We apply Lie algebraic methods of the type developed by Baker, Campbell, Hausdorff, and Zassenhaus to the initial value and eigenvalue problems for certain special classes of partial differential operators which have many important applications in the physical sciences. We obtain detailed information about these operators including explicit formulas for the solutions of the problems of interest...

Van Hoeij’s algorithm for factoring univariate polynomials over the rational integers rests on the same principle as Berlekamp-Zassenhaus, but uses lattice basis reduction to improve drastically on the recombination phase. His ideas give rise to a collection of algorithms, differing greatly in their efficiency. We present two deterministic variants, one of which achieves excellent overall perfo...

The algorithm for factoring polynomials over the integers by Wang and Rothschild is generalized to an algorithm for the irreducible factorization of multivariate polynomials over any given algebraic number field. The extended method makes use of recent ideas in factoring univariate polynomials over large finite fields due to Berlekamp and Zassenhaus. The procedure described has been implemented...

Examples of such algebras are subspaces of associative algebras of characteristic p^O which are closed under the Lie multiplication [ab] =ab — ba and under pih powers. Then one may take a[pl =ap. It is known that every restricted Lie algebra is isomorphic to one of this type. For this reason we may simplify our notation in the sequel and write a" for alp]. We call the mapping a—>ap the p-operat...

We revisit the q-deformed counterpart of the Zassenhaus formula, expressing the Jackson q-exponential of the sum of two non-q-commuting operators as an (in general) infinite product of q-exponential operators involving repeated q-commutators of increasing order, Eq(A + B) = Eqα0 (A)Eqα1 (B) ∏ ∞ i=2 Eqαi (Ci). By systematically transforming the q-exponentials into exponentials of series and usin...