When is Flatness Coherent?

ABSTRACT

We characterize those small categories with the property that flat (contravariant) functors on them are coherently axiomatized in the language of presheaves on them. They are exactly the categories with the property that every finite diagram into them has a finite set of (weakly) initial cocones.     

When is tit-for-tat unbeatable?

We characterize the class of symmetric two-player games in which tit-for-tat cannot be beaten even by very sophisticated opponents in a repeated game. It turns out to be the class of exact potential games. More generally, there is a class of simple imitation rules that includes tit-for-tat but also imitate-the-best and imitate-if-better. Every decision rule in this class is essentially unbeatable in exact potential games. Our results apply to many interesting games including all symmetric 2 $\times $ 2 games, and standard examples of Cournot duopoly, price competition, public goods games, common pool resource games, and minimum effort coordination games.     

When ∏ is Isomorphic to ⊕

For a category , we investigate the problem of when the coproduct ⊕ and the product functor ∏ from  ^I to  are isomorphic for a fixed set I, or, equivalently, when the two functors are Frobenius functors. We show that for an Ab category  this happens if and only if the set I is finite (and even in a much general case, if there is a morphism in  that is invertible with respect to addition). However, we show that ⊕ and ∏ are always isomorphic on a suitable subcategory of  ^I which is isomorphic to  ^I but is not a full subcategory. If  is only a preadditive category, then we give an example that shows that the two functors can be isomorphic for infinite sets I. For the module category case, we provide a different proof to display an interesting connection to the notion of Frobenius corings.     

A Class of Generalized Hausdorff Means

In the research of divergent series the important idea is that choose a suitable trasformation have to be of regularity, of sequence of partial sum of the series. The Hausdorff transformation is H=δμδ in matrix form, in which μ is any diagonal transformation and δ is so-called δ-transformation. In this paper, we construct a class of generalized Housdorff transformation using the Gould-Hsu inverse series, and give necessary and sufficient condition of that it shou ld be regular.     

On the matrix inequality (I + X)−1 ≤ I

In this note we consider the question under which conditions all entries of the matrix I???(I?+?X)^?1 are nonnegative in case matrix X is a real positive definite matrix. Sufficient conditions are presented as well as some necessary conditions. One sufficient condition is that matrix X ^?1 is an inverse M-matrix. A class of matrices for which the inequality holds is presented.     

Min–Max vs. Min–Sum Vehicle Routing: A worst-case analysis

The classical objective function of the Vehicle Routing Problem (VRP) is to minimize the total distance traveled by all vehicles (Min–Sum). In several situations, such as disaster relief efforts, computer networks, and workload balance, the minimization of the longest route (Min–Max) is a better objective function. In this paper, we compare the optimal solution of several variants of the Min–Sum and the Min–Max VRP, from the worst-case point of view. Our aim is two-fold. First, we seek to motivate the design of heuristic, metaheuristic, and matheuristic algorithms for the Min–Max VRP, as even the optimal solution of the classical Min–Sum VRP can be very poor if used to solve the Min–Max VRP. Second, we aim to show that the Min–Max approach should be adopted only when it is well-justified, because the corresponding total distance can be very large with respect to the one obtained by optimally solving the classical Min–Sum VRP.     

When is the Boston mechanism strategy-proof?

Kumano (2013) is the first to investigate the Boston school choice mechanism (BOSM) under restricted priority domains. This paper strengthens and extends his result and shows that the BOSM is strategy-proof, if and only if it is fair, if and only if it is equivalent to the student-optimal stable mechanism (SOSM), and if and only if the number of total seats at any two schools exceeds the number of students.     

When is a Quasi-p-injective Module Continuous?

It is well-known that every quasi-p-injective module has C2-condition. In this note, it is shown that for a quasi-p-injective module M which is a self-generator, if M is projective, duo and semiperfect, then M is continuous. As a special case we re-obtain a result of Puninski-Wisbauer-Yousif saying that, a semiperfect ring R is right continuous if it is right duo, right p-injective.AMS Subject Classifications (1991): 16D50, 16D70, 16D80Supported by The Royal Golden Jubilee Project     

When is a Cleft Extension H-Azumaya?

We show which H ^op -cleft extensions of k for a dual quasi-triangular Hopf algebra (H, r) are H-Azumaya. The result is given in terms of bijectivity of a map defined in terms of the universal r-form r and the 2-cocycle σ, generalizing a well-known result for the commutative and co-commutative case. We illustrate the Theorem with an explicit computation for the Hopf algebras of type E(n).Presented by A. Verschoren     

When is a Coalgebra a Generator?

Let C be a coalgebra over a field k. The aim of this paper is to study the following problem : (P) If C is a k-coalgebra such that C is a generator for the category of left comodules, is C a left quasi-co-Frobenius coalgebra ? The converse always holds. We show that if C has a finite coradical series, the answer is positive.     

When is the Diagonal Functor Frobenius?

Given a complete, cocomplete category 𝒞, we investigate the problem of describing those small categories I such that the diagonal functor Δ: 𝒞 → Functors(I, 𝒞) is a Frobenius functor. This condition can be rephrased by saying that the limits and the colimits of functors I → 𝒞 are naturally isomorphic. We find necessary conditions on I for a certain class of categories 𝒞, and, as an application, we give both necessary and sufficient conditions in the two special cases 𝒞 =Set or _R ?, the category of left modules over a ring R.     

When is the Comaximal Graph Split?

Let R be a commutative ring with non zero unity. Let Ω(R) be a graph with vertices as elements of R whose two distinct vertices x and y are adjacent if and only if Rx + Ry = R. A graph (V, E) is said to be a split graph if V is the disjoint union of two sets K and S where K induces a complete subgraph and S is an independent set. We investigate the properties of R when Ω(R) is split.     

When is an Equally-Weighted Design D-optimal?

We discuss conditions under which an equally weighted design is D-optimal. The concept of a model being maximally invariant with respect to a set is introduced and is shown to be useful in this work. Several illustrative examples are given, including two biomedical applications using compartmental and segmented models.     

When is the Uvarov transformation positive definite?

Let L\cal{L} be a positive definite bilinear functional, then the Uvarov transformation of L\cal{L} is given by  U(p,q) = L(p,q) + m p(a)[`(q)](a<sup>-1</sup>) +[`(m)] p([`(a)]<sup>-1</sup>)\,\mathcal{U}(p,q) = \mathcal{L}(p,q) + m\,p(\alpha)\overline{q}(\alpha^{-1}) + \overline{m}\,p(\overline{\alpha}^{-1}) [`(q)]([`(a)])\overline{q}(\overline{\alpha}) where $|\alpha| > 1, m \in \mathbb{C}$|\alpha| > 1, m \in \mathbb{C}. In this paper we analyze conditions on m for U\cal{U} to be positive definite in the linear space of polynomials of degree less than or equal to n. In particular, we show that m has to lie inside a circle in the complex plane defined by α, n and the moments associated with L\cal{L}. We also give an upper bound for the radius of this circle that depends only on α and n. This and other conditions on m are visualized for some examples.     

When is a symplectic quotient an orbifold?

Let K be a compact Lie group of positive dimension. We show that for most unitary K-modules the corresponding symplectic quotient is not regularly symplectomorphic to a linear symplectic orbifold (the quotient of a unitary module of a finite group). When K is connected, we show that even a symplectomorphism to a linear symplectic orbifold does not exist. Our results yield conditions that preclude the symplectic quotient of a Hamiltonian K  -manifold from being locally isomorphic to an orbifold. As an application, we determine which unitary _SU2${\mathrm{SU}}_2$-modules yield symplectic quotients that are Z⁺${\mathbb{Z}}^{+}$-graded regularly symplectomorphic to a linear symplectic orbifold. We similarly determine which unitary circle representations yield symplectic quotients that admit a regular diffeomorphism to a linear symplectic orbifold.     

When is a natural duality ‘good’?

Building on the most current work in the theory of natural dualities, we continue the study of strong dualities for the quasi-variety generated by a finite algebra. We investigate ten different versions of what we would like to mean by a good duality. Each version concerns, among other things, a specific restriction on the type of the structures in the dual category which insures that the dual structures will in a useful sense be simple. Through each investigation we seek a theorem characterizing, in terms of finitely verifiable conditions, those finite algebras generating a quasi-variety which admits a strong duality meeting the given restrictions. Our study includes a careful treatment of coproducts, logarithmic dualities and strong dualities by various unary structures.Dedicated to the memory of Alan DayPresented by J. Sichler.Research supported by a 1992 ARC Grant (Davey).