Journal articles: 'Modular representation'

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Snaider, Javier, and Stan Franklin. "Modular Composite Representation." Cognitive Computation 6, no. 3 (January 23, 2014): 510–27. http://dx.doi.org/10.1007/s12559-013-9243-y.

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Voskresenskaya, G. V. "Modular forms and group representation." Mathematical Notes 52, no. 1 (July 1992): 649–54. http://dx.doi.org/10.1007/bf01247643.

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Chebolu, Sunil K., J. Daniel Christensen, and Ján Mináč. "Ghosts in modular representation theory." Advances in Mathematics 217, no. 6 (April 2008): 2782–99. http://dx.doi.org/10.1016/j.aim.2007.11.008.

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BRUNETTI, R., D. GUIDO, and R. LONGO. "MODULAR LOCALIZATION AND WIGNER PARTICLES." Reviews in Mathematical Physics 14, no. 07n08 (July 2002): 759–85. http://dx.doi.org/10.1142/s0129055x02001387.

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We propose a framework for the free field construction of algebras of local observables which uses as an input the Bisognano–Wichmann relations and a representation of the Poincaré group on the one-particle Hilbert space. The abstract real Hilbert subspace version of the Tomita–Takesaki theory enables us to bypass some limitations of the Wigner formalism by introducing an intrinsic spacetime localization. Our approach works also for continuous spin representations to which we associate a net of von Neumann algebras on spacelike cones with the Reeh–Schlieder property. The positivity of the energy in the representation turns out to be equivalent to the isotony of the net, in the spirit of Borchers theorem. Our procedure extends to other spacetimes homogeneous under a group of geometric transformations as in the case of conformal symmetries and of de Sitter spacetime.

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CHEN, IMIN, IAN KIMING, and GABOR WIESE. "ON MODULAR GALOIS REPRESENTATIONS MODULO PRIME POWERS." International Journal of Number Theory 09, no. 01 (November 13, 2012): 91–113. http://dx.doi.org/10.1142/s1793042112501254.

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We study modular Galois representations mod pm. We show that there are three progressively weaker notions of modularity for a Galois representation mod pm: We have named these "strongly", "weakly", and "dc-weakly" modular. Here, "dc" stands for "divided congruence" in the sense of Katz and Hida. These notions of modularity are relative to a fixed level M. Using results of Hida we display a level-lowering result ("stripping-of-powers of p away from the level"): A mod pm strongly modular representation of some level Npr is always dc-weakly modular of level N (here, N is a natural number not divisible by p). We also study eigenforms mod pm corresponding to the above three notions. Assuming residual irreducibility, we utilize a theorem of Carayol to show that one can attach a Galois representation mod pm to any "dc-weak" eigenform, and hence to any eigenform mod pm in any of the three senses. We show that the three notions of modularity coincide when m = 1 (as well as in other particular cases), but not in general.

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Kohls, Martin, and Müfi̇t Sezer. "Degree of reductivity of a modular representation." Communications in Contemporary Mathematics 19, no. 03 (April 5, 2017): 1650023. http://dx.doi.org/10.1142/s0219199716500231.

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For a finite-dimensional representation [Formula: see text] of a group [Formula: see text] over a field [Formula: see text], the degree of reductivity [Formula: see text] is the smallest degree [Formula: see text] such that every nonzero fixed point [Formula: see text] can be separated from zero by a homogeneous invariant of degree at most [Formula: see text]. We compute [Formula: see text] explicitly for several classes of modular groups and representations. We also demonstrate that the maximal size of a cyclic subgroup is a sharp lower bound for this number in the case of modular abelian [Formula: see text]-groups.

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Le Bruyn, Lieven. "Bulk irreducibles of the modular group." Journal of Algebra and Its Applications 15, no. 01 (September 7, 2015): 1650006. http://dx.doi.org/10.1142/s0219498816500067.

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As the 3-string braid group B3 and the modular group Γ are both of wild representation type one cannot expect a full classification of all their finite dimensional simple representations. Still, one can aim to describe 'most' irreducible representations by constructing for each d-dimensional irreducible component X of the variety iss n(Γ) classifying the isomorphism classes of semi-simple n-dimensional representations of Γ an explicit minimal étale rational map 𝔸d → X having a Zariski dense image. Such rational dense parametrizations were obtained for all components when n < 12 in [5]. The aim of the present paper is to establish such parametrizations for all finite dimensions n.

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PATTANAYAK, S. K. "ON SOME STANDARD GRADED ALGEBRAS IN MODULAR INVARIANT THEORY." Journal of Algebra and Its Applications 13, no. 01 (August 20, 2013): 1350080. http://dx.doi.org/10.1142/s0219498813500801.

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For a finite-dimensional representation V of a finite group G over a field K we denote the graded algebra R ≔ ⨁d≥0 Rd; where Rd ≔ ( Sym d∣G∣V*)G. We study the standardness of R for the representations [Formula: see text], [Formula: see text], and [Formula: see text], where Vn denote the n-dimensional indecomposable representation of the cyclic group Cp over the Galois field 𝔽p, for a prime p. We also prove the standardness for the defining representation of all finite linear groups with polynomial rings of invariants. This is motivated by a question of projective normality raised in [S. S. Kannan, S. K. Pattanayak and P. Sardar, Projective normality of finite groups quotients, Proc. Amer. Math. Soc.137(3) (2009) 863–867].

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Guzhov, Vladimir I., Ilya O. Marchenko, Ekaterina E. Trubilina, and Dmitry S. Khaidukov. "Comparison of numbers and analysis of overflow in modular arithmetic." Analysis and data processing systems, no. 3 (September 30, 2021): 75–86. http://dx.doi.org/10.17212/2782-2001-2021-3-75-86.

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The method of modular arithmetic consists in operating not with a number, but with its remainders after division by some integers. In the modular number system or the number system in the residual classes, a multi-bit integer in the positional number system is represented as a sequence of several positional numbers. These numbers are the remainders (residues) of dividing the original number into some modules that are mutually prime integers. The advantage of the modular representation is that it is very simple to perform addition, subtraction and multiplication operations. In parallel execution of operations, the use of modular arithmetic can significantly reduce the computation time. However, there are drawbacks to modular representation that limit its use. These include a slow conversion of numbers from modular to positional representation; the complexity of comparing numbers in modular representation; the difficulty in performing the division operation; and the difficulty of determining the presence of an overflow. The use of modular arithmetic is justified if there are fast algorithms for calculating a number from a set of remainders. This article describes a fast algorithm for converting numbers from modular representation to positional representation based on a geometric approach. The review is carried out for the case of a comparison system with two modules. It is also shown that as a result of increasing numbers in positional calculus, they successively change in a spiral on the surface of a two-dimensional torus. Based on this approach, a fast algorithm for comparing numbers and an algorithm for detecting an overflow during addition and multiplication of numbers in modular representation were developed. Consideration for the multidimensional case is possible when analyzing a multidimensional torus and studying the behavior of the turns on its surface.

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Friedlander, Eric M., and Brian J. Parshall. "Modular Representation Theory of Lie Algebras." American Journal of Mathematics 110, no. 6 (December 1988): 1055. http://dx.doi.org/10.2307/2374686.

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Bantay, Peter, and Terry Gannon. "Conformal characters and the modular representation." Journal of High Energy Physics 2006, no. 02 (February 2, 2006): 005. http://dx.doi.org/10.1088/1126-6708/2006/02/005.

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Hanaki, Akihide, Yasuaki Miyazaki, and Osamu Shimabukuro. "Modular representation theory of BIB designs." Linear Algebra and its Applications 514 (February 2017): 174–97. http://dx.doi.org/10.1016/j.laa.2016.10.030.

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Watanabe, Chihiro, Kaoru Hiramatsu, and Kunio Kashino. "Modular representation of layered neural networks." Neural Networks 97 (January 2018): 62–73. http://dx.doi.org/10.1016/j.neunet.2017.09.017.

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14

Willems, Wolfgang. "p∗-Theory and modular representation theory." Journal of Algebra 104, no. 1 (November 1986): 135–40. http://dx.doi.org/10.1016/0021-8693(86)90243-7.

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15

Assadi, Amir H. "Permutation complexes and modular representation theory." Journal of Algebra 144, no. 2 (December 1991): 467–95. http://dx.doi.org/10.1016/0021-8693(91)90117-q.

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Di Vincenzo, O. M., and A. Giambruno. "Modular representation theory and pi-algebras." Communications in Algebra 16, no. 10 (January 1988): 2043–67. http://dx.doi.org/10.1080/00927878808823677.

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Davida, George I., and Bruce Litow. "Fast Parallel Arithmetic via Modular Representation." SIAM Journal on Computing 20, no. 4 (August 1991): 756–65. http://dx.doi.org/10.1137/0220048.

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Franc, Cameron, and Geoffrey Mason. "Constructions of vector-valued modular forms of rank four and level one." International Journal of Number Theory 16, no. 05 (February 13, 2020): 1111–52. http://dx.doi.org/10.1142/s1793042120500578.

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This paper studies modular forms of rank four and level one. There are two possibilities for the isomorphism type of the space of modular forms that can arise from an irreducible representation of the modular group of rank four, and we describe when each case occurs for general choices of exponents for the [Formula: see text]-matrix. In the remaining sections we describe how to write down the corresponding differential equations satisfied by minimal weight forms, and how to use these minimal weight forms to describe the entire graded module of holomorphic modular forms. Unfortunately, the differential equations that arise can only be solved recursively in general. We conclude the paper by studying the cases of tensor products of two-dimensional representations, symmetric cubes of two-dimensional representations, and inductions of two-dimensional representations of the subgroup of the modular group of index two. In these cases, the differential equations satisfied by minimal weight forms can be solved exactly.

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Barnet-Lamb, Thomas, Toby Gee, and David Geraghty. "Serre weights for U ⁢ ( n ) {U(n)}." Journal für die reine und angewandte Mathematik (Crelles Journal) 2018, no. 735 (February 1, 2018): 199–224. http://dx.doi.org/10.1515/crelle-2015-0015.

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Abstract We study the weight part of (a generalisation of) Serre’s conjecture for mod l Galois representations associated to automorphic representations on unitary groups of rank n for odd primes l. Given a modular Galois representation, we use automorphy lifting theorems to prove that it is modular in many other weights. We make no assumptions on the ramification or inertial degrees of l. We give an explicit strengthened result when {n=3} and l splits completely in the underlying CM field.

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MIKI, KEI. "THE REPRESENTATION THEORY OF THE SO(3) INVARIANT SUPERCONFORMAL ALGEBRA." International Journal of Modern Physics A 05, no. 07 (April 10, 1990): 1293–318. http://dx.doi.org/10.1142/s0217751x90000593.

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The representation theory of the SO(3) invariant superconformal algebra is discussed. The necessary and sufficient condition of unitarity, the Kac determinants, the character formulae for unitary representations and their modular transformation laws are obtained under some assumptions.

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Wang, Yilong. "Modular group representations associated to SO(p)2-TQFTS." Journal of Knot Theory and Its Ramifications 28, no. 05 (April 2019): 1950037. http://dx.doi.org/10.1142/s0218216519500378.

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In this paper, we prove that for any odd prime [Formula: see text] greater than 3, the modular group representation associated to the [Formula: see text]-topological quantum field theory can be defined over the ring of integers of a cyclotomic field. We will provide explicit integral bases. In the last section, we will relate these representations to the Weil representations over finite fields.

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EHOLZER, W. "FUSION ALGEBRAS INDUCED BY REPRESENTATIONS OF THE MODULAR GROUP." International Journal of Modern Physics A 08, no. 20 (August 10, 1993): 3495–507. http://dx.doi.org/10.1142/s0217751x93001405.

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Using the representation theory of the subgroups SL 2(ℤp) of the modular group we investigate the induced fusion algebras in some simple examples. Only some of these representations lead to "good" fusion algebras. Furthermore, the conformal dimensions and the central charge of the corresponding rational conformal field theories are calculated. Two series of representations which can be realized by unitary theories are presented. We show that most of the fusion algebras induced by admissible representations are realized in well-known rational models.

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Carroll, Susanne E. "Induction in a modular learner." Second Language Research 18, no. 3 (July 2002): 224–49. http://dx.doi.org/10.1191/0267658302sr205oa.

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This paper presents a theory of inductive learning (i-learning), a form of induction which is neither concept learning nor hypothesis-formation, but rather which takes place within the autonomous and modular representational systems (levels of representation) of the language faculty. The theory is called accordingly the Autonomous Induction Theory. Second language acquisition (SLA) is conceptualized in this theory as:• learning linguistic categories from universal and potentially innate featural primitives;• learning configurations of linguistic units; and• learning correspondences of configurations across the autonomous levels.Here I concentrate on the problem of constraining learning theories and argue that the Autonomous Induction Theory is constrained enough to be taken seriously as a plausible approach to explaining second language acquisition.

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Muñoz Castañeda, Ángel Luis, Noemí DeCastro-García, and Miguel V. Carriegos. "On the State Approach Representations of Convolutional Codes over Rings of Modular Integers." Mathematics 9, no. 22 (November 20, 2021): 2962. http://dx.doi.org/10.3390/math9222962.

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In this study, we prove the existence of minimal first-order representations for convolutional codes with the predictable degree property over principal ideal artinian rings. Further, we prove that any such first-order representation leads to an input/state/output representation of the code provided the base ring is local. When the base ring is a finite field, we recover the classical construction, studied in depth by J. Rosenthal and E. V. York. This allows us to construct observable convolutional codes over such rings in the same way as is carried out in classical convolutional coding theory. Furthermore, we prove the minimality of the obtained representations. This completes the study of the existence of input/state/output representations of convolutional codes over rings of modular integers.

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Jing, Chenchen, Yuwei Wu, Xiaoxun Zhang, Yunde Jia, and Qi Wu. "Overcoming Language Priors in VQA via Decomposed Linguistic Representations." Proceedings of the AAAI Conference on Artificial Intelligence 34, no. 07 (April 3, 2020): 11181–88. http://dx.doi.org/10.1609/aaai.v34i07.6776.

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Most existing Visual Question Answering (VQA) models overly rely on language priors between questions and answers. In this paper, we present a novel method of language attention-based VQA that learns decomposed linguistic representations of questions and utilizes the representations to infer answers for overcoming language priors. We introduce a modular language attention mechanism to parse a question into three phrase representations: type representation, object representation, and concept representation. We use the type representation to identify the question type and the possible answer set (yes/no or specific concepts such as colors or numbers), and the object representation to focus on the relevant region of an image. The concept representation is verified with the attended region to infer the final answer. The proposed method decouples the language-based concept discovery and vision-based concept verification in the process of answer inference to prevent language priors from dominating the answering process. Experiments on the VQA-CP dataset demonstrate the effectiveness of our method.

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Kalinichenko, A. V., I. N. Maliev, and M. A. Pliev. "Modular Sesquilinear Forms and Generalized Stinespring Representation." Russian Mathematics 62, no. 12 (December 2018): 42–49. http://dx.doi.org/10.3103/s1066369x18120034.

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Bessenrodt, C., C. Bowman, and L. Sutton. "Kronecker positivity and 2-modular representation theory." Transactions of the American Mathematical Society, Series B 8, no. 33 (December 10, 2021): 1024–55. http://dx.doi.org/10.1090/btran/70.

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This paper consists of two prongs. Firstly, we prove that any Specht module labelled by a 2-separated partition is semisimple and we completely determine its decomposition as a direct sum of graded simple modules. Secondly, we apply these results and other modular representation theoretic techniques on the study of Kronecker coefficients and hence verify Saxl’s conjecture for several large new families of partitions. In particular, we verify Saxl’s conjecture for all irreducible characters of S n \mathfrak {S}_n which are of 2-height zero.

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Libedinsky, Nicolas, and David Plaza. "Blob algebra approach to modular representation theory." Proceedings of the London Mathematical Society 121, no. 3 (May 2, 2020): 656–701. http://dx.doi.org/10.1112/plms.12333.

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Hamhalter, Jan. "A Representation of Finitely-Modular AC-lattices." Mathematische Nachrichten 147, no. 1 (1990): 281–84. http://dx.doi.org/10.1002/mana.19901470126.

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Yaraneri, Ergün. "A filtration of the modular representation functor." Journal of Algebra 318, no. 1 (December 2007): 140–79. http://dx.doi.org/10.1016/j.jalgebra.2007.06.030.

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Artamonov, V. A. "The magnus representation in congruence modular varieties." Siberian Mathematical Journal 38, no. 5 (October 1997): 842–59. http://dx.doi.org/10.1007/bf02673025.

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Avouris, N. M., R. N. Allan, and M. Lagoudakos. "Modular graphic representation of complex electrical networks." International Journal of Electrical Power & Energy Systems 15, no. 5 (January 1993): 323–32. http://dx.doi.org/10.1016/0142-0615(93)90054-q.

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Papalexandri, Katerina P., and Efstratios N. Pistikopoulos. "Generalized modular representation framework for process synthesis." AIChE Journal 42, no. 4 (April 1996): 1010–32. http://dx.doi.org/10.1002/aic.690420413.

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Saito, Takeshi. "Hilbert modular forms and p-adic Hodge theory." Compositio Mathematica 145, no. 5 (September 2009): 1081–113. http://dx.doi.org/10.1112/s0010437x09004175.

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AbstractFor the p-adic Galois representation associated to a Hilbert modular form, Carayol has shown that, under a certain assumption, its restriction to the local Galois group at a finite place not dividing p is compatible with the local Langlands correspondence. Under the same assumption, we show that the same is true for the places dividing p, in the sense of p-adic Hodge theory, as is shown for an elliptic modular form. We also prove that the monodromy-weight conjecture holds for such representations.

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Pollack, Aaron. "The spin -function on for Siegel modular forms." Compositio Mathematica 153, no. 7 (May 10, 2017): 1391–432. http://dx.doi.org/10.1112/s0010437x17007114.

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We give a Rankin–Selberg integral representation for the Spin (degree eight) $L$-function on $\operatorname{PGSp}_{6}$ that applies to the cuspidal automorphic representations associated to Siegel modular forms. If $\unicode[STIX]{x1D70B}$ corresponds to a level-one Siegel modular form $f$ of even weight, and if $f$ has a nonvanishing maximal Fourier coefficient (defined below), then we deduce the functional equation and finiteness of poles of the completed Spin $L$-function $\unicode[STIX]{x1D6EC}(\unicode[STIX]{x1D70B},\text{Spin},s)$ of $\unicode[STIX]{x1D70B}$.

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Klein, Sulamita, Noemi C. dos Santos, and Jayme L. Szwarcfiter. "A representation for the modular-pairs of a cograph by modular decomposition." Electronic Notes in Discrete Mathematics 18 (December 2004): 165–69. http://dx.doi.org/10.1016/j.endm.2004.06.026.

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Liu, De Fang, Hong Pan Wu, and Li Ying Li. "Research and Development of Intelligent Design System for Machine Tool Fixture Based on KBE." Advanced Materials Research 228-229 (April 2011): 17–22. http://dx.doi.org/10.4028/www.scientific.net/amr.228-229.17.

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In this paper, KBE and some key technologies, such as knowledge representation, knowledge reasoning and knowledge base design, are studied for solving the exiting design problems of modular machine tool fixtures. The methods of hybrid knowledge representation based on ontology and rule-based and case-based hybrid reasoning are introduced to intelligent design of modular machine tool fixtures. By researching the design flow of modular machine tool fixtures, a modular machine tool fixtures intelligent design system (MMTFIDS) is developed based on secondary development technology of UG/NX in the VS.NET integrated development environment. The system realizes the reuse of design knowledge and rapid design for modular machine tool fixture.

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Singh, G. "Diagonal torsion matrices associated with modular data." Algebra and Discrete Mathematics 32, no. 1 (2021): 127–37. http://dx.doi.org/10.12958/adm1368.

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Modular data are commonly studied in mathematics and physics. A modular datum defines a finite-dimensional representation of the modular group SL2(Z). Cuntz (2007) defined isomorphic integral modular data. Here we discuss isomorphic integral and non-integral modular data as well as non-isomorphic but closely related modular data. In this paper, we give some insights into diagonal torsion matrices associated to modular data.

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Evans, David E., and Mathew Pugh. "Spectral measures for G2, II: Finite subgroups." Reviews in Mathematical Physics 32, no. 08 (March 23, 2020): 2050026. http://dx.doi.org/10.1142/s0129055x20500269.

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Joint spectral measures associated to the rank two Lie group [Formula: see text], including the representation graphs for the irreducible representations of [Formula: see text] and its maximal torus, nimrep graphs associated to the [Formula: see text] modular invariants have been studied. In this paper, we study the joint spectral measures for the McKay graphs (or representation graphs) of finite subgroups of [Formula: see text]. Using character theoretic methods we classify all non-conjugate embeddings of each subgroup into the fundamental representation of [Formula: see text] and present their McKay graphs, some of which are new.

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LOEFFLER, DAVID. "IMAGES OF ADELIC GALOIS REPRESENTATIONS FOR MODULAR FORMS." Glasgow Mathematical Journal 59, no. 1 (August 3, 2016): 11–25. http://dx.doi.org/10.1017/s0017089516000367.

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AbstractWe show that the image of the adelic Galois representation attached to a non-CM modular form is open in the adelic points of a suitable algebraic subgroup of GL2 (defined by F. Momose). We also show a similar result for the adelic Galois representation attached to a finite set of modular forms.

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Høverstad, Boye Annfelt. "Noise and the Evolution of Neural Network Modularity." Artificial Life 17, no. 1 (January 2011): 33–50. http://dx.doi.org/10.1162/artl_a_00016.

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We study the selective advantage of modularity in artificially evolved networks. Modularity abounds in complex systems in the real world. However, experimental evidence for the selective advantage of network modularity has been elusive unless it has been supported or mandated by the genetic representation. The evolutionary origin of modularity is thus still debated: whether networks are modular because of the process that created them, or the process has evolved to produce modular networks. It is commonly argued that network modularity is beneficial under noisy conditions, but experimental support for this is still very limited. In this article, we evolve nonlinear artificial neural network classifiers for a binary classification task with a modular structure. When noise is added to the edge weights of the networks, modular network topologies evolve, even without representational support.

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Qi, Ming Long, Luo Zhong, and Qing Ping Guo. "Toward Efficient Multiplication Algorithms over Finite Fields in Lagrange Representation." Applied Mechanics and Materials 20-23 (January 2010): 323–27. http://dx.doi.org/10.4028/www.scientific.net/amm.20-23.323.

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In this paper, we present a representative theory for finite fields called the Lagrange Representation recently initialized by Bajard et al. Our contribution is of introducing a new method for computing the leading coefficient of an arbitrary field polynomial, and establishing a field modular multiplication algorithm. Some concrete examples are given in order to emphasize illustration of the method.

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Bybee, Joan. "Use impacts morphological representation." Behavioral and Brain Sciences 22, no. 6 (December 1999): 1016–17. http://dx.doi.org/10.1017/s0140525x99252223.

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The distinction between regular and irregular morphology is not clear-cut enough to suggest two distinct modular structures. Instead, regularity is tied directly to the type frequency of a pattern. Evidence from experiments as well as from naturally occurring sound change suggests that even regular forms have lexical storage. Finally, the development trajectory entailed by the dual-processing model is much more complex than that entailed by associative network models.

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Voskresenskaya, G. V. "ON REPRESENTATION OF MODULAR FORMS AS HOMOGENEOUS POLYNOMIALS." Vestnik of Samara University. Natural Science Series 21, no. 6 (May 17, 2017): 40–49. http://dx.doi.org/10.18287/2541-7525-2015-21-6-40-49.

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In the article we study the spaces of modular forms such that each element of them is a homogeneous polynomial of modular forms of low weights of the same level. It is a classical fact that it is true for the level 1. N. Koblitz point out that it is true for cusp forms of level 4. In this article we show that the analogous situation takes place for the levels corresponding to the eta-products with multiplicative coeﬃcients. In all cases under consideration the base functions are eta-products. In each case the base functions are written explicitly. Dimensions of spaces are calculated by the Cohen - Oesterle formula, the orders in cusps are calculated by the Biagioli formula.

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INCLEZAN, DANIELA, and MICHAEL GELFOND. "Modular action language." Theory and Practice of Logic Programming 16, no. 2 (July 6, 2015): 189–235. http://dx.doi.org/10.1017/s1471068415000095.

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AbstractThe paper introduces a new modular action language,${\mathcal ALM}$, and illustrates the methodology of its use. It is based on the approach of Gelfond and Lifschitz (1993,Journal of Logic Programming 17, 2–4, 301–321; 1998,Electronic Transactions on AI 3, 16, 193–210) in which a high-level action language is used as a front end for a logic programming system description. The resulting logic programming representation is used to perform various computational tasks. The methodology based on existing action languages works well for small and even medium size systems, but is not meant to deal with larger systems that requirestructuring of knowledge.$\mathcal{ALM}$is meant to remedy this problem. Structuring of knowledge in${\mathcal ALM}$is supported by the concepts ofmodule(a formal description of a specific piece of knowledge packaged as a unit),module hierarchy, andlibrary, and by the division of a system description of${\mathcal ALM}$into two parts:theoryandstructure. Atheoryconsists of one or more modules with a common theme, possibly organized into a module hierarchy based on adependency relation. It contains declarations of sorts, attributes, and properties of the domain together with axioms describing them.Structuresare used to describe the domain's objects. These features, together with the means for defining classes of a domain as special cases of previously defined ones, facilitate the stepwise development, testing, and readability of a knowledge base, as well as the creation of knowledge representation libraries.

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Du, Jie. "The Modular Representation Theory of q-Schur Algebras." Transactions of the American Mathematical Society 329, no. 1 (January 1992): 253. http://dx.doi.org/10.2307/2154087.

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Durr, Peter, Claudio Mattiussi, and Dario Floreano. "Genetic Representation and Evolvability of Modular Neural Controllers." IEEE Computational Intelligence Magazine 5, no. 3 (August 2010): 10–19. http://dx.doi.org/10.1109/mci.2010.937319.

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48

Malakuti, Somayeh, and Mehmet Aksit. "Emergent gummy modules: modular representation of emergent behavior." ACM SIGPLAN Notices 50, no. 3 (May 12, 2015): 15–24. http://dx.doi.org/10.1145/2775053.2658764.

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49

Du, Jie. "The modular representation theory of $q$-Schur algebras." Transactions of the American Mathematical Society 329, no. 1 (January 1, 1992): 253–71. http://dx.doi.org/10.1090/s0002-9947-1992-1022165-3.

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50

Gilmer, Patrick M., and Gregor Masbaum. "An application of TQFT to modular representation theory." Inventiones mathematicae 210, no. 2 (May 13, 2017): 501–30. http://dx.doi.org/10.1007/s00222-017-0734-4.

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Journal articles on the topic 'Modular representation'

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