Cotorsion pairs, model category structures, and representation theory

We make a general study of Quillen model structures on abelian categories. We show that they are closely related to cotorsion pairs, which were introduced by Salce [Sal79] and have been much studied recently by Enochs and coauthors [EJ00]. This gives a method of constructing model structures on abelian categories, which we illustrate by building two model structures on the category of modules over a (possibly noncommutative) Gorenstein ring. The homotopy category of these model structures is a generalization of the stable module category much used in modular representation theory. This stable module category has also been studied by Benson [Ben97]. Received: 14 December 2000; in final form: 17 December 2001 / Published online: 5 September 2002     

Characterization of multivariate stable processes

This paper deals with a characterization of a multivariate stable process using an independence property with a positive random variable. Moreover, we establish a characterization of a multivariate Lévy process based on the notion of cut in a natural exponential family. This allows us to draw some related properties. More precisely, we give the probability density function of this process and the law of the mixture of the Lévy process governed by the convolution semigroup with respect to an exponential random variable. These results are confidentially connected with the univariate case given by [G. Letac and V. Seshadri, Exponential stopping and drifted stable processes, Stat. Probab. Lett., 72:137–143, 2005].     

Weak Equivalence of Internal Categories

Weak equivalence is defined as equivalence in the bicategory of modules between internal categories. It is known that two categories are weakly equivalent if and only if their Cauchy completions are equivalent. We prove that this condition can be generalized to a suitable notion of intermediate category, stable under composition with weak equivalences. Applications to categorical Morita theory are given.     

Fujimoto’s theorem - a further study

In the paper we improve Fujimoto’s sufficient condition for a finite set to be a Unique Range Set under relaxed sharing hypothesis. We introduce a new sharing notion which directly improves one result of the paper [3]. In particular, we rectify the Application Part of [3], and extend and improve all results of [3].     

An upper bound on the independence number of benzenoid systems

Recently, the graph theoretic independence number has been linked to fullerene stability [S. Fajtlowicz, C. Larson, Graph-theoretic independence as a predictor of fullerene stability, Chem. Phys. Lett. 377 (2003) 485-490; S. Fajtlowicz, Fullerene Expanders, A list of Conjectures of Minuteman, Available from S. Fajtlowicz: math0@bayou.uh.edu]. In particular, stable fullerenes seem to minimize their independence numbers. A large piece of evidence for this hypothesis comes from the fact that stable benzenoids—close relatives of fullerenes—do minimize their independence numbers [S. Fajtlowicz, “Pony Express”—Graffiti's conjectures about carcinogenic and stable benzenoids, 〈http://www.math.uh.edu/∼siemion/pony.html〉]. In this paper, an upper bound on the independence number of benzenoids is introduced and proven—giving a limit on how large the independence ratio for benzenoids can be. In conclusion, this bound on independence is correlated to an upper bound on the number of unpaired sites a benzenoid system has with respect to a maximum matching, which is precisely the number of zero eigenvalues in the spectrum of the adjacency matrix (due to a conjecture of Graffiti and its proof by Sachs [S. Fajtlowicz, “Pony Express”—Graffiti's conjectures about carcinogenic and stable benzenoids, 〈http://www.math.uh.edu/∼siemion/pony.html〉; H. Sachs, P. John, S. Fajtlowicz, On Maximum Matchings and Eigenvalues of Benzenoid Graphs, preprint—MATCH]). Thus, since zero eigenvalues and unpaired sites are indicative of instability (reactivity), we get a simple but intuitive bound on how reactive a benzenoid molecule can be.     

De Rham cohomology of rigid spaces

We define de Rham cohomology groups for rigid spaces over non-archimedean fields of characteristic zero, based on the notion of dagger space introduced in [12]. We establish some functorial properties and a finiteness result, and discuss the relation to the rigid cohomology as defined by P. Berthelot [2].     

Locally Stable Grothendieck Categories: Applications

We introduce for any Grothendieck category the notion of stable localizing subcategory, as a localizing subcategory that can be written as an intersection of localizing subcategories defined by indecomposable injectives. A Grothendieck category in which every localizing subcategory is stable is called a locally stable category. As a main result we give a characterization of these categories in terms of the local stability of a localizing subcategory and its quotient category. The locally coirreducible categories (in particular, the categories with Gabriel dimension) and the locally noetherian categories are examples of locally stable categories. We also present some applications to the category of modules over a left fully bounded noetherian ring, to the category of comodules over a coalgebra and to the category of modules over graded rings.     

Localization results for densities associated with stable small-noise diffusions

Summary This paper concerns asymptotic properties of the stationary density associated with small-noise diffusion processes, such as considered in the well-known work of Ventcel and Freidlin [12]. We assume that the origin is a globally attracting asymptotically stable equilibrium point of the underlying deterministic flow. For a bounded domain D, containing the origin, we derive estimates which establish the asymptotic independence, as the size of the noise vanishes, of the equilibrium density in D from the coefficients of the process outside D. These results are applied to generalize a result of Sheu [10] on an asymptotic representation of the equilibrium density.     

On some riesz bases

In this paper we consider three problems concerning systems of vector exponentials. In the first part we prove a conjecture of V. Komornik raised in [14] on the independence of the movement of a rectangular membrane in different points. It was independently proved by M. Horváth [9] and S. A. Avdonin (personal communication). The analogous problem for the circular membrane was partly solved in [3] — the complete solution is given in [10]. In the second part we fill in a gap in the theory of Blaschke-Potapov products developed in the paper [19] of Potapov. Namely we prove that the Blaschke-Potapov product is determined by its kernel sets up to a multiplicative constant matrix. In the third part of the present paper we give a multidimensional generalization of the notion of sine type function developed by Levin [16], [17] and by our generalization we prove the multidimensional variant of the Levin-Golovin basis theorem [16], [6].     

On pallets for Fox colorings of spatial graphs

We introduce the notion of pallets of quandles and define coloring invariants for spatial graphs which give a generalization of Fox colorings studied in Ishii and Yasuhara (1997) [4]. All pallets for dihedral quandles are obtained from the quotient sets of the universal pallets under a certain equivalence relation. We study the quotient sets and classify their elements.     

Divisors,measures and critical functions

In [4] we have introduced a new distance between Galois orbits over ℚ. Using generalized divisors, we have extended the notion of trace of an algebraic number to other transcendental quantities. In this article we continue the work started in [4]. We define the critical function for a class of transcendental numbers, that generalizes the notion of minimal polynomial of an algebraic number. Our results extend the results obtained by Popescu et al [5].     

Schrödinger’s cat

We show that the classical framework of probability spaces, which does not admit a model-theoretical treatment, is equivalent to that of probability algebras, which does. We prove that the category of probability algebras is a stable cat, where non-dividing coincides with the ordinary notion of independence used in probability theory.     

Belief and probability: A general theory of probability cores

This paper considers varieties of probabilism capable of distilling paradox-free qualitative doxastic notions (e.g., full belief, expectation, and plain belief) from a notion of probability taken as a primitive. We show that core systems, collections of nested propositions expressible in the underlying algebra, can play a crucial role in these derivations. We demonstrate how the notion of a probability core can be naturally generalized to high probability, giving rise to what we call a high probability core, a notion that when formulated in terms of classical monadic probability coincides with the notion of stability proposed by Hannes Leitgeb [32]. Our work continues by one of us in collaboration with Rohit Parikh [7]. In turn, the latter work was inspired by the seminal work of Bas van Fraassen [46]. We argue that the adoption of dyadic probability as a primitive (as articulated by van Fraassen [46]) admits a smoother connection with the standard theory of probability cores as well as a better model in which to situate doxastic notions like full belief. We also illustrate how the basic structure underlying a system of cores naturally leads to alternative probabilistic acceptance rules, like the so-called ratio rule initially proposed by Isaac Levi [34].Core systems in their various guises are ubiquitous in many areas of formal epistemology (e.g., belief revision, the semantics of conditionals, modal logic, etc.). We argue that core systems can also play a natural and important role in Bayesian epistemology and decision theory. In fact, the final part of the article shows that probabilistic core systems are naturally derivable from basic decision-theoretic axioms which incorporate only qualitative aspects of core systems; that the qualitative aspects of core systems alone can be naturally integrated in the articulation of coherence of primitive conditional probability; and that the guiding idea behind the primary qualitative features of a core system gives rise to the formulation of lexicographic decision rules.     

On one-sided torsion pair

Motivated by the concept of a torsion pair in a pre-triangulated category induced by Beligiannis and Reiten, the notion of a left (right) torsion pair in the left (right) triangulated category is introduced and investigated. We provide new connections between different aspects of torsion pairs in one-sided triangulated categories, pre-triangulated categories, stable categories and derived categories.     

Linearly perturbed polyhedral normal cone mappings and applications

Under a mild regularity assumption, we derive an exact formula for the Fréchet coderivative and some estimates for the Mordukhovich coderivative of the normal cone mappings of perturbed polyhedra in reflexive Banach spaces. Our focus point is a positive linear independence condition, which is a relaxed form of the linear independence condition employed recently by Henrion et al. (2010) [1], and Nam (2010) [3]. The formulae obtained allow us to get new results on solution stability of affine variational inequalities under linear perturbations. Thus, our paper develops some aspects of the work of Henrion et al. (2010) [1] Nam (2010) [3] Qui (in press) [12] and Yao and Yen (2009) [6] and [7].     

A refined Newton’s mesh independence principle for a class of optimal shape design problems

Shape optimization is described by finding the geometry of a structure which is optimal in the sense of a minimized cost function with respect to certain constraints. A Newton’s mesh independence principle was very efficiently used to solve a certain class of optimal design problems in [6]. Here motivated by optimization considerations we show that under the same computational cost an even finer mesh independence principle can be given.     

Critical Facets of the Stable Set Polytope

Dedicated to the memory of Paul Erdős A facet of the stable set polytope of a graph G can be viewed as a generalization of the notion of an -critical graph. We extend several results from the theory of -critical graphs to facets. The defect of a nontrivial, full-dimensional facet of the stable set polytope of a graph G is defined by . We prove the upper bound for the degree of any node u in a critical facet-graph, and show that can occur only when . We also give a simple proof of the characterization of critical facet-graphs with defect 2 proved by Sewell [11]. As an application of these techniques we sharpen a result of Surányi [13] by showing that if an -critical graph has defect and contains nodes of degree , then the graph is an odd subdivision of . Received October 23, 1998     

On the class of flat stable theories

A new notion of independence relation is given and associated to it, the class of flat theories, a subclass of strong stable theories including the superstable ones is introduced. More precisely, after introducing this independence relation, flat theories are defined as an appropriate version of superstability. It is shown that in a flat theory every type has finite weight and therefore flat theories are strong. Furthermore, it is shown that under reasonable conditions any type is non-orthogonal to a regular one. Concerning groups in flat theories, it is shown that type-definable groups behave like superstable ones, since they satisfy the same chain condition on definable subgroups and also admit a normal series of definable subgroup with semi-regular quotients.     

The Isbell monad

In 1966 [7], John Isbell introduced a construction on categories which he termed the “couple category” but which has since come to be known as the Isbell envelope. The Isbell envelope, which combines the ideas of contravariant and covariant presheaves, has found applications in category theory, logic, and differential geometry. We clarify its meaning by exhibiting the assignation sending a locally small category to its Isbell envelope as the action on objects of a pseudomonad on the 2-category of locally small categories; this is the Isbell monad of the title. We characterise the pseudoalgebras of the Isbell monad as categories equipped with a cylinder factorisation system; this notion, which appears to be new, is an extension of Freyd and Kelly's notion of factorisation system [5] from orthogonal classes of arrows to orthogonal classes of cocones and cones.     

Limits of locally–globally convergent graph sequences

The colored neighborhood metric for sparse graphs was introduced by Bollobás and Riordan [BR11]. The corresponding convergence notion refines a convergence notion introduced by Benjamini and Schramm [BS01]. We prove that even in this refined sense, the limit of a convergent graph sequence (with uniformly bounded degree) can be represented by a graphing. We study various topics related to this convergence notion such as: Bernoulli graphings, factor of i.i.d. processes and hyperfiniteness.