Classification of quintic eutactic forms

From the classical Voronoi algorithm, we derive an algorithm to classify quadratic positive definite forms by their minimal vectors; we define some new invariants for a class, for which several conjectures are proposed. Applying the algorithm to dimension 5 we obtain the table of the 136 classes in this dimension, we enumerate the 118 eutactic quintic forms, and we verify the Ash formula.

     

On the zeros of weakly holomorphic modular forms

In this article, we study the nature of zeros of weakly holomorphic modular forms. In particular, we prove results about transcendental zeros of modular forms of higher levels and for certain Fricke groups which extend a work of Kohnen (Comment Math Univ St Pauli 52:55–57, 2003). Furthermore, we investigate the algebraic independence of values of weakly holomorphic modular forms.     

Minimal weakly isotropic forms

In this article weakly isotropic quadratic forms over a (formally) real field are studied. Conditions on the field are given which imply that every weakly isotropic form over that field has a weakly isotropic subform of small dimension. Fields over which every quadratic form can be decomposed into an orthogonal sum of a strongly anisotropic form and a torsion form are characterized in different ways.     

Totally positive extensions and weakly isotropic forms

The aim of this article is to analyse a new field invariant, relevant to (formally) real fields, defined as the supremum of the dimensions of all anisotropic, weakly isotropic quadratic forms over the field. This invariant is compared with the classical u-invariant and with the Hasse number. Furthermore, in order to be able to obtain examples of fields where these invariants take certain prescribed values, totally positive field extensions are studied.     

Perfect, strongly eutactic lattices are periodic extreme

We introduce a parameter space for periodic point sets, given as unions of m translates of point lattices. In it we investigate the behavior of the sphere packing density function and derive sufficient conditions for local optimality. Using these criteria we prove that perfect, strongly eutactic lattices cannot be locally improved to yield a periodic sphere packing with greater density. This applies in particular to the densest known lattice sphere packings in dimension d?8 and d=24.     

Zagier duality for level p weakly holomorphic modular forms

The Ramanujan Journal - We prove Zagier duality between the Fourier coefficients of canonical bases for spaces of weakly holomorphic modular forms of prime level p with $$11 \le p \le 37$$ with...     

Local normal forms of smooth weakly hyperbolic integrable systems

In the smooth (C∞) category, a completely integrable system near a nondegenerate singularity is geometrically linearizable if the action generated by the vector fields is weakly hyperbolic. This proves partially a conjecture of Nguyen Tien Zung [11]. The main tool used in the proof is a theorem of Marc Chaperon [3] and the slight hypothesis of weak hyperbolicity is generic when all the eigenvalues of the differentials of the vector fields at the non-degenerate singularity are real.     

Congruences for the coefficients of weakly holomorphic modular forms

Recent works have used the theory of modular forms to establishlinear congruences for the partition function and for tracesof singular moduli. We show that this type of phenomenon iscompletely general, by finding similar congruences for the coefficientsof any weakly holomorphic modular form on any congruence subgroup     

The zeros of certain weakly holomorphic Drinfeld modular forms

Duke and Jenkins (Pure Appl Math Q 4(4):1327–1340, 2008) constructed a canonical basis for the space of weakly holomorphic modular forms for \({{\rm SL}_2(\mathbb{Z})}\) and investigated the zeros of the basis elements. In this paper we give an analogy in the Drinfeld setting of the result given by Duke and Jenkins (Pure Appl Math Q 4(4):1327–1340, 2008).     

On weakly clean rings

A ring is called clean if every element is a sum of a unit and an idempotent, while a ring is said to be weakly clean if every element is either a sum or a difference of a unit and an idempotent. Commutative weakly clean rings were first discussed by Anderson and Camillo [2 Anderson, D. D., Camillo, V. P. (2002). Commutative rings whose elements are a sum of a unit and idempotent. Commun. Algebra 30(7):3327–3336.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]] and were extensively investigated by Ahn and Anderson [1 Ahn, M.-S., Anderson, D. D. (2006). Weakly clean rings and almost clean rings. Rocky Mountain J. Math. 36:783–798.[Crossref], [Web of Science ®] , [Google Scholar]], motivated by the work on clean rings. In this paper, weakly clean rings are further discussed with an emphasis on their relations with clean rings. This work shows new interesting connections between weakly clean rings and clean rings.     

On weakly central C-algebras

A unital C^∗-algebra A is weakly central if and only if for every x∈A there exists a sequence of elementary unital completely positive maps αn on A such that the sequence (αn(x)) converges to a central element.     

On weakly nil-clean rings

We obtain the structure of the rings in which every element is either a sum or a difference of a nilpotent and an idempotent that commute. This extends the structure theorems of a commutative weakly nil-clean ring, of an abelian weakly nil-clean ring, and of a strongly nil-clean ring. As applications, this result is used to determine the 2-primal rings R such that the matrix ring \(\mathbb{M}_n (R)\) is weakly nil-clean, and to show that the endomorphism ring End _D (V) over a vector space V _D is weakly nil-clean if and only if it is nil-clean or dim(V) = 1 with D?= ?₃.     

On weakly commutative peo-semigroups

Communicated by Robert McFadden