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4

You could not find a proof because it is not true. In fact $BL(X)$ is separable in the topology of uniform convergence if and only if $X$ is totally bounded. If $X$ is totally bounded, every Lipschitz function can be extended to its completion $\tilde{X}$, which is compact. Thus we obtain an isometric embedding $BL(X) \to C(\tilde{X})$. If $X$ is not ...


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One of the biggest challenges is organization. Deciding how exactly to organize all of the material you want to cover so that it is both, ordered logically, and also provides motivation for an idea before it is introduced. All the while, you also want to draw attention to the various connections between the various concepts presented in the text. A big part ...


3

Your first question is trivial as written: just take $\phi = 0$, $u_1 = v$, $u_2=u_3=0$. Perhaps you meant for the $u_i$ to be independent of $v$? In this case it's almost true, but you need to allow arbitrary linear combinations of the $u_i$: There are three curl-free, divergence-free vector fields $u_i$ on $\mathbb T^3$ such that for every curl-free ...


3

We can see: José Ferreirós' review of C.K. Raju, Cultural Foundations of Mathematics: The Nature of Mathematical Proof and the Transmission of the Calculus from India to Europe in the 16th c. CE, in Philosophia Mathematica Volume 17, Issue 3: In his interest to revise traditional historiography and oppose proofcentred mathematics, Raju devotes a lot ...


2

I've searched up to $10^8$ and the closest you get to your bound is at $230631$ which appears in your first graph. Nothing else is even above $240 + x^{.43}$ let alone your bound. $$\{x \in \Bbb{Z}|\quad0<x\le10^8 , \quad f(x) > 240 + x^{.43} \} = \{230631\}$$ As was mentioned in the comments. There are no known results of the type you are asking. I ...


2

I'm very fond of Forster's book Lectures on Riemann Surfaces. Check it out. There is also a lovely book by Phillip Griffiths called Introduction to Algebraic Curves.


2

The articles by Raju have a conspirational flavor. The history of Indian mathematics is still an uncharted territory. There are more informative unbiased articles, for instance, there are much deeper and less biased studies I've read: A. Seidenberg, “The Origin of Mathematics,” Archive for History of Exact Sciences 18, 301-342 (1978). S.C.Kak, “Science in ...


2

In fact one can! First, if it's all the same to you, let's assume that the base field is algebraically closed so we can say a bit more about representation theory. One should first note that this representation is just $V^* \otimes_{\mathbb{C}} V$ so you can completely describe its character. Second one even has an explicit description of the copy of the ...


2

By Kronecker's formula, the value at $s=1$ of $L(s,\chi)$ functions associated with quadratic characters depends on a class number, so there actually is an algebraic counterpart dealing with reduced binary quadratic forms. The basic problem is to find the sign of a Gauss sum, where $G(\chi)^2 = p$ is almost trivial. However, I am not so sure that "the ...


2

Some quick hints: For (A1), you need to know the algorithms. The Prime pages provide a good start, including the underlying mathematics and references. I recommend following those references too. For (A2), $100$ digits are easily handled with the right algorithms. Primality proving for general numbers below $400$ decimal digits takes less than half a ...


2

(I'm taking the reading that the chain itself is uncountable, not the sets appearing in the chain) Note that $\mathbb{N}^n$ is an infinite discrete subset of $\mathbb{R}^n$, and therefore, so too is any subset. Hence, if we can find an uncountable chain in $\mathcal{P}(\mathbb{N}^n)$ we are done. As $\mathbb{N}^n$ is countable and we no longer care about ...


1

To be precise in what follows, here are what I am assuming are the axioms: $(\beta\rightarrow (\alpha\rightarrow \beta))$ $((\alpha\rightarrow \beta)\rightarrow ((\alpha\rightarrow (\beta\rightarrow \gamma))\rightarrow (\alpha\rightarrow \gamma)))$ $(((\alpha\rightarrow \bot)\rightarrow \bot)\rightarrow \alpha)$ $(\forall x (\varphi\rightarrow \psi) \...


1

With respect to an example about the chain's rule consider that a sequence of maps $\Bbb R\stackrel{\alpha}\to\Bbb R^2\stackrel{\Phi}\to\Bbb R^3$ allows you to control the movement along a curve in a surface: The surface will be the image of $\Phi$, let's dubbed it $\Sigma$, and the curve $C=\Phi\circ\alpha$ in that surface. Now, knowing that $C'$ is a ...


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The only one I am familar with is Dolgachev's Lectures on invariant theory. Much more books can be found in: http://mathoverflow.net/questions/166/resources-on-invariant-theory


1

Something like $f(x) = 10 e^{-100 x^2}$ on $[-1,1]$ The idea is that the Simpson's rule weight heavily the middle point, so if your function is nearly constant but has a very narrow and high spike at the middle point, it will be worse than just taking the endpoints


1

You can find the general statement and proof in Chapter 6 of the book "Sobolev Spaces" by Robert A. Adams and John. J. F. Fournier.


1

An elegant method is to use the Expansion Theorem in Umbral Calculus. Below is a typical statement, from p. 18 of Roman's book The Umbral Calculus.


1

No. Consider the curve $$ \gamma(t) = (s \cos t, s \sin t), 0 \le t \le \frac{L}{2\pi s} $$ where $s < r$. its curvature is $\frac{1}{s}$, which is clearly unbounded. If you don't like that the path intersects itself, just make $s$ a very slowly increasing function of $t$ with mean $S$. Then the curvature will be approximately $\frac{1}{S}$.


1

I think you had the following in mind: The three cycles $$\sigma_i:\quad t\mapsto t e_i\qquad(0\leq t\leq1)$$ form a basis of $H^1(T^3)$. Compute the periods of $v$ along the $\sigma_i$, i.e., the numbers $$\alpha_i:=\int_{\sigma_i}v\cdot dx\ .$$ Use these to define the constant field $$u:=(\alpha_1,\alpha_2,\alpha_3)\ .$$ Then there is a potential $\phi:\&...


1

The obvious way to do this is to let the entries in $F$ be a set of $n^2$ unknowns to be solved for. It is necessary and sufficient for the equations to be satisfied for all $g$ in a generating set $S$ of $G$. So you get a system of $|S|n^2$ linear equations in $n^2$ unknown to solve. That should work up to dimension $22$. It depends a bit on the field. If ...


1

I would say what you're interested in is Fourier multipliers. The Fourier transform has the convenient property that for suitable functions, $\widehat{Df}(\xi)=\xi\hat f(\xi)$ where I've used the French convention $D=\partial/i$. This suggests that a factor of $\xi$ represents one derivative, so a more complicated function of $\xi$ represents a more ...


1

It sounds like you are interested in operator theory. The fact that you are asking this question suggests that you are pretty advanced for where you are but to really understand this, you need a background in functional analysis. I'm also not sure that abstract algebra is going to offer you any answers. Operator algebras generally fall under the umbrella of ...


1

Edward Packel's brief text The Mathematics of Games and Gambling is a basic introduction to the mathematics needed to analyze gambling and game activities. Packel focuses on the mathematics. The book is not intended as a discussion of gambling strategies. https://www.amazon.com/Mathematics-Games-Gambling-Mathematical-Library/dp/0883856468/ref=sr_1_1?ie=...


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This might be too advanced, but as far as I am aware, Dubins and Savage's "How to Gamble if You Must" is the established classic in the field of martingales (a topic in probability theory) as applied to gambling: https://www.amazon.com/How-Gamble-You-Must-Inequalities/dp/0486780643


1

Riemann surfaces is a very standard topics in math, then you can find a lot of books talking about Riemann surfaces under different point of views. I can suggest you: -Compact Riemann surfaces - J.Jost, -Riemann Surfaces - S.Donaldson, -Riemann Surfaces - Farkas and Kra, -Algebraic curves and Riemann surfaces - R.Miranda The first is pretty analytical,...


1

Let's check out the OP's wish list: Numerical analysis Interpolation (Hermite, multidimensional, Newton, Spline ...) The webpage Programming in Delphi contains some material (theory & software) about interpolation, with 2nd order splines only. Numerical methods for solving differential equations My own experience in this field has been summarized as : ...


1

The standard undergraduate text for real analysis is Rudin's Principles of Mathematical Analysis (affectionately referred to as "Baby Rudin" since he wrote it when he was quite young). Another text I enjoyed was Serge Lang's Undergraduate Analysis. I think they're both have their pros and cons but are ultimately both fine books for first learning some ...


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'Diophantine Geometry: An Introduction', by Silverman and Hindry, contains a proof in section A.4.3 (Page 74). A few minor steps are skipped, but it should be fairly easy to follow.


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I don't know exactly if this is what you are looking for or if it's at the right level, but you might want to have a look at the first three chapters of John Ratcliffe's Foundations of Hyperbolic Manifolds. They concern Euclidean, spherical and hyperbolic geometry (with sections about trigonometry), and the book is modern and (in my opinion) well-writen.


1

Let $N = q^k n^2$ be an odd perfect number with Euler prime $q$. In a recent preprint, Brown claims a complete proof for $q < n$, and a partial proof that the inequality $q^k < n$ holds under many cases. (See arXiv.) In particular, since $q < q^k < n$ holds if Brown's proofs are correct (and completed), then the resulting lower bound is $$\gcd(...



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