Let K denote a field and let V denote a vector space over K with finite positive dimension. We consider an ordered pair of linear transformations A : V → V and B : V → V which satisfy both (i), (ii) below. (i) There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing B is diagonal. (ii) There exists a basis for V with respe...

In the paper of Keller and Reiten, it was shown that the quotient of a triangulated category (with some conditions) by a cluster tilting subcategory becomes an abelian category. After that, Koenig and Zhu showed in detail, how the abelian structure is given on this quotient category, in a more abstract setting. On the other hand, as is well known since 1980s, the heart of any tstructure is abel...

Let V denote a vector space with finite positive dimension. We consider a pair of linear transformations A : V → V and A : V → V that satisfy (i) and (ii) below: (i) There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A is diagonal. (ii) There exists a basis for V with respect to which the matrix representing A is irr...

A three-generation SU(5) GUT (three 10 and a single 5–5 pair) is constructed by compactification of the E8 heterotic string. The base manifold is the Z5 × Z5-quotient of the quintic, and the vector bundle is the quotient of a positive monad. The group action on the monad and its bundle-valued cohomology is discussed in detail, including topological restrictions on the existence of equivariant s...

In the present paper, we introduce a class of infinite Lie conformal superalgebras S(p), which are closely related to algebras extended Block type defined in [6]. Then all finite non-trivial irreducible modules over S(p) for p?C? completely classified. As an application, also classifications quotient s(n) n?1 and sh is isomorphic subalgebra algebra N=2 superconformal algebra. Moreover, as gener...

We consider the action of a subtorus of the big torus on a toric variety. The aim of the paper is to define a natural notion of a quotient for this setting and to give an explicit algorithm for the construction of this quotient from the combinatorial data corresponding to the pair consisting of the subtorus and the toric variety. Moreover, we study the relations of such quotients with good quot...

In a recent pair of contributions [2, 3], starting from a set of results about the relationship between the direct sum decomposition of (linear, time-invariant, differential) behaviors and the solvability of certain two-sided Bézout equations, it was shown that every autonomous behavior can be expressed as a direct sum of irreducible components. In this contribution, we further explore this iss...

Abstract. Let n = b1 + · · · + bk = b′1 + · · · + b′k be a pair of compositions of n into k positive parts. We say this pair is irreducible if there is no positive j < k for which b1 + · · · + bj = b′1 + · · · + b′j . The probability that a random pair of compositions of n is irreducible is shown to be asymptotic to 8/n. This problem leads to a problem in probability theory. Two players move al...

In this work, we state and prove Lerch’s theorems for Fermat and Euler quotients over function fields defined analogously to the number fields. 1. Results The Fermat’s little theorem states that if p is a prime and a is an integer not divisible by p, then ap−1 ≡ 1 mod p. This gives rise to the definition of the Fermat quotient of p with base a, q(a, p) = ap−1 − 1 p , which is an integer. This q...

Let D be a division algebra such that D ⊗ D is a Noetherian algebra, then any division subalgebra of D is a finitely generated division algebra. Let ∆ be a finite set of commuting derivations or automorphisms of the division algebra D, then the group Ev(∆) of common eigenvalues (i.e. weights) is a finitely generated abelian group. Typical examples of D are the quotient division algebra Frac(D(X...