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Journal articles on the topic 'Jacobi triple product identity'

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Published: 4 June 2021

Last updated: 1 February 2022

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1

Ewell, John A. "Consequences of a sextuple-product identity." International Journal of Mathematics and Mathematical Sciences 10, no. 3 (1987): 545–49. http://dx.doi.org/10.1155/s0161171287000656.

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A sextuple-product identity, which essentially results from squaring the classical Gauss-Jacobi triple-product identity, is used to derive two trigonometrical identities. Several special cases of these identities are then presented and discussed.

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2

Wenchang, Chu. "Durfee rectangles and the Jacobi triple product identity." Acta Mathematica Sinica 9, no. 1 (March 1993): 24–26. http://dx.doi.org/10.1007/bf02559979.

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3

Chan, Hei-Chi. "Another simple proof of the quintuple product identity." International Journal of Mathematics and Mathematical Sciences 2005, no. 15 (2005): 2511–15. http://dx.doi.org/10.1155/ijmms.2005.2511.

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4

SCZECH, Robert. "Gaussian sums, Dedekind sums and the Jacobi triple product identity." Kyushu Journal of Mathematics 49, no. 2 (1995): 233–41. http://dx.doi.org/10.2206/kyushujm.49.233.

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5

Jun-Ming Zhu. "A Semi-Finite Proof of Jacobi′s Triple Product Identity." American Mathematical Monthly 122, no. 10 (2015): 1008. http://dx.doi.org/10.4169/amer.math.monthly.122.10.1008.

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6

Srivastava, Hari M., M. P. Chaudhary, and Sangeeta Chaudhary. "Some Theta-Function Identities Related to Jacobi’s Triple-Product Identity." European Journal of Pure and Applied Mathematics 11, no. 1 (January 30, 2018): 1. http://dx.doi.org/10.29020/nybg.ejpam.v11i1.3222.

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The main object of this paper is to present some q-identities involving some of the theta functions of Jacobi and Ramanujan. These q-identities reveal certain relationships among three of the theta-type functions which arise from the celebrated Jacobi’s triple-product identity in a remarkably simple way. The results presented in this paper are motivated by some recent works by Chaudhary et al. (see [4] and [5]) and others (see, for example, [1] and [13]).

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7

Bhargava, S., Chandrashekar Adiga, and M. S. Mahadeva Naika. "QUINTUPLE PRODUCT IDENTITY AS A SPECIAL CASE OF RAMANUJAN'S 1ψ1 SUMMATION FORMULA." Asian-European Journal of Mathematics 04, no. 01 (March 2011): 31–34. http://dx.doi.org/10.1142/s1793557111000046.

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In this note we observe an interesting fact that the well-known quintuple product identity can be regarded as a special case of the celebrated 1ψ1 summation formula of Ramanujan which is known to unify the Jacobi triple product identity and the q -binomial theorem.

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8

Cooper, Shaun. "A new proof of the Macdonald identities for An−1." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 62, no. 3 (June 1997): 345–60. http://dx.doi.org/10.1017/s1446788700001051.

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AbstractA new, elementary proof of the Macdonald identities for An−1 using induction on n is given. Specifically, the Macdonald identity for An is deduced by multiplying the Macdonald identity for An−1 and n Jacobi triple product identities together.

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9

Chaudhary, Mahendra. "A family of theta-function identities based upon Rα,Rβ and Rm-functions related to Jacobi’s triple-product identity." Publications de l'Institut Math?matique (Belgrade) 108, no. 122 (2020): 23–32. http://dx.doi.org/10.2298/pim2022023c.

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We establish a set of two new relationships involving R?,R? and Rm-functions, which are based on Jacobi?s famous triple-product identity. We, also provide answer for an open problem of Srivastava, Srivastava, Chaudhary and Uddin, which suggest to find an inter-relationships between R?,R? and Rm(m ? N), q-product identities and continued-fraction identities.

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10

DUVERNEY, DANIEL. "Some arithmetical consequences of Jacobi's triple product identity." Mathematical Proceedings of the Cambridge Philosophical Society 122, no. 3 (November 1997): 393–99. http://dx.doi.org/10.1017/s0305004197001916.

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11

KIM, SUN. "A BIJECTIVE PROOF OF THE QUINTUPLE PRODUCT IDENTITY." International Journal of Number Theory 06, no. 02 (March 2010): 247–56. http://dx.doi.org/10.1142/s1793042110002909.

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12

COOPER, SHAUN. "CONSTRUCTION OF EISENSTEIN SERIES FOR Γ0(p)." International Journal of Number Theory 05, no. 05 (August 2009): 765–78. http://dx.doi.org/10.1142/s1793042109002365.

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A simple construction of Eisenstein series for the congruence subgroup Γ0(p) is given. The construction makes use of the Jacobi triple product identity and Gauss sums, but does not use the modular transformation for the Dedekind eta-function. All positive integral weights are handled in the same way, and the conditionally convergent cases of weights 1 and 2 present no extra difficulty.

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13

Hirschhorn, Michael, Frank Garvan, and Jon Borwein. "Cubic Analogues of the Jacobian Theta Function θ(z, q)." Canadian Journal of Mathematics 45, no. 4 (August 1, 1993): 673–94. http://dx.doi.org/10.4153/cjm-1993-038-2.

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AbstractThere are three modular forms a(q), b(q), c(q) involved in the parametrization of the hypergeometric function analogous to the classical θ2(q), θ3(q), θ4(q) and the hypergeometric function We give elliptic function generalizations of a(q), b(q), c(q) analogous to the classical theta-function θ(z, q). A number of identities are proved. The proofs are self-contained, relying on nothing more than the Jacobi triple product identity

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14

Srivastava, Bhaskar. "Identities for Analogous Ramanujan's Functions by Jacobi's Triple Product Identity." American Journal of Mathematics and Statistics 2, no. 1 (August 31, 2012): 25–28. http://dx.doi.org/10.5923/j.ajms.20120201.06.

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15

Zhou, Jia, Liangyun Chen, Yao Ma, and Bing Sun. "On ω-Lie superalgebras." Journal of Algebra and Its Applications 17, no. 11 (November 2018): 1850212. http://dx.doi.org/10.1142/s0219498818502122.

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Let [Formula: see text] be a finite-dimensional vector space over a field [Formula: see text] of characteristic zero, [Formula: see text] an anti-commutative product on [Formula: see text] and [Formula: see text] a bilinear form on [Formula: see text]. The triple [Formula: see text] is called an [Formula: see text]-Lie algebra if [Formula: see text] (graded [Formula: see text]-Jacobi identity) for all [Formula: see text] In this paper, we introduce the notion of an [Formula: see text]-Lie superalgebra. We study elementary properties and representations of [Formula: see text]-Lie superalgebras. We classify all 3- and 4-dimensional [Formula: see text]-Lie superalgebras over the field of complex numbers.

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16

Adiga, Chandrashekar, and P. S. Guruprasad. "A Note on Four-Variable Reciprocity Theorem." International Journal of Mathematics and Mathematical Sciences 2009 (2009): 1–9. http://dx.doi.org/10.1155/2009/370390.

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We give new proof of a four-variable reciprocity theorem using Heine's transformation, Watson's transformation, and Ramanujan's -summation formula. We also obtain a generalization of Jacobi's triple product identity.

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17

Chaudhary, M. P., and Sangeeta Chaudhary. "On relationships between q-products identities, Ralpha, Rbeta and Rm functions related to Jacobi's triple-product identity." Mathematica Moravica 24, no. 2 (2020): 133–44. http://dx.doi.org/10.5937/matmor2002133c.

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The authors establish a set of two new relationships involving q-product identities, Ralpha, Rbeta, and Rm (m = 1, 2, 3, . . .) functions; and answer a open question of Srivastava et al. [18]. The present work is motivated and based upon recent findings of Chaudhary et al. [8].

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18

Wang, Chun, and Ae Ja Yee. "Truncated Jacobi triple product series." Journal of Combinatorial Theory, Series A 166 (August 2019): 382–92. http://dx.doi.org/10.1016/j.jcta.2019.03.003.

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19

Yee, Ae Ja. "A truncated Jacobi triple product theorem." Journal of Combinatorial Theory, Series A 130 (February 2015): 1–14. http://dx.doi.org/10.1016/j.jcta.2014.10.005.

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20

Girstmair, Kurt. "Triple product identities for the Jacobi symbol." Expositiones Mathematicae 19, no. 2 (2001): 179–85. http://dx.doi.org/10.1016/s0723-0869(01)80028-1.

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21

Balázs, Márton, and Ross Bowen. "Product blocking measures and a particle system proof of the Jacobi triple product." Annales de l'Institut Henri Poincaré, Probabilités et Statistiques 54, no. 1 (February 2018): 514–28. http://dx.doi.org/10.1214/16-aihp813.

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22

Kaneko, Jyoichi. "A Triple Product Identity for Macdonald Polynomials." Journal of Mathematical Analysis and Applications 200, no. 2 (June 1996): 355–67. http://dx.doi.org/10.1006/jmaa.1996.0210.

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23

Milne, S. C. "A triple product identity for Schur functions." Journal of Mathematical Analysis and Applications 160, no. 2 (September 1991): 446–58. http://dx.doi.org/10.1016/0022-247x(91)90317-s.

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24

Kolitsch, Louis W., and Stephanie Kolitsch. "A combinatorial proof of Jacobi’s triple product identity." Ramanujan Journal 45, no. 2 (January 17, 2017): 483–89. http://dx.doi.org/10.1007/s11139-016-9854-5.

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25

Chan, Heng Huat. "Triple product identity, Quintuple product identity and Ramanujan's differential equations for the classical Eisenstein series." Proceedings of the American Mathematical Society 135, no. 07 (July 1, 2007): 1987–93. http://dx.doi.org/10.1090/s0002-9939-07-08723-0.

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26

Hammond, Paul, Richard Lewis, and Zhi-Guo Liu. "Hirschhorn's identities." Bulletin of the Australian Mathematical Society 60, no. 1 (August 1999): 73–80. http://dx.doi.org/10.1017/s0004972700033347.

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We prove a general identity between power series and use this identity to give proofs of a number of identities proposed by M.D. Hirschhorn. We also use the identity to give proofs of a well-known result of Jacobi, the quintuple-product identity and Winquist's identity.

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27

Blanco-Chacón, Iván, and Michele Fornea. "TWISTED TRIPLE PRODUCT -ADIC L-FUNCTIONS AND HIRZEBRUCH–ZAGIER CYCLES." Journal of the Institute of Mathematics of Jussieu 19, no. 6 (February 20, 2019): 1947–92. http://dx.doi.org/10.1017/s1474748019000021.

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Let $L/F$ be a quadratic extension of totally real number fields. For any prime $p$ unramified in $L$, we construct a $p$-adic $L$-function interpolating the central values of the twisted triple product $L$-functions attached to a $p$-nearly ordinary family of unitary cuspidal automorphic representations of $\text{Res}_{L\times F/F}(\text{GL}_{2})$. Furthermore, when $L/\mathbb{Q}$ is a real quadratic number field and $p$ is a split prime, we prove a $p$-adic Gross–Zagier formula relating the values of the $p$-adic $L$-function outside the range of interpolation to the syntomic Abel–Jacobi image of generalized Hirzebruch–Zagier cycles.

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28

Gritsenko, Valery, and Haowu Wang. "Powers of Jacobi triple product, Cohen’s numbers and the Ramanujan $${\Delta }$$ Δ -function." European Journal of Mathematics 4, no. 2 (October 10, 2017): 561–84. http://dx.doi.org/10.1007/s40879-017-0185-x.

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29

Josuat-Vergès, Matthieu, and Jang Soo Kim. "Touchard–Riordan formulas, T-fractions, and Jacobi’s triple product identity." Ramanujan Journal 30, no. 3 (September 13, 2012): 341–78. http://dx.doi.org/10.1007/s11139-012-9403-9.

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30

Ward, A. J. B. "Classroom note: A teaching method for the vector triple product identity." International Journal of Mathematical Education in Science and Technology 35, no. 2 (March 2004): 299–300. http://dx.doi.org/10.1080/00207390310001638403.

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31

Srivastava, H. M., M. P. Chaudhary, and S. Chaudhary. "A Family of Theta-Function Identities Related to Jacobi’s Triple-Product Identity." Russian Journal of Mathematical Physics 27, no. 1 (January 2020): 139–44. http://dx.doi.org/10.1134/s1061920820010148.

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32

Schneider, Robert. "Jacobi’s triple product, mock theta functions, unimodal sequences and the q-bracket." International Journal of Number Theory 14, no. 07 (July 23, 2018): 1961–81. http://dx.doi.org/10.1142/s1793042118501178.

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In Ramanujan’s final letter to Hardy, he listed examples of a strange new class of infinite series he called “mock theta functions”. It turns out all of these examples are essentially specializations of a so-called universal mock theta function [Formula: see text] of Gordon–McIntosh. Here we show that [Formula: see text] arises naturally from the reciprocal of the classical Jacobi triple product—and is intimately tied to rank generating functions for unimodal sequences, which are connected to mock modular and quantum modular forms—under the action of an operator related to statistical physics and partition theory, the [Formula: see text]-bracket of Bloch–Okounkov. Second, we find [Formula: see text] to extend in [Formula: see text] to the entire complex plane minus the unit circle, and give a finite formula for this universal mock theta function at roots of unity, that is simple by comparison to other such formulas in the literature; we also indicate similar formulas for other [Formula: see text]-hypergeometric series. Finally, we look at interesting “quantum” behaviors of mock theta functions inside, outside, and on the unit circle.

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33

Srivastava, Hari Mohan, Rekha Srivastava, Mahendra Pal Chaudhary, and Salah Uddin. "A Family of Theta-Function Identities Based upon Combinatorial Partition Identities Related to Jacobi’s Triple-Product Identity." Mathematics 8, no. 6 (June 5, 2020): 918. http://dx.doi.org/10.3390/math8060918.

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The authors establish a set of six new theta-function identities involving multivariable R-functions which are based upon a number of q-product identities and Jacobi’s celebrated triple-product identity. These theta-function identities depict the inter-relationships that exist among theta-function identities and combinatorial partition-theoretic identities. Here, in this paper, we consider and relate the multivariable R-functions to several interesting q-identities such as (for example) a number of q-product identities and Jacobi’s celebrated triple-product identity. Various recent developments on the subject-matter of this article as well as some of its potential application areas are also briefly indicated. Finally, we choose to further emphasize upon some close connections with combinatorial partition-theoretic identities and present a presumably open problem.

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34

ZOTOV, A. "ON RELATION BETWEEN WEYL AND KONTSEVICH QUANTUM PRODUCTS: DIRECT EVALUATION UP TO THE ℏ3-ORDER." Modern Physics Letters A 16, no. 10 (March 28, 2001): 615–25. http://dx.doi.org/10.1142/s0217732301003693.

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In his celebrated paper Kontsevich has proved a theorem which manifestly gives a quantum product (deformation quantization formula) and states that changing coordinates leads to gauge equivalent star products. To illuminate his procedure, we make an arbitrary change of coordinates in the Weyl (Moyal) product and obtain the deformation quantization formula up to the third order. In this way, the Poisson bivector is shown to depend on ℏ and not to satisfy the Jacobi identity. It is also shown that the values of coefficients in the formula obtained follow from associativity of the star product.

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35

Bhargava, S., Chandrashekar Adiga, and M. S. Mahadeva Naika. "Ramanujan's remarkable summation formula as a $2$-papameter generalization of the quintuple product identity." Tamkang Journal of Mathematics 33, no. 3 (September 30, 2002): 285–88. http://dx.doi.org/10.5556/j.tkjm.33.2002.285-288.

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It is well known that `Ramanujan's remarkable summation formula' unifies and generalizes the $q$-binomial theorem and the triple product identity and has numerous applications. In this note we will demonstrate how, after a suitable transformation of the series side, it can be looked upon as a $2$-parameter generalization of the quintuple product identity also.

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36

BREMNER, MURRAY R., and JUANA SÁNCHEZ-ORTEGA. "LEIBNIZ TRIPLE SYSTEMS." Communications in Contemporary Mathematics 16, no. 01 (January 21, 2014): 1350051. http://dx.doi.org/10.1142/s021919971350051x.

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We define Leibniz triple systems in a functorial manner using the algorithm of Kolesnikov and Pozhidaev which converts identities for algebras into identities for dialgebras; this algorithm is a concrete realization of the white Manin product introduced by Vallette by the permutad Perm introduced by Chapoton. We verify that Leibniz triple systems are natural analogues of Lie triple systems by showing that both the iterated bracket in a Leibniz algebra and the permuted associator in a Jordan dialgebra satisfy the defining identities for Leibniz triple systems. We construct the universal Leibniz envelopes of Leibniz triple systems and prove that every identity satisfied by the iterated bracket in a Leibniz algebra is a consequence of the defining identities for Leibniz triple systems. In the last section, we present some examples of two-dimensional Leibniz triple systems and their universal Leibniz envelopes.

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37

Chaudhary, Mahendra Pal, Sangeeta Chaudhary, and Junesang Choi. "TWO IDENTITIES DERIVABLE FROM THE JACOBI’S TRIPLE-PRODUCT IDENTITY AND THE RAMANUJAN CONTINUED FRACTION." Far East Journal of Mathematical Sciences (FJMS) 102, no. 1 (June 13, 2017): 243–49. http://dx.doi.org/10.17654/ms102010243.

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38

Warnaar, S. Ole. "q-Hypergeometric Proofs of Polynomial Analogues of the Triple Product Identity, Lebesgue?s Identity and Euler?s Pentagonal Number Theorem." Ramanujan Journal 8, no. 4 (January 2005): 467–74. http://dx.doi.org/10.1007/s11139-005-0275-0.

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39

Apagodu, Moa. "New series representations for Jacobiʼs triple product identity and more via the q-Markov method." Advances in Applied Mathematics 48, no. 1 (January 2012): 25–36. http://dx.doi.org/10.1016/j.aam.2011.05.002.

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40

Ostrovsky, Dmitry. "On Barnes beta distributions, Selberg integral and Riemann xi." Forum Mathematicum 28, no. 1 (January 1, 2016): 1–23. http://dx.doi.org/10.1515/forum-2013-0149.

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AbstractThe theory of Barnes beta probability distributions is advanced and related to the Riemann xi function. The scaling invariance, multiplication formula, and Shintani factorization of Barnes multiple gamma functions are reviewed using the approach of Ruijsenaars and shown to imply novel properties of Barnes beta distributions. The applications are given to the meromorphic extension of the Selberg integral as a function of its dimension and the scaling invariance of the underlying probability distribution. This probability distribution in the critical case is described and conjectured to be the distribution of the derivative martingale. The Jacobi triple product is interpreted probabilistically resulting in an approximation of the Riemann xi function by the Mellin transform of the logarithm of a limit of Barnes beta distributions.

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41

Abramov, Viktor. "Matrix 3-Lie superalgebras and BRST supersymmetry." International Journal of Geometric Methods in Modern Physics 14, no. 11 (October 23, 2017): 1750160. http://dx.doi.org/10.1142/s0219887817501602.

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Given a matrix Lie algebra one can construct the 3-Lie algebra by means of the trace of a matrix. In the present paper, we show that this approach can be extended to the infinite-dimensional Lie algebra of vector fields on a manifold if instead of the trace of a matrix we consider a differential 1-form which satisfies certain conditions. Then we show that the same approach can be extended to matrix Lie superalgebras [Formula: see text] if instead of the trace of a matrix we make use of the supertrace of a matrix. It is proved that a graded triple commutator of matrices constructed with the help of the graded commutator and the supertrace satisfies a graded ternary Filippov–Jacobi identity. In two particular cases of [Formula: see text] and [Formula: see text], we show that the Pauli and Dirac matrices generate the matrix 3-Lie superalgebras, and we find the non-trivial graded triple commutators of these algebras. We propose a Clifford algebra approach to 3-Lie superalgebras induced by Lie superalgebras. We also discuss an application of matrix 3-Lie superalgebras in BRST-formalism.

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42

Chaudhary, M. P. "Relations between Rα, Rβ and Rm functions related to Jacobi’s triple-product identity and the family of theta-function identities." Notes on Number Theory and Discrete Mathematics 27, no. 2 (June 2021): 1–11. http://dx.doi.org/10.7546/nntdm.2021.27.2.1-11.

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In this paper, the author establishes a set of three new theta-function identities involving Rα, Rβ and Rm functions which are based upon a number of q-product identities and Jacobi’s celebrated triple-product identity. These theta-function identities depict the inter-relationships that exist among theta-function identities and combinatorial partition-theoretic identities. Here, in this paper we answer a open question of Srivastava et al [33], and established relations in terms of Rα, Rβ and Rm (for m = 1, 2, 3), and q-products identities. Finally, we choose to further emphasize upon some close connections with combinatorial partition-theoretic identities.

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43

Abramov, Viktor. "Nambu–Poisson bracket on superspace." International Journal of Geometric Methods in Modern Physics 15, no. 11 (November 2018): 1850190. http://dx.doi.org/10.1142/s0219887818501906.

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We propose an extension of [Formula: see text]-ary Nambu–Poisson bracket to superspace [Formula: see text] and construct by means of superdeterminant a family of Nambu–Poisson algebras of even degree functions, where the parameter of this family is an invertible transformation of Grassmann coordinates in superspace [Formula: see text]. We prove in the case of the superspaces [Formula: see text] and [Formula: see text] that our [Formula: see text]-ary bracket, defined with the help of superdeterminant, satisfies the conditions for [Formula: see text]-ary Nambu–Poisson bracket, i.e. it is totally skew-symmetric and it satisfies the Leibniz rule and the Filippov–Jacobi identity (fundamental identity). We study the structure of [Formula: see text]-ary bracket defined with the help of superdeterminant in the case of superspace [Formula: see text] and show that it is the sum of usual [Formula: see text]-ary Nambu–Poisson bracket and a new [Formula: see text]-ary bracket, which we call [Formula: see text]-bracket, where [Formula: see text] is the product of two odd degree smooth functions.

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44

Hillman, J. A., and C. Kearton. "Algebraic Invariants of Simple 4-Knots." Journal of Knot Theory and Its Ramifications 06, no. 03 (June 1997): 307–18. http://dx.doi.org/10.1142/s0218216597000212.

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We propose as an algebraic invariant for a simple 4-knot K with exterior X the triple (L, η, [λ]), where L = Z ⊕ π2(X)⊕π3(X) is a commutative graded ring with unit whose multiplication in positive degrees is determined by Whitehead product, η is composition with the Hopf map and [λ] is the orbit of the homotopy class of the longitude in π4(X) under the group of self homotopy equivalences of the universal covering space X′ which induce the identity on L. If K is fibred these invariants determine the fibre, and the natural Z[t,t-1]-module structures on the homotopy groups capture part of the monodromy. Every such triple with L finitely generated as an abelian group (and satisfying the order obviously necessary conditions) may be realized by some fibred simple 4-knot. In certain cases we can show that the triple determines the knot up to a finite ambiguity.

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45

Singer-Krüger, B., H. Stenmark, A. Düsterhöft, P. Philippsen, J. S. Yoo, D. Gallwitz, and M. Zerial. "Role of three rab5-like GTPases, Ypt51p, Ypt52p, and Ypt53p, in the endocytic and vacuolar protein sorting pathways of yeast." Journal of Cell Biology 125, no. 2 (April 15, 1994): 283–98. http://dx.doi.org/10.1083/jcb.125.2.283.

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The small GTPase rab5 has been shown to represent a key regulator in the endocytic pathway of mammalian cells. Using a PCR approach to identify rab5 homologs in Saccharomyces cerevisiae, two genes encoding proteins with 54 and 52% identity to rab5, YPT51 and YPT53 have been identified. Sequencing of the yeast chromosome XI has revealed a third rab5-like gene, YPT52, whose protein product exhibits a similar identity to rab5 and the other two YPT gene products. In addition to the high degree of identity/homology shared between rab5 and Ypt51p, Ypt52p, and Ypt53p, evidence for functional homology between the mammalian and yeast proteins is provided by phenotypic characterization of single, double, and triple deletion mutants. Endocytic delivery to the vacuole of two markers, lucifer yellow CH (LY) and alpha-factor, was inhibited in delta ypt51 mutants and aggravated in the double ypt51ypt52 and triple ypt51ypt52ypt53 mutants, suggesting a requirement for these small GTPases in endocytic membrane traffic. In addition to these defects, the here described ypt mutants displayed a number of other phenotypes reminiscent of some vacuolar protein sorting (vps) mutants, including a differential delay in growth and vacuolar protein maturation, partial missorting of a soluble vacuolar hydrolase, and alterations in vacuole acidification and morphology. In fact, vps21 represents a mutant allele of YPT51 (Emr, S., personal communication). Altogether, these data suggest that Ypt51p, Ypt52p, and Ypt53p are required for transport in the endocytic pathway and for correct sorting of vacuolar hydrolases suggesting a possible intersection of the endocytic with the vacuolar sorting pathway.

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46

HIRSCHHORN, MICHAEL D., and JAMES A. SELLERS. "ELEMENTARY PROOFS OF PARITY RESULTS FOR 5-REGULAR PARTITIONS." Bulletin of the Australian Mathematical Society 81, no. 1 (July 2, 2009): 58–63. http://dx.doi.org/10.1017/s0004972709000525.

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AbstractIn a recent paper, Calkin et al. [N. Calkin, N. Drake, K. James, S. Law, P. Lee, D. Penniston and J. Radder, ‘Divisibility properties of the 5-regular and 13-regular partition functions’, Integers8 (2008), #A60] used the theory of modular forms to examine 5-regular partitions modulo 2 and 13-regular partitions modulo 2 and 3; they obtained and conjectured various results. In this note, we use nothing more than Jacobi’s triple product identity to obtain results for 5-regular partitions that are stronger than those obtained by Calkin and his collaborators. We find infinitely many Ramanujan-type congruences for b5(n), and we prove the striking result that the number of 5-regular partitions of the number n is even for at least 75% of the positive integers n.

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47

Chistoserdova, Ludmila, Gregory J. Crowther, Julia A. Vorholt, Elizabeth Skovran, Jean-Charles Portais, and Mary E. Lidstrom. "Identification of a Fourth Formate Dehydrogenase in Methylobacterium extorquens AM1 and Confirmation of the Essential Role of Formate Oxidation in Methylotrophy." Journal of Bacteriology 189, no. 24 (October 5, 2007): 9076–81. http://dx.doi.org/10.1128/jb.01229-07.

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ABSTRACT A mutant of Methylobacterium extorquens AM1 with lesions in genes for three formate dehydrogenase (FDH) enzymes was previously described by us (L. Chistoserdova, M. Laukel, J.-C. Portais, J. A. Vorholt, and M. E. Lidstrom, J. Bacteriol. 186:22-28, 2004). This mutant had lost its ability to grow on formate but still maintained the ability to grow on methanol. In this work, we further investigated the phenotype of this mutant. Nuclear magnetic resonance experiments with [13C]formate, as well as 14C-labeling experiments, demonstrated production of labeled CO2 in the mutant, pointing to the presence of an additional enzyme or a pathway for formate oxidation. The tungsten-sensitive phenotype of the mutant suggested the involvement of a molybdenum-dependent enzyme. Whole-genome array experiments were conducted to test for genes overexpressed in the triple-FDH mutant compared to the wild type, and a gene (fdh4A) was identified whose translated product carried similarity to an uncharacterized putative molybdopterin-binding oxidoreductase-like protein sharing relatively low similarity with known formate dehydrogenase alpha subunits. Mutation of this gene in the triple-FDH mutant background resulted in a methanol-negative phenotype. When the gene was deleted in the wild-type background, the mutant revealed diminished growth on methanol with accumulation of high levels of formate in the medium, pointing to an important role of FDH4 in methanol metabolism. The identity of FDH4 as a novel FDH was also confirmed by labeling experiments that revealed strongly reduced CO2 formation in growing cultures. Mutation of a small open reading frame (fdh4B) downstream of fdh4A resulted in mutant phenotypes similar to the phenotypes of fdh4A mutants, suggesting that fdh4B is also involved in formate oxidation.

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Burbank, Stephen, and Sean Farhang. "Politics, Identity, and Class Certification on the U.S. Courts of Appeals." Michigan Law Review, no. 119.2 (2020): 231. http://dx.doi.org/10.36644/mlr.119.2.politics.

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This Article draws on novel data and presents the results of the first empirical analysis of how potentially salient characteristics of Court of Appeals judges influence class certification under Rule 23 of the Federal Rules of Civil Procedure. We find that the ideological composition of the panel (measured by the party of the appointing president) has a very strong association with certification outcomes, with all-Democratic panels having dramatically higher rates of procertification outcomes than all-Republican panels—nearly triple in about the past twenty years. We also find that the presence of one African American on a panel, and the presence of two women (but not one), is associated with procertification outcomes. Our results show that, contrary to conventional wisdom in scholarship on diversity on the Courts of Appeals, the impact of diversity extends beyond conceptions of “women’s issues” or “minority issues.” The consequences of gender and racial diversity on the bench, through application and elaboration of certification law, radiate widely across the legal landscape, influencing implementation in such areas as consumer, securities, labor and employment, antitrust, insurance, product liability, environmental, and many other areas of law. In considering possible explanations for our findings on the procertification preferences of women and African Americans, we note that class action doctrine, as transsubstantive procedural law, traverses many policy areas. As strategic actors, it would be rational for judges to take into consideration how class-certification doctrine in a case that does not implicate issues on which they have distinctive preferences might affect certification in cases that do. Alternatively, or in addition, our results may be the first evidence that transsubstantive procedural law affecting access to justice is itself a policy domain in which women and African Americans have distinctive preferences. In either case, the results highlight the importance of exploring the effects of diversity on transsubstantive procedural law more generally. Our findings on gender panel effects in particular are novel in the literature on panel effects and the literature on gender and judging. Past work focusing on substantive antidiscrimination law found that one woman can influence the votes of men in the majority (mirroring what we find with respect to African Americans in class-certification decisions). These results allowed for optimism that the panel structure—which threatens to dilute the influence of underrepresented groups on the bench because they are infrequently in the panel majority—actually facilitates minority influence, whether through deliberation, cue taking, bargaining, or some other mechanism. Our gender results are quite different and normatively troubling. We observe that women have substantially more procertification preferences based on outcomes when they are in the majority. However, panels with one woman are not more likely to yield procertification outcomes. Panels with women in the majority occur at sharply lower rates than women’s percentage of judgeships, and thus certification doctrine underrepresents their preferences relative to their share of judgeships and overrepresents those of male judges.

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Benton, B. M., J. H. Zang, and J. Thorner. "A novel FK506- and rapamycin-binding protein (FPR3 gene product) in the yeast Saccharomyces cerevisiae is a proline rotamase localized to the nucleolus." Journal of Cell Biology 127, no. 3 (November 1, 1994): 623–39. http://dx.doi.org/10.1083/jcb.127.3.623.

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The gene (FPR3) encoding a novel type of peptidylpropyl-cis-trans-isomerase (PPIase) was isolated during a search for previously unidentified nuclear proteins in Saccharomyces cerevisiae. PPIases are thought to act in conjunction with protein chaperones because they accelerate the rate of conformational interconversions around proline residues in polypeptides. The FPR3 gene product (Fpr3) is 413 amino acids long. The 111 COOH-terminal residues of Fpr3 share greater than 40% amino acid identity with a particular class of PPIases, termed FK506-binding proteins (FKBPs) because they are the intracellular receptors for two immunosuppressive compounds, rapamycin and FK506. When expressed in and purified from Escherichia coli, both full-length Fpr3 and its isolated COOH-terminal domain exhibit readily detectable PPIase activity. Both fpr3 delta null mutants and cells expressing FPR3 from its own promoter on a multicopy plasmid have no discernible growth phenotype and do not display any alteration in sensitivity to the growth-inhibitory effects of either FK506 or rapamycin. In S. cerevisiae, the gene for a 112-residue cytosolic FKBP (FPR1) and the gene for a 135-residue ER-associated FKBP (FPR2) have been described before. Even fpr1 fpr2 fpr3 triple mutants are viable. However, in cells carrying an fpr1 delta mutation (which confers resistance to rapamycin), overexpression from the GAL1 promoter of the C-terminal domain of Fpr3, but not full-length Fpr3, restored sensitivity to rapamycin. Conversely, overproduction from the GAL1 promoter of full-length Fpr3, but not its COOH-terminal domain, is growth inhibitory in both normal cells and fpr1 delta mutants. In fpr1 delta cells, the toxic effect of Fpr3 overproduction can be reversed by rapamycin. Overproduction of the NH2-terminal domain of Fpr3 is also growth inhibitory in normal cells and fpr1 delta mutants, but this toxicity is not ameliorated in fpr1 delta cells by rapamycin. The NH2-terminal domain of Fpr3 contains long stretches of acidic residues alternating with blocks of basic residues, a structure that resembles sequences found in nucleolar proteins, including S. cerevisiae NSR1 and mammalian nucleolin. Indirect immunofluorescence with polyclonal antibodies raised against either the NH2- or the COOH-terminal segments of Fpr3 expressed in E. coli demonstrated that Fpr3 is located exclusively in the nucleolus.

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Delatte, H., B. Reynaud, J. M. Lett, M. Peterschmitt, M. Granier, J. Ravololonandrianina, and W. R. Goldbach. "First Molecular Identification of a Begomovirus Isolated from Tomato in Madagascar." Plant Disease 86, no. 12 (December 2002): 1404. http://dx.doi.org/10.1094/pdis.2002.86.12.1404c.

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In April 2001, reduced leaf size, leaf curling, yellowing symptoms, and reduced yield were observed in tomato plants in the southwestern (Toliary, Morondava, Miandrivazo) and northern (Antsiranana) regions of Madagascar. Symptoms were similar to those caused by Tomato yellow leaf curl virus (TYLCV, genus Begomovirus, family Geminiviridae). Large populations of Bemisia tabaci (Gennadius) were observed colonizing tomato, other crops, and weeds. Leaf samples were collected from tomato plants from 14 sites located in northern, central, and southern Madagascar. Two plant samples collected near Antsiranana, one sample near Morondava, and one sample near Toliary were positive in triple-antibody sandwich enzyme-linked immunosorbent assay using a begomovirus-specific antibody purchased from ADGEN (Nellies Gates, Auchincruive, Scotland, UK). A 500-bp product was amplified and cloned (2) from two leaf samples collected near Toliary and one near Morondava using a pair of degenerate primers that are expected to amplify a region of the A component of begomoviruses between the intergenic conserved nonanucleotide sequence and the first 200 nucleotides of the coat protein ORF. The sequences corresponding to the two Toliary samples (GenBank Accession Nos. AJ422123 and AJ422124) and the Morondava sample (GenBank No. AJ422125) showed the most significant alignments (NCBI, BLAST) with begomoviruses, Tobacco leaf curl virus from Zimbabwe (GenBank Accession No. AF 350330) and Tomato leaf curl virus from Tanzania (GenBank Accession No. U73498) with 76 to 77% nucleotide identity (Clustal method, MegAlign, DNASTAR, London) and South African cassava mosaic viruses (SACMV GenBank Accession Nos. AJ422132 and AF155806) and East African cassava mosaic viruses from Malawi (GenBank Accession Nos. AJ006459 and AJ006460) with 74 to 75.5% nucleotide identity. The low nucleotide identity suggests that the begomovirus isolated from tomato in Madagascar is a new species. Since the core region of the coat protein gene is a molecular marker for provisional classification of begomoviruses (1), this region was amplified for the Morondava isolate with degenerate primers. The 519nt core fragment obtained showed the most significant alignments with SACMV (GenBank Accession No. AF329227), Cassava geminivirus from Mozambique (GenBank Accession No. AF329240), and with TYLCV (GenBank Accession Nos. AB014346 and AF105975) with 81 to 82% nucleotide identity. According to the current taxonomic criteria (4), the begomovirus from Madagascar is a new one that is related to begomoviruses from the southern part of Africa and to TYLCV and is provisionally named Tomato yellow leaf curl Morondava virus (TYLCMV). Tomato yellow leaf curl disease was previously described in Madagascar by Reckhaus (3) who presumed that it was caused by TYLCV. Although symptoms in the tomato plant from which TYLCMV was isolated were similar to those induced by TYLCV, TYLCV was not detected in our samples. References: (1) J. K. Brown et al. Arch. Virol. 146:1581, 2001 (2) M. Peterschmitt et al. Plant Dis. 83:303, 1999. (3) P. Reckhaus, Maladies et ravageurs des cultures maraîchères: A l'exemple de Madagascar. GTZ, Weikersem, 1997. (4) M. H. V. van Regenmortel et al. Virus Taxonomy. Seventh Rep. Int. Comm. Taxon. Viruses. Academic Press, San Diego, 2000.

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