Special elements of the lattice of epigroup varieties

We study special elements of eight types (namely, neutral, standard, costandard, distributive, codistributive, modular, lower-modular and upper-modular elements) in the lattice EPI of all epigroup varieties. Neutral, standard, costandard, distributive and lower-modular elements are completely determined. A strong necessary condition and a sufficient condition for modular elements are found. Modular elements are completely classified within the class of commutative varieties, while codistributive and upper-modular elements are completely determined within the wider class of strongly permutative varieties. It is verified that an element of EPI is costandard if and only if it is neutral; is standard if and only if it is distributive; is modular whenever it is lower-modular; is neutral if and only if it is lower-modular and upper-modular simultaneously. We found also an application of results concerning neutral and lower-modular elements of EPI for studying of definable sets of epigroup varieties.Comment: In comparison with the previous version, we slightly optimize the proof of Theorem 1.1, eliminate a few typos and add Question 11.

A class of non-holomorphic modular forms I

This introductory paper studies a class of real analytic functions on the upper half plane satisfying a certain modular transformation property. They are not eigenfunctions of the Laplacian and are quite distinct from Maass forms. These functions are modular equivariant versions of real and imaginary parts of iterated integrals of holomorphic modular forms, and are modular analogues of single-valued polylogarithms. The coefficients of these functions in a suitable power series expansion are periods. They are related both to mixed motives (iterated extensions of pure motives of classical modular forms), as well as the modular graph functions arising in genus one string perturbation theory. In an appendix, we use weakly holomorphic modular forms to write down modular primitives of cusp forms. Their coefficients involve the full period matrix (periods and quasi-periods) of cusp forms.Comment: Based on a talk given at Zagier's 65th birthday conference `modular forms are everywhere'. What was formerly the appendix has now turned into arXiv:1710.0791

A review of modular strategies and architecture within manufacturing operations

This paper reviews existing modularity and modularization literature within manufacturing operations. Its purpose is to examine the tools, techniques, and concepts relating to modular production, to draw together key issues currently dominating the literature, to assess managerial implications associated with the emerging modular paradigm, and to present an agenda for future research directions. The review is based on journal papers included in the ABI/Inform electronic database and other noteworthy research published as part of significant research programmes. The research methodology concerns reviewing existing literature to identify key modular concepts, to determine modular developments, and to present a review of significant contributions to the field. The findings indicate that the modular paradigm is being adopted in a number of manufacturing organizations. As a result a range of conceptual tools, techniques, and frameworks has emerged and the field of modular enquiry is in the process of codifying the modular lexicon and developing appropriate modular strategies commensurate with the needs of manufacturers. Modular strategies and modular architecture were identified as two key issues currently dominating the modular landscape. Based on this review, the present authors suggest that future research areas need to focus on the development and subsequent standardization of interface protocols, cross-brand module use, supply chain power, transparency, and trust. This is the first review of the modular landscape and as such provides insights into, first, the development of modularization and, second, issues relating to designing modular products and modular supply chains

Congruence Property In Conformal Field Theory

The congruence subgroup property is established for the modular representations associated to any modular tensor category. This result is used to prove that the kernel of the representation of the modular group on the conformal blocks of any rational, C_2-cofinite vertex operator algebra is a congruence subgroup. In particular, the q-character of each irreducible module is a modular function on the same congruence subgroup. The Galois symmetry of the modular representations is obtained and the order of the anomaly for those modular categories satisfying some integrality conditions is determined.Comment: References are updated. Some typos and grammatical errors are correcte

Permutation Modular Invariants from Modular Functors

For any finite group G with a finite G-set X and a modular tensor category C we construct a part of the algebraic structure of an associated G-equivariant monoidal category: For any group element g in G we exhibit the module category structure of the g-component over the trivial component. This uses the formalism of permutation equivariant modular functors that was worked out in arXiv:1004.1825. As an application we show that the corresponding modular invariant partition function is given by permutation by g.Comment: 30 pages, several figure