Narrow Results

We continue the study of the spiking neural P systems considered as transducers of binary strings or binary infinite sequences, and we investigate their ability to compute morphisms. The class of computed morphisms is rather restricted: length preserving or erasing, and the so-called 2-block morphisms can be computed; however, non-erasing non-length-preserving morphisms cannot be computed.

As is well-known, Axel Thue constructed an infinite word over a 3-letter alphabet that contains no squares, that is, no nonempty subwords of the form xx. In this paper we consider a variation on this problem, where we try to avoid approximate squares, that is, subwords of the form xx' where |x| = |x'| and x and x' are "nearly" identical.

We prove that the uniform recurrence of morphic sequences is decidable. For this we show that the number of derived sequences of uniformly recurrent morphic sequences is bounded. As a corollary we obtain that uniformly recurrent morphic sequences are primitive substitutive sequences.

Two words are M-equivalent iff they are indistinguishable by Parikh matrices. Even for the ternary alphabet, an incontestable characterization of the M-equivalence relation is long overdue, ever since the introduction of Parikh matrices by Mateescu et al. in 2001. Recent works by Atanasiu attempted to distinguish M-equivalent words by the Parikh matrices of their images under some morphism. This paper addresses various aspects of this approach. In particular, it is shown that no morphism is capable of completely separating M-equivalent words over a given alphabet. However, if the class of words is restricted in length, then such morphism exists, whose codomain is connected to the notion of t-spectrum.

In this paper, we study the geometric equivalence of algebras in several varieties of algebras. We solve some of the problems formulated in [2], in particular, that of geometric equivalence for real-closed fields and finitely generated commutative groups.

Let $\mathcal{𝒞}$ be a category with an involution ∗. Suppose that $\phi :X\to X$ is a morphism and $({\phi }_1,Z,{\phi }_2)$ is an (epic, monic) factorization of $\phi$ through $Z$, then $\phi$ is core invertible if and only if ${({\phi }^{\ast })}^2{\phi }_1$ and ${\phi }_2{\phi }_1$ are both left invertible if and only if $({({\phi }^{\ast })}^2{\phi }_1,Z,{\phi }_2)$, $({\phi }_2^{\ast },Z,{\phi }_1^{\ast }{\phi }^{\ast }\phi )$ and $({\phi }^{\ast }{\phi }_2^{\ast },Z,{\phi }_1^{\ast }\phi )$ are all essentially unique (epic, monic) factorizations of ${({\phi }^{\ast })}^2\phi$ through $Z$. We also give the corresponding result about dual core inverse. In addition, we give some characterizations about the coexistence of core inverse and dual core inverse of an $R$-morphism in the category of $R$-modules of a given ring $R$.

In this paper, given a morhism $\phi$ with its pseudo core inverse ${\phi }^{Ⓓ}$ and a morphism $\eta$ such that $1_X+{\phi }^{Ⓓ}\eta$ is invertible, a necessary and sufficient condition and two sufficient conditions are presented under which the additive property, namely ${(\phi +\eta )}^{Ⓓ}={(1_X+{\phi }^{Ⓓ}\eta )}^{-1}{\phi }^{Ⓓ}$ holds. Several interesting results about additive properties of core inverses of bounded linear operators presented in Huang et al. are generalized to the case of pseudo core inverse of morphism. Also, many results regarding additive properties of core-EP inverses of complex matrices studied by Ma and Stanimirović are extended to the cases of morphism.

In general, topological quantum field theory (TQFT) is studied in detail from the n-dimensional cobordism (nCob) to the Hilbert vector space. However, we study the TQFT in the different way in this paper, that is, the TQFT is studied from the Hilbert vector space to nCob. To do this, the theory called the ϕ-mapping topological current theory is used. The relation between the objects and zero points of the Hilbert states in the Hilbert vector space is studied in this frame. The relation between the morphism and topological current is revealed too.

Let M be a monoid and let θ be an endomorphism of M. We prove that if the Bruck–Reilly extension BR(M, θ) is finitely presented, then the Bruck–Reilly extension BR(M, θ^m) is also finitely presented for all m ⩾ 1.

Graph transformation system, which has the nimble assembly technology and the complete semantics and syntax foundation, is the new methodology about system composition. Ehrig presented a generic composition framework on connector architecture using Petri nets, which provides a powerful support for dynamic and static component assembly in different domains. However, this framework is not flexible enough in practice, because it restricts the disjunction among import interfaces in each reduction step. Therefore, we extend the framework by presenting an unique condition of composition, and prove that the framework is unique on structure and content under this condition.

The purpose of this paper is to make available to the mathematicians and the computer scientists who have limited background in foundations of category theory, an improved essential explanation of ACG and a comprehensible proof of consistency of the systems QM and ZF# in the system ACG.