Kernel feature and derivative of kernel feature linearly independent?
Suppose we have a strictly positive definite symmetric kernel $k$ on an open set $\Omega\subset\mathbb R$. By "strictly" I mean that all kernel matrices $(k(x_i,x_j))_{i,j}$ with distinct $x_i$ are...
View ArticleThe Maximum Principle is valid for piecewise continuous functions?
According to $[2]$ we have:$\textbf{Definition 9. 6. 5 (Maxima and minima).}$ Let $f: X\to \mathbf{R}$ be a function, and let $x_0\in X$. We say that $f$ attains its maximum at $x_0$ if we have...
View ArticleGeneralizing Wilson's Theorem to the real number field [closed]
As we know, Wilson's Theorem shows: $ \forall n \in \mathbb{N} $, if it satisfies $ {(n-1)!\ \equiv{-1} （ \bmod{n} ）} $, it must be a prime number.Could we geralize it to $ \mathbb{R} $,i.e. $ \forall...
View ArticleIs the discretisation formula of high dimension wrong?
In Terence Tao's book "An Introduction to Measure Theory" (Pg.6), the measure of a $d$-dimensional box $B$ is given by$$|B| = \lim_{N\to\infty}\frac{1}{N^d}\#\left(B\cap\tfrac{1}{N}\mathbb...
View ArticleDo we need to take PV to define $1/|x|^{3.5}$ as a tempered distribution?
We know that in 1d, $\frac{1}{|x|^{3.5}}$ is not locally integrable around the origin. I learnt from this post that we could apply the following definition to define a tempered...
View ArticleFinding the inverse function of $g(x,y)$ for fixed $x$, in which the least...
Let $g(x,y)\in C^1(\mathbb{R}^n\times\mathbb{R}^n,\mathbb{R}^n)$ such that the least eigenvalue of $\nabla_y g(x,y)$ is always greater than $1$, $\nabla_y g(x,y)$ is always invertible and $\nabla_y...
View ArticleNested Division in the Ceiling Function
During class, we were introduced to a proof that used the ceiling function. We assumed (without proof) that:$$ \left\lceil{\frac{n}{2^i}}\right \rceil=...
View ArticleWhy do Dedekind cuts have no maximum?
There is a property of Dedekind cuts that states that there exists no maximum $a_0\in A$ for all cuts $A\in\mathbb{R}$.My question is: Why do Dedekind cuts need this property?What would happen if...
View ArticleProof on real decimal fractions [closed]
I'm studying my first course on Real Analysis and I hoped you could help me with: Let c≥ 0 be a real number. Verify that the sequence a0, a1, a2, . . . defined by $a_n := ⌊10^nc⌋− 10⌊10^{n-1}c⌋$,a real...
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Continuity from below property for Hausdorff Content
While studying Hausdorff contents I came across this article by Roy O. Davies. He gives a really nice example in $\mathbb{R}^{2}$ which shows that the 'continuity from below'property...
View ArticleElementary way to evaluate this infinite product [closed]
I would like to evaluate this infinite product for any a>0:$\prod_{i=1}^\infty(1+\frac{a}{i})$.Is it possible? If not can I get a tight upper bound? Thanks!
View ArticleProof of Second Mean Value Theorem for Integrals
This is from wikipedia on MVT.If $G: [a, b] \to \mathbb{R}$ is a positive monotonically decreasing function and $\phi : [a, b]\to \mathbb{R}$ is an integrable function, then there exists a number $x...
View ArticleCalculate the normal cone to $X=\{f: [0,1] \rightarrow [0,1]\mid f \text{...
Let $X:=\{f: [0,1] \rightarrow [0,1]\mid f \text{ increasing}\} $. We endow it with $L_2$ norm, and thus $X \subset L_2([0,1])$.We can show that the set $X$ is closed and thus compact, and also...
View ArticleShow that $S:c_0\to c_0$ given by $S( x_1, x_2,....) = (0,x_1,x_2,..)$ is an...
Let $S:c_{0}\to c_{0}$ be given by $S( x_1, x_2,....) = (0,x_1,x_2,..)$. Show that $S$ is an isometry,where $c_0$ is the subspace of $l_{\infty}$ consisting of sequences which converge to $0$.My...
View ArticleMultinomial theorem for matrices
The multinomial theorem asserts that$$(x_1+\cdots + x_k)^n = \sum_{n_1+\cdots +n_k = n,\ n_1,\dots,n_k\geq0} \frac{n!}{n_1! \cdots n_k!}x_1^{n_1}\cdots x_k^{n_l}.$$How does this fomrula change when we...
View ArticleIn exponent the base and its value can't be negative, then how modulus...
In exponent the base and its value can't be negative, then how modulus function is defined as $y=\sqrt{x^2}$ and $x$ can be any real number. As this modulus function is not obeying exponential definition.
View ArticleDeducing why if $f(x)=2x\mod 1$, then $|x-y|\leq 2^{-n}|f^n(x) - f^n(y)|$ all...
Define$$f(x) = (2x) \mod 1$$ to be the doubling map on $X = (0, 1)$ and the Lebesgue measure mod zero partition $P = \{(0, 1/2), (1/2, 1)\}$. Here, given some finite measure space $(X, \mathcal{F},...
View ArticleFinding the Range of r for a Sequence Avoiding Power of Two Sums
I'm working on a problem and I would appreciate some help.The problem is to determine the set of all real numbers $ r $ for which there exists an infinite sequence of positive integers $ a_1, a_2, ......
View ArticleDetermine if the power series is continuous
I am still getting a hold of power-series, so any help is appreciated. I am being to asked to determine if $g(x)$ is defined/converge/continuous on the following intervals: (I have also included...
View ArticleUnderstanding constructing a 'bad' probability theory example (Williams, 1991)
I am working through the book 'Probability with Martingales' by David Williams. There is a section I would like to check my understanding on. The passage is (in motivating the use of measure in...
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