Narrow Results

The quotient complexity, also known as state complexity, of a regular language is the number of distinct left quotients of the language. The quotient complexity of an operation is the maximal quotient complexity of the language resulting from the operation, as a function of the quotient complexities of the operands. The class of star-free languages is the smallest class containing the finite languages and closed under boolean operations and concatenation. We prove that the tight bounds on the quotient complexities of union, intersection, difference, symmetric difference, concatenation and star for star-free languages are the same as those for regular languages, with some small exceptions, whereas 2ⁿ-1 is a lower bound for reversal.

The quotient of a formal language $K$ by another language $L$ is the set of all strings obtained by taking a string from $K$ that ends with a suffix of a string from $L$, and removing that suffix. The quotient of a regular language by any language is always regular, whereas the context-free languages and many of their subfamilies, such as the linear and the deterministic languages, are not closed under the quotient operation. This paper establishes the closure of the family of languages recognized by input-driven pushdown automata (IDPDA), also known as visibly pushdown automata, under the quotient operation. A construction of automata representing the result of the operation is given, and its state complexity with respect to nondeterministic IDPDA is shown to be exactly $m^{2}n$, where $m$ and $n$ are the numbers of states in the automata recognizing $K$ and $L$, respectively.

The multiarrow formalism for coupled cell networks permits multiple arrows and self-loops. The Lifting Theorem states that any such network is a quotient of a network in which all arrows are single and self-loops do not occur. Previous proofs are inductive, and give no useful estimate of the minimal size of the lift. We give a noninductive proof of the Lifting Theorem, and identify the number of cells in the smallest possible lift. We interpret this construction in terms of the type matrix of the network, which encodes its topology and labeling.

In this paper, we use semigroupoids to describe a notion of algebraic bundles, mostly motivated by Fell ($C^{\ast }$-algebraic) bundles, and the sectional algebras associated to them. As the main motivational example, Steinberg algebras may be regarded as the sectional algebras of trivial (direct product) bundles. Several theorems which relate geometric and algebraic constructions — via the construction of a sectional algebra — are widely generalized: Direct products bundles by semigroupoids correspond to tensor products of algebras; semidirect products of bundles correspond to “naïve” crossed products of algebras; skew products of graded bundles correspond to smash products of graded algebras; Quotient bundles correspond to quotient algebras. Moreover, most of the results hold in the non-Hausdorff setting. In the course of this work, we generalize the definition of smash products to groupoid graded algebras. As an application, we prove that whenever $𝜃$ is a ∧-preaction of a discrete inverse semigroupoid $S$ on an ample (possibly non-Hausdorff) groupoid ${𝒢}$, the Steinberg algebra of the associated groupoid of germs is naturally isomorphic to a crossed product of the Steinberg algebra of ${𝒢}$ by $S$. This is a far-reaching generalization of analogous results which had been proven in particular cases.

This paper summarizes substantive new results derived by a student team (the first three authors) under the direction of the fourth author at the 2008 session of the KSU REU "Brainstorming and Barnstorming". The main results show that the construction of the inner automorphism group of a quandle gives rise to a functor from the category of quandles and surjective quandle homomorphisms to the category of groups, characterize quotient maps of quandles which do not change the group of inner automorphims, and characterize those normal subgroups of the inner automorphism group which arise as kernels of homomorphisms induced by quandle surjections.

In this paper we introduce the concept of a covering of a ffsm by another, admissible partitions and relations of a ffsm, μ-orthogonality of admissible partitions, irreducibile ffsm, and the quotient of a ffsm induced by an admissible partition of the state set.

The paper studies the zeros of 2mI_m/I _{m - 1} - (m+1)I₁/I₀, where I_m are the Bessel functions.

In this paper, we extend the notion of quotient of linear operators to linear relations. We introduce for two given linear relations $A$ and $B$, the linear relation quotient $B/A$ and we give a detailed treatment of some basic algebraic and topological properties of this new notion.

In this paper, we introduce the left and right quotients of a fuzzy language and discuss their algebraic properties. Then, we define the λ-cut of a universal fuzzy automaton, prove that the λ-cut of the universal fuzzy automaton corresponding to a fuzzy language is just the universal automaton corresponding to the λ-cut of this fuzzy language.

We show that every irreducible, simply connected curve on a toric affine surface X over ℂ is an orbit closure of a 𝔾_m-action on X. It follows that up to the action of the automorphism group Aut(X) there are only finitely many nonequivalent embeddings of the affine line 𝔸¹ in X. A similar description is given for simply connected curves in the quotients of the affine plane by small finite linear groups. We provide also an analog of the Jung-van der Kulk theorem for affine toric surfaces, and apply this to study actions of algebraic groups on such surfaces.