scholarly journals Class invariants and cyclotomic unit groups from special values of modular units

2008 ◽  
Vol 20 (2) ◽  
pp. 289-325
Author(s):  
Amanda Folsom
Keyword(s):  
Special Values ◽  
Modular Units ◽  
Class Invariants ◽  
Cyclotomic Unit

scholarly journals Algebraic integers as special values of modular units

2011 ◽  
Vol 55 (1) ◽  
pp. 167-179 ◽  
Author(s):  
Ja Kyung Koo ◽  
Dong Hwa Shin ◽  
Dong Sung Yoon
Keyword(s):  
Quadratic Field ◽  
Algebraic Integer ◽  
Imaginary Quadratic Field ◽  
The Other ◽  
Sufficient Condition ◽  
Algebraic Integers ◽  
Dedekind Eta Function ◽  
Ring Of Integers ◽  
Special Values ◽  
Modular Units

AbstractLet $\varphi(\tau)=\eta(\tfrac12(\tau+1))^2/\sqrt{2\pi}\exp\{\tfrac14\pi\ri\}\eta(\tau+1)$, where η(τ) is the Dedekind eta function. We show that if τ0 is an imaginary quadratic argument and m is an odd integer, then $\sqrt{m}\varphi(m\tau_0)/\varphi(\tau_0)$ is an algebraic integer dividing $\sqrt{m}$ This is a generalization of a result of Berndt, Chan and Zhang. On the other hand, when K is an imaginary quadratic field and θK is an element of K with Im(θK) > 0 which generates the ring of integers of K over ℤ, we find a sufficient condition on m which ensures that $\sqrt{m}\varphi(m\theta_K)/\varphi(\theta_K)$ is a unit.

scholarly journals On some automorphic properties of Galois traces of class invariants from generalized Weber functions of level 5

2019 ◽  
Vol 17 (1) ◽  
pp. 1631-1651
Author(s):  
Ick Sun Eum ◽  
Ho Yun Jung
Keyword(s):  
Modular Forms ◽  
Fourier Coefficients ◽  
Singular Values ◽  
Congruence Subgroups ◽  
Maass Form ◽  
Weakly Holomorphic Modular Forms ◽  
Reciprocity Law ◽  
Class Invariants ◽  
Higher Genus ◽  
Holomorphic Modular Forms

Abstract After the significant work of Zagier on the traces of singular moduli, Jeon, Kang and Kim showed that the Galois traces of real-valued class invariants given in terms of the singular values of the classical Weber functions can be identified with the Fourier coefficients of weakly holomorphic modular forms of weight 3/2 on the congruence subgroups of higher genus by using the Bruinier-Funke modular traces. Extending their work, we construct real-valued class invariants by using the singular values of the generalized Weber functions of level 5 and prove that their Galois traces are Fourier coefficients of a harmonic weak Maass form of weight 3/2 by using Shimura’s reciprocity law.

scholarly journals A reconfiguration strategy for modular robots using origami folding

2018 ◽  
Vol 38 (1) ◽  
pp. 73-89 ◽  
Author(s):  
Meibao Yao ◽  
Christoph H. Belke ◽  
Hutao Cui ◽  
Jamie Paik
Keyword(s):  
Heuristic Algorithm ◽  
Minimum Energy ◽  
Computational Time ◽  
Modular Robots ◽  
Automatic Modeling ◽  
Minimum Energy Consumption ◽  
Modular Units ◽  
Optimal Reconfiguration ◽  
Algorithmic Framework ◽  
Modeling Algorithm

Reconfigurability in versatile systems of modular robots is achieved by changing the morphology of the overall structure as well as by connecting and disconnecting modules. Recurrent connectivity changes can cause misalignment that leads to mechanical failure of the system. This paper presents a new approach to reconfiguration, inspired by the art of origami, that eliminates connectivity changes during transformation. Our method consists of an energy-optimal reconfiguration planner that generates an initial 2D assembly pattern and an actuation sequence of the modular units, both resulting in minimum energy consumption. The algorithmic framework includes two approaches, an automatic modeling algorithm as well as a heuristic algorithm. We further demonstrate the effectiveness of our method by applying the algorithms to Mori, a modular origami robot, in simulation. Our results show that the heuristic algorithm yields reconfiguration schemes with high quality, compared with the automatic modeling algorithm, simultaneously saving a considerable amount of computational time and effort.

Special Values of L-Series

1992 ◽  
Vol 114 (2) ◽  
pp. 337 ◽  
Author(s):  
Rhonda L. Hatcher
Keyword(s):  
Special Values

PROCEDURES AND ALGORITHMS FOR CONTINUOUS INTEGRATION IN AN AGILE SPECIFICATION ENVIRONMENT

2009 ◽  
Vol 19 (01) ◽  
pp. 47-78
Author(s):  
MARTÍN LÓPEZ-NORES ◽  
JOSÉ J. PAZOS-ARIAS ◽  
JORGE GARCÍA-DUQUE ◽  
YOLANDA BLANCO-FERNÁNDEZ ◽  
REBECA P. DÍAZ-REDONDO ◽  
...  
Keyword(s):  
Formal Specification ◽  
Source Code ◽  
Agile Development ◽  
Continuous Integration ◽  
Whole Process ◽  
Modular Units ◽  
Main Ideas

One of the main ideas of agile development is to perform continuous integration, in order to detect and resolve conflicts among several modular units of a system as soon as possible. Whereas this feature is well catered for at the level of programming source code, the support available in formal specification environments is still rather unsatisfactory: it is possible to analyze the composition of several modular units automatically, but no assistance is given to help modify them in case of problems. Instead, the stakeholders who build the specifications are forced to attempt manual changes until reaching the desired functionality, in a process that is far from being intuitive. In response to that, this paper presents procedures and algorithms that automate the whole process of doing integration analyses and generating revisions to solve the diagnosed problems. These mechanisms serve to complete an agile specification environment presented in a previous paper, which was designed around the principle of facilitating the creative efforts of the stakeholders.

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