Evaluation of the convolution sum involving the sum of divisors function for 22, 44 and 52

The convolution sum, (equation presented) where αβ = 22, 44, 52, is evaluated for all natural numbers n. Modular forms are used to achieve these evaluations. Since the modular space of level 22 is contained in that of level 44, we almost completely use the basis elements of the modular space of level 44 to carry out the evaluation of the convolution sums for αβ = 22. We then use these convolution sums to determine formulae for the number of representations of a positive integer by the octonary quadratic forms a(x1 2+x2 2+x3 2+x4 2)+b(x5 2+x6 2+x7 2+x8 2), where (a, b) = (1, 11), (1, 13)

Representations of an integer by some quaternary and octonary quadratic forms

In this paper we consider certain quaternary quadratic forms and octonary quadratic forms and by using the theory of modular forms, we find formulae for the number of representations of a positive integer by these quadratic forms.Comment: 20 pages, 4 tables. arXiv admin note: text overlap with arXiv:1607.0380

A descriptive mode inference for normal logic programs

In general, an n-ary predicate symbol defined by a normal logic program describes the relationship among its arguments, in the sense that no argument has to be of a special mode. The unification and the resolution (SLDNF-resolution) do capture this state of affair. Hence, the aim of logic programming is in some point approximatively achieved if the system is able to automatically determine the descriptive mode of a n-ary predicate symbol. To get a sound descriptive mode for a predicate we consider the abstraction of a set of clauses, the unification of abstract terms and define an NJSLDT-derivation. The descriptive mode inference determines which arguments of an n-ary predicate symbol in a goal become closed or partially instantiated or remain variable. Since the (prescriptive/descriptive) mode of an n-ary predicate symbol does influence the operational semantics of that predicate, this information may be used by a compiler for the purpose of efficiency. The mode inference will also help automatically determine the literal to be selected when constructing an SLDNF-derivation, weaken the condition of allowedness, automatically prove the termination of normal logic programs for a large class of programs, and detect errors at compile time. (orig.)Available from TIB Hannover: RO 6844(1994,7) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman