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We relate the parity of the partition function to the parity of the -series coefficients of certain powers of the modular discriminant using their generating functions. This allows us to make statements about the parity of the initial values of the partition function and to obtain a modified Euler recurrence for its parity.
motivated by examples where and behave like () as . We explore conditions under which such problems have multiple positive solutions, investigate qualitative behavior of these solutions, and discuss computational methods for approximating the solutions.
The semigroup of partial symmetries of a polygon is the collection of all distance-preserving bijections between subpolygons of , with composition as the operation. Around every idempotent of the semigroup there is a maximal subgroup that is the group of symmetries of a subpolygon of . In this paper we construct all of the maximal subgroups that can occur for any regular polygon , and determine for which there exist nontrivial cyclic maximal subgroups, and for which there are only dihedral maximal subgroups.
We consider applications to function fields of methods previously used to study divisibility of class numbers of quadratic number fields. Let be a quadratic extension of , where is an odd prime power. We first present a function field analog to a Diophantine method of Soundararajan for finding quadratic imaginary function fields whose class groups have elements of a given order. We also show that this method does not miss many such fields. We then use a method similar to Hartung to show that there are infinitely many imaginary whose class numbers are indivisible by any odd prime distinct from the characteristic.
A single dyadic orthonormal wavelet on the plane is a measurable square integrable function whose images under translation along the coordinate axes followed by dilation by positive and negative integral powers of 2 generate an orthonormal basis for . A planar dyadic wavelet set is a measurable subset of with the property that the inverse Fourier transform of the normalized characteristic function of is a single dyadic orthonormal wavelet. While constructive characterizations are known, no algorithm is known for constructing all of them. The purpose of this paper is to construct two new distinct uncountably infinite families of dyadic orthonormal wavelet sets in . We call these the crossover and patch families. Concrete algorithms are given for both constructions.
We study the behavior of nonnegative sequences which satisfy certain difference inequalities. Several comparison tests involving difference inequalities are developed for nonnegative sequences. Using the aforementioned comparison tests, it is possible to determine the global stability and boundedness character for nonnegative solutions of particular rational difference equations in a range of their parameters.
Let be a commutative cancellative atomic monoid. We use unions of sets of lengths in to construct the -Delta set of . We first derive some basic properties of -Delta sets and then show how they offer a method to investigate the asymptotic behavior of the sizes of unions of sets of lengths.
Motivated by the result of Rankin for representations of integers as sums of squares, we use a decomposition of a modular form into a particular Eisenstein series and a cusp form to show that the number of ways of representing a positive integer as the sum of triangular numbers is asymptotically equivalent to the modified divisor function .