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Let (g,[p]) be a restricted Lie algebra over an algebraically closed field k of characteristic p ≥ 3. Motivated by the behavior of geometric invariants of the so-called (g,[p])-modules of constant j-rank (j ∈ {1, . . . , p−1}), we study the projective variety E(2,g) of two-dimensional elementary abelian subalgebras. If p ≥ 5, then the topological space E(2,g/C(g)), associated to the factor algebra of g by its center C(g), is shown to be connected. We give applications concerning categories of (g,[p])-modules of constant j-rank and certain invariants, called j-degrees. [ABSTRACT FROM AUTHOR]

We study a block B of a finite group with respect to an algebraically closed field of prime characteristic through the Loewy length llZB of the center ZB. In this paper, we give some upper bounds for llZB in terms of characters, subsections and defect groups associated to B. As a corollary to these results, we characterize some blocks by llZB. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to classify exchange relation planar algebras with 4 dimensional 2-boxes. Besides its skein theory, we emphasize the positivity of subfactor planar algebras based on the Schur product theorem. We will discuss the lattice of projections of 2-boxes, specifically the rank of the projections. From this point, several results about biprojections are obtained. The key break of the classification is to show the existence of a biprojection. By this method, we also classify another two families of subfactor planar algebras: subfactor planar algebras generated by 2-boxes with 4 dimensional 2-boxes and at most 23 dimensional 3-boxes; subfactor planar algebras generated by 2-boxes, such that the quotient of 3-boxes by the basic construction ideal is abelian. They extend the classification of singly generated planar algebras obtained by Bisch, Jones and the author. [ABSTRACT FROM AUTHOR]

We study continuous wavelet transforms associated to matrix dilation groups giving rise to an irreducible square-integrable quasi-regular representation on L²(Rd). It turns out that these representations are integrable as well, with respect to a wide variety of weights, thus allowing to consistently quantify wavelet coefficient decay via coorbit space norms. We then show that these spaces always admit an atomic decomposition in terms of bandlimited Schwartz wavelets. We exhibit spaces of Schwartz functions contained in all coorbit spaces, and dense in most of them. We also present an example showing that for a consistent definition of coorbit spaces, the irreducibility requirement cannot be easily dispensed with. We then address the question of how to predict wavelet coefficient decay from vanishing moment assumptions. To this end, we introduce a new condition on the open dual orbit associated to a dilation group: If the orbit is temperately embedded, it is possible to derive rather general weighted mixed Lp-estimates for the wavelet coefficients from vanishing moment conditions on the wavelet and the analyzed function. These estimates have various applications: They provide very explicit admissibility conditions for wavelets and integrable vectors, as well as sufficient criteria for membership in coorbit spaces. As a further consequence, one obtains a transparent way of identifying elements of coorbit spaces with certain (cosets of) tempered distributions. We then show that, for every dilation group in dimension two, the associated dual orbit is temperately embedded. In particular, the general results derived in this paper apply to the shearlet group and its associated family of coorbit spaces, where they complement and generalize the known results. [ABSTRACT FROM AUTHOR]

We obtain a new upper bound for odd multiperfect numbers. If N is an odd perfect number with k distinct prime divisors and P is its largest prime divisor, we find as a corollary that 1012P2N < 24k . Using this new bound, and extensive computations, we derive the inequality k ⩾ 10. [ABSTRACT FROM AUTHOR]

In 1638, Descartes showed that 32 . 72 . 112 . 132 . 220211 would be an odd perfect number if 22021 were prime. We give a formal definition for such "spoof" odd perfect numbers, and construct an algorithm to find all such integers with a given number of distinct quasi-prime factors. We show that Descartes' example is the only spoof with less than seven such factors. [ABSTRACT FROM AUTHOR]