Reciprocity Law Research Articles

A reciprocity law in function fields

A reciprocity law in function fields

Study of the Bunsen–Roscoe Reciprocity Law in Solar Water Disinfection (Optical Effect) for E. coli, E. faecalis and C. perfringens

Water stress and water quality represent major environmental challenges in the 21st century. In response, wastewater management and its potential reuse emerge as strategies to mitigate these problems. This research aims to verify the law of reciprocity in the solar disinfection process of real secondary wastewater effluents for different faecal microorganisms. Flat disinfection reactors, subjected only to natural and continuous UV radiation, were used. The study focused on the optical effect of UV radiation, eliminating the significant influence of the thermal effect and its synergy in solar disinfection at temperatures above 45 °C, by controlling the temperatures of the water samples to levels below 20 °C. Three experimental tests were carried out on sunny days. Each test comprised two trials, under the following conditions: (a) low solar irradiance over a prolonged time (duration approximately: 2.6 h) and (b) high solar irradiance and a shorter period of time (approximately 2 h), with each receiving the same UV dose. Inactivation kinetics was analysed for E. coli, E. faecalis, and C. perfringens (including spores). The results validated the reciprocity law for E. coli in all tests for UV doses > 20 Wh/m2, showing no significant deviations, with inactivation rates of 0.44 to 0.51 m2/Wh for initial concentrations of 106–107 CFU/100 mL. In contrast, for E. faecalis, the reciprocity was only valid at intensities < 700 W/m2, with rates of 0.04 and 0.035 m2/Wh for 105–106 CFU/100 mL; above this irradiance value, the law varied significantly and was not valid. C. perfringens did not show significant disinfection results during the experiments to verify this law, mainly due to the resistance of its spores. Additional experimentation with C. perfringens is necessary, by extending the length of the experiments and/or conducting them at higher irradiance values, in order to reach bacterial inactivation to enable the analysis of the reciprocity law. In general, the main conclusion from these results is that the reciprocity law in solar disinfection would be difficult to use for the estimation of water solar disinfection based on the irradiance and exposure times, as there are deviations from it at least in one specie (E. faecalis). Mores studies should be carried out to fully understand and determine the validity of this law and its potential application for forecasting solar water disinfection.

Higher reciprocity law and an analogue of the Grunwald–Wang theorem for the ring of polynomials over an ultra-finite field

In this paper, we establish an explicit higher reciprocity law for the polynomial ring over a nonprincipal ultraproduct of finite fields. Such an ultraproduct can be taken over the same finite field, which allows to recover the classical higher reciprocity law for the polynomial ring Fq[t] over a finite field Fq that is due to Dedekind, Kühne, Artin, and Schmidt. On the other hand, when the ultraproduct is taken over finite fields of unbounded cardinalities, we obtain an explicit higher reciprocity law for the polynomial ring over an infinite field in both characteristics 0 and p>0 for some prime p. We then use the higher reciprocity law to prove an analogue of the Grunwald–Wang theorem for such a polynomial ring in both characteristics 0 and p>0 for some prime p.

Arthropods and ultraviolet radiation: Survival, growth and development

Abstract Ultraviolet radiation (UVR) has positive and negative impacts on organisms and affects entire ecosystems. It is currently receiving attention because of the interactions of UVR with changes in climate that are likely to affect agriculture and food security. A review is provided on the effects of different types of UVR on the survival, development and reproduction of insects, collembolans and mites. The foci are on how UVR is used in pest control and how it affects arthropod coloration and distribution. Direct exposure to UV‐A, UV‐B and UV‐C delays development and reduces survival and host‐seeking ability of arthropods; immature stages are more sensitive than the adults. Increased exposure of crops to UVR has indirect effects on herbivory, and regions with high UVR levels may permit crop production in the absence of certain pest insect species. The Bunsen‐Roscoe law of reciprocity was upheld for Diaphorina citri and Tetranychus urticae under the range of irradiances and durations of exposure used in two studies, which showed that, for a constant dose, the effects of low irradiances of long durations were equivalent to those of higher irradiances of shorter durations. DNA damage caused by UV‐C and UV‐B can be repaired by photoreactivation; both white light and UV‐A are effective.

Wide-angle camouflage detectors by manipulating emissivity using a non-reciprocal metasurface array.

Camouflage detectors that can detect incoming radiation from any angle without being detected are extremely important in stealth, guided missile, and heat-seeking missile industries. In order to accomplish this, the absorption and emission processes must be manipulated simultaneously. However, Kirchhoff's fundamental law suggests that absorption and emission are always in the same direction α(θ) = ε(θ), i.e., absorption and emission are reciprocal. This means that the emission from the detector always points back to the source, for example towards a laser source in a guided missile. Thus, detector emission serves as a complementary measure to hide an object. Here, we present a novel camouflage detector that uses a nonreciprocal metasurface array to independently detect the direction of the incoming radiation as well as manipulate its emissivity response. This is accomplished by using a magneto-optical material called indium arsenide (InAs), which breaks Lorentz reciprocity and Kirchhoff's fundamental law such that α(θ) ≠ ε(θ). This design results in the following absorption and emission: α(θ) = ε(-θ). Nine metasurfaces were designed, optimized, and operated at different incident angles from +50° to -50° at a wavelength of 13 μm. Furthermore, by keeping all metasurfaces in a pixilated array form, one could make a device that works over the full ±50° range. Potentially, this array of nonreciprocal metasurfaces can be used to fabricate thermal emitters or solar-harvesting systems.

Density of Selmer ranks in families of even Galois representations, Wiles' formula, and global reciprocity

This paper concerns the distribution of Selmer ranks in a family of even Galois representations in residual characteristic p=2 obtained by allowing ramification at auxiliary primes. The main result is a Galois cohomological analogue of a theorem of Friedlander, Iwaniec, Mazur and Rubin on the distribution of Selmer ranks in a family of twists of elliptic curves. The Selmer groups are constructed as prescribed by the Galois cohomological method for GL(2): At each ramified place, the local Selmer condition is the tangent space of a smooth quotient of the local deformation ring. By methods of global class field theory, the Selmer group at the minimal level is computed explicitly. The infinitude of primes for which the Selmer rank increases by one is proved, and the density of such primes is shown to be 1/192. The proof combines Wiles' formula and the global reciprocity law. The result has implications for the algebraic structure of even deformation rings and the distribution of their presentations in families.

Photo-induced bending of chalcogenide glass fiber

Optical fibers of a typical chalcogenide glass, As2S3, can be sharply bent only by sideward irradiation of polarized light. The bending direction depends on the polarization; the fiber bends in the forward/backward directions of light propagation for bandgap light that is linearly-polarized orthogonal/parallel to the fiber axis. A reciprocity law between the intensity and the exposure time holds, which suggests that thermal effects are irrelevant. Alternatively, the bending appears to arise from photo-induced optomechanical forces and fluidity. This phenomenon could be developed to a photo-manipulation method of chalcogenide-glass fibers, promising for all-optical switching/modulation devices working at infrared wavelengths.

RECIPROCITY LAWS FOR OSCILLATIONS OF DISSIPATIVE SYSTEMS

) is presented. The method is based on the use of the algebraic theorem of P.L. Pasternak and on the new properties of the Duhamel integral, which are obtained for a dissipative system with internal friction of the material, which is taken into account on the basis of the non-proportional damping model. For displacements, velocities and accelerations, the dynamic reaction equations are written in the form of systems of linear equations and their symmetrical structure is shown. The functional dependence of the force parameters of the calculation model and the corresponding kinematic parameters of the reaction is determined by an arbitrary scalar function of time. An extended interpretation of the reciprocity theorems is given and sufficient conditions for their fulfillment are formulated, which consist in the requirement that the matrix differential operator of the equation of motion be symmetrical. New laws of reciprocity in dissipative systems are formulated and proved. The reciprocity of the product between the velocities / accelerations of masses and nodal forces is established. In contrast to the well-known theorem on the reciprocity of possible work, these laws are theorems on the 1st / 2nd derivative of possible work with respect to time and therefore go beyond the Betti principle. For particular cases of these theorems, the reciprocity of velocities and reciprocity of accelerations is shown. Expressions of general and particular theorems have a fairly simple mathematical form that does not require recourse to integral transformations, and are presented in an analytical form.

The Folk Belief and Cultural Heritage in the Syncretic Theravada Buddhism : The Cases of Human-Spirits Relationship in Indonesia, Cambodia and Myanmar

This sociological study of religion attempts to explain the relationship between humans and spirits in the syncretism of Theravada Buddhism tradition with case studies in Indonesia, Cambodia, and Myanmar. In these countries, before the influence of the Indian Dharmic Religion, people practiced Theravada Buddhism side by side with folk beliefs. This study uses the theory of correlational theology from Paul Tillich as the search for common ground in reflection of the syncretic folk belief of worshipping Danghyang among Javanese people in Indonesia, Neak Ta among Khmer people in Cambodia, and Nat among Burmese people in Myanmar. The method used literature research and field observations in three countries in 2020. The results of this study are in the exploration of doctrinal concepts from religious texts and the acculturation practices on cultural heritage between the folk belief with the view of wholeness theology in Theravada Buddhism cosmology. Humans live in harmony with other creatures in this world according to the law of attachment and reciprocity, or Paticcasamuppada, consisting of 31 realms of existence. In the context of the three countries, these spirits articulate three relations: the relationship between guardian spirits and the people, the relationship between the deceased spirits and the living, and the relationship between spirits and the social environment, as in line with spiritual ecology (215 words)

Erythematous reactions to two ultraviolet excimer therapy devices with different irradiance levels: Reconsidering the reciprocity law.

Few studies have examined the effectiveness of the reciprocity law in ultraviolet excimer therapy. This study aimed to examine the difference in erythematous reaction in human skin when the irradiance of ultraviolet excimer treatment devices differed while the irradiation dose was constant. This study, conducted at the Department of Dermatology, Chiba University, included 15 healthy adults aged 20-65 years (mean age, 46.3 years; seven men). Using ultraviolet excimer treatment devices with different irradiances (50 or 150mW/cm2 ), the upper abdomen of each participant was irradiated with ultraviolet light at set irradiation doses (80, 100, 120, 140, 160, 180, and 200 mJ/cm2 ). The erythema index of each irradiated site was measured using a melanin- and erythema-measuring device, and the difference in erythema index before and 24h after irradiation was the primary endpoint. The change in erythema index was significantly higher for an irradiance of 150mW/cm2 . Significant differences (p<0.05) were observed between these irradiance levels at irradiation doses of 100-200 mJ/cm2 . Even for the same irradiation dose, stronger erythematous reactions occurred at higher irradiances in ultraviolet excimer treatment. This suggests that the reciprocity law may not always hold true in excimer therapy.

Identities for a Special Finite Sum Related to the Dedekind Sums and Fibonacci Numbers

The origin of this article is to achieve original equations related to the special finite sum C(μ,β;1), which is connected with Dedekind, Hardy, Simsek, and many other finite sums. By using the analytic properties of this sum, many useful identities are established between the C(μ,β;1) sum and other well-known finite sums. Through the use of these identities, the reciprocity law of this sum is obtained. Furthermore, another reciprocity law of the sum C(μ,β;1) is presented for μ and β are particular Fibonacci numbers. This remarkable result establishes a connection between number theory and analysis.

Exponential-type extended Riordan arrays and reciprocity law for generalized Stirling numbers

We propose an exponential-type extended Riordan array (or exponential recursive matrix) with permitted negative indices using the Roman factorial. In this context, we establish a fundamental theory of the generalized Stirling numbers introduced by Hsu and Shiue [11]. Subsequently, we provide a lucid explanation for the reciprocity law among these numbers found by Choi et al. [5] and Maltenfort [19]. Moreover, our method reveals that interesting numbers emerge in the fourth quadrant of the proposed exponential recursive matrix.

Basic theorems in local class field theory and some further exploration

The relation between the abelian extension of a field and the topological groups of the field itself can be constructed using class field theory. In this piece of writing, the author will introduce the fundamental theorems of local class field theory by conducting a method of literature study. These theorems are the Reciprocity Law and the Existence Theorem, respectively. In addition to that, the author will discuss several unresolved issues in class field theory and provide examples of their applications in number theory. In class field theory, the results can be shown in two different ways. Both can be considered broad strokes. The first step is to demonstrate that the local case is true. By employing the methodology of cohomology and the theory of Lubin and Tate, one can demonstrate the Local Reciprocity Law and the Local Existence Theorem. The fundamental theorems in the global case are going to be demonstrated by utilizing the local results in conjunction with cohomology. Directly demonstrating the Gocal Reciprocity Law is another viable option.

Chow dilogarithm and strong Suslin reciprocity law

We prove a conjecture of A. Goncharov concerning strong Suslin reciprocity law. The main idea of the proof is the construction of the norm map on so-called lifted reciprocity maps. This construction is similar to the construction of the norm map on Milnor K K -theory. As an application, we express Chow dilogarithm in terms of Bloch-Wigner dilogarithm. Also, we obtain a new reciprocity law for four rational functions on an arbitrary algebraic surface with values in the pre-Bloch group.

Double Dirichlet series associated with arithmetic functions II

This paper is a continuation of our previous work on double Dirichlet series associated with arithmetic functions such as the von Mangoldt function, the Möbius function, and so on. We consider the analytic behaviour around the non-positive integer points on singularity sets which are points of indeterminacy. In particular, we show a certain reciprocity law of their residues. Also on this occasion we correct some inaccuracies in our previous paper.

Legendre–Teege Quadratic Reciprocity

Legendre published the first attempted proof of the law of Quadratic Reciprocity. In its final form (1797), however, it had a gap in the form of an unproven hypothesis. Some 125 years later, Herman Teege published the first rigorous proof of that hypothesis. Then, 48 years later, Kenneth Rogers published a second (but implicit) proof. These proofs elevated Legendre’s attempt to the list of complete proofs. No detailed exposition of these proofs appears in the literature. Our paper fills that gap.

The &amp;lt;i&amp;gt;m&amp;lt;/i&amp;gt;Θ Quadratic Character in the &amp;lt;i&amp;gt;m&amp;lt;/i&amp;gt;Θ Set &amp;lt;img width=&amp;quot;40&amp;quot; height=&amp;quot;20&amp;quot; src=&amp;quot;http://article.sciencepublishinggroup.com/journal/247/2471199/image002.png&amp;quot; /&amp;gt;

The modal Θ-valent logic is a logic that contains all the thesis of the classical logical calculus and, besides allows to express notions of possibility, of necessity, and more others. The modal Θ-valent sets are the supports in term of the structure of the Θ-valent rings. A Θ chr (<i>m</i>Θ) is a structure which is rich at the same time of inheritance in the meaning of the romanian academician Gr. C. Moisil, as the algebraic model of a such logic. The set <img width="20" height="10" src="http://article.sciencepublishinggroup.com/journal/247/2471199/image002.png" /> contains the set <img width="10" height="10" src="http://article.sciencepublishinggroup.com/journal/247/2471199/image007.png" /> and the elements <img width="25" height="10" src="http://article.sciencepublishinggroup.com/journal/247/2471199/image003.png" /> such that the support of <i>x</i> is not congruent to 0 modulo <i>n</i>. In this paper the purpose is to define on <img width="40" height="10" src="http://article.sciencepublishinggroup.com/journal/247/2471199/image008.png" />, <i>p</i> prime, a notion of quadratic residues and quadratic character which respects its structure of <i>m</i>Θs. Hoping that this approach will bring something of interest to the notion of quadratic residues. First of all, we construct the modal Θ-valent congruences of (<img width="20" height="10" src="http://article.sciencepublishinggroup.com/journal/247/2471199/image002.png" />, <i>F_α</i>). We characterize the <i>m</i>Θ set (<img width="20" height="10" src="http://article.sciencepublishinggroup.com/journal/247/2471199/image002.png" />, <i>F_α</i>) and we then give some arithmetical and intrinsic <i>m</i>Θ parameters of <img width="20" height="10" src="http://article.sciencepublishinggroup.com/journal/247/2471199/image002.png" /> which lead us to the notion of factorial of <i>m</i> without <i>n</i> in <img width="20" height="10" src="http://article.sciencepublishinggroup.com/journal/247/2471199/image002.png" />, the <i>m</i>Θ quotient of (<img width="20" height="10" src="http://article.sciencepublishinggroup.com/journal/247/2471199/image002.png" />, <i>F_α</i>) modulo (<img width="30" height="10" src="http://article.sciencepublishinggroup.com/journal/247/2471199/image004.png" />) and a complete system of <i>m</i>Θ residues modulo <img width="30" height="10" src="http://article.sciencepublishinggroup.com/journal/247/2471199/image004.png" />, <img width="25" height="10" src="http://article.sciencepublishinggroup.com/journal/247/2471199/image005.png" />. After that, we define a <i>p</i>-valent modal quadratic residue, <i>p</i> prime. We characterize some properties of <i>p</i>-valent modal quadratic character and <i>p</i>-valent modal quadratic residue of <i>p^k</i> which establish the difference between the <i>m</i>Θ Euler’s theorem and the Euler’s theorem in the classical arithmetic. Later, we establish the theorem for determining the p-valent modal quadratic character of <img width="65" height="10" src="http://article.sciencepublishinggroup.com/journal/247/2471199/image006.png" /> with respect to <i>p^k</i>. This theorem is a non-classical version of Gauss’s lemma. Finally, we establish an example introducing the law of quadratic reciprocity of Gauss.

Analysing the Reciprocity Law for UV-LEDs in Water Disinfection of Escherichia coli, Enterococcus faecalis, and Clostridium perfringens

The aim of this study is to verify the reciprocity law in the wastewater disinfection process using UV light. The optical power UV-LEDs used were 1.6 mW and 50 mW, and the wavelengths were 265 nm and 275 nm. E. coli, Enterococcus faecalis, and Clostridium perfringens were the three microorganisms analysed in the study. The results showed lower inactivation rates around 0.063–0.065 cm2/mJ for 265 nm and 0.047–0.049 cm2/mJ for 275 nm for the Clostridium perfringens compared with the other two bacteria. For E. coli and Enterococcus faecalis, the inactivation rate was almost identical; 0.28 and 0.21 cm2/mJ, respectively, using 265 nm wavelength. There was a slightly better inactivation performance using the medium-power 275 nm UV-LEDs of 0.39 cm2/mJ and 0.29 cm2/mJ for E. coli and Enterococcus faecalis, respectively, and 0.33 cm2/mJ and 0.26 cm2/mJ using the low-power 275 nm UV-LEDs. The analysed data justify the reciprocity law for UV-LEDs disinfection using 265 nm and 275 nm UV-LEDs with two optical powers of 1.6 mW and 50 mW.

Obstructions to reciprocity laws related to division fields of Drinfeld modules

Let q q be an odd prime power, denote by F q \mathbb {F}_q the finite field with q q elements, and set A ≔ F q [ T ] A ≔\mathbb {F}_q[T] , F ≔ F q ( T ) F ≔\mathbb {F}_q(T) . Let ψ : A → F { τ } \psi : A \to F\{\tau \} be a Drinfeld A A -module over F F , of rank r ≥ 2 r \geq 2 , with E n d F ¯ ( ψ ) = A End_{\overline {F}}(\psi ) = A . For a non-zero ideal n \mathfrak {n} of A A , denote its unique monic generator by n n , denote the degree of n n as a polynomial in T T by deg ⁡ n \deg n , and denote the n \mathfrak {n} -division field of ψ \psi by F ( ψ [ n ] ) F(\psi [\mathfrak {n}]) . A reciprocity law for ψ \psi asserts that, if gcd ( c h a r F , r ) = 1 \gcd (charF, r) = 1 or if n \mathfrak {n} is prime, then a non-zero prime ideal p ∤ n \mathfrak {p} \nmid \mathfrak {n} of A A splits completely in F ( ψ [ n ] ) F(\psi [\mathfrak {n}]) if and only if the Frobenius trace a 1 , p ( ψ ) a_{1, \mathfrak {p}}(\psi ) of ψ \psi at p \mathfrak {p} and the first component b 1 , p ( ψ ) b_{1, \mathfrak {p}}(\psi ) of the Frobenius index of ψ \psi at p \mathfrak {p} satisfy the congruences a 1 , p ( ψ ) ≡ − r ( mod n ) a_{1, \mathfrak {p}}(\psi ) \equiv -r \pmod n and b 1 , p ( ψ ) ≡ 0 ( mod n ) b_{1, \mathfrak {p}}(\psi ) \equiv 0 \pmod n . We find the Dirichlet density of the set of non-zero prime ideals p \mathfrak {p} for which the latter congruence never holds, that is, for which b 1 , p ( ψ ) = 1 b_{1, \mathfrak {p}}(\psi ) = 1 . Using similar methods, we prove an asymptotic formula for the function of x x defined by the average 1 # { p : deg ⁡ p = x } ∑ p : deg ⁡ p = x τ A ( b 1 , p ( ψ ) ) \frac {1}{\#\{\mathfrak {p}: \ \deg p = x \}} \sum _{\mathfrak {p}: \ \deg p = x} \tau _A(b_{1, \mathfrak {p}}(\psi )) , where p = A p \mathfrak {p} = A p denotes an arbitrary non-zero prime ideal of A A whose monic generator p ∈ A p \in A has degree x x and where τ A ( b 1 , p ( ψ ) ) \tau _A(b_{1, \mathfrak {p}}(\psi )) denotes the number of monic divisors of b 1 , p ( ψ ) b_{1, \mathfrak {p}}(\psi ) .

Time-dose reciprocity mechanism for the inactivation of Escherichia coli explained by a stochastic process with two inactivation effects

There is a great demand for developing and demonstrating novel disinfection technologies for protection against various pathogenic viruses and bacteria. In this context, ultraviolet (UV) irradiation offers an effective and convenient method for the inactivation of pathogenic microorganisms. The quantitative evaluation of the efficacy of UV sterilization relies on the simple time-dose reciprocity law proposed by Bunsen-Roscoe. However, the inactivation rate constants reported in the literature vary widely, even at the same dose and wavelength of irradiation. Thus, it is likely that the physical mechanism of UV inactivation cannot be described by the simple time-dose reciprocity law but requires a secondary inactivation process, which must be identified to clarify the scientific basis. In this paper, we conducted a UV inactivation experiment with Escherichia coli at the same dose but with different irradiances and irradiation durations, varying the irradiance by two to three orders of magnitude. We showed that the efficacy of inactivation obtained by UV-light emitting diode irradiation differs significantly by one order of magnitude at the same dose but different irradiances at a fixed wavelength. To explain this, we constructed a stochastic model introducing a second inactivation rate, such as that due to reactive oxygen species (ROS) that contribute to DNA and/or protein damage, together with the fluence-based UV inactivation rate. By solving the differential equations based on this model, the efficacy of inactivation as a function of the irradiance and irradiation duration under the same UV dose conditions was clearly elucidated. The proposed model clearly shows that at least two inactivation rates are involved in UV inactivation, where the generally used UV inactivation rate does not depend on the irradiance, but the inactivation rate due to ROS does depend on the irradiance. We conclude that the UV inactivation results obtained to date were simply fitted by one inactivation rate that superimposed these two inactivation rates. The effectiveness of long-term UV irradiation at a low irradiance but the same dose provides useful information for future disinfection technologies such as the disinfection of large spaces, for example, hospital rooms using UV light, because it can reduce the radiation dose and its risk to the human body.