# Tag Info

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### Reference request for studying product measure.

"REAL ANALYSIS " by H.L.Royden and P.M.Fitzpatrick (4th edition) See the chapter $20$ : The Construction of Particular Measures. I think you will enjoy to study from here. Let's give a try.
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### Resources for finding the dimension of the orbits

For any matrix $X\in M_{n\times p}\mathbb{R}$ there is an upper triangular matrix $Y$ with nonnegative diagonal entries such that $Y=UX$ for some $U\in O(n)$. Indeed, the diagonal entries of $Y$ will ...
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### What's the definition of $C^\alpha$ norm of a tensor?

For a definition of $C^{k,\alpha}$ as well as references see my answer here: Definition of Hölder Space on Manifold, where a function $f:M\to N$ between two $C^\infty$ manifolds (of finite or ...
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### Formalizing Natural Languages

So, here are a few directions you may wish to explore for formalizing natural language, which is a broad topic. From looking a bit, this question does not appear to be an exact duplicate of an earlier ...
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### What is this concept called (differentiating a matrix, NOT talking about Jacobians)

What you have written is not the derivative $\frac{\partial A}{\partial X}$, which would be zero for a fixed matrix $A$. What you want is $\frac{\partial f}{\partial X} \mid_{A}$, i.e. the derivative ...
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### Generalised inclusion-exclusion principle

Here are some additional references: Enumerative Combinatorics, Vol. I by R. P. Stanley: From chapter 2 Sieve Methods', Exercise 3: Let $S=\{P_1,\ldots,P_n\}$ be a set of properties, and let $f_k$ (...
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### Resources, references, or examples for logics with finitely many sentences

I've come up with a semantics for a simple logic that captures some of the intuition behind having finitely many statements. It does this by keeping track of the length of formulas and declaring all ...
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### Reference for Analysis book in which natural numbers constructed from sets

I think that you should turn more to Set Theory / Logic books than Real analysis books. One example is Basic set theory from Azriel Levy. What you're looking for is the definition of ordinals and ...
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### Covariant and exterior covariant derivative of a bundle-valued $n$-form.

Actually by digging around more deeply I found this quite interesting and detailed answer of Ribeiro to an MO question Ribeiro_answer that generalizes this result to higher order iteration of the ...
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### Why is positional number system natural?

Why binary? With $0$ characters in our alphabet, we can't have nonempty strings, or represent more than one object; with $1$ character, strings of length $\le n$ can represent only $O(n)$ objects; ...
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### Why is positional number system natural?

This is something that's recently made me curious: I also wonder if the choice of representation is somehow arbitrary, or whether maybe positional notation satisfies a kind of universal property with ...

### Erdős-Straus conjecture

As a note, when $p$ is a prime of the form $p=4a^2+2a-1$, the conjecture holds. More generally, if $p$ is of the form $p=4a^2+4ak-2a-k$ where $a$ is a positive integer $k$ is another integer ...
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### Any good alternatives to Inverse Symbolic Calculator?

The site is down indefinitely, however, it is still accessible at the original site.
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### Optimality results for Fitch-style natural deduction proofs

I can sort of answer the second question, and say something about the first. It would be kind of meaningless to ask for the shallowest proof, for two reasons. Firstly, every FOL tautology can be ...
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Lemma. If $\phi:\Bbb R_{\geq 0}^n\to\Bbb R_{\geq 0}$ is continuous and additive, then it is linear. Proof. For any nonnegative integer $n$, $\phi(nx)=\phi((n-1)x+x)=\phi((n-1)x)+\phi(x)=\cdots = n\phi(... • 49.7k 2 votes Accepted ### On the function$n \mapsto |a_n|^{\frac 1n}$for a given power series$\sum_{n} a_n z^n$At the beginning of its mathematical career, Shmuel Agmon wrote several complex analysis papers dealing with the singularities of Taylor series, possibly under the influence of its advisor Szolem ... • 8,358 2 votes Accepted ### Who should be owed to the Morita equivalence theorem on modules over algebras? (First, note that this is only true for all$k$-algebras if$k$is algebraically closed.) I believe that it is (essentially, at least) due to P. Gabriel. At least, the quiver$Q$is often called the &... • 24.8k 3 votes ### A property of product forcing First argue that it is enough to show that if$x\in V[G_1]\cap V[G_2]$is a set of ordinals then$x\in V$. Now suppose$\dot x$is a$\mathbb P\times\mathbb P$-name for a subset of some$\alphaand $$... • 2,763 2 votes ### Anyone recommend a fairly modern/new textbook on functional analysis/PDE's to be used as a reference for a graduate level course? [1] Lieb, Elliott H., and Michael Loss. Analysis. Vol. 14. American Mathematical Soc., 2001. is a graduate analysis book that has some functional analysis and more PDE than other analysis books. I ... • 10k 1 vote ### Is there an introduction to probability and statistics that balances frequentist and bayesian views? Too long for a comment. I am an applied statistician, I don't distinguish much between the two. Bayes theorem is non-controversial, it is a theorem anyway. The problem arises when people use Bayes ... • 6,092 1 vote Accepted ### Rational functions with a special symmetry Geometry g:z\mapsto -1/z is a reflection at the unit circle S, and it maps S to itself. It's an isometry on the Riemann sphere \Bbb C\cup\{\infty\} that maps the southern hemisphere (interior ... • 3,785 0 votes ### On the general relationship between automata, expressions, and grammars Automata Theory was (at least in the US) much more popular in the 60's and 70's than it is today. There are still applications today in Programming Language Theory (aka "Theory B"). If you ... • 13.4k 1 vote ### Is it sufficient to prove collatz conjecture doing it for 3+6k, k \geq 0? This paper proves your idea to be correct: Kenneth M. Monks, The sufficiency of arithmetic progressions for the 3x+1 conjecture (2006) The result of this paper is more general and directly proves your ... • 301 1 vote ### S is the set of words generated by an alphabet. A\subset S , x\in S. How to find if x is generated by concatenating elements of A? Partial answer: For your example - you can build a non-deterministic FSM in the spirit of Aho-Corasick algorithm (transitions = elements of A), and feed x to that FSM to get an answer. However, ... • 22.3k 1 vote ### Calculus of Variations text my Mark Kot I am currently studying Classical mechanics, so naturally, I felt the need to refer to the Calculus of Variations, and unexpectedly I came across this book. So far, I can say this book has been the ... 0 votes ### Numerical methods to minimize a matrix function \def\a{\alpha}\def\b{\beta}\def\g{\theta}\def\l{\lambda} \def\p{\partial} \def\A{\|A\|_{S_p}} \def\LR#1{\left(#1\right)} \def\trace#1{\operatorname{Tr}\LR{#1}} \def\qiq{\quad\implies\quad} \def\grad#... • 27.4k 1 vote ### Solving system of delay differential equations Have a look at the ddeint package. Some further explanation can be found here. It is not very fast, but very flexible, and coded in just a few lines on top of Scipy’s differential equations solver, ... • 146 0 votes Accepted ### Explicit matrices h_\alpha that correspond to the long roots \alpha in a classical compact simple Lie algebra over the reals Over on MathOverflow, Konrad Waldorf supplied the reference Gawȩdzki, Krzysztof; Reis, Nuno, Basic gerbe over non-simply connected compact groups, J. Geom. Phys. 50, No. 1-4, 28-55 (2004). ZBL1067.... 0 votes ### How to compute \int_0^{\pi/2}\frac{\sin^3 t}{\sin^3 t+\cos^3 t}dt? Dividing both the numerator and denominator by \cos ^n x converts$$ \begin{aligned} I_{n} =&\int_{0}^{\frac{\pi}{2}} \frac{1}{1+\tan ^{n} t} d t \\ \begin{aligned} \\ \end{aligned} \\\stackrel{... • 5,066 2 votes ### On dimension of the Lie groupSL(n,\mathbb{C})\$

It depends on what they meant by dimension: If they meant "dimension as a complex manifold" (or a complex Lie group), then (b) is the right answer, but if they meant "dimension as a ...
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### Rigorous Development of Relational Algebra

I liked the book "Database Systems" by Hector Garcia Molina et al. It's free online https://www.pearson.com/us/higher-education/program/Garcia-Molina-Database-Systems-The-Complete-Book-2nd-...
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