wofsy said:If one has a simplicial complex how does one define the adjoint boundary operator from C_k into C_k+1?
This is an alternative to defining cohomology in terms of the dual space.
An adjoint boundary operator is a mathematical concept that is used in the study of boundary value problems. It is essentially the dual operator to the original boundary operator, and it allows for the solution of boundary value problems through the use of Green's functions and other mathematical techniques.
The main difference between an adjoint boundary operator and a regular boundary operator is that the adjoint operator operates on a different space. While the regular boundary operator acts on the space of functions, the adjoint operator acts on the space of distributions, which allows for more flexibility in solving boundary value problems.
The adjoint boundary operator has various applications in physics, engineering, and other fields that involve the study of boundary value problems. It is commonly used in the finite element method, optimization problems, and the study of partial differential equations.
The adjoint boundary operator plays a crucial role in solving boundary value problems using Green's functions. It allows for the representation of the solution to a boundary value problem as an integral involving the adjoint operator and the Green's function, making it a powerful tool in solving such problems.
There is no general method for finding the adjoint boundary operator, as it depends on the specific boundary value problem at hand. However, there are techniques and algorithms that can be used to determine the adjoint operator for a given problem, such as the method of variation of parameters or the method of images.