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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1answer
86 views

Find data to perform regression analysis

I'm trying to find some data (two continuous variables that I believe are correlated) online for which I can perform a regression anaylsis, my assignment sheet says: The data may be found anywhere ...
1
vote
1answer
25 views

Sketching Semi-Cubical Graphs

I'm having trouble sketching these kind of graphs, for example, I have the equation: $$(1-y_2^2)^3=\frac{9}{4}y_1^2$$ Note: the map is over the real numbers. Could somebody please give me some ...
2
votes
0answers
45 views

Determinant of Jacobi of $C^3$ function

If I have a function $f:\mathbb{R}^n \to \mathbb{R}^n$ that is $C^3$, then the elements of the Jacobian matrix is $C^2.$ Is the determinant of the Jacobian matrix also $C^2$?
0
votes
1answer
199 views

Permitted value of epsilon in Discrete Metric Space

If we define a Metric Space with Discrete Metric, say $(\mathbb{R},d)$. Then whenever we talk about epsilon or delta, such as talking about neighbouringhood, limitpoint, can we take epsilon to be not ...
1
vote
0answers
88 views

Compute significance of Kendall tau-b?

I have so-far tried all ways of computing kendall tau significance (where there are ties) described here. However, none of them works good, even for relatively large vectors. I think the problem is ...
2
votes
1answer
56 views

Question about a theorem of integration

In Darboux integration, If $f$ is a bounded function on $[a,b]$, then $L(f)\le U(f)$ where $L(f),U(f)$ are the lower and upper Darboux integral. My question is why it requires $f$ to be bounded?
4
votes
1answer
165 views

Is the zero set of a non zero real valued harmonic function discrete?

It is a basic fact that the zero set of a non zero holomorphic function defined on a open set $A$ is discrete. By a result in Rudin's textbook on "Real and Complex Analysis", we know that any real ...
1
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2answers
1k views

Triangle Inequality for supremum metric

Edited Heavily Here all functions are from $[0,1]$ to $\mathbb{R}$ and are bounded. Prove the following Triangle inequality in following case: ...
2
votes
4answers
141 views

Understanding Set Notation

I'm having some trouble understanding a definition and explanation in my textbook Introduction to Analysis by Edward Gaughan 5th edition. The book begins with some preliminary information about sets ...
1
vote
2answers
107 views

How to express this convolution by the sum of integrals

If $$f\left(x\right)=\begin{cases} f_{1}\left(x\right), & x\in[0,1]\\ f_{2}\left(x\right), & x\in[1,\sqrt{5}]\\ 0, & \mbox{elsewhere} \end{cases}$$ what does the piecewise-defined function ...
2
votes
1answer
50 views

Multivariable Jacobian Problem: tranpose of the Jacobian times $f(x)$ is zero

Here's a little problem I ran across and could use a hint or two on: Let $U \subset \mathfrak{R}^{n}$ be open, $f:U \rightarrow \mathfrak{R}^{m}$ differential on $U$ and satisfying $\|f(X)\|=1$ on ...
0
votes
2answers
229 views

Why trigonometric polynomials form an inner product space.

Trigonometric polynomial is defined as a function $$ f(x)=\sum_{n=1}^ka_ne^{i\lambda_nx}$$ for some positive integer $k$, complex coefficients $\{a_n\}$ and real coefficients $\{\lambda_n\}$. Let TP ...
4
votes
3answers
213 views

The value of $(2^n+3^n+4^n)^{1/n}$ as $n \rightarrow \infty?$

I was thinking about the following problem: How can i find the value of $(2^n+3^n+4^n)^{1/n}$ as $n \rightarrow \infty?$ Can someone point me in the right direction? Thanks in advance for your time. ...
2
votes
1answer
83 views

Is there a homeomorphism from the space of sequences to [0,1]?

If I consider the norm for the space of sequences of digits {0-9} to mimic the norm for real numbers. $|\left\{x_n\right\}| = \sum_{n=1}^{\infty} \frac{x_n}{10^n}$ shouldn't I now have a space ...
2
votes
1answer
120 views

uniform continuity of $f$ on $(a,b) \implies \exists \lim_{x \to a^+} f(x)$?

Does $\lim_{x \to a^+} f(x)$ necessarily exist for all $f$ uniformly continuous on $(a,b)$? This has no relation whatsoever to homework. It occurred to me after looking at another question (which ...
2
votes
1answer
44 views

Invertible idempotent in a C-star algebra question

Let $J$ be an idempotent element in a unital $C^*$ algebra. Why is $I+(J-J^*)(J^*-J)$ invertible? I have been trying to show that $\|(J-J^*)(J^*-J)\|<1$, but I could not do this.
2
votes
1answer
294 views

Similar proof of Peano's Existence Theorem

As many of you will know, Peano's theorem states that if $f(x,y)$ is continuous and bounded in the strip $T: |x-x_0| \le a, |y|\le\infty $. Then the intitial value problem $y'=f(x,y), y(x_0)=y_0$, ...
2
votes
1answer
68 views

How to prove that an algebraic structure can be embedded into another?

What is the general method to prove that an algebraic structure can be embedded into another? If this is too general a question, how can I prove that $\mathbb{Z}$, or $\mathbb{Q}$ can be embedded in ...
2
votes
0answers
126 views

Lipschitz Vectors

I am trying to understand why a vector valued function where it is Lipschitz in each dimension, is also lipschitz. In particular, I have a probability density function $p(x)= \prod p_{j}(x_{j})$ ...
0
votes
2answers
2k views

Illustration Proof that every sequence of real numbers has monotone subsequence

While proving every sequence of real numbers has a monotone subsequence, we take two cases, either there are infinitely many "peaks" or else "finitely many" peaks. However, I am unable to grasp from ...
1
vote
1answer
80 views

Analytic extension in several variables

Let $D \subset \mathbb{C}^{n}$ be an open domain and $K \subset D$ be a compact subset such that $D - K$ is simply connected. Let furthermore $f$ be a analytic function defined on $D-K$. Is it ...
4
votes
1answer
487 views

Metrizability of weak convergence by the bounded Lipschitz metric

Why is the weak convergence of probability measures on $\mathbb{R}$ with respect to bounded continuous test functions $C^0_b(\mathbb{R})$ metrizable by the bounded Lipschitz metric $$d(\mu, \nu) = ...
2
votes
2answers
376 views

Let $X$ and $Y$ be Banach spaces, show that if they are isomorphic, then $X$ is reflexive iff $Y$ is reflexive.

I want to show that if $X$ and $Y$ are two Banach spaces, and $T : X \to Y$ is an isomorphism, then $$ X \textrm{ reflexive} \iff Y \textrm{ reflexive}. $$ I saw several proofs, but I cannot ...
5
votes
1answer
163 views

How to prove $ \sum_{n=1}^{\infty}\left|\frac{a_{1}+\cdots+a_{n}}{n}\right|^{p}\leq\left(\frac{p}{p-1}\right)^{p}\sum_{n=1}^{\infty}|a_{n}|^{p} $

Let define $(a_n)_{n\geq1}$ as real series. Prove, that $$ \sum_{n=1}^{\infty}\left|\frac{a_{1}+\cdots+a_{n}}{n}\right|^{p}\leq\left(\frac{p}{p-1}\right)^{p}\sum_{n=1}^{\infty}|a_{n}|^{p} $$ (*) ...
5
votes
1answer
290 views

Complex infinite sum convergence problem

Suppose that the complex infinite sum $ \sum_{n=1}^{\infty}(-1)^{n}Z_n$ converges. Define $A \subset \mathbb{C}$ by $A=\{{z\in\mathbb{C}\mid\exists ...
2
votes
2answers
90 views

Complex function inequality

Let $\lambda \in (0, 1)$ . Suppose, that function $f : \mathbb{C} \rightarrow \mathbb{C}$ satisfy the inequality $|f(u) - f(v)| \leq \lambda|u-v|$ Prove, that for all $a \in \mathbb{C}$ $z = f(z) + ...
2
votes
1answer
158 views

Convergence test with polynomials

Let $P, Q \colon \mathbb{R} \rightarrow \mathbb{R}$ are polynomials, and $Q(n) \neq 0$ for $n\in\mathbb{N}$ Suppose, that $\deg (P) < \deg (Q)$. Prove, that infinite sum ...
3
votes
1answer
101 views

Index of a Fredholm Operator on Paths

I'm a novice to analysis but I need to understand the following example. Any help would be greatly appreciated. This might be of interest to some because it gives a way of quantifying changes in ...
2
votes
1answer
181 views

A question on Corollary of Lusin's Theorem in Rudin's Real and Complex analysis

I have a question on Corollary of Lusin's Theorem in Rudin's Real and Complex analysis (3rd edition, page 56). Here Rudin explicitly requires that $|f| \leq 1$. But I can not see why this requirement ...
5
votes
0answers
143 views

Linear algebra estimates

Here is something that has been troubleing me lately. I don't know if it is true of not. I suspect it is. Suppose that $A,B$ are two $n \times n$ matrices with complex entries. $A^t = A$, $\bar B^t = ...
4
votes
1answer
103 views

Trading localisation for regularity

When reading about Schrödinger's fundamental solution in 1D, $$u(t,x)=\frac{1}{\sqrt{4\pi it}} \int_\mathbb{R} u_0(y) e^{\frac{i(x-y)^2}{4t}}dy$$ the author says thus Schrödinger evolution is ...
3
votes
1answer
154 views

calculating a multivariate integral via level sets

I'm considering the possibility of calculating an integral of the form $\int_{S_n} f(x_1,\dots,x_n) dx_1\dots dx_n$ via level sets, where $S_n$ is the domain of integration. In my problem everything ...
2
votes
1answer
173 views

Proving a theorem about continuity property with contradiction.

Proving that if $f$ is continuous on an interval $I$, any $a,b\in I,a<b$ and for any $y$,where $f(a) < y < f(b)$, there exist a $x\in (a,b)$ s.t $f(x)=y$. I have seen a prove using the ...
12
votes
9answers
1k views

Does $\bigcap_{n=1}^{+\infty}(-\frac{1}{n},\frac{1}{n}) = \varnothing$?

When I learn the below theorem: If $I_n$ is closed interval, and $I_{n+1} \subset I_n$,then $$\bigcap I_n \not= \varnothing$$ and someone says if we replace closed interval with open interval, can ...
2
votes
2answers
122 views

calculate the integral

Compute $$I=\int_C\frac{e^{zt}}{1+z^2}dz$$ where $t>0$, a real number, and $C$ is the line $\{z| Re(z)=1\}$ with direction of increasing imaginary part. I tried to integral along the boundary of ...
0
votes
1answer
82 views

question on analytic extension

Suppose $f$ is analytic in the annulus $1<|z|<2$ and there exists a sequence of polynomials $p_n$ converging to $f$ uniformly on every compact subset of this annulus. Show that $f$ has an ...
0
votes
1answer
130 views

Examples of dictionaries between two distinct fields of mathematics (or between “differents” structures of math).

I'd like to meet explicit examples of dictionaries between two distinct fields of Mathematics (or between two "different" structures of Mathematics). I'm not interested in the usual sense dictionary ...
4
votes
1answer
303 views

Method of isoclines

I have this exercise and I do not know how to solve it. By using the method of isoclines represent the integrals of equation corbes nonautonomous $x'=x^2-t$. There are some indications: Let $P = ...
4
votes
2answers
146 views

Proving Inequality

I stumbled upon the following inequality and I need to prove it. $$\left(1+\frac{a}{b}\right)^x+\left(1+\frac{b}{a}\right)^x\ge 2^{x+1}$$ I am expected to use Holder's Inequality but there seem to ...
10
votes
2answers
169 views

Bolzano-Weierstrass for sequences of sets

Let $\mathcal{A}_n,\,n\in\mathbb{N}$ be a sequence of subsets of, say, $\mathbb{R}$. Let $\limsup_{n\rightarrow\infty} \mathcal{A}_n = \{x:x\in\mathcal{A}_n\mbox{ for infinitely many } n\}$, and ...
2
votes
3answers
212 views

Why is arctangent smooth?

I just read a proof by Spivak that $\arctan(x)$ has the $(2n+1)$-th Taylor polynomial at zero $$x - \frac{x^3}{3} + \cdots + (-1)^n\frac{x^{2n+1}}{2n+1}$$ The proof relied on the assumption that ...
26
votes
1answer
762 views

A question about series with a strange property.

Does there exist a sequence $\left(a_n\right)_{n\ge1}$ with $a_n < a_{n+1}+a_{n^2}, \forall n=1,2,3,\ldots$ such that the series $\displaystyle{\sum_{n=1}^{\infty}a_n}$ converges? This is the ...
3
votes
0answers
350 views

Riemann integral vs Lebesgue integral

Let $f$ be analytic on a domain $\Omega$ of the complex plane, such that the closed disc $\overline{D(0,R)}$ is contained in $\Omega$. What is the difference between $$ \int_{D(0,R)}|f(w)|dm(w)$$ and ...
2
votes
0answers
121 views

a function in the unit ball of C(X) where X is a compact space is a limit of convex combinations of extreme points

Suppose $f(x)$ is a function in $C(X)$ such that $\|f\|<1-\frac{2}{n}$. Then there exist n extreme points of the unit ball of $C(X)$, $g_1,\ldots,g_n$ such that ...
2
votes
4answers
139 views

Addition of points on Metric Space

Well, I was not quite aware that addition of points is not defined in metric spaces but is defined only on linear spaces and others. Could anyone elaborate why is this? Is the addition of intervals ...
5
votes
3answers
2k views

How to prove differentiability implies continuity with $\epsilon-\delta$ definition?

I know that's a very common theorem in calculus but when i try to prove it with $\epsilon-\delta$ definition of continuity, i found that it is not so obvious. Attempts:Let $f:\mathbb{R}\to\mathbb{R}$ ...
2
votes
1answer
108 views

Why can't $\theta(0)$ be $0$? (tempered distributions)

I'm reading about the Littlewood-Paley decomposition, but there is a definition I can't understand, it says: We denote by $S'h(\mathbb{R}^d)$ the space of tempered distributions $u$ such that ...
4
votes
1answer
332 views

non constant bounded holomorphic function on some open set

this is an exercise I came across in Rudin's "Real and complex analysis" Chapter 16. Suppose $\Omega$ is the complement set of $E$ in $\mathbb{C}$, where $E$ is a compact set with positive Lebesgue ...
3
votes
2answers
287 views

a problem relate to analytic function

Suppose $f$ is an analytic function on $|z|\leq 1$ with $f(0)=0$, and let $|f(z)|$ have a maximum for $|z|\leq 1$ at 1, show that $f'(1)\neq 0$ unless $f$ is a constant. Remarks: 1, At first ...
3
votes
1answer
347 views

non-continuous function satisfies $f(x+y)=f(x)+f(y)$

As mentioned in this link, it shows that For any $f$ on the real line $\mathbb{R}^1$, $f(x+y)=f(x)+f(y)$ implies $f$ continuous $\Leftrightarrow$ $f$ measurable. But how to show there exists such ...