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4

The e+33 means $\times 10^{33}$, so this number has a 4 at the front and is followed by 33 more digits, 16 of which you know. The fancy name for $10^{33}$ is a 'decillion', so your number is approximately four decillion.


3

Short answer: $(f_1, f_2)$ is just a pair of arrows $f_1$ and $f_2$. $\langle f_1, f_2 \rangle$ is actually just one arrow, and it is the induced one into a product. $f_1 \times f_2$ is also just one arrow, but now between products and it is constructed using projections of the one and the universal mapping property of the other. Before making this more ...


3

$x_1\in\Bbb R^n$ and $x_1\in\mathcal X$ means $x_1\in\Bbb R^n \cap \mathcal X. $ That is essentially the definition of set intersection.


2

Apparently you allow $95$ characters, for $$ 95^{17}=4\,181\,203\,352\,191\,774\,128\,676\,605\,224\,609\,375$$


2

$C_0(\Omega)$ is the completion of $C_c(\Omega)$ in the sup-norm, i.e., $f\in C_0(\Omega)$ if $f$ is continuous on $\overline{\Omega}\subset\mathbb{R}^n\cup\{\infty\}$ and vanishes at the boundary (including both the finite part and the $\infty$ if $\Omega$ is unbounded). Similarly, $C_0^k(\Omega)$ is the completion of $C_c^k(\Omega)$ in the $C^k$-uniform ...


2

In your quote, $L$ is a functional, which takes a function and returns a real number. It absolutely makes sense to talk about optimizing the function argument $f$ for given operators like $L$, and the field of Functional Analysis is dedicated to studying these sorts of optimization problems, among other things.


2

Since $K$-linear maps are the (homo)morphisms of the category of $K$-vector spaces, it is perfectly valid (though not very common) to write $\operatorname{Hom}(V,W)$ for the set (in fact $K$-vector space) of linear maps from $V$ to $W$.


2

Notation is always a matter of context and of convention. That being said ... No. $f_{\Bbb N}(n)$ looks mor like a single function $f_{\Bbb N}$ (so just happening to have the set $\Bbb N$ as an index), evaluated at $n$. This differs from $\{f_1(n),f_2(n),f_3(n),\ldots\}$ which suggests (though I am no fan of "$\ldots$" occuring anywhere) an infinite set (...


1

I strongly recommand Calculus of Variations, from Gelfand & Fomin Given a functional, by example: $$ J[y]=\int_a^b F(x,y(x),y'(x))dx $$ then $\delta J$ is nothing more than the differential $dJ$. In the cited book you can read, page 12: is called the variation (or differential) of $J[y]$ and is denoted by $\delta J[h]$ More details: $$\delta J[...


1

Defining $K^3$ as the set of tuples $(x,y,z)$ with $x,y,z\in K$ and then later on writing these as columns $\left(\begin{smallmatrix}x\\y\\z\end{smallmatrix}\right)$ is common practice to allow multiplying elements of $K^3$ with $3\times 3$ matrices from the left. However, it doesn't hurt to write $(x,y,z)^t$ explicitly when you want to point out to the ...


1

In mathematical notation, your formula is $$ x = \operatorname{sgn}(v_{\arg\max\{\lvert v_i\rvert : v_i\in V\}}) \max V. $$ Notice that if the most negative element of $V$ has a larger magnitude than the most positive element, your formula uses the sign of the most negative element to determine the result, but it does not use the magnitude of the most ...


1

It is also written $\operatorname{card}D$. So it is natural to call the operator card, following the convention of calling the logarithm, cosine, tangent (etc.) functions after their standard abbreviations log, cos, tan, and so on.


1

I don’t think there’s any standard notation for this. One possibility that’s clunky but not wordy is this: $$\displaystyle\sum_{\substack{i\in{\mathbb Z^+\cup \{0\}}\\\left(\lfloor{x\over 2^i}\rfloor\!\!\!\!\mod\!\!2\right)=1} }{\text {(summand)}}.$$ You can change the $2$ and $1$ for the particular base and digit you want. If I were to put something like ...


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