Building and Environment

Abstract This paper addresses the question: given a scalar group, can we determine all the additions that transform this scalar group into a (near-)field? A key approach to addressing this problem involves transporting (near-)field structures via multiplicative automorphisms. We compute the set of continuous multiplicative automorphisms of the real and complex fields and analyze their structures. Additionally, we characterize the endo-bijections on the scalar group that define these additions.

Abstract The main aim of this note is to introduce and study the properties of cohesive frames. We study the extent to which some classical results may be extended to a larger terrain of pointfree topology and point out some interesting observations and results in this context. In particular, we show that the class of all cohesive frames contains all nontrivial pointless connected frames; a result that has no interpretation in classical topology. We introduce the notion of strong cohesion and study conditions under which cohesive frames are strongly cohesive.

Abstract Let $$\mathcal {A}_{p}$$ A p be the class of functions f(z) of the form $$ f(z)=z^{p}+a_{p+1}z^{p+1}+a_{p+2}z^{p+2}+\cdots , (p\in \mathbb {N}=\{1,2,3,\ldots \}) $$ f ( z ) = z p + a p + 1 z p + 1 + a p + 2 z p + 2 + ⋯ , ( p ∈ N = { 1 , 2 , 3 , … } ) that are analytic in the open unit disc $$\mathbb {U}=\big \{ z\in \mathbb {C}: |z| <1\big \}$$ U = { z ∈ C : | z | < 1 } . For $$f(z)\in \mathcal {A}_{p}$$ f ( z ) ∈ A p , Nunokawa considered some conditions such that f(z) is $$p-$$ p - valent in $$\mathbb {U}$$ U . Applying the results by Nunokawa, we discuss some interesting properties for functions $$f(z)\in \mathcal {A}_{p}$$ f ( z ) ∈ A p . Also, we give some examples for our results.

Abstract Robotics and networking have transformed the world in numerous ways. Nowadays, computer networks play a vital role not only in business, education, and medical treatments, but also in entertainment. Graph theory has wide-ranging applications in robotics, tricky games, computer networking, map formations, image processing, Loran or Sonar models, pattern recognition, artificial intelligence, medical networks (such as oxygen or other treatment wiring), navigation problems, electrical networks, and more. Metric dimension has left its mark on artificial intelligence, map navigation, image recognition, pattern formation, facility location problems, and resource management (cost, time, labor, and resources). Robots are used in almost every field of life, including saving lives in the medical field, and metric dimension plays a crucial role in ensuring their accuracy. In this recent work, we enhance the concept of reflection of graphs and introduce new graphs, which we have called horizontal reflection graphs. We also compute the metric dimension of $$h-Vrl(mid(Tower_{p}))$$ h - V r l ( m i d ( T o w e r p ) ) and h-Vrl(Middle tower Path Graph) and found it to be constant.

For a commutative ring R with identity, the zero-divisor graph of R, denoted $$\Gamma (R)$$ , is the simple graph whose vertices are the nonzero zero-divisors of R where two distinct vertices x and y are adjacent if and only if $$xy=0$$ . In this paper, we are interested in partitioning the vertex set of $$\Gamma (R)$$ into global defensive alliances for a finite commutative ring R. This problem has been well investigated in graph theory. Here we connected it with the ring theoretical context. We characterize various commutative finite rings for which the zero-divisor graph is partitionable into global defensive alliances. We also give several examples to illustrate the scopes and limits of our results.

In this article, with the aim of further investigating BL-algebras, the concepts of a prime less filter is introduced and discussed. Also, for the faster study of prime less filters in BL-algebras, some equivalent conditions are obtained. We prove some relations between this filter and other types of filters in BL-algebras. Also, some results that proven for prime filters in BL-algebras, to the case of prime less filters are proven. In this way, we find new properties for the primary filters in BL-algebras. Also, the notions of prime less BL-algebras (Pls–BL-algebras) are introduced and investigated.

In this paper, two-fluid cosmological models have been investigated in five-dimensional Kaluza–Klein spacetime, where one fluid represents the barotropic fluid of the universe and another fluid is chosen to model dark energy. The Einstein’s field equations have been solved for interacting as well as the non-interacting two-fluid scenario and physical behaviour of parameters are discussed in view of results obtained from current observations.The cosmological models presented in the article are best described by quintessence dark energy cosmological models of the universe. The total density parameter ( $$\varOmega $$ ) value remains the same either two-fluid are interacting or non-interacting. Our models are consistent with the current lambda-CDM model of the universe and show that $$q\to -1$$ and $$j\left(t\right)\to 1$$ as $$t\to \infty $$ .

Abstract In the present paper, we introduce the subclasses $$\sum _{b}^{*}\left( q,\phi \right) $$ ∑ b ∗ q , ϕ and $$\sum _{b}^{*}\left( \alpha ,q,\phi \right) $$ ∑ b ∗ α , q , ϕ of meromorphic functions $$f\left( z\right) $$ f z satisfying $$1+\frac{1}{b}\left[ -\frac{qzD_{q}^{*}f(z)}{f(z)}-1\right] \prec \phi (z)$$ 1 + 1 b - q z D q ∗ f ( z ) f ( z ) - 1 ≺ ϕ ( z ) and $$1+\frac{1}{b}\left[ \frac{-\left( 1-\frac{\alpha }{q}\right) qzD_{q}^{*}f\left( z\right) +\alpha qzD_{q}^{*}\left[ zD_{q}^{*}f\left( z\right) \right] }{\left( 1-\frac{\alpha }{q}\right) f\left( z\right) -\alpha zD_{q}^{*}f\left( z\right) }-1\right] \prec \phi (z)\ (b\in \mathbb {C} ^{*}=\mathbb {C}\backslash \left\{ 0\right\} ,\ $$ 1 + 1 b - 1 - α q q z D q ∗ f z + α q z D q ∗ z D q ∗ f z 1 - α q f z - α z D q ∗ f z - 1 ≺ ϕ ( z ) ( b ∈ C ∗ = C \ 0 , $$\alpha \in \mathbb {C}\backslash (0,1],\ \operatorname {Re}(\alpha )\ge 0,\ 0<q<1)$$ α ∈ C \ ( 0 , 1 ] , Re ( α ) ≥ 0 , 0 < q < 1 ) , respectively. Sharp bounds for the Fekete-Szegö functional $$\left| a_{1}-\mu a_{0}^{2}\right| $$ a 1 - μ a 0 2 are obtained.

Abstract We interpret in the setting of modules over schemes classical results pertaining to involutions of the first kind on algebras $${{\,\textrm{End}\,}}_R(V)$$ End R ( V ) , where V is a faithfully projective R-module.

We initiate the algebraic study of the semigroup of one-to-one order-preserving partial contraction mappings of a totally ordered set $$\{1,2,\ldots ,n\}$$ , which we denote by $$\mathcal {OCI}_{n}$$ . In particular, we characterise the Green’s relations and their starred analogues in $$\mathcal {OCI}_{n}$$ . We also compute the rank of $$\mathcal {OCI}_{n}$$ as $$2n-1$$ .

Abstract Let $$\mathbb {F}_q$$ F q be the finite field with q elements and $$\mathbb {F}_q[x_1,\ldots , x_n]$$ F q [ x 1 , … , x n ] the ring of polynomials in n variables over $$\mathbb {F}_q$$ F q . In this paper we consider permutation polynomials and local permutation polynomials over $$\mathbb {F}_q[x_1,\ldots , x_n]$$ F q [ x 1 , … , x n ] , which define interesting generalizations of permutations over finite fields. We are able to construct permutation polynomials in $$\mathbb {F}_q[x_1,\ldots , x_n]$$ F q [ x 1 , … , x n ] of maximum degree $$n(q-1)-1$$ n ( q - 1 ) - 1 and local permutation polynomials in $$\mathbb {F}_q[x_1,\ldots , x_n]$$ F q [ x 1 , … , x n ] of maximum degree $$n(q-2)$$ n ( q - 2 ) when $$q>3$$ q > 3 , extending previous results.

Abstract A shear-free relativistic fluid with nonzero heat flux is studied by examining the higher dimensional spherically symmetric shear-free interior spacetime. The model is a heat conducting fluid undergoing dissipation. The Einstein field equations are generated in arbitrary dimensions for such a fluid and the assumption of isotropic fluid pressure realises a coupled partial differential equation in the gravitational potentials. We generate exact solutions to the isotropic pressure condition equation which represents a generalisation of the known four dimensional solutions. We also illustrate that the matter variables are physically well behaved and that the energy conditions are satisfied which is essential for constructing a complete and physically viable model in the context of radiating stars with vanishing shear.

Abstract M. Ward and R.P. Dilworth were the first to describe the commutative residuated lattices as a generalization of ideal ring lattices. Complete studies on residuated lattices were developed by H. Ono, T. Kowalski, P. Jipsen and C. Tsinakis. Furthermore, Y. Xu is credited with the invention of lattice implication algebra. The aim of the paper was to link up the structures used in foundations of quantum logic and arising in many-valued reasoning. It is shown that implication algebra with unity can be described as residuated lattice.

Abstract It is well known that the closure operator on a lattice is an extensive, isotone, and idempotent map. In this paper, we extend this concept by introducing the notion of an expansion map on lattices, which serves as a generalization of closure operators. The focus is to explore the properties of the collection of all expansion maps on a lattice, which forms a lattice. We delve into the discussion of their covering relation and present a comprehensive characterization of the atoms and dual atoms within the lattice obtained from these expansion maps.

Using the Brzdȩk fixed point theorem, we establish the Hyers–Ulam stability of the following generalized Drygas functional equation $$\begin{aligned} \mathcal {F}(x+\sigma (y))+\mathcal {F}(x+\tau (y))=2\mathcal {F}(x)+\mathcal {F}(\sigma (y))+\mathcal {F}(\tau (y)) \end{aligned}$$ for all $$x,y\in X$$ , where $$\sigma ,\tau : X\rightarrow X$$ are additive isometries.