p-Adic q-Twisted Dedekind-Type Sums

The main purpose of this paper is to define p-adic and q-Dedekind type sums. Using the Volkenborn integral and the Teichmüller character representations of the Bernoulli polynomials, we give reciprocity law of these sums. These sums and their reciprocity law generalized some of the classical p-adic Dedekind sums and their reciprocity law. It is to be noted that the Dedekind reciprocity laws, is a fine study of the existing symmetry relations between the finite sums, considered in our study, and their symmetries through permutations of initial parameters.

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

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.

Fermat’s Last Theorem and Related Problems

Empirical evidence in support of generalizations of Fermat’s equation is presented. The empirical evidence consists mainly of results for the p = 3 case where Fermat’s Last Theorem is almost false. The empirical evidence also consists of results for general p values. The \pth power with respect to" concept (involving congruences) is introduced and used to derive these generalizations. The classical Furtw¨angler theorems are reformulated. Hasse used one of his reciprocity laws to give a more systematic proof of Furtw¨angler’s theorems. Hasse’s reciprocity law is modified to deal with a certain condition. Vandiver’s theorem is reformulated and generalized. The eigenvalues of 2p x 2p matrices for the p = 3 case are investigated. (There is a relationship between the modularity theorem and a re-interpretation of the quadratic reciprocity theorem as a system of eigenvalues on a finite-dimensional complex vector space.) A generalization involving generators and \reciprocity" has solutions for every p value.

The strong Suslin reciprocity law

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.

Light Transmittance and Depth of Cure of a Bulk Fill Composite Based on the Exposure Reciprocity Law

The objective of this study was to evaluate the effect of the exposure reciprocity law of a multi-wave light-emitting diode (LED) on the light transmittance (LT), depth of cure (DOC) and degree of conversion in-depth (DC) of a bulk fill composite. A bulk fill composite (EvoCeram® bulk fill, Ivoclar Vivadent) was photoactivated using the multi-wave LED (VALO™ Cordless, Ultradent). The LED was previously characterized using a spectrophotometer to standardize the time of exposure when using the Standard or Xtra-Power modes with the same radiant exposure of 20J/cm2. LT was evaluated through samples of the bulk fill composite every millimeter till 4 mm in-depth. DOC was evaluated according to the ISO 4049. DC of the central longitudinal cross-section from each sample of the DOC test was mapped using FT-NIR microscopy. Data were statistically analyzed according to the experimental design (α=0.05; ß=0.2). The radiant exposure in the violet wavelength range for Standard and Xtra-Power was 4.5 and 5.0 J/cm2, respectively; for the blue wavelength range the radiant exposure for Standard and Xtra-Power was 15.5 and 15.0 J/cm2, respectively. There was no statistical difference in the DOC using Standard or Xtra-Power light-curing modes, but the DOC was lower than the claimed by the manufacturer (4 mm). The DC was not significantly affected by the light-curing mode up to 4 mm in depth (p>0.05). According to exposure reciprocity law, the reduction in exposure time using the same radiant exposure did not affect the depth of cure of the bulk fill composite.

Using decomposition groups to prove theorems about quadratic residues

In this text we elaborate on the modern viewpoint of the quadratic reciprocity law via methods of alge- braic number theory and class field theory. We present original short and simple proofs of so called addi- tional quadratic reciprocity laws and of the multiplicativity of the Legendre symbol using decompositon groups of primes in quadratic and cyclotomic extensions of Q.