Questions tagged [analysis]
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).
44,196 questions
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Checking Uniform Continuity of a function [closed]
Let $f(x)=\sum_{n=1}^{\infty} f_n(x)$, where
\begin{align}
{
f_n(x)= n\left(x-n+\frac{1}{n}\right) \text{ for } x \in \left[n-\frac{1}{n},n\right] \\
f_n(x)= n\left(n+\frac{1}{n}-x\right) \text{ for ...
0
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51
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How to solve this function with banach's fixed theorem? [closed]
Consider the function $f: \mathbb R \to \mathbb R$ defined by
$$f(x)=1+x+x³.$$
Define iteratively the sequence $\{x(k)\}$ by
\begin{align}
x(0) &= 0, \\
x(k+1) &= F[x(k)], \text{ where } F(x)=...
- 9
2
votes
0
answers
32
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How do I compute the Laplace transform of a reaction-diffusion PDE with a Heaviside source?
I'm working with the following linear reaction-diffusion PDE, defined on a finite spatial domain $( x \in [-L_s, L_p] )$:
\begin{equation}
\frac{\partial c(x,t)}{\partial t} = D \frac{\partial^2 c(x,t)...
- 31
0
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0
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25
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Functional Equation of Eisenstein series using (or not?) symmetry of Laplacian (possible error in Garrett's book)
I am following the proof of the functional equation of the (non-holomorphic) Eisenstein series $E_s$ in Garrett's book "Modern Analysis of Automorphic Forms", Corollary 1.10.5. Let $f=E_{1-s}...
- 106
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votes
1
answer
78
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Prove That $F(y)$ is Continuous
The Following Question is from "Pugh" chapter 5.
If $Y$ is a metric space and $f : [a, b] × Y → \mathbb{R}$ is continuous, show that
$$F(y) = \int_{a}^{b}
f(x, y) \, dx$$
is continuous.
I ...
- 338
1
vote
1
answer
34
views
Does separately right continuous imply Borel measurability?
Let $X=\mathbb R^2$ endowed with the Borel sigma field. Let $f:X\to\mathbb R$ be a function.
Assume that for each $x$, the mapping $f_x:y\mapsto f(x,y)$ is right continuous, and for each $y$, the ...
- 3,416
1
vote
0
answers
29
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Are the Nested Intervals Theorem and Cauchy Completeness Equivalent in Ordered Fields?
Wikipedia says,
Nested intervals theorem shares the same logical status as Cauchy completeness in this spectrum of expressions of completeness. In other words, nested intervals theorem by itself is ...
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0
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45
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Cosec formula without Fourier series [closed]
I would like to obtain a proof of
$$
\frac{\pi}{\sin \pi x} = \lim_{n\to\infty} \sum_{m=-n}^n \left( 1 - \frac{|m|}{n}\right) \frac{(-1)^m}{x + m}
$$
without using Fourier analysis, since my students ...
- 1,329
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42
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Are solutions to $a^x=b$ where a and b are algebraic numbers, countable? Is there a real number that is not a solution to such an equation? [closed]
Algebraic numbers are defined as solutions to polynomial equations with rational coefficients and are countable. Ordinals are often expressed using exponentiation and are uncountable, so I was ...
0
votes
1
answer
40
views
If a limited function $g$ disagrees with an integrable function $f$ only on a zero content set, then $g$ is integrable?
I'm trying to solve the following problem from Elon Lima's "Análise Real":
If a bounded function $g \colon [a, b] \to \mathbb{R}$ coincides with an integrable function $f \colon [a, b] \to \...
- 423
-1
votes
1
answer
86
views
Partitioning $\Bbb R$ into sets $A$ & $B$, where measures of $A$ & $B$ in every non-empty open interval of constant length has a “near” constant ratio
Motivation: (If you don't need the motivation or attempts, skip to the question.)
Suppose, we partition $\mathbb{R}$ into sets $A$ and $B$ with a positive measure in each non-empty, open interval. I ...
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1
vote
0
answers
53
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Is there a subelementary surjection $\mathbb{R} \to \mathbb{R}^2$?
I am trying to figure out if there is a nice formula for a space-filling curve.
Define a subelementary function as one composed of:
addition, subtraction, multiplication,
$\sin$,
the absolute value ...
- 3,252
-1
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1
answer
102
views
Partitioning $\Bbb R$ into sets $A$ and $B$, where the measures of $A$ and $B$ in every non-empty interval of constant length have a constant ratio
Motivation: (If you don't need the motivation, skip it.)
I've tried partitioning $\mathbb{R}$ into sets $A$ and $B$, where the Lebesgues measures in every non-empty open interval have a non-zero ...
- 53
1
vote
0
answers
40
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Constructing a function in $L^2$ which outgrows the gaussian. Nica, Spreicher Free Probability. Exercise $7.23$
Hi I would like to solve the following exercise from Nica, Speicher Combinatorics of Free Probability, 7.23
Important note:
\begin{align*}
||\hat{ab}||=\phi(b^*a)
\end{align*}
Exercise 7.23 (from Nica–...
- 23
1
vote
0
answers
45
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Every region is made up of simple pieces in a suitable coordinate system
I’ve been thinking about this question for a while now, but I do not seem to able to answer it. A region $D$ in the plane is said to be of $\textit{type 1}$ if it can be written as $ D = \{(x, y) \in {...
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