Definite integration and summation are #P-hard

Deenite Integration and Summation Are #p-hard

| We show that the common symbolic manipulation tasks of computing multiple partial derivatives , deenite integration, and deenite summation, are #P-hard, i.e., at least as hard as counting the accepting input strings for any Turing machine that halts in polynomial time.

ON AN EXPRESSION FOR THE NUMBERS OF BERNOULLI, BY MEANS OF A DEFINITE INTEGRAL, AND ON SOME CONNECTED PROCESSES OF SUMMATION AND INTEGRATION By

An Implementation of Karr’s Summation Algorithm in Mathematica∗

Implementations of the celebrated Gosper algorithm (1978) for indefinite summation are available on almost any computer algebra platform. We report here about an implementation of an algorithm by Karr, the most general indefinite summation algorithm known. Karr’s algorithm is, in a sense, the summation counterpart of Risch’s algorithm for indefinite integration. This is the first implementation...

A conjecture in the problem of rational definite summation

A conjecture is given that, if true, could lead to an algorithm for computing definite sums of rational functions.

Vibrotactile temporal summation: probability summation or neural integration?

Temporal summation, a decrease in the detection threshold that occurs when either the duration of a stimulus or the number of stimuli in a sequence is increased, has been attributed to the operations of either the mechanism of neural integration or of probability summation. Our experiments indicate that under certain conditions, both mechanisms may operate, but that the process of neural integr...