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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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4 votes
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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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4 votes
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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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  • 6,406
0 votes

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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6 votes
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Cohen reals satisfying a formula

No; consider for example $\varphi(x$) = "Every other bit of $x$ is $0$." No Cohen-generic real has this property, but if $c$ is Cohen-generic then the "spaced out" version of $c$ $$...
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1 vote
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A new (?) infinitely nested radical equals $1$

Consider the function $f:[-b/a,\infty)$ with $a\gt0,$ $b\in\mathbb{R}$ such that $f(x)=\sqrt{ax+b}.$ We can define the sequence family $s[f]$ such that $s[f]_{n+1}=f(s[f]_n).$ Define $f^n$ such that $...
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maximal number of independent subsets (bitstrings with uniform distribution)?

Here is a constructive converse of the accepted answer for completeness ($P=\{0,1,\ldots,p-1\}^n$, $p$ prime). As suggested in the question, the $n$ cylinder sets $P_i=\{x=(x_1,\ldots,x_n)\in P : x_i=...
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2 votes
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Natural Deduction: An unusual(?) presentation

Long comment Based on the previous Def.1.1 the tensor $⊗$ "works like" AND: $A ⊗ \top \to A$, and thus the pair of rules above are exactly those for $∧$ in Sequent Calculus. The first one ...
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2 votes

Good applied differential geometry books

Personally I like differential geometry a lot and also like to find more about their applications. Not sure if there is an overview book talking about all the applications, but for each field there ...
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1 vote

Name for sum of diagonals for Hilbert Schmidt normalized matrix.

Use an unconstrained matrix $(U)$, the identity matrix $(I)$, the Hadamard product $(U\odot V)$ and the matrix inner product $\big(U:V={\rm tr}(U^TV)\big)$ to construct the constrained matrix $(A)$ ...
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maximal number of independent subsets (bitstrings with uniform distribution)?

Since $|\Omega|=|\{0,1\}^n|=2^n$, there can be at most $n$ non-trivial independent events in $2^{\Omega}$. Thus, excluding the constant functions $f\equiv 1$ and $f\equiv 0$, there can be at most $n$ ...
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1 vote
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maximal number of independent subsets (bitstrings with uniform distribution)?

Here's a proof that if the sample space $B$ has size $p^n$ where $p$ is prime, then we can have at most $n$ non-trivial independent events. (The simpler answer suggested by d.k.o. gives the same ...
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3 votes
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A Reference From Andrej Bauer's Recent Talk on Countable Reals

The constructive news google group doesn’t appear in Google search results, it seems. The post was from June 10, 2018, and is available at this link: https://groups.google.com/g/constructivenews/c/...
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1 vote
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Prerequisite mathematics for nonlinear systems

You will need some (linear) algebra, analysis, and dynamical systems for that book. For linear algebra and analysis, any book would do the job. For dynamical systems you may look at the books by ...
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Literature Request: Koszul-Tate Resolutions

If you are not familiar with the usual Koszul resolution, a purely classical (as opposed to derived) treatment is in Eisenbud-Harris's Commutative Algebra. You might also look at Toen's EMS survey ...
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Proof a graph is bipartite if and only if it contains no odd cycles

One direction is clear. For the other direction you can show this by induction on the number of vertices of $G$. Suppose $G$ is a graph on $n$ vertices and that it does not contain any odd cycle. Pick ...
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Closure in the strong dual topology.

I think that this is true for any locally convex space $F$. For a subspace $E$ of the topological dual $F^*$ we want to prove $$ \overline{E}^{\beta(F^*,F)}= \{f\in F^*: f|_B \text{ $\sigma(F,E)$-...
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0 votes
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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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3 votes

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 ...
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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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3 votes

Book/Note recommendation for topology on spaces of matrices

What you might want to do is to familiarise yourself with the properties of $S_1\cap S_2\cap\cdots\cap S_k$, where each $S_i$ is of the form $p(x_1,x_2,\ldots,x_{n^2})=0$ or of the form $p(x_1,x_2,\...
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Euler characteristic of an $n$-sphere is $1 + (-1)^n$.

In Hatcher's algebraic topology, he has a formula $$\chi(M)=\sum_n(-1)^n dim_{\mathbb{Z}}(H_n(M))$$ with $H_n$ being the $n$-th singular homology group. For an $n$-sphere $S^n$, we have $H_0(S^n)=H_n(...
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1 vote
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Multi-dimensional Dirichlet-Dini criterion for Fourier series

In the multidimensional case the condition for a continuous function $f$ with the modulus of continuity $\omega(t)$ to have pointwise Pringsheim convergence it is enough that $\omega(t)\log^d\frac1t\...
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1 vote
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Completeness of weighted $L^p$ spaces

When you write $(L^p(\mathbb{R}^n),\omega\,dx)$ I would guess you mean the space of all measurable functions $f$ such that $|f|^p\omega\in L^1$? A better notation might be $L^p(\mathbb{R}^n,\omega\,...
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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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1 vote

Classifying the Hessian Matrix of a Morphism

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(...
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2 votes
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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 ...
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2 votes
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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 &...
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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 $\alpha$ and $$...
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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 ...
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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 ...
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1 vote
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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 ...
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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 ...
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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 ...
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$S$ 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, ...
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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 ...
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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#...
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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, ...
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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....
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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{...
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  • 5,066
2 votes

On dimension of the Lie group $SL(n,\mathbb{C})$

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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  • 507
2 votes
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Analytic works versus synthetic works in mathematical research

Professional mathematicians do not classify research as synthetic or analytic. The analytic/synthetic distinction comes from philosophy, and it is very rare for mathematicians to use these words in ...
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