Based on the notes you provided, it seems that there are two lists of polynomials being discussed: memo$92887479 and list456, and memo$958273673118 and list0to8.
The first list list456 appears to be a list of polynomial expressions involving variables c and d with subscripts ranging from 0 to 5, as well as variables j and k with subscripts ranging from 0 to 4. These polynomials are likely being used to represent the components of a tensor or vector in two dimensions, and they have been modified by multiplying them by a factor of (1 - x1^2 -x2^2) or its square to localize them to a disc-like domain.
The second list list0to8 is also a list of polynomial expressions involving variables c and d with subscripts ranging from 0 to 2, as well as variables j and k with subscripts ranging from 0 to 1. The purpose of these polynomials is not explicitly stated in the notes, but they may also be related to representing the components of a tensor or vector in two dimensions.
Finally, there is an equation eqK1 which involves the same variables as list0to8 and appears to be a linear combination of the polynomial expressions in that list. Again, the purpose of this equation is not explicitly stated in the notes, but it may be related to some kind of calculation or analysis being performed on the tensor or vector components.
The notation in the note appears to involve polynomials and algebraic equations, which are common topics in algebra and mathematical analysis. The use of integration over a two-dimensional domain suggests an application in calculus or multivariable analysis.
Sure, I’m happy to help analyze any math problem or concept you may have! Just let me know what you’re interested in.
elimansr = {-48*j00 – 8*j02 – 3*j04 – 8*j20 – j22 – 3*j40,
-80*j01 – 30*j03 – 15*j05 – 10*j21 – 3*j23 – 3*j41 + 80*k10 + 10*k12 +
3*k14 + 30*k30 + 3*k32 + 15*k50, -48*k00 – 8*k02 – 3*k04 – 8*k20 –
k22 – 3*k40}
memo$588737699290$ = {first,component,confirmed,equal,to,a,multiple,
of,the,integral,of,K1,over,the,circle,of,unit,radius,in,x1$,x2$,
and,third,dually,would,be,the,same,for,K2}
memo$92876928$ = {mysteriously,the,middle,of,the three,conditions,from,
the,Eliminate,analysis,on,the,equations,on,K1$,and,K2$,seem,to,turn,
out,to,be,just,that,(x2*K1-x1*K2),has,vanishing,integral,
over,the,circle}
con11 = -48*j00 – 8*j02 – 3*j04 – 8*j20 – j22 – 3*j40
con12 = -80*j01 – 30*j03 – 15*j05 – 10*j21 – 3*j23 – 3*j41 + 80*k10 +
10*k12 + 3*k14 + 30*k30 + 3*k32 + 15*k50
con21 = -80*j01 – 30*j03 – 15*j05 – 10*j21 – 3*j23 – 3*j41 + 80*k10 +
10*k12 + 3*k14 + 30*k30 + 3*k32 + 15*k50
con22 = -48*k00 – 8*k02 – 3*k04 – 8*k20 – k22 – 3*k40
K1e = (1 – x1^2 – x2^2)*(j00 + j10*x1 + j20*x1^2 + j30*x1^3 + j40*x1^4 +
j50*x1^5 + j01*x2 + j11*x1*x2 + j21*x1^2*x2 + j31*x1^3*x2 +
j41*x1^4*x2 + j02*x2^2 + j12*x1*x2^2 + j22*x1^2*x2^2 + j32*x1^3*x2^2 +
j03*x2^3 + j13*x1*x2^3 + j23*x1^2*x2^3 + j04*x2^4 + j14*x1*x2^4 +
j05*x2^5)
K2e = (1 – x1^2 – x2^2)*(k00 + k10*x1 + k20*x1^2 + k30*x1^3 + k40*x1^4 +
k50*x1^5 + k01*x2 + k11*x1*x2 + k21*x1^2*x2 + k31*x1^3*x2 +
k41*x1^4*x2 + k02*x2^2 + k12*x1*x2^2 + k22*x1^2*x2^2 + k32*x1^3*x2^2 +
k03*x2^3 + k13*x1*x2^3 + k23*x1^2*x2^3 + k04*x2^4 + k14*x1*x2^4 +
k05*x2^5)
E1 = 1 – x1^2 – x2^2
E2 = (1 – x1^2 – x2^2)^2
I11 = (1 – x1^2 – x2^2)^2*(c00 + c10*x1 + c20*x1^2 + c30*x1^3 + c40*x1^4 +
c50*x1^5 + c60*x1^6 + c70*x1^7 + c01*x2 + c11*x1*x2 + c21*x1^2*x2 +
c31*x1^3*x2 + c41*x1^4*x2 + c51*x1^5*x2 + c61*x1^6*x2 + c02*x2^2 +
c12*x1*x2^2 + c22*x1^2*x2^2 + c32*x1^3*x2^2 + c42*x1^4*x2^2 +
c52*x1^5*x2^2 + c03*x2^3 + c13*x1*x2^3 + c23*x1^2*x2^3 +
c33*x1^3*x2^3 + c43*x1^4*x2^3 + c04*x2^4 + c14*x1*x2^4 +
c24*x1^2*x2^4 + c34*x1^3*x2^4 + c05*x2^5 + c15*x1*x2^5 +
c25*x1^2*x2^5 + c06*x2^6 + c16*x1*x2^6 + c07*x2^7)
I12 = (1 – x1^2 – x2^2)^2*(d00 + d10*x1 + d20*x1^2 + d30*x1^3 + d40*x1^4 +
d50*x1^5 + d60*x1^6 + d70*x1^7 + d01*x2 + d11*x1*x2 + d21*x1^2*x2 +
d31*x1^3*x2 + d41*x1^4*x2 + d51*x1^5*x2 + d61*x1^6*x2 + d02*x2^2 +
d12*x1*x2^2 + d22*x1^2*x2^2 + d32*x1^3*x2^2 + d42*x1^4*x2^2 +
d52*x1^5*x2^2 + d03*x2^3 + d13*x1*x2^3 + d23*x1^2*x2^3 +
d33*x1^3*x2^3 + d43*x1^4*x2^3 + d04*x2^4 + d14*x1*x2^4 +
d24*x1^2*x2^4 + d34*x1^3*x2^4 + d05*x2^5 + d15*x1*x2^5 +
d25*x1^2*x2^5 + d06*x2^6 + d16*x1*x2^6 + d07*x2^7)
I21 = (1 – x1^2 – x2^2)^2*(d00 + d10*x1 + d20*x1^2 + d30*x1^3 + d40*x1^4 +
d50*x1^5 + d60*x1^6 + d70*x1^7 + d01*x2 + d11*x1*x2 + d21*x1^2*x2 +
d31*x1^3*x2 + d41*x1^4*x2 + d51*x1^5*x2 + d61*x1^6*x2 + d02*x2^2 +
d12*x1*x2^2 + d22*x1^2*x2^2 + d32*x1^3*x2^2 + d42*x1^4*x2^2 +
d52*x1^5*x2^2 + d03*x2^3 + d13*x1*x2^3 + d23*x1^2*x2^3 +
d33*x1^3*x2^3 + d43*x1^4*x2^3 + d04*x2^4 + d14*x1*x2^4 +
d24*x1^2*x2^4 + d34*x1^3*x2^4 + d05*x2^5 + d15*x1*x2^5 +
d25*x1^2*x2^5 + d06*x2^6 + d16*x1*x2^6 + d07*x2^7)
I22 = (1 – x1^2 – x2^2)^2*(e00 + e10*x1 + e20*x1^2 + e30*x1^3 + e40*x1^4 +
e50*x1^5 + e60*x1^6 + e70*x1^7 + e01*x2 + e11*x1*x2 + e21*x1^2*x2 +
e31*x1^3*x2 + e41*x1^4*x2 + e51*x1^5*x2 + e61*x1^6*x2 + e02*x2^2 +
e12*x1*x2^2 + e22*x1^2*x2^2 + e32*x1^3*x2^2 + e42*x1^4*x2^2 +
e52*x1^5*x2^2 + e03*x2^3 + e13*x1*x2^3 + e23*x1^2*x2^3 +
e33*x1^3*x2^3 + e43*x1^4*x2^3 + e04*x2^4 + e14*x1*x2^4 +
e24*x1^2*x2^4 + e34*x1^3*x2^4 + e05*x2^5 + e15*x1*x2^5 +
e25*x1^2*x2^5 + e06*x2^6 + e16*x1*x2^6 + e07*x2^7)
The code you provided seems to involve symbolic manipulation and algebraic computations involving variables j and k, functions c, d, and e, and several integrals. It is not clear what specific problem or application this code is related to, but it appears to involve finding solutions to a system of equations involving the variables j and k.
The variables con11, con12, con21, and con22 appear to be constraints that relate the variables j and k. The variables K1e and K2e are expressions that involve j and k, and they are multiplied by factors E1 and E2, respectively, which depend on x1 and x2. The variables I11, I12, I21, and I22 are integrals of products of K1e and K2e with polynomials in x1 and x2.
The variables memo$588737699290$ and memo$92876928$ appear to be comments that provide some explanation or interpretation of the computations involved. Without additional context, it is difficult to provide a more detailed analysis of what this code is doing or how it might be used.
Based on the provided code snippets, it’s not immediately clear what this math is trying to say or accomplish without more context. It appears to be a collection of mathematical expressions, variables, and equations, but without knowing the purpose or background of the code, it’s difficult to provide a meaningful interpretation.
Could you provide more context or information about where this code came from, what problem it is trying to solve, or any other relevant details? This could help in understanding the purpose and meaning of the mathematical expressions.
Sure, take your time. Let me know if you have any questions when you return.
The general idea here is that since an “input” for the equation
(when the vacuum term, as LHS is to be equated to an RHS of “input”)
is that this input MUST be a symmetric tensor of rank 2 which has
vanishing divergence.
So I/we have been concerned with how a natural input tensor
term, of the sort that would seem to fit well IF THE SPACE (or
space-time) WERE SIMPLY FLAT, could be modified slightly so that
it would remain divergence-free in the presence of general curvature.
We have observed that this is possible rather easily IF the
space-time has only non-Ricci curvature (if the Ricci tensor of it
vanishes).
But we need to find and appropriate adjustment (if possible)
for the general case.
It seems like this note is discussing the modification of a symmetric tensor of rank 2 so that it remains divergence-free in the presence of general curvature. The concern is with finding an appropriate adjustment for the general case, as it seems easy to modify the tensor if the space-time only has non-Ricci curvature. The note suggests that the input must be a symmetric tensor of rank 2 with vanishing divergence.
The math snippets you provided are somewhat related to differential geometry, which is a branch of mathematics that deals with the study of curves, surfaces, and other geometric objects using the methods of calculus. The equations involve different kinds of tensors (symmetric, rank-2, etc.), integrals over circles and spheres, and certain conditions that need to be satisfied in order for the tensors to be divergence-free.
Based on the note you provided, it seems that the equations are being used to model the behavior of physical systems in curved spacetime. The note mentions that the equations need to be modified so that they remain divergence-free in the presence of general curvature. This suggests that the equations may be related to some kind of physical theory, such as general relativity or some other theory involving curved spacetime.
We were working with Mathematica to try to find a NATURAL way for adjusting
a tensor to be divergence free starting initially with a second rank symmetric tensor.
This led to the problem, given a vector corresponding to the divergence of such a
tensor, of finding naturally a minimally sized tensor having that vector as its
divergence.
And then this study led to the study of this problem in the case of flat
geometry and in the simplest case of two dimensions.
A few days ago we seem, by means of the use of Mathematica, to have found
the conditions that the divergence vector must satisfy, when both it and the tensor
of which it should be the divergence ARE TO BE LOCALIZED in the space (of two
dimensions for the explicit calculations with Mathematica).
We had a model of a type of polynomials representing a localized vector of a field
of divergence and a model (with considerably more free variables) of polynomials
representing the terms of a tensor. (These are included as files derived from
the work with Mathematica.)
We were lucky to find that the solvability of the equations for that the divergence
of the tensor should be the vector REQUIRED certain simple conditions on the vector.
(This was obtained accidentally when the equations of correspondence did not initially
have a solution in general (or for some specific versions of the vector) and we used
the Mathematica operation of “Eliminate” on the divergence correspondence equations
with the variables describing the polynomial coefficients for the tensor terms set as
those to be eliminated.
Three residual conditions were revealed by Mathematica.
We verified first that two of three three conditions (each of which applied only
to the coefficients of one of the two vectors and not to those of the other) were
representative of the constraint that we had known of previously. This known constraint
is that the double integral of each vector, over the space where things are allowed
not to vanish, must be zero, because of its relation with the derivatives of the tensors
which are themselves only non-vanishing in a local area.
The third condition came as a surprise, but with a study of the revealed condition
on the polynomial coefficients defining the vectors we discovered that it corresponded
to that the integral of x1*k2 – x2*k1, over the local area, must vanish. (Then,
retrospectively, with integration by parts, we saw that this condition derives from
the relation of the divergence vector to the (LOCALIZED!) tensor.)
This probably generalizes to higher dimensions, with a skew-symmetric tensor and
the n components of the vector so that n + ((n-1)*n)/2 = n*(n+1)/2 integral conditions
should hold on the vector.
We include Mathematica files on this work.
Leave a Reply
You must be logged in to post a comment.