Reciprocity Law Research Articles

On a class of generalized Fermat equations of signature (2,2n,3)

We consider the Diophantine equation 7x2+y2n=4z3. We determine all solutions to this equation for n=2,3,4 and 5. We formulate a Kraus type criterion for showing that the Diophantine equation 7x2+y2p=4z3 has no non-trivial proper integer solutions for specific primes p>7. We computationally verify the criterion for all primes 7<p<109, p≠13. We use the symplectic method and quadratic reciprocity to show that the Diophantine equation 7x2+y2p=4z3 has no non-trivial proper solutions for a positive proportion of primes p. In the paper [10] we consider the Diophantine equation x2+7y2n=4z3, determining all families of solutions for n=2 and 3, as well as giving a (mostly) conjectural description of the solutions for n=4 and primes n≥5.

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

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. 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. 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. 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.

Gravitational-wave cosmological distances in scalar-tensor theories of gravity

We analyze the propagation of high-frequency gravitational waves (GW) in scalar-tensor theories of gravity, with the aim of examining properties of cosmological distances as inferred from GW measurements. By using symmetry principles, we first determine the most general structure of the GW linearized equations and of the GW energy momentum tensor, assuming that GW move with the speed of light. Modified gravity effects are encoded in a small number of parameters, and we study the conditions for ensuring graviton number conservation in our covariant set-up. We then apply our general findings to the case of GW propagating through a perturbed cosmological space-time, deriving the expressions for the GW luminosity distance dL (GW) and the GW angular distance dA (GW). We prove for the first time the validity of Etherington reciprocity law dL (GW) = (1+z)2 dA (GW) for a perturbed universe within a scalar-tensor framework. We find that besides the GW luminosity distance, also the GW angular distance can be modified with respect to General Relativity. We discuss implications of this result for gravitational lensing, focussing on time-delays of lensed GW and lensed photons emitted simultaneously during a multimessenger event. We explicitly show how modified gravity effects compensate between different coefficients in the GW time-delay formula: lensed GW arrive at the same time as their lensed electromagnetic counterparts, in agreement with causality constraints.

Investigating photodegradation of antibodies governed by the light dosage

The present study investigated the photodegradation of three different monoclonal antibodies (mAb) by visible light. Several chromatographic techniques, such as size-exclusion and hydrophobic interaction chromatography as well as mass spectrometry were used to measure relative changes of various oxidation related monoclonal antibody species. The results show that visible light is indeed capable of inducing the formation of protein photo-oxidation products, such as acidic, basic, hydrophilic, and several other protein species with altered physicochemical properties. Although, the formation rate of degradants of these three protein species was dependent on the light source’s intensity (I), their yield is clearly correlated to the applied light dosage (ld), which is defined as the product of light intensity I and irradiation time t (light dosage = I·t). Hence, our findings indicate that the degradation of monoclonal antibodies can be described according to the Bunsen-Roscoe reciprocity law. This correlation can be useful to assess the impact of photodegradation of biologics with regards to changes in light intensity and/or duration of light exposure of the protein, e.g. during the manufacturing of biologics.

The triple reciprocity law for the twisted second moments of Dirichlet $L$-functions over function fields

The triple reciprocity law for the twisted second moments of Dirichlet $L$-functions over function fields

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.

Congruence and metaplectic covariance: Rational biquadratic reciprocity and quantum entanglement

The purpose of the paper is to elucidate the cyclotomographic applications of the coadjoint orbit methodology to the Legendre-Hilbert-Artin symbolic tower of class field theory in the sense of the theories of Chevalley, Hasse, Weil and Witt. The Witt arithmetics concludes with the law of rational biquadratic reciprocity and quantum entanglement. The purpose of the paper is to elucidate the cyclotomographic applications of the coadjoint orbit methodology to the Legendre-Hilbert-Artin symbolic tower of class field theory in the sense of the theories of Chevalley, Hasse, Weil and Witt. The Witt arithmetics concludes with the law of rational biquadratic reciprocity and quantum entanglement.

Special Values of L-functions for GL(n) Over a CM Field

Abstract We prove a Galois-equivariant algebraicity result for the ratios of successive critical values of $L$-functions for ${\textrm GL}(n)/F,$ where $F$ is a totally imaginary quadratic extension of a totally real number field $F^+$. The proof uses (1) results of Arthur and Clozel on automorphic induction from ${\textrm GL}(n)/F$ to ${\textrm GL}(2n)/F^+$, (2) results of my work with Harder on ratios of critical values for $L$-functions of ${\textrm GL}(2n)/F^+$, and (3) period relations amongst various automorphic and cohomological periods for ${\textrm GL}(2n)/F^+$ using my work with Shahidi. The reciprocity law inherent in the algebraicity result is exactly as predicted by Deligne’s conjecture on the special values of motivic $L$-functions.

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.

Arbitrary polarization-independent backscattering or reflection by rotationally symmetric reciprocal structures

We study the backward scatterings of plane waves by reciprocal scatterers and reveal that $n$-fold ($n\geq3$) rotation symmetry is sufficient to secure invariant backscattering for arbitrarily-polarized incident plane waves. It is further demonstrated that the same principle is also applicable for infinite periodic structures in terms of reflection, which simultaneously guarantees the transmission invariance if there are neither Ohmic losses nor extra diffraction channels. At the presence of losses, extra reflection symmetries (with reflection planes either parallel or perpendicular to the incident direction) can be incorporated to ensure simultaneously the invariance of transmission and absorption. The principles we have revealed are protected by fundamental laws of reciprocity and parity conservation, which are fully independent of the optical or geometric parameters of the photonic structures. The optical invariance obtained is intrinsically robust against perturbations that preserve reciprocity and the geometric symmetries, which could be widely employed for photonic applications that require stable backscatterings or reflections.

Near-Infrared Light and Skin: Why Intensity Matters.

Infrared light (760 nm-1 mm) constitutes approximately 40% of the solar radiation reaching the ground at sea level. Shortest wavelength near-infrared (NIR) photons (NIR or IR-A: 760-1,400 nm) can penetrate the epidermis, dermis, and subcutaneous tissue with numerous biological effects. NIR used to have a bad reputation on the basis of past studies using high-intensity artificial light sources (above the solar IR-A irradiance threshold) at high doses leading to detrimental effects (i.e., upregulation of matrix metalloproteinase-1). However, when looking at the other side of the coin and what we can learn from the sun, NIR intensity matters. Hence, mimicking sunlight NIR intensity (30-35 mW/cm2) will rather trigger beneficial cutaneous effects. It is likely that intensity is more important than the fluence (dose) delivered. Moreover, the law of reciprocity (i.e., the biological effect is directly proportional to the total dose irrespective of intensity) does not always apply when considering tissue response in photobiology. In fact, the biphasic dose curve (Arndt-Schulz curve) of photobiomodulation establishes that if irradiance is lower than the physiological threshold value for a given target, it does not produce beneficial effects, even when -irradiation duration is extended. Also, photo-inhibitory deleterious effects may occur at higher irradiances. Remarkably, the beneficial "sweet spot" in between corresponds to the irradiance of the sun. NIR might even precondition the skin from an evolutionary standpoint as exposure to early morning NIR wavelengths in sunlight may prepare the skin for upcoming mid-day harmful UVR. Consequently, NIR light appears to be the solution, not the problem.

A proof of quadratic reciprocity via linear recurrences

In a 1999 preprint, A.&nbsp;Nakhash uses the recurrence relation for the Fibonacci numbers and their closed form over $\mathbb Q (\sqrt {5})$ to provide a proof of quadratic reciprocity specifically for the prime $5$. In this note, we construct a similar

ON ℓ-ADIC GALOIS POLYLOGARITHMS AND TRIPLE ℓTH POWER RESIDUE SYMBOLS

The $\ell$-adic Galois polylogarithm is an arithmetic function on an absolute Galois group with values in $\ell$-adic numbers, which arises from Galois actions on $\ell$-adic \'etale paths on ${\mathbb P}^1 \backslash \{0,1,\infty\}$. In the present paper, we discuss a relationship between $\ell$-adic Galois polylogarithms and triple $\ell$-th power residue symbols in some special cases studied by a work of Hirano-Morishita. We show that a functional equation of $\ell$-adic Galois polylogarithms by Nakamura-Wojtkowiak implies a reciprocity law of triple $\ell$-th power residue symbols.

On the quadratic reciprocity law

In the number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. It noticed by Euler and Legendre and proved by Gauss. In this paper, we will study the quadratic reciprocity law theorem where the Euler Criterion and Legendre Symbol are involved. The application of quadratic reciprocity law theorem is given in cryptography, where the Quadratic Residuosity Problem considered as a hard mathematical problem for Goldwasser Micali Randomized Public Key Cryptosystem. This system will be discussed with the details in this paper.

Genus 3 hyperelliptic curves with CM via Shimura reciprocity

Up to isomorphism over C, every simple principally polarized abelian variety of dimension 3 is the Jacobian of a smooth projective curve of genus 3. Furthermore, this curve is either a hyperelliptic curve or a plane quartic. Given a sextic CM field K, we show that if there exists a hyperelliptic Jacobian with CM by K, then all principally polarized abelian varieties that are Galois conjugated to it are hyperelliptic. Using Shimura's reciprocity law, we give an algorithm for computing approximations of the invariants of the initial curve, as well as their Galois conjugates. This allows us ton define and compute class polynomials for genus 3 hyperelliptic curves with CM.

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.

Even points on an algebraic curve

We show that the points of a global function field, whose classes are 2-divisible in the Picard group, form a connected regular infinite graph, with the incidence relation generalizing the well known quadratic reciprocity law. We prove that for every global function field the diameter of this graph is precisely 2. In addition we develop an analog of global square theorem that concerns points 2-divisible in the Picard group.

Extended Galbraith’s test on the anonymity of IBE schemes from higher residuosity

At PKC 2019, Clear and McGoldrick presented the first identity-based encryption (IBE) scheme that is group homomorphic for addition modulo a poly-sized prime e. Assuming that deciding solvability of a special system of multivariate polynomial equations is hard, they proved that their scheme for $$e>2$$ is anonymous. In this paper, we review the classical Galbraith’s test on the anonymity of the first pairing-free IBE scheme due to Cocks. With the eye of the reciprocity law for $$\mathbb {F}_\mathtt {q}[x]$$ , we can have a profound understanding of the test and naturally extend it to give a practical attack on the anonymity of the Clear–McGoldrick IBE scheme. Furthermore, we believe that our technique plays a crucial role in anonymizing IBE schemes from higher residuosity.

A note on additive twists, reciprocity laws and quantum modular forms

We prove that the central values of additive twists of a cuspidal $L$-function define a quantum modular form in the sense of Zagier, generalizing recent results of Bettin and Drappeau. From this we deduce a reciprocity law for the twisted first moment of multiplicative twists of cuspidal $L$-functions, similar to reciprocity laws discovered by Conrey for the twisted second moment of Dirichlet $L$-functions. Furthermore we give an interpretation of quantum modularity at infinity for additive twists of $L$-functions of weight 2 cusp forms in terms of the corresponding functional equations.

Menemukan Universalitas Pesan Alquran Sebagai Keniscayaan dalam Penafsiran Multi-Interdisipliner

This article aims to lay a philosophical foundation on the universality of the message of the Qur’an as a necessity in a multi-interdisciplinary approach to interpret the Qur’an. The study departs from the thoughts of Qur’anic scholars from the classical to the contemporary period as set out in their works. The “universality” here means two senses: first, compatibility of the message, owing to its universal and perennial ethical contents, to be applied across boundaries of time and space; second, various aspects of contents of the Qur’an, such as theological, legal, ethical, and scientific issues. This article also suggests methods in exploring the universality of the Qur’anic doctrines and gives one example of its application, especially in understanding Quranic values in its doctrine of warfare. The authors conclude that an integrative approach to the Qur’an is needed to be accompanied by an awareness of the universality of the message of the Qur’an which lies in the spirit of ethical principles and the need for meaning in a broader spectrum in the modern era. Neglecting its universal message will cause interpretations losing this spirit, such as classical Moslem interpreters’ understanding on jiha&gt;d-verses by insisting that the jihad as warfare is a permanent obligation upon Moslems. The falsehood of the interpretation is due to neglecting the ethical principle i.e. humanity behind the law of reciprocity that underlies particular verses on warfare.