous function. Let $\lambda_1(t) \ge \cdots \ge \lambda_n(t)$ is all of the ...
1
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1answer
20 views

is the real root of a higher order equation continuous with the parameters?

I come across one equation that $a_{0}x^{2}+x(a_{1}\sqrt{f(x)}+a_{2})+a_{3}\sqrt{f(x)}+a_{4}=0$, in which $f(x)=b_{2}x^{2}+b_{1}x+b_{0}$ and $b_{1}^{2}-4b_{2}b_{0}\leq 0$ and $b_{2}>0$. I cannot ...
1
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3answers
78 views

Continuity of $f(z) = \sin(\theta)$ - how to prove?

If $f: \mathbb{C}\to\mathbb{C}$ is defined by $f[r(\cos(\theta)+i\sin(\theta)]=\sin\theta$ if $r>0$, and $f(0) = 0$, then how does one prove that $f$ is discontinuous at $0$ and continuous ...
2
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3answers
82 views

Prove $\lim_{x\to 2}4x^2-1 =15$ .

Prove $$ \lim_{x \to 2}\mathrm{f}\left(x\right) = \lim_{x \to 2}\left(4x^{2} - 1\right) =15 $$ I am getting stuck about the procedure of things. As $x$ gets infinitely closer to $2$, I can see ...
0
votes
1answer
19 views

Derivative of the Root of a Positive Matrix [duplicate]

Suppose that the map $t \mapsto A(t)$ from some open subset of $\mathbb{R}$ to the set of positive matrices is differentiable. It is known that the map $t \mapsto \sqrt{A(t)}$ is differentiable, ...
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0answers
4 views

Analytical Representation of a Sparse Matrix.

I have a sparse, rectangular array where sparse means that missing entries in the table are non-existant, not zero. The array is "patterned" such that on any one line or column, there is one and only ...
1
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1answer
25 views

Stuck with epsilon-delta proof of existence of a limit

Given $f(z):=f(x+iy) := u(x,y) + iv(x,y)$, I'm trying to prove that if $\lim\limits_{z\to z_0} f(z) = L$, where $L\in \mathbb{C}$, $\lim\limits_{(x,y)\to(x_0,y_0)} u(x,y) = a$ and ...
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0answers
40 views

Can we use sequences to test continuity of a weak$^*$-continuous operator?

Let $X,Y$ be Banach spaces. Now assume we have a map $T:X'\rightarrow Y'$ where $X'$ and $Y'$ are equipped with the weak$^*$ topology and not the norm topology. Can I infer from this that an operator ...
0
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0answers
33 views

For a linear function the following are equivalent: continuity and Lipschitz continuity

Let $(X,||\cdot ||_X)$ and $(Y,||\cdot ||_Y)$ be normed Vectorspaces over a common field $\Bbb K$. Let $A:X \to Y$ be a linear function. I have to show that the following statements are equivalent: ...
0
votes
1answer
43 views

Definition of limit point: Is the superset necessary?

Consider: (A) The definition of a limit point from wiki: "Let S be a subset of a topological space X. A point x in X is a limit point of S if every neighbourhood of x contains at least one point of S ...
3
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0answers
40 views

Any bounded region $G \subseteq \mathbb{C}$ with transitive automorphism group and sufficiently “smooth” edges is biholomorphic to the unit ball

Let $G$ be a bounded region in $\mathbb{C}$ (i.e. we have $G ≠ \emptyset$, and $G$ is open and connected), and let $G$ have a transitive automorphism group (that is, for each two points $z_1, z_2 \in ...
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0answers
30 views

Interesting Analysis

What is the difference between adherent point and limit point please someone explain this with proper example
0
votes
1answer
70 views

Urysohn's Lemma, Stone-Weierstrass

Let $X$ be a compact space. Show that the following statements are equivalent: a) $X$ is homeomorphic to a compact subset of $\mathbb{R}^n$ b) There are functions $f_1,\dotso, f_n\in ...
1
vote
1answer
15 views

Continuity of Lipchitz constant of local lipschitz function

Suppose $f:\mathbb{R}\to \mathbb{R}$ be local lipschitz, which is equivalent to Lipschitz on compact sets. That is, for any $R>0$, there exists some $L >0$ such that $$\sup_{|x|,|y|\le ...
1
vote
1answer
44 views

Show continuity or uniform continuity of $\phi: (C([0,1];\Bbb R ),||\cdot||_\infty )\to (\Bbb R, |\cdot | )$

$\phi: (C([0,1];\Bbb R ),||\cdot||_\infty )\to (\Bbb R, |\cdot | ); \: \: \: \: \: \: \phi(u):=\int_0^1 u^2(t) dt $ Is this function continuous or even uniformly continuous? (I know that the ...
2
votes
2answers
95 views

Computation of $\int _{-\pi} ^\pi \frac {e^{in\theta} - e^{i(n-1)\theta}} {\mid \sin {\theta} \mid} d\theta .$

I need to compute $$\int \limits _{-\pi} ^\pi \frac {e^{in\theta} - e^{i(n-1)\theta}} {\mid \sin {\theta} \mid} d\theta .$$ Does anyone see any good strategy? Thanks.