Integral Categories and Calculus Categories

Differential categories are now an established abstract setting for differentiation. The paper presents the parallel development for integration by axiomatizing an integral transformation in a symmetric monoidal category with a coalgebra modality. When integration is combined with differentiation, the two fundamental theorems of calculus are expected to hold (in a suitable sense): a differential category with integration which satisfies these two theorem is called a calculus category. Modifying an approach to antiderivatives by T. Ehrhard, it is shown how examples of calculus categories arise as differential categories with antiderivatives in this new sense. Having antiderivatives amounts to demanding that a certain natural transformation K, is invertible. We observe that a differential category having antiderivatives, in this sense, is always a calculus category and we provide examples of such categories

Degenerate cyclotomic Hecke algebras and higher level Heisenberg categorification

We associate a monoidal category H-lambda to each dominant integral weight lambda of sl(p) or sl(infinity). These categories, defined in terms of planar diagrams, act naturally on categories of modules for the degenerate cyclotomic Hecke algebras associated to lambda. We show that, in the sl infinity case, the level d Heisenberg algebra embeds into the Grothendieck ring of H-lambda, where d is the level of lambda. The categories H-lambda can be viewed as a graphical calculus describing induction and restriction functors between categories of modules for degenerate cyclotomic Hecke algebras, together with their natural transformations. As an application of this tool, we prove a new result concerning centralizers for degenerate cyclotomic Hecke algebras.info:eu-repo/semantics/publishedVersio

On Differential Rota-Baxter Algebras

A Rota-Baxter operator of weight $\lambda$ is an abstraction of both the integral operator (when $\lambda=0$) and the summation operator (when $\lambda=1$). We similarly define a differential operator of weight $\lambda$ that includes both the differential operator (when $\lambda=0$) and the difference operator (when $\lambda=1$). We further consider an algebraic structure with both a differential operator of weight $\lambda$ and a Rota-Baxter operator of weight $\lambda$ that are related in the same way that the differential operator and the integral operator are related by the First Fundamental Theorem of Calculus. We construct free objects in the corresponding categories. In the commutative case, the free objects are given in terms of generalized shuffles, called mixable shuffles. In the noncommutative case, the free objects are given in terms of angularly decorated rooted forests. As a byproduct, we obtain structures of a differential algebra on decorated and undecorated planar rooted forests.Comment: 21 page

Peningkatan Peran Aktif Mahasiswa pada Kalkulus Integral Menggunakan Metode Pembelajaran Kooperatif Tipe Student Teams-Achievement Division

Problems in teaching integral calculus that requires attention in the motivation of learning alongside students is the active role of the student. The active role of the student during the teaching and learning process will affect on the results of his studies. As for the purpose of this research is to improve the students active role by using cooperative learning method type STAD. The subject of research as many as 20 students PGMIPA education courses Math UAD who take Courses Integral Calculus semester three academic year 2012/2013. The cycle is done as much as 3 times the cycle. Data collection is done using sheets of observation, interviewing, documentation, diagnostic tests and field notes. Furthermore the data analyzed by qualitative descriptive. Based on the results of data analysis revealed that the use of cooperative learning methods type STAD can enhance the active role of students in cycle I of 48.53% in category enough, cycle II increased by 59.21% in the category of pretty and cycle III increased by 71.76% in both categories

Peningkatan Peran Aktif Mahasiswa pada Kalkulus Integral Menggunakan Metode Pembelajaran Kooperatif Tipe Student Teams-Achievement Division

Problems in teaching integral calculus that requires attention in the motivation of learning alongside students is the active role of the student. The active role of the student during the teaching and learning process will affect on the results of his studies. As for the purpose of this research is to improve the students ' active role by using cooperative learning method type STAD.  The subject of research as many as 20 students PGMIPA education courses Math UAD who take Courses Integral Calculus semester three academic year 2012/2013. The cycle is done as much as 3 times the cycle. Data collection is done using sheets of observation, interviewing, documentation, diagnostic tests and field notes. Furthermore the data analyzed by qualitative descriptive.   Based on the results of data analysis revealed that the use of cooperative learning methods type STAD can enhance the active role of students in cycle I of 48.53% in category enough, cycle II increased by 59.21% in the category of pretty and cycle III increased by 71.76% in both categories

From triangulated categories to module categories via localisation II: Calculus of fractions

We show that the quotient of a Hom-finite triangulated category C by the kernel of the functor Hom(T, -), where T is a rigid object, is preabelian. We further show that the class of regular morphisms in the quotient admit a calculus of left and right fractions. It follows that the Gabriel-Zisman localisation of the quotient at the class of regular morphisms is abelian. We show that it is equivalent to the category of finite dimensional modules over the endomorphism algebra of T in C.Comment: 21 pages; no separate figures. Minor changes. To appear in Journal of the London Mathematical Society (published version is different

Abelian categories from triangulated categories via Nakaoka-Palu's localization

The aim of this paper is to provide an expansion to Abe-Nakaoka's heart construction of the following two different realizations of the module category over the endomorphism ring of a rigid object in a triangulated category: Buan-Marsh's localization and Iyama-Yoshino's subfactor. Our method depends on a modification of Nakaoka-Palu's HTCP localization, a Gabriel-Zisman localization of extriangulated categories which is also realized as a subfactor of the original ones. Besides of the heart construction, our generalized HTCP localization involves the following phenomena: (1) stable category with respect to a class of objects; (2) recollement of triangulated categories; (3) recollement of abelian categories under a mild assumption.Comment: 28 page