We consider the state complexity of extensions of the Kleene star and quotient operations to unranked tree languages. Due to the nature of the tree structure, there are two distinct ways to define the star operation for trees, we call these operations, respectively, bottom-up and top-down star. We show that (n+ 3 2 )2 states are sufficient and necessary in the worst case to recognize the bottom...

The aim of this note is to provide a quick proof of the Borel-Weil-Bott theorem, which describes the cohomology of line bundles on flag varieties. Let G denote a reductive algebraic group over the field C of complex numbers. We let B denote a Borel subgroup of G, and X = G/B be quotient of G by B, acting by right multiplication. The quotient X is a compact complex manifold. The anticanonical bu...

0 //K ⊂ //E q //Q //1. Here K stands for kernel and Q stands for quotient. She assumed that K is abelian (= commutative), so she wrote its product as + with identity element 0. She did not assume that Q is abelian, so she wrote its product as · with identity element 1. That explains the joke notation with 0 at the left and 1 at the right. The sequence is exact, which means that K is a normal su...

In this paper we derive a new algorithm for constructing a unitary decomposition of a sequence of matrices in product or quotient form. The unitary decomposition requires only unitary left and right transformations on the individual matrices and amounts to computing the generalized singular value decomposition of the sequence. The proposed algorithm is related to the classical Golub–Kahan proce...

Based on the general operation ◦ of words, called bw-operation, the notions of ◦-primitive words, ◦-closed languages, ◦-bases of languages and operationleft-quotient-closed languages are defined and investigated. These notions turn out to be generalizations of the classical notions of primitive words, plus-closed (star-closed) languages, minimal generating sets and deletion-closed languages. Pr...

The quotient complexity of a regular language L, which is the same as its state complexity, is the number of left quotients of L. An atom of a non-empty regular language L with n quotients is a non-empty intersection of the n quotients, which can be uncomplemented or complemented. An NFA is atomic if the right language of every state is a union of atoms. We characterize all reduced atomic NFAs ...

The Hunter–Saxton equation is the Euler equation for the geodesic flow on the quotient space of the infinite-dimensional group of orientation preserving diffeomorphisms of the unit circle modulo the subgroup of rigid rotations equipped with a right-invariant metric. We establish several properties of this quotient space: it has constant sectional curvature equal to 1, the Riemannian exponential...

We report the case of a 31-year-old man who had mild traumatic brain injury as a result of an accident at the age of 24 years. Seven years after the trauma, at the age of 31 years, he had a lower verbal intelligence quotient than performance intelligence quotient by the Wechsler Adult Intelligence Scale - Revised, and frontal lobe dysfunction, for example, difficulty in maintaining or changing ...

We study the state complexity of regular operations in the class of ideal languages. A language L ⊆ Σ∗ is a right (left) ideal if it satisfies L = LΣ∗ (L = Σ∗L). It is a two-sided ideal if L = Σ∗LΣ∗, and an all-sided ideal if L = Σ∗ L, the shuffle of Σ∗ with L. We prefer the term “quotient complexity” instead of “state complexity”, and we use derivatives to calculate upper bounds on quotient co...