Narrow Results

Let ${{𝒪}{𝒫}{ℰ}}_n$ be the monoid of all orientation-preserving and extensive full transformations on $\{1,\dots ,n\}$. In this paper, first we discuss left and right abundance of the principal ideals of ${{𝒪}{𝒫}{ℰ}}_n$. Second, we give necessary and sufficient conditions for the principal ideals of ${{𝒪}{𝒫}{ℰ}}_n$ to be abundant and regular. Finally, we study abundance of the ideals in ${{𝒪}{𝒫}{ℰ}}_n$.

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 study of the strange nonchaotic attractors is an interesting topic, where the dynamics are neither regular nor chaotic (the word chaotic means the positive Lyapunov exponents), and the shape of the attractors has complicated geometry structure, or fractal structure. It is found that in a class of planar first-order nonautonomous systems, it is possible that there exist attractors, where the shape of the attractors is regular, the orbits are transitive on the attractors, and the dynamics are not chaotic. We call this type of attractors as regular nonchaotic attractors with positive plural, which are different from the strange nonchaotic attractors, attracting fixed points, or attracting periodic orbits. Several examples with computer simulations are given. The first two examples have annulus-shaped attractors. Another two examples have disk-shaped attractors. The last two examples with externally driven terms at two incommensurate frequencies have regular nonchaotic attractors with positive plural, implying that the existence of externally driven terms at two incommensurate frequencies might not be the sufficient condition to guarantee that the system has strange nonchaotic attractors.

We investigate the dynamics of two types of nonautonomous ordinary differential equations with quasi-periodic time-varying coefficients and nonlinear terms. The vector fields for the nonautonomous systems are written as $ẇ=a{\mathrm{\Phi }}_1(w)+b{\mathrm{\Phi }}_2(t,w)$, $w\in {ℝ}^3$, where $a{\mathrm{\Phi }}_1$ is the spacial part and $b{\mathrm{\Phi }}_2$ is the time-varying part, and $a$ and $b$ are real parameters. The first type has a polynomial as the nonlinear term, another type has a continuous periodic function as the nonlinear term. The polynomials and periodic functions have simple zeros. Several examples with numerical experiments are given. It is found by numerical calculation that there might exist only one attractor for the systems with polynomials as nonlinear terms and $\left|b\right|\gg max\{1,|a|\}$, and there might exist infinitely many attractors for systems with periodic functions as nonlinear terms and $\left|b\right|\gg max\{1,|a|\}$. For $\left|b\right|$ sufficiently small, the parameter regions for $(a,b)$ are roughly divided into three parts: the spacial region ($\left|a\right|\gg \left|b\right|$), the balance region ($\left|a\right|\approx \left|b\right|$), and the time-varying region ($\left|a\right|\ll \left|b\right|$); (i) for $\left|a\right|\gg \left|b\right|$, the orbits approach some planes depending on the zeros of the polynomials or the periodic functions; (ii) for $\left|a\right|\approx \left|b\right|$, there exist attractors with the number no less than the number of zeros of the polynomials or the periodic functions, implying the existence of infinitely many attractors for systems with periodic functions as nonlinear terms; (iii) for $\left|a\right|\ll \left|b\right|$, the orbits wind around some region depending on the choice of the initial position. The shape of the attractors might be strange or regular for different parameters, and we obtain the existence of ball-like (regular) attractors, two-wings (strange) attractors, and other attractors with different shapes. The Lyapunov exponents are negative. These results reveal an intrinsic relationship between the existence of attractors (or strange dynamics) and the parameters $a$ and $b$ for nonautonomous systems with quasi-periodic coefficients. These results will be very useful in the understanding of the dynamics of general nonautonomous systems, nonautonomous control theory and other related fields.

A regular set of words is (k-)locally testable if membership of a word in the set is determined by the nature of its subwords of some bounded length k. In this article we study groups for which the set of all geodesic words with respect to some generating set is (k-)locally testable, and we call such groups (k-)locally testable. We show that a group is 1-locally testable if and only if it is free abelian. We show that the class of (k-)locally testable groups is closed under taking finite direct products. We show also that a locally testable group has finitely many conjugacy classes of torsion elements.

Our work involved computer investigations of specific groups, for which purpose we implemented an algorithm in GAP to compute a finite state automaton with language equal to the set of all geodesics of a group (assuming that this language is regular), starting from a shortlex automatic structure. We provide a brief description of that algorithm.

We study properties determined by idempotents in the following families of matrix semigroups over a semiring $S$: the full matrix semigroup $M_n(S)$, the semigroup $U{T}_n(S)$ consisting of upper triangular matrices, and the semigroup $U_n(S)$ consisting of all unitriangular matrices. Il’in has shown that (for $n>2$) the semigroup $M_n(S)$ is regular if and only if $S$ is a regular ring. We show that $U{T}_n(S)$ is regular if and only if $n=1$ and the multiplicative semigroup of $S$ is regular. The notions of being abundant or Fountain (formerly, weakly abundant) are weaker than being regular but are also defined in terms of idempotents, namely, every class of certain equivalence relations must contain an idempotent. Each of $M_n(S)$, $U{T}_n(S)$ and $U_n(S)$ admits a natural anti-isomorphism allowing us to characterise abundance and Fountainicity in terms of the left action of idempotent matrices upon column spaces. In the case where the semiring is exact, we show that $M_n(S)$ is abundant if and only if it is regular. Our main interest is in the case where $S$ is an idempotent semifield, our motivating example being that of the tropical semiring $𝕋$. We prove that certain subsemigroups of $M_n(S)$, including several generalisations of well-studied monoids of binary relations (Hall relations, reflexive relations, unitriangular Boolean matrices), are Fountain. We also consider the subsemigroups $U{T}_n(S^{\ast })$ and $U_n(S^{\ast })$ consisting of those matrices of $U{T}_n(S)$ and $U_n(S)$ having all elements on and above the leading diagonal non-zero. We prove the idempotent generated subsemigroup of $U{T}_n(S^{\ast })$ is $U_n(S^{\ast })$. Further, $U{T}_n(S^{\ast })$ and $U_n(S^{\ast })$ are families of Fountain semigroups with interesting and unusual properties. In particular, every $\widetilde{{ℛ}}$-class and $\widetilde{{ℒ}}$-class contains a unique idempotent, where $\widetilde{{ℛ}}$ and $\widetilde{{ℒ}}$ are the relations used to define Fountainicity, but yet the idempotents do not form a semilattice.

In this paper, we present a regular black hole with a positive cosmological constant. The regular black hole considered is the well known Bardeen black hole and it is a solution to the Einstein equations coupled to nonlinear electrodynamics with a magnetic monopole. The paper discusses the properties of the Bardeen–de Sitter black hole. We have computed the gray body factors and partial absorption cross-sections for massless scalar field impinges on this black hole with the third-order WKB approximation. A detailed discussion on how the behavior of the gray body factors depend on the parameters of the theory such as the mass, charge and the cosmological constant is given. Possible extensions of the work is discussed at the end of the paper.

In this paper, the dynamics of a Filippov system with logistically growing prey and Holling Type II predation is explored. The selective nonlinear harvesting of prey is carried out when the ratio of prey to predator is above a specified threshold value. In order to prevent the extinction of both the species and sustainable economic gains, no harvesting is allowed when the ratio is below a threshold value. The equilibrium states and their stability for the individual smooth are analyzed. Filippov convex method is applied to determine the dynamics on the boundary. The sliding mode dynamics is scrutinized, and it is observed that there is no escaping region. The existence of boundary points, tangent points and pseudo-equilibrium points are established. The conditions are obtained for the visibility of the tangent points. The complex behavior within the sliding region is studied by experimenting with different initial conditions. The increase in harvesting effort may lead to extinction of both the species through the sliding region of the boundary. The change in the stability behavior of both the interior equilibrium states is investigated with respect to model parameters. Border cross bifurcates with threshold value as the bifurcation parameter is shown for both the interior equilibrium states.

In this work, we present several ways to obtain different types of weighting triangles, due to these types characterize some interesting properties of Extended Ordered Weighted Averaging operators, EOWA, and Extended Quasi-linear Weighted Mean, EQLWM, as well as of their reverse functions. We show that any quantifier determines an EOWA operator which is also an Extended Aggregation Function, EAF, i.e., the weighting triangle generated by a quantifier is always regular. Moreover, we present different results about generation of weighting triangles by means of sequences and fractal structures. Finally, we introduce a degree of orness of a weighting triangle associated with an EOWA operator. After that, we mention some results on each class of triangle, considering each one of these triangles as triangles associated with their corresponding EOWA operator, and we calculate the ornessof some interesting examples.

Let R be a ring. Modules satisfying ascending or descending chain conditions (respectively, acc and dcc) on non-summand submodules belongs to some particular classes , such as the class of all R-modules, finitely generated, finite-dimensional and cyclic modules, are considered. It is proved that a module M satisfies acc (respectively, dcc) on non-summands if and only if M is semisimple or Noetherian (respectively, Artinian). Over a right Noetherian ring R, a right R-module M satisfies acc on finitely generated non-summands if and only if M satisfies acc on non-summands; a right R-module M satisfies dcc on finitely generated non-summands if and only if M is locally Artinian. Moreover, if a ring R satisfies dcc on cyclic non-summand right ideals, then R is a semiregular ring such that the Jacobson radical J is left t-nilpotent.

The rings in the title were called UN rings by Călugăreanu in [G. Călugăreanu, UN-rings, J. Algebra Appl.15(9) (2016) 1650182]. He gave two examples of simple UN rings: matrix rings over a skew field and a ring, which is the filtered union of such rings. We give new examples of simple UN rings as endomorphism rings of ‘vector space like’ modules and determine the structure of UN rings, which satisfy a polynomial identity or have Krull dimension. We also answer some questions in [G. Călugăreanu, UN-rings, J. Algebra Appl.15(9) (2016) 1650182] about Morita equivalence of UN rings and show that this question is related to Köthe’s conjecture. Finally a complete characterization is given of modules over a Dedekind domain (in particular Abelian groups) and modules of finite length with a UN endomorphism ring.

Let $R$ be a Cohen–Macaulay local ring. We prove that the $n$th syzygy module of a maximal Cohen–Macaulay $R$-module cannot have a semidualizing direct summand for every $n\ge 1$. In particular, it follows that $R$ is Gorenstein if and only if some syzygy of a canonical module of $R$ has a nonzero free direct summand. We also give a number of necessary and sufficient conditions for a Cohen–Macaulay local ring of minimal multiplicity to be regular or Gorenstein. These criteria are based on vanishing of certain Exts or Tors involving syzygy modules of the residue field.

Semiabelian rings, defined by the property that each of their idempotents is either left semicentral or right semicentral, are one among several natural generalizations of abelian rings. In this semi-expository paper, we review a number of interesting properties of semiabelian (and other closely allied) rings that are so far well known only for abelian rings. For instance, semiabelian rings $R$ are always “J-abelian” in the sense that each idempotent of $R$ maps onto a central idempotent in $R$/rad($R$). On the other hand, J-abelian rings turn out to be precisely the “strongly perspective rings” as well as the “strongly IC rings”, and the von Neumann regular elements in such rings are automatically strongly regular and are closed under taking $n$-th powers. In addition, all J-abelian exchange rings have idempotent stable range one, and are in particular clean (although not necessarily strongly clean) rings.

Protein secondary structure prediction (PSSP) is an important and challenging task in protein bioinformatics. Protein secondary structures (SSs) are categorized in regular and irregular structure classes. Regular SSs, representing nearly 50% of amino acids consist of helices and sheets, whereas the remaining amino acids represent irregular SSs. $\beta$-turns and $\gamma$-turns are the most abundant irregular SSs present in proteins. Existing methods are well developed for separate prediction of regular and irregular SSs. However, for more comprehensive PSSP, it is essential to develop a uniform model to predict all types of SSs simultaneously. In this work, using a novel dataset comprising dictionary of secondary structure of protein (DSSP)-based SSs and PROMOTIF-based $\beta$-turns and $\gamma$-turns, we propose a unified deep learning model consisting of convolutional neural networks (CNNs) and long short-term memory networks (LSTMs) for simultaneous prediction of regular and irregular SSs. To the best of our knowledge, this is the first study in PSSP covering both regular and irregular structures. The protein sequences in our constructed datasets, RiR6069 and RiR513, have been borrowed from benchmark CB6133 and CB513 datasets, respectively. The results are indicative of increased PSSP accuracy.

It is well known that a discounting function describes the price of a default free bond. In this paper, some mathematical properties of this function are studied when it is subadditive. Subadditive discounting means that the total discount is minor when the year is divided into months and implies that the more divided the interval is into subintervals, the smaller the total discount will be. First, two sufficient conditions for a discounting function being subadditive (resp. superadditive) are shown; more precisely, the increase (resp. decrease) of the instantaneous discount rate and certain integral inequality are proposed as such sufficient conditions. On the other hand, the convexity (resp. concavity) of the Napierian logarithm of the discounting function is shown as a particular case of another more general sufficient condition. Finally, under certain conditions, subadditive discounting functions are necessarily regular.

Any semigroup S can be embedded into a semigroup, denoted by ΨS, having some remarkable properties. For general semigroups there is a close relationship between local submonoids of S and of ΨS. For a number of usual semigroup properties , we prove that S and ΨS simultaneously satisfy or not. For a regular semigroup S, the relationship of S and ΨS is even closer, especially regarding the natural partial order and Green's relations; in addition, every element of ΨS is a product of at most four idempotents. For completely regular semigroups S, the relationship of S and ΨS is still closer. On the lattice of varieties of completely regular semigroups regarded as algebras with multiplication and inversion, by means of ΨS, we define an operator, denoted by Ψ. We compare Ψ with some of the standard operators on and evaluate it on a small sublattice of .

If X and Y are sets, we let P(X, Y) denote the set of all partial transformations from X into Y (that is, all mappings whose domain and range are subsets of X and Y, respectively). If θ ∈ P(Y, X), then P(X, Y) is a so-called "generalized semigroup" of transformations under the "sandwich operation": α * β = α ◦ θ ◦ β, for each α, β ∈ P(X, Y). We denote this semigroup by P(X, Y, θ) and, in this paper, we characterize Green's relations on it: that is, we study equivalence relations which determine when principal left (or right, or 2-sided) ideals in P(X, Y, θ) are equal. This solves a problem raised by Magill and Subbiah in 1975. We also discuss the same idea for important subsemigroups of P(X, Y, θ) and characterize when these semigroups satisfy certain regularity conditions.

It is well-known that the transformation semigroup $T(X)$ is regular, but their subsemigroups need not be. A fence is an ordered set that the order forms a path with alternating orientation. Consider $X$ as the base set of a fence $X=(X;\le )$. Two subsemigroups of $T(X)$ are studied. Namely, the semigroup $DT(X)$ of all order-decreasing self-mappings of $X$ and the semigroup $OT(X)$ of all order-preserving self-mappings of $X$. In this paper, we obtain that $DT(X)$ is a coregular subsemigroup of $T(X)$. A characterization of regular subsemigroups $OT(X)$ of $T(X)$ is given, that is, $OT(X)$ is regular if and only if $\left|X\right|\le 4$. Finally, we discuss the regularity of elements in $OT(X)$.

In this paper, we introduce the notion of derivation of pseudo BCI-algebras and investigate some related properties.

In this paper, nil extensions of simple regular ordered semigroups, left simple and right regular ordered semigroups, etc. have been characterized. Also, we describe the ordered semigroups which are complete semilattices of nil extensions of left simple and right regular ordered semigroups, left group like ordered semigroups, etc.