This paper studies FA-presentable structures and gives a complete classification of the finitely generated FA-presentable cancellative semigroups. We show that a finitely generated cancellative semigroup is FA-presentable if and only if it is a subsemigroup of a virtually abelian group.

In this paper, we first introduce the notion of $c$-affine functions for $c> 0$. Then we deal with some properties of strongly convex functions in real inner product spaces by using a quadratic support function at each point which is $c$-affine. Moreover, a Hyers–-Ulam stability result for strongly convex functions is shown.

For an $n$-gon with vertices at points $1,2,cdots,n$, the Betti numbers of its suspension, the simplicial complex that involves two more vertices $n+1$ and $n+2$, is known. In this paper, with a constructive and simple proof, wegeneralize this result to find the minimal free resolution and Betti numbers of the $S$-module $S/I$ where  $S=K[x_{1},cdots, x_{n}]$ and $I$ is the associated ideal to ...

Affine semigroups – discrete analogues of convex polyhedral cones – mark the crossroads of algebraic geometry, commutative algebra and integer programming. They constitute the combinatorial background for the theory of toric varieties, which is their main link to algebraic geometry. Initiated by the work of Demazure [De] and Kempf, Knudsen, Mumford and Saint-Donat [KKMS] in the early 70s, toric...

‎highest weight modules of the double affine lie algebra $widehat{widehat{mathfrak{sl}}}_{2}$ are studied under a‎ ‎new triangular decomposition‎. ‎singular vectors of verma modules are‎ ‎determined using a similar condition with horizontal affine lie‎ ‎subalgebras‎, ‎and highest weight modules are described under the‎ ‎condition $c_1>0$ and $c_2=0$.

Continuous, piecewise affine (CPA) functions can be algorithmically parameterized to deliver Lyapunov functions for compact invariant sets. We discuss flexible structures and algorithms to manipulate CPA functions for these purposes and discuss their implementation in C++11 using the Armadillo linear algebra library. Especially, we discuss some of the new language features in C++11 that lead to...

Let V be a real vector space of dimension d and V ∗ its dual space. By a cone in V ∗ we will always mean a closed polyhedral cone σ with apex 0 such that σ ∩ −σ = {0}. Let Σ be a fan in V ∗, i.e., a collection of cones such that (1) if σ ∈ Σ then any face of σ belongs to Σ, (2) if σ1, σ2 ∈ Σ then σ1 ∩ σ2 is a face in both. We will assume that Σ is complete, that is ∪Σ = V ∗. The elements of Σ a...

In [7] we have shown that special types of simplicial arrangements of d lines contain simple arrangements which are related to a class of bivariate polynomials Jd(x,y) having many critical points with few critical values. The polynomials have been used in the construction of algebraic surfaces with many A and D singularities [4, 5, 6, 7]. Tilings exhibiting non crystallographic symmetries have ...

‎Let $R$ be a ring‎ ‎with the Jacobson radical $J(R)$ and let $picolon Rto R/J(R)$ be‎ the canonical map‎. ‎Then $pi$ induces an order preserving group homomorphism‎ ‎$K_0picolon K_0(R)to K_0(R/J(R))$ and an‎ ‎affine continuous map $S(K_0pi)$ between the state space $St(R/J(R))$ and the‎ ‎state space $St(R).$‎ ‎In this paper‎, ‎we consider the natural affine map $S(K_0pi).$ We give a condition ...