We discuss inference for additive models with random scaling factors. The additive effects are of the form (1 + γ)f (z) where f is a nonlinear function of the continuous covariate z modeled by P(enalized)splines and 1 + γ is a random scaling factor. Additionally, monotonicity constraints on the nonlinear functions are possible. Our work is motivated by the situation of a retailer analyzing the ...

In an arbitrary Grothendieck category, we find necessary and sufficient conditions for the class of $\text{FP}_n$-injective objects to be a torsion class. By doing so, propose notion $n$-hereditary categories. We also define study $\text{FP}_n$-flat in categories with generating set small projective objects, provide several equivalent this torsion-free. end, present applications examples contex...

One of the earliest theorems in category theory stated that an abelian category could be represented faithfully by exact functors into the category Ab of abelian groups [Freyd, 1964], [Lubkin, 1961] and [Heron, unpublished]. Then Mitchell [1965] showed that every such category had a full exact embedding into a module category. An equivalent formulation is that every abelian category into a cate...

This thesis deals mainly with hereditary coreflective subcategories of the category Top of topological spaces. After preparing the basic tools used in the rest of thesis we start by a question which coreflective subcategories of Top have the property SA = Top (i.e., every topological space can be embedded in a space from A). We characterize such classes by finding generators of the smallest cor...

1 Brown Representability First we introduce a portly abelian category A(S) for any preadditive category S. This will be a certain full subcategory of the category of all contravariant additive functors S −→ Ab. Modulo the fact that S is not assumed to be small, this is precisely what we call a right module over a ringoid in our Rings With Several Objects (RSO) notes. For background on portly ab...

We exhibit a surprising connection between the following two concepts: Brown representability which arises in stable homotopy theory, and flat covers which arise in module theory. It is shown that Brown representability holds for a compactly generated triangulated category if and only if for every additive functor from the category of compact objects into the category of abelian groups a flat c...

We calculate the Grothendieck group K0(A), where A is an additive category, locally finite over a Dedekind ring and satisfying some additional conditions. The main examples are categories of modules algebras stable homotopy category SW CW-complexes. show that this direct sum free arising from localizations analogous to groups ideal classes maximal orders. As corollary, we obtain new simple proo...

An additive category in which each object has a Krull-Remak-Schmidt decomposition—that is, finite direct sum decomposition consisting of objects with local endomorphism rings—is known as Krull-Schmidt category. A Hom-finite is an for there commutative unital ring k, such that Hom-set length k-module. The aim this note to provide proof Krull-Schmidt, if and only it split idempotents, indecomposa...

Following work of Ehrhard and Regnier, we introduce the notion of a differential category: an additive symmetric monoidal category with a comonad (a " coalgebra modality ") and a differential combinator, satisfying a number of coherence conditions. In such a category, one should imagine the morphisms in the base category as being linear maps and the morphisms in the coKleisli category as being ...