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).
44,433 questions
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True or false? If $\lim n|x_n-x_{n+1}|=0$ then $\{x_n\}$ converges. [duplicate]
I want to know whether it is true that if a real sequence $\{x_n\}_{n=1}^\infty$ satisfies $\lim\limits_{n\to\infty} n|x_n-x_{n+1}|=0$ then it converges.
I guess it is false but I can't find a ...
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Demonstration Written in Formal Language
I was working on my homework and I got stuck on this exercise:
We define $f$ as type $A$ if: $\forall x\in \mathbf{R}\ \exists y\in \mathbf{R}(y\geq x \land |f(y)|\geq 1) $
We define $f$ as $B$ ...
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Classical solution for fractional laplacian
I have this problem
\begin{equation*}
\begin{cases}
\frac{\partial u}{\partial t}+(-\Delta_N)^{s}u=f(t,u),
\quad x\in\Omega,\quad t>0,\\
\frac{\partial u}{\partial\eta}=0,\quad x\in\partial\Omega,\...
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Does definition of empty set use universal specification
In chapter 3 of Analysis I by Terence Tao, the following definition of empty set is given:
(Empty set). There exists a set $\phi$, known as the empty set, which contains no elements, i.e., for every ...
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Sufficient condition at infinity for uniqueness of a non-linear parabolic PDE
Let me consider the harmonic map heat flow from $\mathbb R^2$ onto $S^2 \subset \mathbb R^3$, given by
\begin{equation}
\begin{cases}
\partial_t u = \Delta u + |\nabla u|^2 u & \text{in } \mathbb ...
- 4,546
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What is the relationship between the two different definitions of Concave Function? [duplicate]
Some articles indicate the definition of a concave function $f(x)$ as follows:
$$\forall x_1,x_2\in D_f, \forall\lambda\in(0,1): f\left((1-\lambda)x_1+\lambda x_2\right) > (1-\lambda)f(x_1) + \...
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Simple functions of $ L^p(\Omega,\mathcal F_\infty)$ is in the closure of $\cup_{n\geq 1}L^p(\Omega,\mathcal F_n)$ in $L^p(\Omega,\mathcal F_\infty.)$
Consider a probability space $(\Omega,\mathcal F,\Bbb P)$, a family of sub sigma algebras $\{\mathcal F_n\}_{n\geq 1}$, and $\mathcal F_\infty:=\sigma(\cup_n \mathcal F_n).$ Assume $\{\mathcal F_n\}_{...
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$f(0)=1$, $f(x) \ge 0 \ge f'(x)$, $f''(x)\le f(x)$ for $x\ge 0$
Problem
Let $f$ be a twice differentiable function on the open interval $(-1,1)$ such that $f(0)=1$. Suppose $f$ also satisfies $f(x) \ge 0, f'(x) \le 0$ and $f''(x) \le f(x)$, for all $x\ge 0$. Show ...
- 3,478
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Induced Topological Vector Space Structure?
Suppose $V$ is a real normed vector space where we denote addition on $V$ by
$$+_V:V\times V\to V,$$
we denote left scalar multiplication on $V$ by
$$\cdot_V:\mathbb{R}\times V\to V,$$
and we denote ...
- 237
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Does uniformly continuous functions apply to something like "sandwich theorem"? [closed]
Suppose $f,g$ are two uniformly continuous functions on $\mathbb R$, and $h$ is a continuous function on $\mathbb R$ that satisfies:$$f(x)\le h(x) \le g(x)$$Does that mean $h$ is a uniformly ...
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Generalization of Cauchy's functional equation. What are the general solutions, $f$?
Consider a function $f : (0,\infty) \to (0,\infty)$ satisfying the identity
$$
f(x^a y^b) \;=\; f(x)^{1/a}\, f(y)^{1/b}
\qquad\text{for all } x,y>0 \text{ and all real } a,b\neq 0.
$$
This can be ...
- 1,209
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Formal Justification of Alternating Harmonic Series: $1-\frac{1}{2}+\frac{1}{3} - \cdots = \ln(2)$ [duplicate]
It's well known that $\sum_{n=1}^\infty \frac{(-1)^n}{n}=\ln(2)$, which can most easily be seen by the following derivation:
$$\frac{1}{1-x} = \sum_{n=0}^\infty x^n \tag{Geometric Series}$$
$$\frac{1}{...
- 5,547
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How do you think about uniform continuity?
My background is in physics, so I never had a proper course in either real or complex analysis; topics like uniform convergence weren't touched upon. I really like analysis though, so for the last few ...
- 2,354
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Representation of this indicator function.
Let $(S,\Sigma)$ be a measurable space, $f_1,\cdots,f_n:S\to\Bbb R$ be measurable functions, and $f:=(f_1,\cdots,f_n)$. For $A\in \mathcal F:=\{f^{-1}(B)\mid B\in\mathcal B(\Bbb R^n)\}$, I want to ...
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The intuition behind defining trigonometric functions in the complex plane as special combinations of exponential functions [closed]
I was studying Complex Analysis from "A First Course of Complex Analysis" and the authors stated directly that sine and cosine are defined as follows (without any intuition):
$$ \sin\left(z\...
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