User:Markzwatinger

In http://math.stackexchange.com/questions/364018/can-all-programs-be-modeled-as-operations-of-elementary-arithmetic-operations-onmathematics and computabiltiy theory, I asked:

> we treat all inputs and intermediate results and final outputs as natural number. While algorithms/programs themselves are considered natural numbers, here we treat these programs/functions/algorithms as just computable functions. The question is, when the function operates on an input to produce an output, can we consider the operation of function as using only a number of arithmetic operations (addition, subtraction, multiplication and division) on an input? Or does the use of if/else make the aforementioned not true? If this is true, is the number of arithmetic operations polynomially proportional to the lowest time complexity bound possible for solving a problem? (That is, if the lowest time complexity is $text{O(whatever)}$, then the number of arithmetic operations is $\text{O(whatever)}^k$ where $k$ is some rational number.)

I learned an answer to this, and now I would like to present variation: If we limit our scope to programs that can be modeled as operations of arithmetic operations on inputs, can these program be simulated by a machine that can only do basic arithmetic processes on inputs (multiplciation, division, subtraction, addition) with polynomial overhead (That is, if the lowest time complexity is $text{O(whatever)}$, then the number of arithmetic operations is $\text{O(whatever)}^k$ where $k$ is some rational number.) to the lowest possible time complexity for solving a problem?

Can all programs reducible to ones with only arithmetic operations on inputs be simulated with polynomial overhead by arithmetic machine?

Regarding orientation and orientation-reversing in local diffeomorphism

I am confused about orientation and orientation reversing in local diffeomorphism $f$ from manifold $X$ to $Y$ at some points. So, what does $f$ orientation-reversing at a point mean?