arithmetic functions – Grade 6 math Gymi problem

This is the original problem in german below. Can someone tell me what does it mean and what resources should my daughter read to help translate these problems by herself? She is a2-b1 level in German. But seems mathematical German is a different thing.

Many thanks for any help.

Berechne die diferenz des produktes der Zahlen 4 und 18 von ihrem Quotienten,wobei die grössere Zahl der Dividend ist

nt.number theory – Ellenberg and Gijswijt’s result on arithmetic progressions in subsets of $mathbb{F}_q^n$ and a generalisation to sets of linear equations

Ellenberg and Gijswijt showed that the largest subset of $mathbb{F}_q^n$ with no three terms in arithmetic progression has size $c^n$ where $c<q$.

Ellenberg and Gijswijt actually proved a generalisation:

Let $x_1, x_2, x_3in S subset mathbb{F}_q^n$ then

$a_1x_1+a_2x_2+a_3x_3=0$ with $a_1+a_2+a_3=0$ has no solutions (excluding the trivial $x_1=x_2=x_3$) $implies$ $|S|<c^n$ where $c<q$.

(Choosing $(a_1,a_2,a_3)=(1,-2,1)$ gives the case of AP’s)

My question is:

How do the above bounds for |S| improve if we add additional equations and ask that all of them individually have no solutions?

Specifically consider a set of equations $$a_{i1}x_1+a_{i2}x_2+a_{i3}x_3=0,$$ $1 leq i leq m$ with $a_{i1}+a_{i2}+a_{i3}=0$, distinct (up to multiplication by a constant) and the $a_{ij}$‘s all non-zero.

If $x_iin S subset mathbb{F}_q^n$. How large can $|S|$ be if none of the above equations have a (non-trivial) solution?

Note that my condition is not that the system of equations as a whole has no solution but rather the much stronger assertion that none of the equations of the system individually can be satisfied non-trivially.


Any guesses, heuristics, plausible conjectures are welcome.

arithmetic progressions – Question from AP

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arithmetic – Adder-Subtractor Circuit With Negative Results

So, I understand how binary arithmetic works, and I understand how an adder-subtractor works for signed numbers. There is only one thing I am not sure about:

All the cases work ok in the circuit I have, except if the result of a subtraction is negative, I need to take the two’s complement of the output byte to get the actual result. What can I do about it? Do I need an extra array of adders to compute the two’s complement only in that specific way, or is there any smarter solution I can apply?

Thanks in advance.

lo.logic – Disjunction in weakened Robinson arithmetic

Let $ T $ denote the theory obtained by removing the axiom $ forall x ( x = 0 lor exists y , S y = x ) $ and restricting double negation elimination to disjunction-free formulas of Robinson arithmetic. In other words, $ T $ is axiomatized over intuitionistic logic by arithmetical axioms of Robinson arithmetic other than $ forall x ( x = 0 lor exists y , S y = x ) $, and double negation elimination for disjunction-free formulas.

I’ve been playing around with $ T $ for quite a while (actually months, on and off), and it seems that no genuine theorems containing a positive disjunction can be proven in $ T $. By genuine I mean those theorems that are not proven by mere logic, like those of the form $ A rightarrow A lor B $, or $ A lor B rightarrow C lor D $ where $ A $ implies $ C $ and $ B $ implies $ D $. For example, I couldn’t find any way of proving $ forall x ( x < 2 rightarrow x = 0 lor x = 1 ) $. As neither induction nor classical logic is at hand, non of the usual ways of thinking seems to work, at least as far as I could check. I’m really not sure about unprovability of this sentence, as I couldn’t find any useful method to show that it’s not provable.

So to avoid any possible complexity that could come from considering more general theorems containing positive disjunctions, I ask my question like this:

Is there a method for showing that $ forall x ( exists y , x + S y = 2 rightarrow x = 0 lor x = 1 ) $ is provable/unprovable in $ T $?

sequences and series – question on arithmetic mean with high order thinking skill

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Dereferencing pointer and storing reference to the pointer, then getting address of reference and performing pointer arithmetic on it. c++

In order to make a diagram(of my own, to understand pointers/pointers-to-pointers/arrays) that is out of the scope this question, I want to know if it is adequate to assume the following:

int *intptr = new int(10){0};
int &intref = *intptr;
int thirdInt = *((&intref)+2);//is this defined behavior and always equivalent to intptr(2)?????
delete () intptr;

as the only comment in the code asks: does this code invoke undefined behavior or not work as expected by the comment?

lo.logic – What subsystem of third order arithmetic proves the real numbers are Dedekind complete?

Reverse mathematics is mainly about subsystems of second-order arithmetic, but in recent years it’s expanded to cover subsystems of third-order arithmetic as well. Now the fact that the real numbers are Dedekind complete is (encodable as) a statement in the language of third order arithmetic. And I think it’s probably provable using full third order arithmetic.

But my question is, what is the weakest subsystem of third-order arithmetic capable of proving that the real numbers are Dedekind complete?

By the way, the fact that the real numbers form a real closed field is provable even in $RCA_0$, so my question is really about the interpretability of the second-order theory of real numbers.

reference request – arithmetic del Pezzo surfaces in comparison with del Pezzo surfaces over a field

A del Pezzo surface is a smooth, 2-dimensional projective variety $X$ with ample anticanonical divisor, i.e. a 2-dimensional Fano variety.

I am interested in the arithmetic analogue, a 2-dimensional projective scheme $X$ with a relative 1-dimensional morphism to a Dedekind scheme $X to S$, such that the anticanonical divisor is ample.

It would be sensible to require $X$ to be a regular scheme, but I would prefer not to require the structure morphism to $S$ to be smooth, as I would like to coexist peacefully with fibers of bad reduction.

I am typically thinking of $S = mathrm{Spec}(mathbb{Z}), mathrm{Spec}(mathbb{F}_q(t)), mathbb{P}^1_{mathbb{F}_q}$, or finite extensions.

Question: In what way are such objects analogous (or not) to del Pezzo surfaces over a field?
By this I mean in terms of their intrinsic properties, and in how they may be distinguished from other surfaces in their respective area. Classical del Pezzo surfaces are quite special.

Is there a specific name for such schemes? Some good references to read about them?

I hoped to find something in Qing Liu’s nice book, but have not found anything yet.

Cancellation of inequalities in floating point arithmetic

In finite precision floating point arithmetic the associative property of addition is not satisfied. This is, it is not always the case that $$(a+b)+c=a+(b+c)$$
Even $a=(a+b)-b$ is not always true.

To prove that $x+y<z$ is equivalent to $x<z-y$ with real numbers we can add $-y$ on both sides of $x+y<z$ to get $(x+y)-y<z-y$ and then from this $x=x+(y-y)<z-y$. But I can’t repeat the last step for floating point.

Question: Are the inequalities $x+y<z$ and $x<z-y$ equivalent in finite precision floating point arithmetic?