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 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.     

Kendall's τ is equal to the correlation coefficient for the BVE distribution

For the BVE (BiVariate Exponential) distribution, defined and explained by Marshall and Olkin (1967), the population Kendall's τ (1948), which is the probability of concordance minus the probability of discordance for a bivariate distribution, is shown to be equal to the population correlation coefficient. Moreover, Kendall's t statistic is superior to Pearson's r for estimating the BVE correlation. Some other bivariate distributions, due to Gumbel, Morgenstern and Freund, are also considered.     

Two Orbits: When is one in the closure of the other?

Let G be a connected linear algebraic group, let V be a finite dimensional algebraic G-module, and let and be two G-orbits in V. We describe a constructive way to find out whether or not lies in the closure of . Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2009, Vol. 264, pp. 152–164. In memory of V.A. Iskovskikh     

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 the value of public information positive in a game?

The value of public information is studied by considering the equilibrium selections that maximize the weighted sum of players' payoffs. We show that the value of information can be deduced from the deterministic games where the uncertain parameters have given values. If the maximal weighted sum of equilibrium payoffs in deterministic games is convex then the value of information in any Bayesian game derived from the deterministic games is positive with respect to the selection. We also show the converse result that positive value of information implies convexity. Hence, the convexity of maximal weighted sum of payoffs in deterministic games fully characterizes the value of information with respect to considered selections. We also discuss the implications of our results when positive value of information means that for any equilibrium in a game with less information there is a Pareto dominant equilibrium in any game with more information.     

When is the Isbell topology a group topology?

Conditions on a topological space X under which the space C(X,R) of continuous real-valued maps with the Isbell topology κ is a topological group (topological vector space) are investigated. It is proved that the addition is jointly continuous at the zero function in Cκ(X,R) if and only if X is infraconsonant. This property is (formally) weaker than consonance, which implies that the Isbell and the compact-open topologies coincide. It is shown the translations are continuous in Cκ(X,R) if and only if the Isbell topology coincides with the fine Isbell topology. It is proved that these topologies coincide if X is prime (that is, with at most one non-isolated point), but do not even for some sums of two consonant prime spaces.     

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.     

When is rounding allowed in integer nonlinear optimization?

In this paper we consider nonlinear integer optimization problems. Nonlinear integer programming has mainly been studied for special classes, such as convex and concave objective functions and polyhedral constraints. In this paper we follow an other approach which is not based on convexity or concavity. Studying geometric properties of the level sets and the feasible region, we identify cases in which an integer minimizer of a nonlinear program can be found by rounding (up or down) the coordinates of a solution to its continuous relaxation. We call this property rounding property. If it is satisfied, it enables us (for fixed dimension) to solve an integer programming problem in the same time complexity as its continuous relaxation. We also investigate the strong rounding property which allows rounding a solution to the continuous relaxation to the next integer solution and in turn yields that the integer version can be solved in the same time complexity as its continuous relaxation for arbitrary dimensions.     

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 the Bloch–Okounkov q-bracket modular?

The Ramanujan Journal - We obtain a condition describing when the quasimodular forms given by the Bloch–Okounkov theorem as q-brackets of certain functions on partitions are actually modular....