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
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2answers
101 views

Is there such a thing as function decomposability?

I am not a mathematician, so what I ask might be trivial, however I couldn't find something relevant in the web. My question is the following: Is there a formal notation for functions that comply the ...
4
votes
1answer
509 views

Continuous partials at a point but not differentiable there?

In Question on differentiability at a point, it is mentioned (and in Equivalent condition for differentiability on partial derivatives it is cited from Apostol) that for $f:\mathbb{R}^2\to\mathbb{R}$ ...
7
votes
1answer
638 views

Wrong Wolfram alpha result?

I have this function $$ f(x,y) = \left\{ \begin{array}{ll} \frac{x^3}{x^2 + y^2} & \mbox{if } (x,y) \neq (0,0) \\ 0 & \mbox{if } (x,y) = (0,0) \end{array} \right. $$ And I want to ...
0
votes
1answer
201 views

Continuous function with Continous right derivative must be differentiable on [0,T]?

I think a continuous function with continuous right derivative must be differentiable on a closed bounded interval [0,1] but I do not know how to prove it. If this is not true, can any one give me a ...
3
votes
2answers
39 views

Lipschitz maps on $[0,1]$

Here is just a little curiosity. Assume the $f : [0,1] \to [0,1]$ is a Lipschitz function that maps 0 to 0 and 1 to 1. If we impose that the Lipschitz constant of $f$ is $\le 1$, can $f$ be anything ...
0
votes
2answers
293 views

Show that the closure of an open ball is exactly $B(0,R) := \{ x \in X : ||x|| \le R\}$

Let $X$ be a normed space and $$ B(0,R) := \{ x \in X : ||x|| < R \} \qquad \overline{B}(0,R) := \{ x \in X : ||x|| \le R \}. $$ I want to show that $\overline{B(0,R)} = \overline{B}(0,R)$. I ...
2
votes
0answers
271 views

Generalization of the Jordan decomposition theorem for functions of bounded variation.

The famous ( ? ) Jordan decomposition Theorem for Bounded variation functions is generally stated as below. Theorem. If $f:[a,b] \to\mathbb R$ is a function of bounded variation then there is ...
1
vote
0answers
44 views

Ergodic transformation [duplicate]

Possible Duplicate: Showing a Transformation increases measure (Ergodic Theory) Hoi, i want to show that the piece-wise linear map $T: [0,1)\to[0,1)$ given by $Tx=3x$ for $x\in [0,1/3)$ ...
1
vote
1answer
64 views

Show that a certain point lies outside a ball, might be simple but i am stuck…

Consider the ball $$ B(0, R) := \{ x | ||x|| \le R \} $$ and consider a point $x$ outside of the ball, that is $||x|| > R$. Now i construct another ball of radius $\frac{1}{2}(||x|| - R)$ around ...
2
votes
1answer
239 views

Existence and uniqueness of solutions to a system of non-linear equations

Consider an arbitrary system of non-linear equations $F(x)=0$ where $F:\mathbb{R}^n \rightarrow \mathbb{R}^m$. Are there any properties to check in order to study whether solutions exist, are unique ...
19
votes
3answers
597 views

Why can't $\sqrt{2}^{\sqrt{2}{^\sqrt{2}{^\cdots}}}>2$?

So we have$$\sqrt{2}^{\sqrt{2}{^\sqrt{2}{^\cdots}}}=x\\\sqrt{2}^x=x$$where $x=2$ heuristically seems like a good solution. However, $x=4$ seems like an equally good solution. I was told in passing ...
1
vote
2answers
86 views

what does it mean to say a space is norm separable?

I came across in my textbook the term: norm separable. I looked in the textbook and online and could not find a definition.
1
vote
0answers
198 views

Special solution of Helmholtz equation separated Phase

I'm searching for some special solutions for the Helmholtz-equation in 2D: $(\partial_x^2 + \partial_y^2 + a) f(x,y) = 0$ where $f(x,y) \in \mathbb{C}$ (boundary condition: $\lim_{x,y\to \infty} ...
1
vote
1answer
78 views

Implicit Function

Let $f:R^{k+n} \rightarrow R^n$ be of class $C^1$; suppose that $f(a)=0$ and that $Df(a)$ has rank n. Show that if $c$ is a point of $R^n$ sufficiently close to $0$, then the equation $f(x)=c$ has ...
0
votes
1answer
60 views

$\epsilon$ closeness proof , elementary analysis

Definition: Let $\epsilon > 0$, and $x,y$ be rational numbers. We say that $y$ is $\epsilon$ close to $x$ iff we have $d(y,x) \leq \epsilon.$ Question: Let $\epsilon > 0.$ If $x$ and ...
1
vote
1answer
187 views

Finding lower bound sequence for squeeze theorem

I need to find: $$\lim_{n\to \infty} a_n =\lim_{n\to \infty} \frac{1}{n^2 +n} , for: \forall n \in \Bbb N \setminus \{ 0 \}$$ By using the Sandwich a.k.a. squeeze theorem.My ideas so far: when i ...
0
votes
1answer
64 views

Prove if one set is complete then another set is complete

Let $X$ be a set. Let $l^{\infty}(X,N)$ be all bounded functions on the form $f: X\longrightarrow N$. Let $d(f,g)=\sup\{n(f(x),g(x): x\in X)\}$ be a metric on $l^{\infty}(X,N)$, where $n$ is metric on ...
1
vote
3answers
72 views

The infinite sum $ \sum_{m=2}^\infty \space \frac {1} {p_m \space \log\space m} $

Let $p_n$ denote the $n$th prime , for example $p_1$ = $2$ , $p_2 = 3 $ etc. Then is the sum $$ \sum_{m=2}^\infty \space \frac {1} {p_m \space \log\space m} $$ convergent ?
1
vote
0answers
38 views

A certain type of points in the plane

In the plane a point $O$ and a sequence of points $ P_1 , P_2 , P_3 , ... $ are given. The distances $ OP_1 , OP_2 , OP_3 , ...$ are $ r_1 , r_2 , r_3 , ....$ , where $ r_1 ≤ r_2 ≤ r_3 ≤ $ ... . ...
1
vote
2answers
43 views

Query on supremums of sets of real numbers.

I would like to know if the following reasoning is correct: If we have $a, b \in \mathbb{R}$ and $a < b$ then $\exists c \in \mathbb{R}$ such that $a < c < b$. From this, it follows that if ...
1
vote
1answer
67 views

Nowhere dense notation confusion

The text Foundations of Mathematical Analysis by Johnsonbaugh and Pfaffenberger defines nowhere dense as $X$ is nowhere dense in $M$ if $X^{-,-} = M$. What does this mean?
0
votes
1answer
613 views

show that: $f$ is injective $\iff$ there exists a $g: Y\rightarrow X$ such that $g \circ f = idX$

** proof under construction - will post when done and more or less confident it's true. ** also please easy with the downgrades.. i don't understand why i'm getting them. what is meant by show ...
0
votes
2answers
98 views

Does X always has a finite open cover?

Does X always has a finite open cover? X is a metric space. I think it is false because there is a possibility to have closed neighborhood near a point in X.
1
vote
1answer
249 views

Open Cover (0,1)

Find and proof an open cover of (0; 1) that has no finite subcover. I need to find an example and also proof the example. Thank you.
4
votes
3answers
230 views

Questions about convergence in Lp

If $X_n$ converges to $X$ in $L^p$, do we have $X_n^p$ converges to $X^p$ in $L^1$? We can prove that it is true when p=1,2 easily. I am curious whether this is true for all $p>0$.
2
votes
1answer
140 views

Is there any Elliptic Operator of first order in $U\subset \mathbb R^n$?

Suppose $P=\sum_i a_i \partial/\partial x^i$. It seems for me that there always exists a nonzero $\xi=\{\xi_1, \cdots, \xi_n\}\in \mathbb R^n\setminus 0$ such that the principle symbol $\sigma(P)(x, ...
1
vote
1answer
127 views

Continuity of the orthogonal projection into tangent space.

Let $\mathcal M \subset \mathbb R^d$ be a smooth manifold, and for each $s \in \mathcal M$ let $T_s[\mathcal M]$ denote the tangent space of $\mathcal M$ at $s$. For each $s \in \mathcal M$ let $P_s$ ...
3
votes
2answers
776 views

Extreme points of unit ball in $C(X)$

Let $X$ be a compact Hausdorff space and $C(X)$ be the space of continuos functions in sup-norm. I read in Douglas' Banach algebra techniques in operator theory that the followings are equivalent: ...
2
votes
0answers
50 views

Tight bounds for harmonic measure

I recently came across a question concerning harmonic measure here, and was wondering if there is a good reference summarizing different methods of estimating harmonic measure? Specifically, I would ...
0
votes
1answer
230 views

Initial value problem: Solution blows up conditions

Consider the differential equation in $\mathbb R$: $$x' = x^2-\lambda x^3; \,\,\,\,\,\, x(0) = x_0; \,\,\,\,\, t\geq 0$$ where $\lambda $ is a parameter. For which initial conditions is the solution ...
0
votes
0answers
59 views

Can all algebraic numbers be expressed as infinite sums whose summands never permanently disappear?

Can all algebraic numbers (i.e. quantities such as $3/5$, $\sqrt{2}$, $\sqrt{3}$, etc.) be expressed as an infinite sum whose summands never permanently vanish? A well known example is ...
3
votes
1answer
162 views

Norm of operator $g\mapsto \int fg$

Let $T_f:(C([a,b],\mathbb C), \lVert \cdot \rVert_1) \to \mathbb C$ with $g\mapsto \int_a^b f(x)g(x) dx$ for any given $f\in C([a,b],\mathbb C)$. I have to find the norm of $T_f$. I started with: ...
3
votes
1answer
75 views

Limit involving Khinchin's constant for continued fractions

Hoi, i wish to show a few things... Suppose we know $\lim_{n\to\infty}(a_1a_2\cdots a_{n})^{1/n} = \prod_{k=1}^{\infty}\left(1+\frac{1}{k(k+2)}\right)^{\frac{\log k}{\log 2}}$ I hope to show this ...
4
votes
1answer
91 views

When does a measure have a density?

Consider a measure space $(X, \Sigma, \mu)$ and another measure $\nu$ on the same space. I'm interested in the conditions under which $\nu$ can be represented by a density function $f$ on $X$, so for ...
1
vote
3answers
97 views

Supremum of the set $M_x$ for $x\in \mathbb{R}$

Let $x\in \mathbb{R}$ then the set $M_x=\{n\in \mathbb{Z}\;|\; n\leq x\}\neq \varnothing$ is bounded from above, then $M_x$ have a supremum. Donote $[x]=\sup(M_x)$. On my own, I proved that $[x]\leq ...
2
votes
2answers
156 views

Questions on the Nature of Nonmeasurable Sets and Functions

Me and some classmates are stuck on a problem from our real analysis textbook (Royden/Fitzpatrick 4th Ed.), which we've given some thought. The question is: "Suppose $f$ is a real valued function on ...
0
votes
1answer
40 views

integrable funcion and measurabiliy

Let $(X, \mathcal{S},\mu)$ be a measure space, and let $f: X \to \mathbb{C}$ be a function. Then $f$ is integrable if Re$f$ and Im$f$ are integrable and $\int f := \int$Re$f+i\int$Im$f$. It is easy ...
1
vote
1answer
127 views

Why $ F(u_\gamma) = \int_{\Omega} | \nabla u (D \tau_{\gamma})^{-1} |^2 \det (D \tau_\gamma) ?$ Is this by change of variables?

I'd like to understand the following: Let \begin{equation} F(u) = \int_{\Omega } |\nabla u|^2 \end{equation} where $ \Omega $ is adomain in $ \mathbb{R}^{n} \cdots $ For $ | \gamma | $ small enough ...
-4
votes
2answers
176 views

Show that $f$ is a linear transformation

Let $f: \mathbb{R}^m \to \mathbb{R}^n$, where $f(tx)=tf(x)$, for all $x \in \mathbb{R}^m$ and $t \in \mathbb{R}$. Show that $f$ is a linear transformation.
0
votes
3answers
58 views

Prove $\forall K > 0: \lim_{n\rightarrow\infty} \sqrt[n]{K} = 1$

Alright, so I've already proven that both $\forall n \in \mathbb{N}:\lim_{n\rightarrow\infty}\sqrt[n]{n} = 1$ and $\forall K\geq 1:\lim_{n\rightarrow\infty}\sqrt[n]{K}=1$. I got the feeling, that I ...
4
votes
0answers
97 views

Existence of a sequence which is good for mean convergence but not good for pointwise convergence

The ergodic theorems talk about the limit behaviour of the ergodic averages. For example, if $(X,\chi,\mu,T)$ is a measure preserving system, where $\mu$ is a probability measure on $X$, then we have ...
2
votes
1answer
151 views

Compactness in $C([0,1])$

I read in a paper (pp8) that the set $$A=\left\{f\in W^{1,1}(0,1):\sup |f|\leq C,\ \int_0^1|f'(t)|dt\leq M \right\}$$ is compact in $C([0,1])$, where $C$ and $M$ are fixed constants. I understand that ...
0
votes
0answers
73 views

I need to show that $x(t)=0$ is a solution for all $t∈ \mathbb{ R}$

Assume that $f(t,x)$ is locally Lipschitz continuous in x, uniformly w.r.t. $t$ and that $f(t,0)=0$ for all $t$. Consider the equation $x'(t)=f(t,x)$ on $\mathbb R\times \mathbb R^n$. If $x(t)$ is a ...
0
votes
1answer
312 views

Find conditions on $\alpha$ so that f satisfies a Local Lipschitz condition.

Let $ f(x)=x^{\alpha}$ for $\alpha \in \mathbb {R}$. Determine the conditions on $\alpha$ such that f satisfies a local Lipschitz condition on $\mathbb R$. For the local Lipschitz case, I need to find ...
1
vote
1answer
339 views

Integrability of a piecewise function on $[0,1]$

Consider the function $f(x)=\begin{cases} x & x\in \mathbb{Q}\cap [0,1] \\ -x & x\in [0,1]-\mathbb{Q} \\ \end{cases}$ I argue that this function is not integrable since ...
1
vote
0answers
98 views

How do I prove this function is not continuous?

Let $\alpha$ a path from $[0,1]$ to a topological space X. Let $\alpha(0)=\alpha(1)=c$, where $c\in X$. The standard function to prove that $\alpha\cdot\bar \alpha$ is homotopic to the constant map ...
1
vote
1answer
222 views

Limit of a sequence bounded below but has no cluster points

Question is this: Let $a_n$ be a sequence of real numbers. Prove that if $a_n$ is bounded below and has no cluster points then $a_n$ → ∞. I could not really find a way to prove it. Could you give me ...
0
votes
1answer
210 views

parametrizing quarter of a circle

I am given the circle whose equation is: $(x-\frac{1}{2})^{2}+(y+\frac{1}{2})^{2}=\frac{1}{2}$. So, the coordinates of the origin of the circle are: $(\frac{1}{2},-\frac{1}{2})$ and the radius of the ...
0
votes
1answer
71 views

Inequality from Von Neumann entropy.

I am looking over some old course notes. First, Von Neumann entropy is defined. The Von Neumann entropy of a system described by a density matrix $\rho$ is defined by $S(\rho)\equiv ...
2
votes
1answer
107 views

Convergence of a matrix

Let $A_j$ be a sequence in $\mathbb {C}^{n\times n}$. Show that $\displaystyle \sum_{j=0}^\infty A_j$ converges if $\displaystyle \sum_{j=0}^\infty ||A_j||$ does. Note that $\displaystyle \Vert ...