The Reciprocity Law from Euler to Eisenstein

1994 ◽  
pp. 67-90 ◽  
Author(s):  
Günther Frei
Keyword(s):  
Reciprocity Law

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 Analysis of the holographic reciprocity law for dichromated gelatin

1992 ◽  
Vol 31 (17) ◽  
pp. 3200 ◽  
Author(s):  
I. Pascual ◽  
A. Beléndez ◽  
A. Fimia
Keyword(s):  
Reciprocity Law ◽  
Dichromated Gelatin

Applicability of Exposure Reciprocity Law for Fast Polymerization of Restorative Composites Containing Various Photoinitiating Systems

2021 ◽  
Author(s):  
A Tichy ◽  
P Bradna
Keyword(s):  
High Power ◽  
Irradiation Time ◽  
Spectral Characteristics ◽  
Degree Of Polymerization ◽  
High Irradiance ◽  
Bottom Surface ◽  
Surface Microhardness ◽  
Radiant Exposure ◽  
Reciprocity Law ◽  
Restorative Composites

SUMMARY Objectives: The exposure reciprocity law (ERL) has been used to calculate the optimal irradiation time of dental composites. This study examined the applicability of ERL for fast polymerization of restorative composites containing various photoinitiating systems using a high-power multi-peak light-emitting diode (LED) lamp. Methods: Three commercial composites differing in photoinitiating systems were tested: Filtek Ultimate Universal Restorative (FU) with a camphorquinone-amine (CQ-A) photoinitiating system, Tetric EvoCeram (TEC) with CQ-A and (2,4,6-trimethylbenzoyl)phosphine oxide (TPO), and Estelite Σ Quick (ESQ) with CQ and a radical amplified photopolymerization (RAP) initiator. Specimens 2-mm thick were polymerized using a high-power multipeak LED lamp (Valo) at 3 pairs of radiant exposures (referred to as low, moderate, and high) ranging from 15.8–26.7 J/cm2. They were achieved by different combinations of irradiation time (5–20 seconds) and irradiance (1300–2980 mW/cm2) as determined with a calibrated spectrometer. Knoop microhardness was measured 1, 24, and 168 hours after polymerization on specimen top (irradiated) and bottom surfaces to characterize the degree of polymerization. The results were statistically analyzed using a three-way analysis of variance and Tukey’s post hoc tests, α = 0.05. Results: Microhardness increased with radiant exposure and except for ESQ, top-surface microhardness was significantly higher than that on bottom surfaces. Combinations of high irradiance and short irradiation time significantly increased the top-surface microhardness of TEC at low and moderate radiant exposures, and the bottom-surface microhardness of FU at a low radiant exposure. In contrast, the microhardness of ESQ on both surfaces at high radiant exposure increased significantly when low irradiance and long irradiation time were used. With all tested composites, bottom-surface microhardness obtained at low radiant exposure was below 80% of the maximum top-surface microhardness, indicating insufficient polymerization. Conclusion: Combinations of irradiance and irradiation time had a significant effect on microhardness, which was affected by photoinitiators and the optical properties of composites as well as spectral characteristics of the polymerization lamp. Therefore, ERL cannot be universally applied for the calculation of optimal composite irradiation time. Despite high irradiance, fast polymerization led to insufficient bottom-surface microhardness, suggesting the necessity to also characterize the degree of polymerization on the bottom surfaces of composite increments when assessing the validity of ERL.

scholarly journals The photographic action of β-rays

1927 ◽  
Vol 114 (767) ◽  
pp. 266-276 ◽  
Keyword(s):  
Direct Method ◽  
Photographic Plate ◽  
Characteristic Curve ◽  
Total Charge ◽  
X Rays ◽  
Reciprocity Law ◽  
Ordinary Light ◽  
General Experience ◽  
High Degree ◽  
The Relationship

The study of β-ray spectra has now advanced to that stage at which, for the majority of radioactive substances, the velocities of the homogeneous electrons forming the “lines” have been measured with a fair accuracy. The relative intensities of the “lines” have in the past been obtained by visual estimation of the photographic blackening of the plates on which they have been recorded, and it has become important to obtain more precise information on this subject. The most direct method of determining the relative intensities would be to count the number of, or measure the total charge carried by, the particles forming the lines. This is not practicable, to any high degree of accuracy, because of the small effects which are obtainable, and it is obvious that the photographic plate, in giving quite intense and sharp lines, in addition to a permanent record, presents many advantages. The use of this method, however, necessitates the calibration of the plate both for the variation of the blackening with exposure and also with velocity of the rays. The corresponding calibrations for ordinary light have now become a matter of routine, but since there has, as yet, been little systematic work on the behaviour of the photographic plate to β-rays, we have thought it best to record in this paper such experiments as we have found necessary before undertaking the main intensity problem. It is the dependence of blackening on exposure which is mainly treated in this paper, although we have in addition obtained some interesting results on the effect of β-particles of different velocities. In an investigation on the relationship between blackening and exposure i. e ., the characteristic curve of a type of plate, it is first necessary to find the dependence of the blackening D on the time of exposure t , when the product of intensity I and time of exposure t is kept constant. If D is independent of t , when I. t is constant, then the Reciprocity Law of Bunsen and Roscoe is said to be valid for the plate. This law, although it has to be slightly modified for luminous radiation, has been shown to be true in the case of X-rays and also for heterogeneous β-rays. In all previous work on the photographic action of β-rays the particles were of heterogeneous velocities, but in view of the ultimate object of this investigation it was thought unsatisfactory to employ heterogeneous particles, and although the procedure was thereby made rather more laborious, beams of practically homogeneous particles, taken from the continuous β-ray spectrum, were used. It is in general important, when dealing with an unknown type of plate, after investigating the validity of the Reciprocity Law, to determine the variation of the characteristic curve with time of development, and thereby find the optimum value. Our general experience in photographing β-ray spectra had led us to employ a particular type of plate and method of development, and the main value of our experiments on the variation of the characteristic curve with time of development was that it indicated the error introduced into the value of the density for a given uncertainty in the time of development.

A reciprocity law in non-solvable extensions

2002 ◽  
pp. 695-706
Author(s):  
Goro Shimura
Keyword(s):  
Reciprocity Law

scholarly journals The strong Suslin reciprocity law

2021 ◽  
Vol 157 (4) ◽  
pp. 649-676
Author(s):  
Daniil Rudenko
Keyword(s):  
Rational Functions ◽  
Projective Curve ◽  
Main Ingredient ◽  
Homotopy Invariance ◽  
Reciprocity Law ◽  
Hyperbolic Volume ◽  
Scissors Congruence ◽  
Dehn Invariant ◽  
Contracting Homotopy ◽  
Invariance Theorem

We prove the strong Suslin reciprocity law conjectured by A. Goncharov. The Suslin reciprocity law is a generalization of the Weil reciprocity law to higher Milnor $K$ -theory. The Milnor $K$ -groups can be identified with the top cohomology groups of the polylogarithmic motivic complexes; Goncharov's conjecture predicts the existence of a contracting homotopy underlying Suslin reciprocity. The main ingredient of the proof is a homotopy invariance theorem for the cohomology of the polylogarithmic motivic complexes in the ‘next to Milnor’ degree. We apply these results to the theory of scissors congruences of hyperbolic polytopes. For every triple of rational functions on a compact projective curve over $\mathbb {C}$ we construct a hyperbolic polytope (defined up to scissors congruence). The hyperbolic volume and the Dehn invariant of this polytope can be computed directly from the triple of rational functions on the curve.

scholarly journals Measurement of near-UV irradiance taking into consideration reciprocity law failure for irradiated materials.

1981 ◽  
Vol 5 (1) ◽  
pp. 11-17 ◽  
Author(s):  
Kanji KINAMERI ◽  
Mamoru NONAKA ◽  
Masahiro NISHIZAWA ◽  
Hiroshi YOKOMIZO
Keyword(s):  
Reciprocity Law ◽  
Uv Irradiance

scholarly journals A generalization of Scholz’s reciprocity law

2007 ◽  
Vol 19 (3) ◽  
pp. 583-594 ◽  
Author(s):  
Mark Budden ◽  
Jeremiah Eisenmenger ◽  
Jonathan Kish
Keyword(s):  
Reciprocity Law

scholarly journals On the sum of two integral squares in certain quadratic fields

2015 ◽  
Vol 27 (4) ◽  
Author(s):  
Dasheng Wei
Keyword(s):  
Quadratic Fields ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Reciprocity Law ◽  
Necessary And Sufficient

AbstractIn this note, we give a necessary and sufficient condition for determining which integers can be written as a sum of two integral squares for certain quadratic fields by using the integral Brauer–Manin obstruction. The condition is computable and originally from the reciprocity law.

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