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Journal articles on the topic 'Quadrati reciprocity'

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Published: 10 December 2022

Last updated: 28 January 2023

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1

HAMBLETON, S., and V. SCHARASCHKIN. "QUADRATIC RECIPROCITY VIA RESULTANTS." International Journal of Number Theory 06, no. 06 (September 2010): 1413–17. http://dx.doi.org/10.1142/s179304211000354x.

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2

Rousseau, G. "On the quadratic reciprocity law." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 51, no. 3 (December 1991): 423–25. http://dx.doi.org/10.1017/s1446788700034583.

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AbstractA version of Gauss's fifth proof of the quadratic reciprocity law is given which uses only the simplest group-theoretic considerations (dispensing even with Gauss's Lemma) and makes manifest that the reciprocity law is a simple consequence of the Chinese Remainder Theorem.

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3

Hambleton, S., and V. Scharaschkin. "Pell conics and quadratic reciprocity." Rocky Mountain Journal of Mathematics 42, no. 1 (February 2012): 91–96. http://dx.doi.org/10.1216/rmj-2012-42-1-91.

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4

Kronheimer, P. B., M. J. Larsen, and J. Scherk. "Casson's invariant and quadratic reciprocity." Topology 30, no. 3 (November 1991): 335–38. http://dx.doi.org/10.1016/0040-9383(91)90018-y.

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5

Virgil Barnard. "A Proof of Quadratic Reciprocity." American Mathematical Monthly 122, no. 6 (2015): 588. http://dx.doi.org/10.4169/amer.math.monthly.122.6.588.

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6

Lemmermeyer, F. "Selmer groups and quadratic reciprocity." Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 76, no. 1 (December 2006): 279–93. http://dx.doi.org/10.1007/bf02960869.

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7

Perutka, Tomas. "Using decomposition groups to prove theorems about quadratic residues." Journal of the ASB Society 1, no. 1 (December 28, 2020): 12–20. http://dx.doi.org/10.51337/jasb20201228002.

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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.

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8

Cox, David A. "Quadratic Reciprocity: Its Conjecture and Application." American Mathematical Monthly 95, no. 5 (May 1988): 442. http://dx.doi.org/10.2307/2322482.

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9

Duke, William, and Kimberly Hopkins. "Quadratic Reciprocity in a Finite Group." American Mathematical Monthly 112, no. 3 (March 1, 2005): 251. http://dx.doi.org/10.2307/30037441.

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10

Cox, Darrell, Sourangshu Ghosh, and Eldar Sultanow. "Quadratic, Cubic, Biquadratic, and Quintic Reciprocity." International Journal of Pure and Applied Mathematics Research 2, no. 1 (April 5, 2022): 15–39. http://dx.doi.org/10.51483/ijpamr.2.1.2022.15-39.

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11

Merrill, Kathy, and Lynne Walling. "On quadratic reciprocity over function fields." Pacific Journal of Mathematics 173, no. 1 (March 1, 1996): 147–50. http://dx.doi.org/10.2140/pjm.1996.173.147.

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12

Duke, William, and Kimberly Hopkins. "Quadratic Reciprocity in a Finite Group." American Mathematical Monthly 112, no. 3 (March 2005): 251–56. http://dx.doi.org/10.1080/00029890.2005.11920190.

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13

Cox, David A. "Quadratic Reciprocity: Its Conjecture and Application." American Mathematical Monthly 95, no. 5 (May 1988): 442–48. http://dx.doi.org/10.1080/00029890.1988.11972029.

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14

Sander, J. W. "A reciprocity formula for quadratic forms." Monatshefte f�r Mathematik 104, no. 2 (June 1987): 125–32. http://dx.doi.org/10.1007/bf01326785.

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15

Richard Steiner. "Modular Divisor Functions and Quadratic Reciprocity." American Mathematical Monthly 117, no. 5 (2010): 448. http://dx.doi.org/10.4169/000298910x485978.

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16

Swan, Richard G. "Another Proof of the Quadratic Reciprocity Theorem?" American Mathematical Monthly 97, no. 2 (February 1990): 138. http://dx.doi.org/10.2307/2323916.

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17

Czogała, Alfred. "On reciprocity equivalence of quadratic number fields." Acta Arithmetica 58, no. 1 (1991): 27–46. http://dx.doi.org/10.4064/aa-58-1-27-46.

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18

Yap, Audrey. "Gauss’ quadratic reciprocity theorem and mathematical fruitfulness." Studies in History and Philosophy of Science Part A 42, no. 3 (September 2011): 410–15. http://dx.doi.org/10.1016/j.shpsa.2010.09.002.

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19

Swan, Richard G. "Another Proof of the Quadratic Reciprocity Theorem?" American Mathematical Monthly 97, no. 2 (February 1990): 138–39. http://dx.doi.org/10.1080/00029890.1990.11995563.

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20

Gauthier, Yvon. "A quadratic reciprocity theorem for arithmetical logic." International Journal of Algebra 13, no. 8 (2019): 445–55. http://dx.doi.org/10.12988/ija.2019.91039.

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21

Hashim, Hayder R. "Solutions of the Diophantine Equation 7X2 + Y7 = Z2 from Recurrence Sequences." Communications in Mathematics 28, no. 1 (June 1, 2020): 55–66. http://dx.doi.org/10.2478/cm-2020-0005.

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AbstractConsider the system x2 − ay2 = b, P (x, y) = z2, where P is a given integer polynomial. Historically, the integer solutions of such systems have been investigated by many authors using the congruence arguments and the quadratic reciprocity. In this paper, we use Kedlaya’s procedure and the techniques of using congruence arguments with the quadratic reciprocity to investigate the solutions of the Diophantine equation 7X2 + Y7 = Z2 if (X, Y) = (Ln, Fn) (or (X, Y) = (Fn, Ln)) where {Fn} and {Ln} represent the sequences of Fibonacci numbers and Lucas numbers respectively.

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22

Meyer, Jeffrey L. "A reciprocity congruence for an analogue of the Dedekind sum and quadratic reciprocity." Journal de Théorie des Nombres de Bordeaux 12, no. 1 (2000): 93–101. http://dx.doi.org/10.5802/jtnb.268.

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23

Tangedal, Brett A. "Eisenstein's Lemma and Quadratic Reciprocity for Jacobi Symbols." Mathematics Magazine 73, no. 2 (April 1, 2000): 130. http://dx.doi.org/10.2307/2691084.

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24

Mack, Thomas. "A proof of quadratic reciprocity via linear recurrences." Acta Arithmetica 199, no. 4 (2021): 433–40. http://dx.doi.org/10.4064/aa210213-21-3.

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25

Kim, Sey Y. "An Elementary Proof of the Quadratic Reciprocity Law." American Mathematical Monthly 111, no. 1 (January 2004): 48. http://dx.doi.org/10.2307/4145015.

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26

Czogała, Alfred, and Przemysław Koprowski. "Yet another proof of the quadratic reciprocity law." Acta Arithmetica 185, no. 3 (2018): 297–300. http://dx.doi.org/10.4064/aa180321-10-7.

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27

Tangedal, Brett A. "Eisenstein's Lemma and Quadratic Reciprocity for Jacobi Symbols." Mathematics Magazine 73, no. 2 (April 2000): 130–34. http://dx.doi.org/10.1080/0025570x.2000.11996820.

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28

Kim, Sey Y. "An Elementary Proof of the Quadratic Reciprocity Law." American Mathematical Monthly 111, no. 1 (January 2004): 48–50. http://dx.doi.org/10.1080/00029890.2004.11920046.

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29

Brunyate, Adrian, and Pete L. Clark. "Extending the Zolotarev–Frobenius approach to quadratic reciprocity." Ramanujan Journal 37, no. 1 (November 11, 2014): 25–50. http://dx.doi.org/10.1007/s11139-014-9635-y.

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30

Tascı, Dursun, and Gul Ozkan Kızılırmak. "On the Periods of Biperiodic Fibonacci and Biperiodic Lucas Numbers." Discrete Dynamics in Nature and Society 2016 (2016): 1–5. http://dx.doi.org/10.1155/2016/7341729.

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This paper is concerned with periods of Biperiodic Fibonacci and Biperiodic Lucas sequences taken as modulo prime and prime power. By using Fermat’s little theorem, quadratic reciprocity, many results are obtained.

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31

Laubenbacher, Reinhard C., and David J. Pengelley. "Eisenstein's Misunderstood Geometric Proof of the Quadratic Reciprocity Theorem." College Mathematics Journal 25, no. 1 (January 1994): 29. http://dx.doi.org/10.2307/2687081.

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32

Castryck, Wouter. "A Shortened Classical Proof of the Quadratic Reciprocity Law." American Mathematical Monthly 115, no. 6 (June 2008): 550–51. http://dx.doi.org/10.1080/00029890.2008.11920561.

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33

Veklych, Bogdan. "A Minimalist Proof of the Law of Quadratic Reciprocity." American Mathematical Monthly 126, no. 10 (November 20, 2019): 928. http://dx.doi.org/10.1080/00029890.2019.1655331.

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34

Laubenbacher, Reinhard C., and David J. Pengelley. "Eisenstein's Misunderstood Geometric Proof of the Quadratic Reciprocity Theorem." College Mathematics Journal 25, no. 1 (January 1994): 29–34. http://dx.doi.org/10.1080/07468342.1994.11973576.

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35

Steven H. Weintraub. "On Legendre’s Work on the Law of Quadratic Reciprocity." American Mathematical Monthly 118, no. 3 (2011): 210. http://dx.doi.org/10.4169/amer.math.monthly.118.03.210.

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36

Eichler, Martin. "The quadratic reciprocity law and the elementary theta function." Glasgow Mathematical Journal 27 (October 1985): 19–30. http://dx.doi.org/10.1017/s0017089500006042.

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This note points out a new aspect of the well-known relationship between the subjects mentioned in the title. The following result and its generalization in totally real algebraic number fields is central to the discussion. Let denote the Legendre symbol for relatively prime numbers a and b ℇ ℤ and a substitution of the modular subgroup Γ0(4). Then, if γ>0 and b≡1 mod 2,withandAccording to (1), the Legendre symbol behaves somewhat like a modular function ﹙apart from the known behaviour under and ﹚. (1) follows (see below) from the functional equationwithprovided thatHere we used and always will use the abbreviationand ℇδ means the absolutely least residue of δ mod 4. In the proof, Hecke [4] assumed γ>0 (see also Shimura [5]).

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37

Starkl, R., and L. del Re. "On the reciprocity of bilinear systems and their quadratic extensions." IFAC Proceedings Volumes 37, no. 13 (September 2004): 249–54. http://dx.doi.org/10.1016/s1474-6670(17)31231-4.

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38

Konstantinou, Elisavet, and Aristides Kontogeorgis. "Computing Polynomials of the Ramanujan tn Class Invariants." Canadian Mathematical Bulletin 52, no. 4 (December 1, 2009): 583–97. http://dx.doi.org/10.4153/cmb-2009-058-6.

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AbstractWe compute the minimal polynomials of the Ramanujan values tn, where n ≡ 11 mod 24, using the Shimura reciprocity law. These polynomials can be used for defining the Hilbert class field of the imaginary quadratic field and have much smaller coefficients than the Hilbert polynomials.

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39

Peter, Manfred. "Dirichlet series and automorphic functions associated to a quadratic form." Nagoya Mathematical Journal 171 (2003): 1–50. http://dx.doi.org/10.1017/s0027763000025502.

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AbstractStarting from the reciprocity law for Gaussian sums attached to an integral quadratic form we prove functional equations for a new kind of Dirichlet series in two variables. For special values of one variable they are of Hecke type with respect to the other variable. With Weil’s converse theorem we derive automorphic functions which generalize Siegel’s genus invariant and the automorphic functions of Cohen and Zagier.

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40

Lu, Hui, and Ruiyao Niu. "Generation method of GPS L1C codes based on quadratic reciprocity law." Journal of Systems Engineering and Electronics 24, no. 2 (April 2013): 189–95. http://dx.doi.org/10.1109/jsee.2013.00024.

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41

Gannon, Terry. "Postcards from the Edge, or Snapshots of the Theory of Generalised Moonshine." Canadian Mathematical Bulletin 45, no. 4 (December 1, 2002): 606–22. http://dx.doi.org/10.4153/cmb-2002-056-6.

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AbstractWe begin by reviewing Monstrous Moonshine. The impact of Moonshine on algebra has been profound, but so far it has had little to teach number theory. We introduce (using ‘postcards’) a much larger context in which Monstrous Moonshine naturally sits. This context suggests Moonshine should indeed have consequences for number theory. We provide some humble examples of this: new generalisations of Gauss sums and quadratic reciprocity.

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42

Schinzel, A., and M. Skałba. "An improvement of a lemma from Gauss’s first proof of quadratic reciprocity." Bulletin Polish Acad. Sci. Math. 65, no. 1 (2017): 29–33. http://dx.doi.org/10.4064/ba8109-4-2017.

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43

Ji, Chun-Gang, and Yan Xue. "An elementary proof of the law of quadratic reciprocity over function fields." Proceedings of the American Mathematical Society 136, no. 09 (April 30, 2008): 3035–39. http://dx.doi.org/10.1090/s0002-9939-08-09327-1.

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44

Aljohani, Abdulah Jeza, Muhammad Moinuddin, Ubaid M. Al-Saggaf, Mohammed El-Hajjar, and Soon Xin Ng. "Statistical Beamforming for Multi-Set Space–Time Shift-Keying-Based Full-Duplex Millimeter Wave Communications." Mathematics 11, no. 2 (January 13, 2023): 433. http://dx.doi.org/10.3390/math11020433.

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Full-duplex (FD) communication has been shown to provide an increased achievable rate, while millimeter wave (mmWave) communications benefit from a large available bandwidth that further improves the achievable rate. On the other hand, the concept of multi-set space-time shift keying (MS-STSK) has been proposed to provide a flexible design trade-off between throughput and performance. Hence, in this work, we consider the design of an FD-aided MS-STSK transceiver for millimeter wave communications. However, a major challenge is that channel reciprocity is not valid in mmWave communications due to shorter channel coherence time. Thus, the uplink (UL) pilots cannot be utilized to estimate the downlink (DL) channel. To overcome this challenge, we propose a beamforming technique based on channel statistics without assuming channel reciprocity. For this purpose, a closed-form expression for the outage probability of the system is derived by employing the characterization of the ratio of the Indefinite Quadratic Form (IQF). The derived analytical expression is then utilized to design optimum beamforming weights using the Sequential Quadratic Programming (SQP)-based heuristic method. Moreover, an Iterative Statistical Method (ISM) of joint transmit and receive beamforming algorithm is also developed by utilizing Principle Eigenvector (PE) and Generalized Rayleigh Quotient (G-RQ) optimization techniques. Finally, we verify our simulation results with the theoretical analysis.

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45

Laubenbacher, Reinhard C., and David J. Pengelley. "Gauss, eisenstein, and the “third” proof of the quadratic reciprocity theorem:ein kleines schauspiel." Mathematical Intelligencer 16, no. 2 (March 1994): 67–72. http://dx.doi.org/10.1007/bf03024287.

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46

Fenster, Della Dumbaugh. "Why Dickson Left Quadratic Reciprocity Out of HisHistory of the Theory of Numbers." American Mathematical Monthly 106, no. 7 (August 1999): 618–27. http://dx.doi.org/10.1080/00029890.1999.12005095.

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47

Fenster, Della Dumbaugh. "Why Dickson Left Quadratic Reciprocity out of His History of the Theory of Numbers." American Mathematical Monthly 106, no. 7 (August 1999): 618. http://dx.doi.org/10.2307/2589491.

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48

HABLICSEK, MÁRTON, and GUILLERMO MANTILLA-SOLER. "POWER MAP PERMUTATIONS AND SYMMETRIC DIFFERENCES IN FINITE GROUPS." Journal of Algebra and Its Applications 10, no. 05 (October 2011): 947–59. http://dx.doi.org/10.1142/s0219498811005051.

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Let G be a finite group. For all a ∈ ℤ, such that (a, |G|) = 1, the function ρa: G → G sending g to ga defines a permutation of the elements of G. Motivated by a recent generalization of Zolotarev's proof of classic quadratic reciprocity, due to Duke and Hopkins, we study the signature of the permutation ρa. By introducing the group of conjugacy equivariant maps and the symmetric difference method on groups, we exhibit an integer dG such that [Formula: see text] for all G in a large class of groups, containing all finite nilpotent and odd order groups.

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49

Hayashi, Heima. "Note on a product formula for the Bayad function and a law of quadratic reciprocity." Acta Arithmetica 149, no. 4 (2011): 321–36. http://dx.doi.org/10.4064/aa149-4-2.

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50

Dasgupta, Samit. "Computations of Elliptic Units for Real Quadratic Fields." Canadian Journal of Mathematics 59, no. 3 (June 1, 2007): 553–74. http://dx.doi.org/10.4153/cjm-2007-023-0.

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AbstractLet K be a real quadratic field, and p a rational prime which is inert in K. Let α be amodular unit on Γ0(N). In an earlier joint article with Henri Darmon, we presented the definition of an element u(α, τ) ∊ Kxp attached to α and each τ ∈ K. We conjectured that the p-adic number u(α, τ) lies in a specific ring class extension of K depending on τ , and proposed a “Shimura reciprocity law” describing the permutation action of Galois on the set of u(α, τ). This article provides computational evidence for these conjectures. We present an efficient algorithm for computing u(α, τ), and implement this algorithm with the modular unit α(z) = Δ(z)2Δ(4z)/Δ(2z)3. Using p = 3, 5, 7, and 11, and all real quadratic fields K with discriminant D < 500 such that 2 splits in K and K contains no unit of negative norm, we obtain results supporting our conjectures. One of the theoretical results in this paper is that a certain measure used to define u(α, τ) is shown to be Z-valued rather than only Zp ∩ Q-valued; this is an improvement over our previous result and allows for a precise definition of u(α, τ), instead of only up to a root of unity.

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