## ac.commutative algebra – Reverse mathematics of Noetherian rings over $mathbb{Q}$

Take the Hilbert Basis Theorem over the rational numbers in this form in the language of Second Order Arithmetic:  For every $$nin N$$ every ideal of the polynomial ring $$mathbb{Q}(x_1,dots,x_n)$$ is finitely generated.  Simpson has shown this is stronger than Exponential Function Arithmetic (EFA).  He proved much more on the subject but this is what interests me.

I expect the statement gets weaker if you specify $$n$$.   Is that right?  So for example it would take less to prove every ideal of  $$mathbb{Q}(x_1,dots,x_4)$$ is finitely generated.

Maybe the statement does not get weaker when you specify $$n$$, since the number of generators of ideals remains unbounded. See answers to the question Number of generators of an ideal in a polynomial ring over a Noetherian ring.

Is it known how strong the theorem is for specific $$n$$, or at least for some low specific $$n$$?

The reference for Simpson is

• “Ordinal Numbers and the Hilbert Basis Theorem,” The Journal of Symbolic Logic, Sep., 1988, Vol. 53, No. 3 (Sep., 1988), pp. 961-974. doi:10.2307/2274585, JSTOR

## discrete mathematics – How to solve this proof

Proof: Suppose r and s are rational numbers. If r + s is rational, then by definition of a rational r + s = a/b for some integers a and b with b $$neq$$ 0. Also since r and s are rational, r = i/j and s = m/n for some integers i, j, m and n with j $$neq$$ 0 and n $$neq$$ 0. It follows that

r + s = $$frac ij$$ + $$frac mn$$ = $$frac ab$$

which is a quotient of two integers with a nonzero denominator. Hence it is a rational number. This is what was to be shown.

What are the steps to solve this, find the mistake and write it down correctly?

## Undergraduate student in mathematics studying on his own

Hello everyone for a long time I want to start studying mathematics for a bachelor’s degree myself but do not know from which book I should study I realized I should start from calculus 1 and calculus 2 but I need a recommendation from you for good textbooks
I would be happy if you would answer my request.

## discrete mathematics – Does a function exist that maps the natural numbers to the natural numbers excluding multiples of 2 and 3 and starting at an arbitrary point?

This question is a generalization of this other question. To recap, I was looking for a function that mapped the set of natural numbers to the set of natural numbers excluding multiples of 2 or 3 or both. So a function $$f: x mapsto y$$ where $$x = {0, 1, 2, 3,…,m}$$ and $$y={1,5,7,11,13,17,…,n}$$ for some arbitrary limit $$m$$. @A.J. gave a great solution

$$y=qquad 6left lceil dfrac{x}{2} right rceil + (-1)^x$$

Now I’m wondering about a generalization that allows for the set $$y$$ to start at an arbitrary point (that is not a multiple of 2 nor 3). For example instead of $$x=0$$ mapping to $$y=1$$ it would instead map to $$y=5$$ and the formula would then be $$y=6left lceil dfrac{x+3}{2} right rceil + (-1)^{x+3}$$ in other words translated 3 units. Or if we wanted $$x=0$$ to may to $$y=41$$ then it would be $$y=6left lceil dfrac{x+13}{2} right rceil + (-1)^{x+13}$$. But I only know this because I precomputed the set starting at 1 and translated the number per the number of skipped members. If the number is bigger this would be non-trivial. For example, $$101$$ is not a multiple of 2 nor 3 so what function would start with $$x=0$$ mapping to $$y=101$$ and then continue by skipping multiples of 2 and 3 e.g. $$103, 107, 109$$.

Is there a generalization for a function that given the “starting point” $$a$$ that is not a multiple of 2 nor 3, and the parameter $$x$$, it maps $$x={0,1,2,3,…,m}$$ to $$y$$ where each member of $$y$$ is $$ge$$ $$a$$ and each subsequent term is the next non-multiple of 2 nor 3?

## Differential equation prove – Mathematics Stack Exchange

I’m starting the differential equation course and I would really appreciate if anyone can help me with ideas solving this prove:

Let a, b:$$Ito R$$ continuous functions with $$I subseteq R$$ and $$U: I to R$$ the solution of the differential equation $$u”= a(t)cdot u + b(t)cdot u’$$ with $$u(t_0)=0$$ and $$u'(t_0)=1$$ for some $$t_0$$ belonging to $$I$$. Show that if a>0 then $$u$$ is increasing in $${t in I: tgeq t_0}$$

## Measure preserving transformation exercise – Mathematics Stack Exchange

im trying to solve this exercise but I need some hints because my teacher didn’t give me the theory necessary to solve it.

Let (X, M, µ) be a measurement space such that µ (X) = 1. Suppose
that T: X → X is measurable and µ (T^−1 (E)) = µ (E) for all E ∈ M. Prove that
(1) For all E ∈ M such that µ (E)> 0 there exists a natural n such that µ (E ∩
T^n (E))> 0. Here T^0 is the identity map in X and T^n = T ◦T^(n − 1) for
n ≥ 1.
(2) For every E the set of points for which there is a
natural n0 such that T^n(x) doesnt belong to E for all n ≥ n0 has measure 0.

## Statistic and Probability – Mathematics Stack Exchange

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• Making statements based on opinion; back them up with references or personal experience.

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## math – Mathematics Economics Accountings and Business Laws

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Mathematics is the Science in which one learns how to transcript any information in usual numeric functions

Economics is the Science in which one learns how to define, determine and allocate resources

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## Measure abstract – Mathematics Stack Exchange

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But avoid

• Making statements based on opinion; back them up with references or personal experience.

Use MathJax to format equations. MathJax reference.

## What research level mathematics do you talk about? [closed]

I am new to this sort of thing.