Bounding sup-norms of cusp forms of large level

Let f be an $L^2$-normalized weight zero Hecke-Maass cusp form of square-free level N, character $\chi$ and Laplacian eigenvalue $\lambda\geq 1/4$. It is shown that $\| f \|_{\infty} \ll_{\lambda} N^{-1/37}$, from which the hybrid bound $\|f \|_{\infty} \ll \lambda^{1/4} (N\lambda)^{-\delta}$ (for some $\delta > 0$) is derived. The first bound holds also for $f = y^{k/2}F$ where F is a holomorphic cusp form of weight k with the implied constant now depending on k.Comment: version 3: substantially revised versio

Non-vanishing of LL-functions associated to cusp forms of half-integral weight

In this article, we prove non-vanishing results for $L$-functions associated to holomorphic cusp forms of half-integral weight on average (over an orthogonal basis of Hecke eigenforms). This extends a result of W. Kohnen to forms of half-integral weight.Comment: 8 pages, Accepted for publication in Oman conference proceedings (Springer

Pinwheel patterns and powder diffraction

Pinwheel patterns and their higher dimensional generalisations display continuous circular or spherical symmetries in spite of being perfectly ordered. The same symmetries show up in the corresponding diffraction images. Interestingly, they also arise from amorphous systems, and also from regular crystals when investigated by powder diffraction. We present first steps and results towards a general frame to investigate such systems, with emphasis on statistical properties that are helpful to understand and compare the diffraction images. We concentrate on properties that are accessible via an alternative substitution rule for the pinwheel tiling, based on two different prototiles. Due to striking similarities, we compare our results with the toy model for the powder diffraction of the square lattice.Comment: 7 pages, 4 figure

The third moment of quadratic Dirichlet L-functions

We study the third moment of quadratic Dirichlet L-functions, obtaining an error term of size $O(X^{3/4 + \varepsilon})$.Comment: 27 pages. v2: modified a remark on p.

Pilot Study: Unique Response of Bone Tissue During an Investigation of Radio-Adaptive Effects in Mice

PURPOSE: We obtained bone tissue to evaluate the collateral effects of experiments designed to investigate molecular mechanisms of radio-adaptation in a mouse model. Radio-adaptation describes a process by which the prior exposure to low dose radiation can protect against the toxic effect of a subsequent high dose exposure. In the radio-adaptation experiments, C57Bl/6 mice were exposed to either a Sham or a priming Low Dose (5 cGy) of Cs-137 gamma rays before being exposed to either a Sham or High Dose (6 Gy) 24 hours later. ANALYSIS: Bone tissue were obtained from two experiments where mice were sacrificed at 3 days (n=3/group, 12 total) and at 14 days (n=6/group, 24 total) following high dose exposure. Tissues were analyzed to 1) evaluate a radio-adaptive response in bone tissue and 2) describe cellular and microstructural effects for two skeletal sites with different rates of bone turnover. One tibia and one lumbar vertebrae (LV2), collected at the 3-day time-point, were analyzed by bone histomorphometry and micro-CT to evaluate the cellular response and any evidence of microarchitectural impact. Likewise, tibia and LV2, collected at the 14-day time-point, were analyzed by micro-CT alone to evaluate resulting changes to bone structure and microarchitecture. The data were analyzed by 2-way ANOVA to evaluate the effects of the priming low dose radiation, of the high dose radiation, and of any interaction between the priming low and high doses of radiation. Bone histomorphometry was performed in the cancellous bone (aka trabecular bone) compartments of the proximal tibial metaphysis and of LV2. RESULTS: Cellular Response @ 3 Days The priming Low Dose radiation decreased osteoblast-covered bone perimeter in the proximal tibia and the total cell density in the bone marrow in the LV2. High Dose radiation, regardless of prior exposure to priming dose, dramatically reduced total cell density in bone marrow of both the long bone and vertebra. However, in the proximal tibia, High Dose radiation increased the osteoclast-covered bone perimeters, the density of adipocytes in bone marrow, and the area of bone marrow occupied by fat cells -- while in the LV2, adipocytes were rare and not stimulated by High Dose radiation. In an unexpected response, High Dose radiation dramatically increased (10-fold) osteoblast-covered bone perimeter in the LV2