Combinatorics and Graph Theory

2606 Submissions

[2] viXra:2606.0078 [pdf] submitted on 2026-06-21 02:57:24

Definite Integration and Summation Are #P-Hard

Authors: Leonid Gurvits, Warren D. Smith
Comments: 4 Pages.

We show that the common symbolic manipulation tasks of computing multiple partial derivatives, definite integration, and definite 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. (The "multiple partial derivatives" part was previously known.)
Category: Combinatorics and Graph Theory

[1] viXra:2606.0055 [pdf] submitted on 2026-06-15 16:20:10

Recurrence Relations in the Tessellation of the Upper Half-Plane by the Full Modular Group

Authors: Hans Montanus
Comments: 11 Pages.

The successive actions of the generators S and T of the full modular group SL(2,Z) on the fundamental domain F lead to a tessellation of the upper half-plane H. Each image can be obtained by acting on the fundamental domain with a word consisting of solely S's and T's. For instance STTSTTT(F) is the image when the word STTSTTT acts on F. There is no shorter word that produces the image STTSTTT(F). A recurrence relation will be derived for the number of images for a given minimal word length.
Category: Combinatorics and Graph Theory