Reciprocity Law Research Articles

Nonreciprocal transmission of electromagnetic waves with nonlinear active plasmonic metasurfaces

Nonreciprocity is important for optical information processing, full-duplex communications, and protection of sensitive laser equipment from unwanted reflections. However, it is very challenging to obtain strong nonreciprocal response in optical frequencies, especially when nanoscale configurations are considered. In this work, we solve this problem by demonstrating a compact bifacial plasmonic metasurface that acts as an ultrathin nonreciprocal transmission filter at near-infrared frequencies. The proposed nanostructure is simple to be practically implemented, since it is composed of two silver nanostripes with different dimensions placed on both sides of an ultrathin active dielectric spacing layer. The introduced gain leads to an exceptional point formation in the linear operation regime, where unidirectional perfect transmission due to loss compensation is achieved from the presented non-Hermitian parity-time symmetric nanoscale system. The demonstrated plasmonic metasurface breaks the Lorentz reciprocity law at the nanoscale when its nonlinear response is considered, mainly due to its spatially asymmetric geometry combined with enhanced nonlinearity. Strong and broadband nonreciprocity with unity transmission contrast from opposite directions is realized under relatively low-input laser intensity values. Furthermore, asymmetric nonlinear third-harmonic generation with a pronounced contrast is also illustrated by the same nonreciprocal metasurface design. The findings of this work can lead to the realization of various self-induced nonreciprocal nanophotonic configurations, such as integrated ultrafast switches, ultrathin protective coatings, and asymmetric directional imaging devices.

A Hilbert reciprocity law on 3-manifolds

Based on our homological idelic class field theory, we formulate an analogue of the Hilbert reciprocity law on a rational homology 3-sphere endowed with an infinite link, in the spirit of arithmetic topology; We regard the intersection form on the unitary normal bundle of each knot as an analogue of the Hilbert symbol at each prime ideal to formulate the Hilbert reciprocity law, ensuring that cyclic covers of links are analogues of Kummer extensions.

Fermat’s Last Theorem over and

AbstractWe prove Fermat’s Last Theorem over $\mathbb {Q}(\sqrt {5})$ and $\mathbb {Q}(\sqrt {17})$ for prime exponents $p \ge 5$ in certain congruence classes modulo $48$ by using a combination of the modular method and Brauer–Manin obstructions explicitly given by quadratic reciprocity constraints. The reciprocity constraint used to treat the case of $\mathbb {Q}(\sqrt {5})$ is a generalization to a real quadratic base field of the one used by Chen and Siksek. For the case of $\mathbb {Q}(\sqrt {17})$ , this is insufficient, and we generalize a reciprocity constraint of Bennett, Chen, Dahmen, and Yazdani using Hilbert symbols from the rational field to certain real quadratic fields.

AN AMAZING IDENTITY OF GAUSS AND JENKINS’ LEMMA

AbstractWe prove several finite product-sum identities involving the q-binomial coefficient, one of which is used to prove an amazing identity of Gauss. We then use this identity to evaluate certain quadratic Gauss sums and, together with known properties of quadratic Gauss sums, we prove the quadratic reciprocity law for the Jacobi symbol. We end our article with a new proof of Jenkins’ lemma, a lemma analogous to Gauss’ lemma. This article aims to show that Gauss’ amazing identity and the properties of quadratic Gauss sums are sufficient to establish the quadratic reciprocity law for the Jacobi symbol.

English

In this note, we give an elementary proof of the following classical fact. Any positive definite ternary quadratic form over the rational numbers fails to represent infinitely many positive integers. For any ternary quadratic form (positive definite or indefinite), our method constructs certain congruence classes whose elements, up to a square factor, are the only elements not represented over the rational numbers by that form. In the case of a positive definite ternary form, we show that these classes are non-empty. This shows that the minimum number of variables in a positive definite quadratic form representing all positive integers is four. Our proof is very elementary and only uses quadratic reciprocity of Gauss.

Tensor decomposition, parafermions, level-rank duality, and reciprocity law for vertex operator algebras

For a semisimple Lie algebra s l n \frak {sl}_n , the basic representation L s l n ^ ( 1 , 0 ) L_{\widehat {\frak {sl}_{n}}}(1,0) of the affine Lie algebra s l n ^ \widehat {\frak {sl}_{n}} is a lattice vertex operator algebra. The first main result of the paper is to prove that the commutant vertex operator algebra of L s l n ^ ( l , 0 ) L_{\widehat {\frak {sl}_{n}}}(l,0) in the l l -fold tensor product L s l n ^ ( 1 , 0 ) ⊗ l L_{\widehat {\frak {sl}_{n}}}(1,0)^{\otimes l} is isomorphic to the parafermion vertex operator algebra K ( s l l , n ) K(\frak {sl}_{l},n) , which is the commutant of the Heisenberg vertex operator algebra L h ^ ( n , 0 ) L_{\widehat {\frak {h}}}(n,0) in L s l l ^ ( n , 0 ) L_{\widehat {\frak {sl}_l}}(n,0) . The result provides a version of level-rank duality. The second main result of the paper is to prove more general version of the first result that the commutant of L s l n ^ ( l 1 + ⋯ + l s , 0 ) L_{\widehat {\frak {sl}_{n}}}(l_1+\cdots +l_s, 0) in L s l n ^ ( l 1 , 0 ) ⊗ ⋯ ⊗ L s l n ^ ( l s , 0 ) L_{\widehat {\frak {sl}_{n}}}(l_1,0)\otimes \cdots \otimes L_{\widehat {\frak {sl}_{n}}}(l_s, 0) is isomorphic to the commutant of the vertex operator algebra generated by a Levi Lie subalgebra of s l l 1 + ⋯ + l s \frak {sl}_{l_1+\cdots +l_s} corresponding to the composition ( l 1 , ⋯ , l s ) (l_1, \cdots , l_s) in the rational vertex operator algebra L s l ^ l 1 + ⋯ + l s ( n , 0 ) L_{\widehat {\frak {sl}}_{l_1+\cdots +l_s}}(n,0) . This general version also resembles a version of reciprocity law discussed by Howe in the context of reductive Lie groups. In the course of the proof of the main results, certain Howe duality pairs also appear in the context of vertex operator algebras.

On the Tits indices of absolutely almost simple algebraic groups over local and global fields

We present a new, detailed and uniform proof of the validity of those parts of the classification results regarding absolutely almost simple algebraic groups over p -adic and number fields, announced earlier by Tits, which also covers the case of local and global function fields. The main tools used in this paper are Prasad–Rapinchuk's reciprocity law for Tits indices over global fields and also some (new) Hasse principles for homogeneous spaces over global function fields and for forms in characteristic 2.

Strong Suslin Reciprocity Law and the Norm Map

Strong Suslin Reciprocity Law and the Norm Map

Hermite’s identity and the quadratic reciprocity law

In this note, we give a proof of the quadratic reciprocity law based on Gauss’s Lemma and Hermite’s identity, which is related to other proofs using distribution relations (see Kubota [3]).

The law of reciprocity holds (more or less) for circadian-effective lighting

Delivering effective lighting in buildings to support circadian entrainment of the occupants is of growing interest and importance to lighting specifiers. Circadian-effective lighting is not simply about how much light to provide; it also depends upon how long that light is provided. Recent research suggests that the amount of light and the duration of light exposure are reciprocally related for entrainment in the morning hours (i.e. 08:00–12:00). This opens up a number of design options heretofore unaddressed by lighting specifiers.

English

For any number field K with non-elementary 3-class group Cl(3,K) = C(3^e) x C(3), e >= 2, the punctured capitulation type kappa(K) of K in its unramified cyclic cubic extensions Li, 1 <= i <= 4, is an orbit under the action of S3 x S3. By means of Artin's reciprocity law, the arithmetical invariant kappa(K) is translated to the punctured transfer kernel type kappa(G2) of the automorphism group G2 = Gal(F(3,2,K)/K) of the second Hilbert 3-class field of K. A classification of finite 3-groups G with low order and bicyclic commutator quotient G/G' = C(3^e) x C(3), 2 <= e <= 6, according to the algebraic invariant kappa(G), admits conclusions concerning the length of the Hilbert 3-class field tower F(3,infty,K) of imaginary quadratic number fields K.

Quadratic, Cubic, Biquadratic, and Quintic Reciprocity

Quadratic, Cubic, Biquadratic, and Quintic Reciprocity

Importance of Duration, Duty-Cycling and Thresholds for the Implementation of Ultraviolet C in Marine Biofouling Control

The introduction of a surface into the marine environment begins a process known as biofouling, which increases the weight and hydrodynamic drag of the fouled structure. This process is detrimental to maritime vessels and costs the industry ∼$150B in fuel and maintenance spending annually. Preventing the settlement of fouling organisms mitigates these issues and limits the spread of non-indigenous species (NIS). This is primarily achieved via antifouling paints. Ultraviolet light is a sterilization method used in water purification, food storage packaging, and within medical fields. Ultraviolet C (UV-C) radiation interacts with DNA to prevent growth, proliferation, and survival of bacteria, and biofilm formation. Recent progress in microelectronics technology has advanced the range of commercially available light-emitting diodes (LEDs) to include the UV wavelengths, and the reduced size and cost has allowed their integration into previously inaccessible locales. This study builds on recent progress in integrating UV-C LEDs into UV-lucent silicone tiles for fouling control. The operational cycle needed to prevent growth of Navicula incerta cells was determined. Constant irradiance at a peak of 5.77 μW/cm2 resulted in a significant reduction in diatoms within 2 h, and a 2 log and 3 log reduction after 48 h and 5 days, respectively. Duty cycling (pulsing) in all variations from 50 to 2.5%, indicated significant reductions in cell densities, and the lowest cycle could effectively reduce biofouling growth and increase the longevity of the LEDs for up to 45.6 years. Irradiance and exposure were altered over a set duration and indicated a restriction in growth between 0.01–0.82 J/cm2 and an increased mortality at irradiances &amp;gt; 2.65 J/cm2, suggesting an effective antifouling threshold between these dosages. The effective dosage for 1 log reduction in fouling was estimated to be 25 J/cm2 but varied according to irradiance delivery method. Effective dosage for a 1 log reduction between experimental methods was variable indicating that UV treatment of N. incerta departed from the Bunsen-Roscoe reciprocity law expectancy. The variation in densities at similar dosages could be explained with further investigation of DNA repair mechanisms. In conclusion, UV-B/C use was effective at all irradiances, including as low as 0.01 J/cm2, and holds considerable promise for marine biofouling control.

BGG categories in prime characteristics

Let \({\mathfrak {g}}\) be a simple complex Lie algebra. In this paper we study the BGG category \({\mathcal {O}}_q\) for the quantum group \(U_q({\mathfrak {g}})\) with q being a root of unity in a field K of characteristic \(p >0\). We first consider the simple modules in \({\mathcal {O}}_q\) and prove a Steinberg tensor product theorem for them. This result reduces the problem of determining the corresponding irreducible characters to the same problem for a finite subset of finite dimensional simple modules. Then we investigate more closely the Verma modules in \({\mathcal {O}}_q\). Except for the special Verma module, which has highest weight \(-\rho \), they all have infinite length. Nevertheless, we show that each Verma module has a certain finite filtration with an associated strong linkage principle. The special Verma module turns out to be both simple and projective/injective. This leads to a family of projective modules in \({\mathcal {O}}_q\), which are also tilting modules. We prove a reciprocity law, which gives a precise relation between the corresponding family of characters for indecomposable tilting modules and the family of characters of simple modules with antidominant highest weights. All these results are of particular interest when \(q = 1\), and we have paid special attention to this case.

Principles of corneal cross-linking : Presentation based on the development of the various treatment protocols

Corneal cross-linking (CXL) is used to treat corneal ectatic diseases. The aim is to improve the reduced consolidation of the cornea in order to halt further corneal protrusion and therefore subsequent deterioration of the optical imaging proportions. In this article the principles of corneal cross-linking based on riboflavin and UV light are presented including recent research results. Furthermore, the most important treatment protocols including standard CXL (S-CXL), accelerated CXL (A-CXL), transepithelial CXL (TE-CXL) and the approach of the CXL procedure for thin corneas are explained. The CXL method depends on four major components, the riboflavin solution, oxygen, UV light and the availability of cross-linking sites on the collagen tissue. According to the present state of knowledge, the photochemical process of the CXL method induces covalent bonds between the fibrils and proteoglycans and thus stabilizes the collagen fibers, resulting in corneal consolidation. In addition to the S‑CXL, which has proven its effectiveness and safety in alarge number of studies, there are other treatment protocols that have been developed based on the Bunsen-Roscoe law of reciprocity. The A‑CXL protocol has the advantage of having ashorter irradiation time but it seems to be less effective than the S‑CXL protocol concerning the increase in corneal stiffness. The use of TE-CXL has so far not yet gained acceptance in the clinical practice. The CXL procedures primarily aim to stabilize the cornea. In the future, in addition to stabilization of the cornea, simultaneous improvement of visual acuity will be the main focus.

Reprint of: Endomorphism rings of reductions of Drinfeld modules

Let A=Fq[T] be the polynomial ring over Fq, and F be the field of fractions of A. Let ϕ be a Drinfeld A-module of rank r≥2 over F. For all but finitely many primes p◁A, one can reduce ϕ modulo p to obtain a Drinfeld A-module ϕ⊗Fp of rank r over Fp=A/p. The endomorphism ring Ep=EndFp(ϕ⊗Fp) is an order in an imaginary field extension K of F of degree r. Let Op be the integral closure of A in K, and let πp∈Ep be the Frobenius endomorphism of ϕ⊗Fp. Then we have the inclusion of orders A[πp]⊂Ep⊂Op in K. We prove that if EndFalg(ϕ)=A, then for arbitrary non-zero ideals n,m of A there are infinitely many p such that n divides the index χ(Ep/A[πp]) and m divides the index χ(Op/Ep). We show that the index χ(Ep/A[πp]) is related to a reciprocity law for the extensions of F arising from the division points of ϕ. In the rank r=2 case we describe an algorithm for computing the orders A[πp]⊂Ep⊂Op, and give some computational data.

Local abelian Kato-Parshin reciprocity law: A survey

Let $K$ denote an $n$-dimensional local field. The aim of this expository paper is to survey the basic arithmetic theory of the $n$-dimensional local field $K$ together with its Milnor $K$-theory and Parshin topological $K$-theory; to review Kato's ramification theory for finite abelian extensions of the $n$-dimensional local field $K$, and to state the local abelian higher-dimensional $K$-theoretic generalization of local abelian class field theory of Hasse, which is developed by Kato and Parshin. The paper is geared toward non-abelian generalization of this theory.

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.

Ultraviolet-B irradiance and cumulative dose combine to determine performance and survival.

Despite decades of research, the role of elevated solar ultraviolet-B radiation (UVBR; 280-315nm) in shaping amphibian populations remains ambiguous. These difficulties stem partly from a poor understanding of which parameters of UVBR exposure - dose, irradiance, and time interval - determine UVBR exposure health risk, and the potentially erroneous assumption that effects are proportional to the dose of exposure, irrespective of the administered regime (Bunsen-Roscoe Law of Reciprocity; BRL). We tested if the BRL holds with respect to UVBR-induced physiological effects in amphibians by acutely exposing tadpoles of the Australian green tree frog (Litoria caerulea) to a combination of different UVBR irradiances and doses in a fully factorial experiment. The BRL was invalid across all metrics assessed, with UVBR irradiance influencing the effects of a given dose on growth, coloration and burst swimming performance of larvae. We demonstrated some of the first empirical evidence for beneficial physiological effects of UVBR exposure in a larval amphibian, with improvements in growth, burst swimming performance and survival at the highest UVBR doses, contrary to hypotheses. Our findings demonstrate the species-specific nature of amphibian responses to UVBR, and the importance of UVBR irradiance in influencing the long-term physiological effects of a given dose of radiation. This work enhances our understanding of which parameters of complex UVBR exposures determine amphibian health risk.

Lettuce growth, morphology and critical leaf trait responses to far-red light during cultivation are low fluence and obey the reciprocity law

The mediating role of far-red (FR) light at the end of day (EOD) during cultivation on plant growth, morphology and critical leaf traits was analyzed. Three experiments were conducted on lettuce. Firstly, it was clarified whether or not the FR light effect depends on the presence of photosynthetic active radiation (PAR). Secondly, dose-response curves were obtained by applying six FR light intensities. Eventually, three FR light intensities were applied for different durations in a way that daily light integral (DLI) remained constant. Assessments included specific leaf area, dry mass partitioning, leaf optical properties, net photosynthesis rate (Pn), as well as leaf photosynthetic pigment, soluble sugar, starch, cellulose, soluble protein, nitrate, total nitrogen and total carbon contents. Pn was not significantly affected by FR light during growth, despite decreased leaf light absorption and photosynthetic pigment content (2 and 16%, respectively). Starch and nitrate contents were not affected by FR light, whereas cellulose, soluble protein and total nitrogen contents were generally decreased. When applied alone, FR light better promoted growth, radiation use efficiency (∼22%) and soluble sugar content (∼34%). The FR light-induced effect was intensity-dependent, while both qualitative and quantitative threshold responses were apparent depending on the trait. Five minutes of daily exposure to FR light (90 μmol m − 2s − 1) were adequate in eliciting the photomorphogenic responses. Notably, no significant differences were observed in all traits between FR light intensities under the same DLI. In conclusion, FR light at the EOD better stimulates growth when applied alone, through improved light interception. FR light response is low fluence and depends on the DLI, obeying the reciprocity low.