Narrow Results

The Folded Hypercube (FHC) has been proven to be an attractive hypercube-based network. This paper closely compares the FHC to its standard hypercube counterpart from the subcube allocation viewpoint. It is shown that the FHC(n) outperforms the n-dimensional hypercube (n-cube for short) in offering subcubes of size k by a factor of . In an environment where subcubes of the original network must be allocated to incoming tasks, the FHC achieves an excellent processor utilization by assigning subcubes in an efficient and compact manner. Using the concept of virtual hypercubes, an efficient way is suggested to recognize the available subcubes in the FHC by adapting the already developed subcube recognition algorithms. An alternative approach to the subcube recognition problem is also given.

A subdirect product $A\le \mathrm{\Pi }〈A_i:i\in I〉$ is global if there is a topology on $I$ which makes of $A$ the algebra of all global sections of a sheaf whose stalks are the algebras $A_i$ with $i\in I$. A quasivariety ${𝒬}$ has BL-global representations if there is a class ${ℱ}$ containing but close to the class of all relatively subdirectly irreducible members of ${𝒬}$, satisfying that every member of ${𝒬}$ is isomorphic to a global subdirect product whose factors are in ${ℱ}$. The adjective “BL” refers to “Birkhoff-like” since this type of representations are analogous to the classical Birkhoff subdirect representation by relatively subdirectly irreducibles. This paper has two main contributions. The first one is a generalization of a theorem proved in [D. Vaggione, Sheaf representation and Chinese remainder theorems, Algebra Universalis 29 (1992) 232–272] which characterizes the existence of a global representation of an algebra in terms of the solvability of certain congruence systems. The second one is a theorem assuring the existence of BL-global representations for quasivarieties with a near unanimity term.

We give a description of Khovanov's knot homology theory in the language of sheaves. To do this, we identify two cohomology theories associated to a commutative diagram of abelian groups indexed by elements of the cube {0, 1}ⁿ. The first is obtained by taking the cohomology groups of the chain complex constructed by summing along the diagonals of the cube and inserting signs to force d² = 0. The second is obtained by regarding the commutative diagram as a sheaf on the cube (in the order-filter topology) and considering sheaf cohomology with supports. Included is a general study of sheaves on finite posets, and a review of some basic properties of knot homology in the language of sheaves.

Since topoi were introduced, there have been efforts putting mathematics into the context of topoi. Amongst known topoi, the topoi of sheaves or presheaves over a small category are of special interest. We have here as the base topos that of sheaves over a monoid $M$ as a one object category. By means of closure operators we then obtain categories of sheaves related to the right ideals of $M$. These categories have already been studied but we give these categories a more thorough treatment and reveal some additional properties. Namely, for a weak topology determined by a right ideal $I$ of $M$, we show that the category of sheaves associated to this topology is a subtopos of $MSet$ (the presheaves over $M$) and determine the Lawvere–Tierney topology yielding the same subtopos, which is the Lawvere–Tierney topology associated to the idempotent hull of the (not necessarily idempotent) closure operator associated to $I$. We will then find conditions under which the subcategory of separated objects turns out to be a topos, and in the last section, we find conditions under which the category of sheaves becomes a De Morgan topos.

We explain and motivate Stefan–Sussmann singular foliations, and by replacing the tangent bundle of a manifold with an arbitrary Lie algebroid, we introduce singular subalgebroids. Both notions are defined using compactly supported sections. The main results of this note are an equivalent characterization, in which the compact support condition is removed, and an explicit description of the sheaf associated to any Stefan–Sussmann singular foliation or singular subalgebroid.

In this paper we prove that the category of locally sliceable fiber spaces over the fixed topological space X and the category of sheaves of sets over the same topological space X are equivalent.

The fundamental approach toward matter, space and time is that particles (either objects of macrocosm or microcosm), space and time are all presheafified. Namely, the concept of a presheaf is most fundamental for matter, space and time. An observation of a particle is represented by a morphism from the observed particle (its associated presheaf) to the observer (its associated presheaf) over a specified object (called a generalized time period) of a t-site (i.e. a category with a Grothendieck topology). This formulation provides a scale independent and background space-time free theory (since, for the t-topos theoretic formulation, space and time are discretely defined by the associated particle, whose particle-dependency is a consequence of quantum entanglement.). It is our basic scheme that the method of t-topos may provide a device for understanding and concrete formulation of macro-object and micro-object interconnection as morphisms in the sense of t-topos.