All Questions
Tagged with examples ca.classical-analysis-and-odes
14
questions
0
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1
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167
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What can be an interesting problem of differential equations involving the definition of the Gudermannian function? [closed]
In this post I denote the Gudermannian function as $$\operatorname{gd}(x)=\int_0^x\frac{dt}{\cosh t}$$
and its inverse as $\operatorname{gd}^{-1}(x)$, please see if you need it the definitions, ...
3
votes
2
answers
198
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Example of convex functions fulfilling a (strange) lower bound
I am reading a preliminary version of a paper which focuses on some minimization problems connected to a class of integral functionals. Reading the assumptions of one of the theorems I cannot convince ...
- 391
2
votes
2
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954
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Function with all but mixed second partial derivatives twice differentiable?
Let $f(x,y)$ be a a real valued function on an open subset of $\mathbf{R}^2$ with continuous partial derivatives $\frac{\partial^2 f}{\partial x^2}$ and $\frac{\partial^2}{\partial y^2}$. Is $f$ twice ...
- 97
9
votes
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answer
738
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Mean value property with fixed radius
Let $f$ be a continuous function defined on $\mathbb{R^n}$. It is well known that both the spherical mean value property (MVP) of $f$, i.e.
$$f(x)=\frac{1}{|\partial B(x,r)|}\int_{\partial B(x,r)}f,\ ...
- 981
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220
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Example of function with a certain behavior.
Let $f: R \rightarrow R$. Consider the following properties:
$(1)$ - There are positive constants $a$ and $r$ such that $\forall x, y$
$$|f(x)-f(y)|\leq a(1 + |x|^r+|y|^r)|x-y|.$$
$(2)$ - There is a ...
- 93
6
votes
1
answer
675
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On the Existence of Certain Fourier Series
Is there an $f\in L^{1}(T)$ whose partial sums of Fourier series $S_{n}(f)$ satisfies $\|S_{n}(f)\|_{L^{1}(T)} \rightarrow \|f\|_{L^{1}(T)}$ but $S_{n}(f)$ fails to converge to $f$ in $L^1$-norm ?
- 623
3
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2
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708
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Is there a non-trivial example for a 1-homogeneous function satisfying a specific inequality of second order?
Let $\mathbb{R}^n$ be the $n$-dimensional real vector space with Cartesian coordinates $x=(x^1,\ldots, x^n)\in \mathbb{R}^n$. I'm searching for a non-trivial example of a function $A:\mathbb{R}^n \...
- 31
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votes
3
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808
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Holomorphic function with a.e. vanishing radial boundary limits
Hello everybody.
I'm looking for an "easy" example of a (non-zero) holomorphic function $f$ with almost everywhere vanishing radial boundary limits: $\lim\limits_{r \rightarrow 1-} f(re^{i\phi})=0$.
...
- 732
9
votes
1
answer
2k
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Sequence that converge if they have an accumulation point
I am looking for classes of sequence, that converge iff they contain a converging sub-sequence.
The basic example of such sequences are monotone sequences of real numbers.
A more interesting examples ...
- 757
7
votes
4
answers
8k
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Good example of a non-continuous function all of whose partial derivatives exist
What's a good example to illustrate the fact that a function all of whose partial derivatives exist may not be continuous?
- 1,429
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3
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2k
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What are some interesting sequences of functions for thinking about types of convergence?
I'm thinking about the basic types of convergence for sequences of functions: convergence in measure, almost uniform convergence, convergence in Lp and point wise almost everywhere convergence. I'm ...
29
votes
12
answers
6k
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When does 'positive' imply 'sum of squares'?
Does anyone have examples of when an object is positive, then it has (or does not have) a square root? Or more generally, can be written as a sum of squares?
Example. A positive integer does not ...
9
votes
1
answer
1k
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"Vector bundle" with non-smoothly varying transition functions
I'm working my way through Lang's Fundamentals of Differential Geometry, and when he introduces vector bundles, he states that for finite dimensional bundles, the third axiom is redundant. I'm hoping ...
- 787
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3
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988
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Does the "continuous locus" of a function have any nice properties?
Suppose $f:\mathbf{R}\to\mathbf{R}$ is a function. Let $S=\{x\in \mathbf{R}|f\text{ is continuous at }x\}$. Does $S$ have any nice properties?
Here are some observations about what $S$ could be:
$S$ ...
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