Heights of points on elliptic curves over Q\mathbb Q

In this note we obtain effective lower bounds for the canonical heights of non-torsion points on $E(\mathbb{Q})$ by making use of suitable elliptic curve ideal class pairings $$\Psi_{E,-D}: E(\mathbb{Q})\times E_{-D}(\mathbb{Q})\mapsto \mathrm{CL}(-D).$$ In terms of the class number $H(-D)$ and $T_E(-D)$, a logarithmic function in $D$, we prove $$\widehat{h}(P)> \frac{|E_{\mathrm{tor}}(\mathbb{Q})|^2}{\left( H(-D)+ |E_{\mathrm{tor}}(\mathbb{Q})|\right)^2}\cdot T_E(-D).$$Comment: The revision includes a new remark on the proportion of curves for which Proposition 1.3 applie

Understanding pet food consumers with various sensory analysis methods

Master of ScienceFood Science InstituteKadri KoppelThe pet food industry has been fast-growing over the past decades, which makes the pet food market very competitive. Understanding pet food consumers plays an important role in this competition to investigate purchase decisions. There are two aspects of pet food consumers, which includes pets that consume products, and pet owners who buy products. This research focused on two factors, palatability and emotion, that may affect purchase decisions from the perspectives of pet and pet owner. The objective of the first study was to investigate dog preference for various meats using the preference ranking test. Twelve beagle dogs performed the preference ranking procedure to rank five different meats: beef, chicken, lamb, pork, and turkey. Dog preference for cooked and raw meats were both tested during the study. In addition, a descriptive analysis of cooked meat aroma characteristics was conducted with four highly trained panelists to profile the meats. The results showed that dog preference for cooked and raw meats was similar, in which beef was preferred over chicken and pork. Specific cooked meat aroma characteristics such as meaty overall and brown/roasted might be intrinsic drivers of liking in dogs. The goals of the second study were to generate emotion terminology from both pets and pet owners and to investigate body languages and signs perceived by pet owners related to pet emotion. Four focus group sessions were conducted to gather insights from pet owners according to emotions they and their pets experienced. Lists of 38 and 55 emotion terms were generated for dogs and cats, respectively. In addition, lists of 33 and 62 emotion terms were created for dog owners and cat owners, respectively. Examples of emotion terms generated for pets and pet owners included excited, happy, anxious, and fearful. The validation of these emotion terms was conducted in the third study. The objective of the third study was to understand consumer (dog and dog owner) acceptance of granola-bar-like dog treats made with various sources of crisp and binder. Fifteen different kinds of treats were baked with three sources of crisp (rice crisp, white sorghum crisp, and red sorghum crisp) and five sources of binder (corn syrup, spray dried plasma, gelatin, albumin, and egg white). A home-use-test was conducted with 39 dog owners and their pets to evaluate the treats for overall liking, dog’s liking, appearance, aroma, and texture. In addition, emotion terms were probed during the test. The results showed that there was no significant difference for the owner’s overall liking and the dog’s liking among the treats. Positive emotions such as content/satisfied and happy were experienced frequently by the owners when dogs consumed the treats. Conversely, dogs were reported excited, happy, and eager when eating the treats. These studies demonstrated that a combination of palatability test and sensory analysis provided insights from both and pet and pet owners, which may be helpful for the pet food industry to understand the pet food consumers further

Variants of Lehmer's speculation for newforms

In the spirit of Lehmer's unresolved speculation on the nonvanishing of Ramanujan's tau-function, it is natural to ask whether a fixed integer is a value of $\tau(n)$ or is a Fourier coefficient $a_f(n)$ of any given newform $f(z)$. We offer a method, which applies to newforms with integer coefficients and trivial residual mod 2 Galois representation, that answers this question for odd integers. We determine infinitely many spaces for which the primes $3\leq \ell\leq 37$ are not absolute values of coefficients of newforms with integer coefficients. For $\tau(n)$ with $n>1$, we prove that $$\tau(n)\not \in \{\pm 1, \pm 3, \pm 5, \pm 7, \pm 13, \pm 17, -19, \pm 23, \pm 37, \pm 691\},$$ and assuming GRH we show for primes $\ell$ that $$\tau(n)\not \in \left \{ \pm \ell\ : \ 41\leq \ell\leq 97 \ {\textrm{with}}\ \left(\frac{\ell}{5}\right)=-1\right\} \cup \left \{ -11, -29, -31, -41, -59, -61, -71, -79, -89\right\}.$$ We also obtain sharp lower bounds for the number of prime factors of such newform coefficients. In the weight aspect, for powers of odd primes $\ell$, we prove that $\pm \ell^m$ is not a coefficient of any such newform $f$ with weight $2k>M^{\pm}(\ell,m)=O_{\ell}(m)$ and even level coprime to $\ell,$ where $M^{\pm}(\ell,m)$ is effectively computable.Comment: The paper arXiv:2005.10345 written for the Proceedings of Modular forms and the Arithmetic of Function Fields (Pisa, 2018), is an exposition of some special cases of the general results obtained in this pape

Even values of Ramanujan’s tau-function

https://arxiv.org/abs/2102.00111First author draf

Fictive Motion in Chinese and English Tourist Guidebooks

This study focuses on the usage of fictive motion in tourist guidebooks. The analysis for the study draws on the theories of image schema and metaphorical extension (Johnson, 1987; Lakoff, 1987 & 1989). Fictive motion is often used to depict the features of natural scenery and the movement of time. We concentrate on the spatial description of fictive motion with data taken from official tourist guidebooks for seven National Parks in Taiwan. Both the narrations in Chinese and in English versions are analyzed. From the comparison, we attempt to assist tourist comprehension of the narratives in tourist guidebooks. The research results indicate that fictive motion description is often used in the depiction of linear movement or for the location of scenic spots, as for example with The river starts from the mountain in English and yan2 shi2 huan2 rao4 si4 zhou1 ‘the rock surrounded’ in Chinese. There are varied applications of fictive motion in Chinese and English, but fictive motion in both languages also shares common characteristics in spatial description.Key words: Fictive motion; Tourist guidebooks; Image schema; Metaphorical extension; Spatial descriptio