Let S be a locally compact topological foundation semigroup with identity and Ma(S) be its semigroup algebra. In this paper, we give necessary and sufficient conditions to have abounded approximate identity in closed codimension one ideals of the semigroup algebra $M_a(S)$ of a locally compact topological foundationsemigroup with identity.

Suppose that to every non-degenerate simplex ∆ ⊂ Rn a ‘center’ C(∆) is assigned so that the following assumptions hold: (i) The map ∆ → C(∆) commutes with similarities and is invariant under the permutations of the vertices of the simplex; (ii) The map ∆ → Vol(∆)C(∆) is polynomial in the coordinates of the vertices of the simplex. Then C(∆) is an affine combination of the center of mass CM(∆) a...

We formulate an integration scheme for Lagrangian mechanics, referred to as the force-stepping scheme, which is symplectic, energy conserving, time-reversible, and convergent with automatic selection of the time-step size. The scheme also conserves approximately all the momentum maps associated with the symmetries of the system. The exact conservation of momentum maps may additionally be achiev...

The translation equivariance of convolutions can make convolutional neural networks equivariant or invariant. Equivariance to other transformations (e.g. rotations, affine transformations, scalings) may also be desirable as soon we know a priori that transformed versions the same objects appear in data. semigroup cross-correlation, which is linear operator actions, was recently proposed and app...

Let W ⋉L be an irreducible affine Weyl group with Coxeter complex Σ, where W denotes the associated finite Weyl group and L the translation subgroup. The Steinberg torus is the simplicial complex obtained by taking the quotient of Σ by the lattice L. We show that the ordinary and flag h-polynomials of the Steinberg torus (with the empty face deleted) are generating functions over W for a descen...

Let ~ C be a category whose objects are semigroups with topology and morphisms are closed semigroup relations, in particular, continuous homomorphisms. An object X of the category ~ C is called ~ C-closed if for each morphism Φ ⊂ X×Y in the category ~ C the image Φ(X) = {y ∈ Y : ∃x ∈ X (x, y) ∈ Φ} is closed in Y. In the paper we survey existing and new results on topological groups, which are ~...

We initiate the study of interval-valued fuzzy quasi-ideal of a semigroup. In Section 2, we list some basic definitions in the later sections. In Section 3, we investigate interval-valued fuzzy subsemigroups and in Section 4, we define intervalvalued fuzzy quasi-ideals and establish some of their basic properties. In Section 5, we obtain characterizations of regular and intraregular semigroups ...

It has been well known that the band of idempotents of a naturally ordered orthodox semigroup satisfying the “strong Dubreil-Jacotin condition” forms a normal band. In the literature, the naturally ordered orthodox semigroups satisfying the strong Dubreil-Jacotin condition were first considered by Blyth and Almeida Santos in 1992. Based on the name “epigroup” in the paper of Blyth and Almeida S...

We study equivariant resolutions and local cohomologies of toric sheaves for affine toric varieties, where our focus is on the construction of new examples of indecomposable maximal Cohen-Macaulay modules of higher rank. A result of Klyachko states that the category of reflexive toric sheaves is equivalent to the category of vector spaces together with a certain family of filtrations. Within th...

There is a growing demand for engineering applications which need a sophisticated treatment of geometric properties. Implementations of Euclidian geometry, commonly used in current commercial Geographic Information Systems and CAD/CAM, are impeded by the finiteness of computers and their numbering systems. To overcome these deficiencies a spatial data model is proposed which is based upon the m...