user35959
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Show that this set is a topology of X.
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6 votes

Your TeX didn't render on my screen and it looks a bit off. I believe you're asking about the topology that consists of sets $\tau = \{O : O \subset X, p \notin O \text{ or } O = X\}$. This is ...

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1 answers
1 votes
166 views
Using the TPS to interpolate between points
3 votes

Given data sites (centers) $\{x_i\}_{i=1}^N$ and data values to interpolate $\{y_i\}_{i=1}^N$, the thin plate spline interpolant is the function $$s(x) = \sum_{i=1}^n c_i \varphi(\|x-x_i\|) + \sum_{l=...

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1 answers
2 votes
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Confusion related to reproducible kernel hilbert space
3 votes

In a reproducing kernel Hilbert space $K$, the reproducing kernel $\phi$ "reproduces" functions pointwise by $$(f, \phi(\cdot, x)_K = f(x)$$ You can actually build your own reproducing kernel ...

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2 answers
1 votes
87 views
Interpolation in 2-D co-ordinate system
3 votes

For a general scattered data interpolation problem with $N$ known data points in $\mathbb{R}^d$, I recommend trying a radial basis function approach. Let the data sites be denoted $\{x_1,...,x_N\}$ ...

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1 answers
3 votes
205 views
A question concerning the Hilbert space trace
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3 votes

I believe that since $T$ is a positive operator, it has a positive square root. A positive operator is self-adjoint, so $T^{\frac{1}{2}} = (T^{\frac{1}{2}})^*$. Therefore, you can decompose $T$ into $...

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1 answers
1 votes
83 views
Bases and Dimension
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3 votes

What have you tried so far? I'll give some hints to help you get started and more details if you need. For 1: To show that $W_1 \cap W_2$ is a subspace, use your definitions! Remember that a subset ...

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1 answers
2 votes
436 views
Notation: Representer Theorem for Reproducing kernel hilbert spaces
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2 votes

I might be misunderstanding the question here, but it seems that you're a bit confused about what $\langle f,k(x_i,\cdot)\rangle $ means. Yes, $f$ is a vector, but it's not necessarily a column vector ...

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2 answers
2 votes
208 views
Recommendations for website/journal/magazine in applied mathematics
2 votes

For multivariate/spatial interpolation (I'm interested in RBFs and meshfree methods), I see things published in SIAM Journal of Numerical Analysis, Mathematics of Computation (Math. Comp), Foundations ...

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3 answers
-1 votes
59 views
antiderivative of $\frac{4}{\sqrt{u}}$
2 votes

Hint: You can re-write this as $4 u^{-\frac{1}{2}}$. Now, use the rule for integrating functions of the form $\int u^n du$. I'll give more details if you need.

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2 answers
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antiderivative of $8t^{-{1/2}}$
2 votes

Hint: $\int x^\alpha dx = \frac{1}{\alpha + 1} x^{\alpha + 1} + C$, as long as $\alpha \neq -1$. Also, note that $\int 8 t^{-\frac{1}{2}} dt = 8 \int t^{-\frac{1}{2}} dt$.

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1 answers
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Refinement of a finite union of measurable sets.
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2 votes

I think you might want a slight change. In your construction, you can't say $E^*_2 = E^*_2 - (E_1 \cup E_2)$, because you have $E_2^*$ on both sides. But, you're close. Define $E_k^* = E_k - [\...

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3k views
Derivatives of trig functions
2 votes

There are a few ways to do this. I'm going to assume you know the chain rule and how to differentiate sine and cosine. Then, $$\frac{d}{dx} \csc(x) = \frac{d}{dx} \frac{1}{\sin(x)} = \frac{d}{dx} (\...

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2 answers
2 votes
326 views
Approximation with sets of functions not forming a vector space?
2 votes

I might be misinterpreting your question, but here is one approach (which Leonid Kovalev discussed a bit above). Let's say we have some samples of a function $f: \mathbb{R}^n \to \mathbb{R}$ at some ...

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1 answers
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565 views
Proving a Sequence is Cauchy
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1 votes

Here is a hint to help you get started. First, recall the definition of a Cauchy sequence. We need to show that for each $\epsilon > 0$, there exists an $N$ such that for all $m,n > N$, we have $...

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1 answers
3 votes
215 views
Laplacian on Sphere of Function Only Depending on Angle Between Points
1 votes

I've ben informed that on the sphere, the Laplacian is rotation invariant. Therefore, the claim that was given to me in the original question appears to be true. I also found the following .pdf that ...

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3 answers
3 votes
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Two distinct Eigenvectors corresponding to the same Eigenvalue are always linearly dependent.
1 votes

The two vectors you list are linearly dependent, as one is just a scalar multiple of the other. This matrix has one eigenvector corresponding to $\lambda = 1$ , given by the vector $(1 \; 0)^T$ and ...

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1 answers
1 votes
168 views
Condition for point wise convergence of a function to be uniform
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1 votes

I can't remember Egorov's Theorem exactly to be honest, so here's another approach if you're curious. I'll give some hints and more details if you want. With the additional assumption that the $f_n$ ...

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1 answers
1 votes
247 views
Subsequence - is it a mapping of the original sequence or the sequence of elements?
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1 votes

This may not be exactly what you're looking for, but I'm suspicious of your function definition of a subsequence that you mention. One can view a sequence $(a_n)$ as a function $f: \mathbb{N} \to \...

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2 answers
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122 views
Bounded , monotone sequence - help with proof
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To start on a problem like this, you should probably start with the definitions. Do you know what bounded and monotone mean? Let's consider the bounded situation for a moment. A sequence is said to ...

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