If A is infinite and well-ordered, then |2^A|<=|Part(A)|<=|A^A|.
We show how to reduce the problem of solving members of a certain family of nonlinear differential equations to that of solving some corresponding linear differential equations.
If the coefficients of polynomials are selected by some random process, the zeros of the resulting polynomials are in some sense random. In this paper the author rephrases the above in more precise language, and calculates the joint conditional densities of a random vector whose values determine almost surely the zeros of a random reduced cubic.
We introduce the concept of a consistency space. The idea of the consistency space is motivated by the question, Given only the collection of sets of sentences which are logically consistent, is it possible to reconstruct their lattice structure?
We define the eigenderivatives of a linear operator on any real or complex Banach space, and give a sufficient condition for their existence.
We generalize the concept of disjunction.
In this paper we show how to approximate (learn) a function f, where X and Y are metric spaces.
Suppose we want to find the eigenvalues and eigenvectors for the linear operator L, and suppose that we have solved this problem for some other nearby operator K. In this paper we show how to represent the eigenvalues and eigenvectors of L in terms of the corresponding properties of K.
Note that the family of closed curves C_N={(x,y)in R^2;x^(2N)+y^(2N)=1} for N=1,2,3,... approaches the boundary of [-1,1]^2 as N to infty. In this paper we exhibit a natural parameterization of these curves and generalize to a larger class of equations.
In this paper, we exhibit the creation of the maximal integral domain mid(R) generated by a nonzero commutative ring R.
