If x satisfies equation x^2+x+f=0 (modulo 2), then x^2+x = f (modulo 2).

Alec

What do you mean by a "polynomial function"?

Here is an example.

alias(f=RootOf(_Z^2+_Z+1));

with calculations mod 2 defines GF(4) with elements 0, 1, f, f+1. Now,

alias(k=RootOf(_Z^2+_Z+f));

with calculations mod 2 defines GF(16) with elements 0, 1, f, f+1, k, k+1, k+f, k+f+1, kf, kf+1, kf+f, kf+f+1, kf+k, kf+k+1, kf+k+f, kf+k+f+1.

The x->x is a homomorphism (inclusion) from GF(4) to GF(16).

There is no a non-zero homomorphism from GF(16) to GF(4) - otherwise, what would be its kernel? It should be an ideal, and the only ideals in fields are 0 and the entire field.

One can construct a mapping from GF(16) to GF(4) in many ways. For example, by substituting k=0. Such mappings don't have good algebraic properties, but can be used in cryptography.

Alec

What do you mean by a "polynomial function"?

Here is an example.

alias(f=RootOf(_Z^2+_Z+1));

with calculations mod 2 defines GF(4) with elements 0, 1, f, f+1. Now,

alias(k=RootOf(_Z^2+_Z+f));

with calculations mod 2 defines GF(16) with elements 0, 1, f, f+1, k, k+1, k+f, k+f+1, kf, kf+1, kf+f, kf+f+1, kf+k, kf+k+1, kf+k+f, kf+k+f+1.

The x->x is a homomorphism (inclusion) from GF(4) to GF(16).

There is no a non-zero homomorphism from GF(16) to GF(4) - otherwise, what would be its kernel? It should be an ideal, and the only ideals in fields are 0 and the entire field.

One can construct a mapping from GF(16) to GF(4) in many ways. For example, by substituting k=0. Such mappings don't have good algebraic properties, but can be used in cryptography.

Alec

I would replace "between" with "in".

One can either say "No roots between a and b", or "No roots in the interval [a,b]", but not "between the interval".

Alec

Great!

The formula for the sum for trivial zeros can be obtained by changing the order of summation (in the double sum), replacing 1+1+1+... with Zeta(0)=-1/2, 1+2+3+... with Zeta(-1) etc. and using the functional equation for zeta-function.

It doesn't make much of a difference, it is a monotonic (not oscillating) function with values between -1/2 and 0, approaching 0 when x increases. It can be also written as -arccot(ln(x)/Pi)/Pi.

Alec

Great!

The formula for the sum for trivial zeros can be obtained by changing the order of summation (in the double sum), replacing 1+1+1+... with Zeta(0)=-1/2, 1+2+3+... with Zeta(-1) etc. and using the functional equation for zeta-function.

It doesn't make much of a difference, it is a monotonic (not oscillating) function with values between -1/2 and 0, approaching 0 when x increases. It can be also written as -arccot(ln(x)/Pi)/Pi.

Alec

Almost identiical code :-)

Took me about a minute to write - but I spent more time earlier on such things in this thread.

Alec

Almost identiical code :-)

Took me about a minute to write - but I spent more time earlier on such things in this thread.

Alec

If you write in C, then you could write R in C or C++ (either finding a library for zeta function, or calculating in Maple, say, 100 values of it at 2, 3, etc.), put it in a dll (or so), and then call from Maple using external calling. That would produce plots fast.

Alec

If you write in C, then you could write R in C or C++ (either finding a library for zeta function, or calculating in Maple, say, 100 values of it at 2, 3, etc.), put it in a dll (or so), and then call from Maple using external calling. That would produce plots fast.

Alec

That is just the difference in the direction. Sage is usable for research and teaching, while neither research nor teaching was included in the poll at the left hand side.

Creating reports and documents - I use LaTeX for that and Microsoft Office, as well as for presentations. Modeling and simulation - specialized programs, C, C++, Matlab and R. Programming - are they joking? And I use calcuratorium as a scratchpad.

I don't often need to factor polynomials or solve differential equations, and when I need that, the usual situation is that I do it faster by hand than in Maple. And there is a long list of things that Sage can do and Maple can not, many of them more important to me than differential equation solving.

Alec

That is just the difference in the direction. Sage is usable for research and teaching, while neither research nor teaching was included in the poll at the left hand side.

Creating reports and documents - I use LaTeX for that and Microsoft Office, as well as for presentations. Modeling and simulation - specialized programs, C, C++, Matlab and R. Programming - are they joking? And I use calcuratorium as a scratchpad.

I don't often need to factor polynomials or solve differential equations, and when I need that, the usual situation is that I do it faster by hand than in Maple. And there is a long list of things that Sage can do and Maple can not, many of them more important to me than differential equation solving.

Alec

It is a known bug in the Standard Maple. You can either use proc instead, or use Classic Maple, or command line Maple.

Alec

It is a known bug in the Standard Maple. You can either use proc instead, or use Classic Maple, or command line Maple.

Alec

That can be done as above, or, say, as

f:=(m::integer,n::integer)->if (m+n)::posint then m+n fi;

f(3,5);

                                  8

f(3,-5);

It is not very good for plotting though. Plots in Maple don't work like that.

Alec