## Optimization – Modeling a Minimal NOR Circuit Problem with CP

I am currently working on a constraint programming problem that I find difficult to model.

Here is the definition of the problem:

Given a specification of a Boolean function f (x1, …, xn) in the form of a truth table to find a NOR circuit (meaning only NOR gates) that meets the specification that minimizes the depth (and in the Case of a deep tie with minimal size).
To simplify the problem, various assumptions are made:

• Only NOR gates with 2 inputs and 1 output can be used: more general
NOR gates with more inputs are not allowed (ie the fan-in of NOR.)
Gates is always 2).

• The output of a NOR gate can only be the input of a single gate:
Outputs can not be considered inputs of more than one gate (i.e.
Fan-out of NOR gates is always 1).

• In addition to the input signals of the Boolean function to be implemented, constant 0 signals can also be used as inputs to NOR gates.
Constant 0 signals as well as input signals can be used as often as desired
as needed as inputs to NOR gates. On the other hand, the circuit does
do not need to use all input signals. Similar is the constant 0 signal
does not need to be used if it is not needed. "

Source: https://www.cs.upc.edu/~erodri/webpage/cps/projects/Q2-16-17/proj.pdf

Since this is the first time I have encountered such a problem, I find it difficult to find "good" decision variables. Maybe I miss something trivial …

Do you have an idea / hint how to model this problem?

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## p adic – Automorphims of \$ mathbb C_p \$ with limitations

In the link Automorphisms of \$ mathbb C_p \$, K. Conrad showed that there are countless numbers $$mathbb Q_p$$-automorphim of $$mathbb C_p$$, I ask a very similar question
To let $$(a_n) _ {n in mathbb N}$$ and $$(b_n) _ {n in mathbb N}$$ To be consequences of $$mathbb Q_p$$, Is there an innumerable amount? $$S$$ from uninterrupted automorphomorphs of $$mathbb C_p$$ that remains invariable $${b_n mid n in mathbb N }$$ and such that the set of sequences are the pictures of the sequence $$(a_n) _ {n in mathbb N}$$ through the elements of $$S$$ is innumerable. I assume that everything is forever $$n in mathbb N$$, $$b_n notin mathbb Q (a_j) _ { midj in mathbb N}$$,

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## Is the cyclic group \$ Bbb Z / 4 Bbb Z \$ free on \$ S ^ {2k} times Bbb CP ^ n \$?

I was wondering if the cyclical group $$mathbb Z / 4 Bbb Z$$ acts freely $$S ^ {2k} times Bbb CP ^ n$$ from where $$n> 1$$? It seems to me that it is not free. When it works freely, the induced effect on cohomology need not be trivial, since the Euler characteristic is not zero. I tried to prove the Lefschetz fixed point theorem. However, I could not derive any contradiction.

## at.algebraic topology – Does the cyclic group \$ Bbb Z / 4 { Bbb Z} \$ work free on \$ S ^ {2k} times Bbb CP ^ n \$, where \$ n> 1 \$?

Does the cyclic group work? $$Bbb Z / 4 { Bbb Z}$$ act freely $$S ^ {2k} times Bbb CP ^ n$$, from where $$n> 1$$? It seems to me that it is not free. If it works freely, the induced effect on cohomology need not be trivial, since the Euler characteristic is not zero. I tried to prove the Lefschetz fixed point theorem. However, I could not derive any contradiction.

To let $$f colon mathbb {CP} ^ n an mathbb {CP} ^ n$$ be a continuous function that induces a non-zero map $$f _ *$$ on every homology group $$H_ {2k} ( mathbb {CP} ^ n)$$, Show that $$f$$ is surjective.
It is a standard exercise in embedment theory to show this $$S ^ 3 to mathbb {R} ^ 4$$ given by $$(x, y, z) mapsto (x ^ 2-y ^ 2, xy, xz, yz)$$ causes an embedding $$mathbb {R} P ^ 2 after mathbb {R} ^ 4$$, Since $$mathbb {R} P ^ 2 / , mathbb {R} P ^ 1 cong mathbb {R} P ^ 2$$The previous card contains an embedding of $$mathbb {R} P ^ 2 / , mathbb {R} P ^ 1$$ in $$mathbb {R} ^ 4$$,
Is there a nice embedding of $$mathbb {C} P ^ 2 / , mathbb {C} P ^ 1$$ in $$mathbb {R} ^ 8$$?