Cup Product Research Articles

Commuting symplectomorphisms on a surface and the flux homomorphism

Let $$(S,\omega )$$ be a closed connected oriented surface whose genus l is at least two equipped with a symplectic form. Then we show the vanishing of the cup product of the fluxes of commuting symplectomorphisms. This result may be regarded as an obstruction for commuting symplectomorphisms. In particular, the image of an abelian subgroup of $$\textrm{Symp}^c_0(S, \omega )$$ under the flux homomorphism is isotropic with respect to the natural intersection form on $$H^1(S;{\mathbb {R}})$$ . The key to the proof is a refinement of the non-extendability result, previously given by the first-named and second-named authors, for Py’s Calabi quasimorphism $$\mu _P$$ on $$\textrm{Ham}^c(S, \omega )$$ .

Arithmetic Chern–Simons theory with real places

The goal of this paper is two-fold: we generalize the arithmetic Chern–Simons theory over totally imaginary number fields studied in [H.-J. Chung, D. Kim, M. Kim, J. Park and H. Yoo, Arithmetic Chern–Simons theory II, preprint (2016), arXiv:1609.03012v3; M. Kim, Arithmetic Chern–Simons theory I, preprint (2015) arXiv:1510.05818v4] to arbitrary number fields (with real places) and provide new examples of non-trivial arithmetic Chern–Simons invariant with coefficient [Formula: see text] [Formula: see text] associated to a non-abelian gauge group. The main idea for the generalization is to use cohomology with compact support (see [17]) to deal with real places. Before the results of this paper, non-trivial examples were limited to some non-abelian gauge group with coefficient [Formula: see text] in [H.-J. Chung, D. Kim, M. Kim, J. Park and H. Yoo, Arithmetic Chern–Simons theory II, preprint (2016), arXiv:1609.03012v3] and the abelian cyclic gauge group with coefficient [Formula: see text] in [H.-J. Chung, D. Kim, M. Kim, J. Park and H. Yoo, Arithmetic Chern–Simons theory II, preprint (2016), arXiv:1609.03012v3]. Our non-trivial examples (with non-abelian gauge group and general coefficient [Formula: see text]) will be given by a simple twisting argument based on examples of [F. Bleher, T. Chinburg, R. Greenberg, M. Kakde, G. Pappas and M. Taylor, Cup products in the étale cohomology of number fields, New York J. Math. 24 (2018) 514–542].

THE EISENSTEIN IDEAL OF WEIGHT k AND RANKS OF HECKE ALGEBRAS

AbstractLet p and $\ell $ be primes such that $p> 3$ and $p \mid \ell -1$ and k be an even integer. We use deformation theory of pseudo-representations to study the completion of the Hecke algebra acting on the space of cuspidal modular forms of weight k and level $\Gamma _0(\ell )$ at the maximal Eisenstein ideal containing p. We give a necessary and sufficient condition for the $\mathbb {Z}_p$ -rank of this Hecke algebra to be greater than $1$ in terms of vanishing of the cup products of certain global Galois cohomology classes. We also recover some of the results proven by Wake and Wang-Erickson for $k=2$ using our methods. In addition, we prove some $R=\mathbb {T}$ theorems under certain hypotheses.

Cup product in bounded cohomology of negatively curved manifolds

Let M M be a negatively curved compact Riemannian manifold with (possibly empty) convex boundary. Every closed differential 2 2 -form ξ ∈ Ω 2 ( M ) \xi \in \Omega ^2(M) defines a bounded cocycle c ξ ∈ C b 2 ( M ) c_\xi \in C_b^2(M) by integrating ξ \xi over straightened 2 2 -simplices. In particular Barge and Ghys [Invent. Math. 92 (1988), pp. 509–526] proved that, when M M is a closed hyperbolic surface, Ω 2 ( M ) \Omega ^2(M) injects this way in H b 2 ( M ) H_b^2(M) as an infinite dimensional subspace. We show that the cup product of any class of the form [ c ξ ] [c_\xi ] , where ξ \xi is an exact differential 2-form, and any other bounded cohomology class is trivial in H b ∙ ( M ) H_b^{\bullet }(M) .

Codimension 2 transfer of higher index invariants

This paper is devoted to the study of the higher index theory of codimension 2 submanifolds originated by Gromov–Lawson and Hanke–Pape–Schick. The first main result is to construct the ‘codimension 2 transfer’ map from the Higson–Roe analytic surgery exact sequence of a manifold M to that of its codimension 2 submanifold N under some assumptions on homotopy groups. This map sends the primary and secondary higher index invariants of M to those of N. The second is to establish that the codimension 2 transfer map is adjoint to the co-transfer map in cyclic cohomology, defined by the cup product with a group cocycle. This relates the Connes–Moscovici higher index pairing and Lott’s higher $$\rho $$ -number of M with those of N.

Use of menstrual cup and awareness of environmental effects of menstrual hygiene products

ABSTRACT Besides the health problems and economic losses, not managing the menstrual cycle successfully also leads to environmental pollution and affects millions of individuals around the world. Among the reusable menstrual hygiene products, menstrual cups come to the forefront, thanks to their minor environmental effects. This study aims to compare the menstrual cup and single-use menstrual hygiene products regarding the level of user knowledge and awareness “being green” on the environmental effects. The study is carried out between April and May of 2021 on menstruating individuals aged 18–49 years. Among 3365 individuals reached within the scope of the prior study, 298 individuals using menstrual cup and 596 individuals using single-use products were selected. Based on the literature, a survey form consisted of 16 items is prepared by authors questioning the “being green” about the environmental effects of menstrual hygiene products (EKMP). The survey is published online on Twitter and responds are collected using Google Forms. In all the items with statistically significant differences between responded questions in terms of “green” answer percentages in EKMP, it was found that menstrual cup users have a higher percentage of green answers (p < 0.001). In further analysis, as the aged advanced, EKMP scores were higher among menstrual cup users and those expressing concerns about the negative effects of menstrual hygiene products on environmental health. Increasing knowledge and awareness on the environmental effects of menstrual hygiene products, and especially the environmental effects of single-use menstrual hygiene products, might raise the demand for reusable products.

Conjectural Positivity of Chern–Schwartz–MacPherson Classes for Richardson Cells

Abstract Following some work of Aluffi–Mihalcea–Schürmann–Su for the CSM classes of Schubert cells and some elaborate computer calculations by Rimanyi and Mihalcea, I conjecture that the CSM classes of the Richardson cells expressed in the Schubert basis have nonnegative coefficients. This conjecture was principally motivated by a new product $\square $ coming from the Segre-SM classes in the cohomology of flag varieties (such that the associated Gr of this product is the standard cup product) and the conjecture that the structure constants of this new product $\square $ in the standard Schubert basis have alternating sign behavior. I prove that this conjecture on the sign of the structure constants of $\square $ would follow from my above positivity conjecture about the CSM classes of Richardson cells.

Analysis of Work System Design to Increase Production Productivity Using the MOST (Maynard Operation Sequence Technique) Method (Case Study of PT. Segar Murni Utama)

PT. Segar Murni Utama is a company engaged in the mineral water industry, especially the production of MOJO TRAS demineralized water. PT. Segar Murni Utama introduces demineralized water products widely and is marketed among the general public to meet the needs of the community. This company prioritizes the quality of demineralized water very well.The purpose of this study was to optimize the production operator time for the cup section and optimize the packing process time for the cup section using the MOST (Maynard Operation Sequence of Technique) method. MOST (Maynard Operation Sequence Tecnique) is one of several ways of timing sub-operations or movements. The BEST method includes basic sort patterns.The results of this study in the cup production operator section showed that the right hand and left hand maps were known to have 8 activities, and 8 activities on the process flow map. The total time is 1410, the normal time (WN) is 0.85 minutes, the standard time (WS) is 0.017 hours, the standard output (OS) is 59 cartons/hour, and the final calculation result is 413 cartons/day. While the results of improvements in the packing cup section obtained the following results: Right hand and left hand maps are known to have 11 activities, and 11 activities on the process flow map. The total time is 2110, the normal time (WN) is 1.26 minutes, the standard time (WS) is 0.023 hours, the standard output (OS) is 43 cartons/hour, and the final calculation result is 301 cartons/day.

Penggunaan kapang Rhizopus oligosporus dalam pembuatan biofoam cup berbahan dasar sabut kelapa dan tepung kedelai

Biofoam is an alternative packaging to replace styrofoam because it is renewable and biodegradable. In this study, the coconut fibre were used in the production of biofoam cup with addition of commercial tempe yeast (Rhizopus oligosporus). The purpose of this study was to produce biofoam cup which has the characteristics of water resistant, not easily crushed, and easily decomposed. This study was conducted with a factorial Block Randomized Design (BRD) consisting of 2 factors (the amount of soybean flour (T) amd the length of fermentation (S)). The first factor consisted of three levels (20 g, 24 g, and 28 g). The second factor consisted of three levels (3 days, 4 days, and 5 days). All treatments were carried out with 3 replications so there were 27 experimental units. The biofoam cup was analysed for physical (water absorption and porosity), mechanical (puncture strength and compressive strength), and biodegrability. The results showed that the best biofoam cup were obtained from a combination of using 28 g soybean flour and 3 days fermentation. The biofoam cup has the following characteristics: 73 % of water absorption capacity, 72.3 % of porosity value, 3.27 kg/cm2 of puncture strength value, 3.17 kPa of compressive strength value. The average value of biodegradability value of biofoam cup made from coconut fibre was 48 % suggesting that the analysis should be done more than 14 days

Almost-formality and deformations of representations of the fundamental groups of Sasakian manifolds

For a \(2n+1\)-dimensional compact Sasakian manifold, if \(n\ge 2\), we prove that the analytic germ of the variety of representations of the fundamental group at every semi-simple representation is quadratic. To prove this result, we prove the almost-formality of de Rham complex of a Sasakian manifold with values in a semi-simple flat vector bundle. By the almost-formality, we also prove the vanishing theorem on the cup product of the cohomology of semi-simple flat vector bundles over a compact Sasakian manifold.

Completed K-theory and equivariant elliptic cohomology

Kitchloo and Morava give a strikingly simple picture of elliptic cohomology at the Tate curve by studying a completed version of S1-equivariant K-theory for spaces. Several authors (cf. [2],[10],[11]) have suggested that an equivariant version ought to be related to the work of Freed-Hopkins-Teleman ([5],[6],[7]). However, a first attempt at this runs into apparent contradictions concerning twist, degree, and cup product. Several authors (cf. [3],[8],[9]) have solved the problem over the complex numbers by interpreting the S1-equivariant parameter as a complex variable and using holomorphicity as the technique for completion. This paper gives a solution that works integrally, by constructing a carefully completed model of K-theory for S1-equivariant stacks which allows for certain “convergent” infinite-dimensional cocycles.

New Invariant Quantity To Measure The Entanglement In The Braids

In this work, we demonstrate that the integral formula for a generalised Sato-Levine invariant is consistent in certain situations with Evans and Berger's formula for the fourth-order winding number. Also, we found that, in principle, one can derive analogous high-order winding numbers by which one can calculate the entanglement of braids. The winding number for the Brunnian 4-braid is calculated algebraically using the cup product on the cohomology of a finite regular CW-space which is the complement $\mathbb{R}^3\backslash \mathcal{B}_4$.

Mini Oreo Cup Production Plan with Aggregate Method to Minimize Production Costs

Manufacturing companya PT.Sinar Jatimulia Gemilang produces many types of plastic packaging, one of which is mini oreo cups.The production system in this company is based on job shop, changing product demand and limited product capacity make it difficult for the company to make a productin plan. The purpose of this research is to make a mini cup production plan to meet future demand with the aim of minimize production costs. At the initial stage a field study was conducted to obtain to required data. The data obtained through interviews with production employees from the company. Then do the literature study through books and articles from related journal as the theoretical basis used in this research. Production planning research using exponential smoothing forecasting method with α=0.6 and standard error of 617814.3 was selected with the criteria of the smallest mean square error and aggregate planning.The result of analysis I strategy of inventory level inventory cost is IDR. 441,654,900. In analysis II the strategy for variation in the number workers costs is IDR. 747,373,500. The cost for analysis III subcontracting strategy is IDR. 79,065,843,000. From the research that production cost before the calculation is IDR. 52,002,000,000 and the number of workers is 4. After the result of the research on optimization the number of workers that will be used by 2.

Determinants of Attitudes Toward Green Product and Purchase Intention of Zero-Waste Product: A Case Study of Menstrual Cup

Waste, currently, is one of the biggest concerns in Indonesia. As one of the solutions, innovation in the zero-waste product can help in reducing waste because it can be used repeatedly. A menstrual cup is one of zero waste products that substitute conventional sanitary napkins. However, the use of menstrual cups in Indonesia is still taboo. The purpose of this study is to determine the effect of attitudes towards green products on the purchase intention of menstrual cup products based on the factors that influence it. The data are taken from 295 Indonesian women who know about menstrual cup products through online questionnaires. The data analysis process is carried out using the SEM-PLS method with several supporting applications. The results showed that environmental attitudes, perceived health benefits, perceived value, and e-wom attitudes influenced the desire of having products that are environmentally friendly and increased the willingness to pay and purchase intentions for menstrual cup products.

Multiplicativity and nonrealizable equivariant chain complexes

Let G be a finite p -group and F a field of characteristic p . We filter the cochain complex of a free G -space with coefficients in F by powers of the augmentation ideal of F G . We show that the cup product induces a multiplicative structure on the arising spectral sequence and compute the E 1 -page as a bigraded algebra. As an application, we prove that recent counterexamples of Iyengar and Walker to an algebraic version of Carlsson's conjecture can not be realized topologically.

On the cup product of singular Hochschild cohomology

On the cup product of singular Hochschild cohomology

Trivial cup products in bounded cohomology of the free group via aligned chains

Abstract We prove that the cup product of Δ-decomposable quasimorphisms, Brooks quasimorphisms or Rolli quasimorphisms with any bounded cohomology class of arbitrary positive degree is trivial.

Gerstenhaber algebra of ℤ2 × ℤ2-Galois covering of quantum exterior algebra

Let [Formula: see text] be the quantum exterior algebras over a field [Formula: see text] with char [Formula: see text], [Formula: see text] be the [Formula: see text]-Galois coverings of [Formula: see text]. In this paper, a basis of each degree of Hochschild cohomology of [Formula: see text], the cup product and the Gerstenhaber bracket on Hochschild cohomology of [Formula: see text] are given.

Global zeolite market set for modest growth: projected to grow at a CAGR of 3.3% by 2027

Let G be a finite p-group and F a field of characteristic p. We filter the cochain complex of a free G-space with coefficients in F by powers of the augmentation ideal of FG. We show that the cup product induces a multiplicative structure on the arising spectral sequence and compute the E1-page as a bigraded algebra. As an application, we prove that recent counterexamples of Iyengar and Walker to an algebraic version of Carlsson's conjecture can not be realized topologically.

Cyclic Gerstenhaber–Schack cohomology

We show that the diagonal complex computing the Gerstenhaber–Schack cohomology of a bialgebra (that is, the cohomology theory governing bialgebra deformations) can be given the structure of an operad with multiplication if the bialgebra is a (not necessarily finite dimensional) Hopf algebra with invertible antipode; if the antipode is involutive, the operad is even cyclic. Therefore, the Gerstenhaber–Schack cohomology of any such Hopf algebra carries a Gerstenhaber (resp. Batalin–Vilkovisky) algebra structure; in particular, one obtains a cup product and a cyclic boundary B that generate the Gerstenhaber bracket, and that allows to define cyclic Gerstenhaber–Schack cohomology. In case the Hopf algebra in question is finite dimensional, the Gerstenhaber bracket turns out to be zero in cohomology and hence the interesting structure is not given by this e_2 -algebra structure but rather by the resulting e_3 -algebra structure, which is expressed in terms of the cup product and B .