The Annals of Communications in Mathematics (ACM) is an international, interdisciplinary, open-access journal which provides an advanced forum for studies related to mathematical sciences that has been fully refereed since 2018. It devotes exclusively to the publication of high-quality reviews, regular research papers and short communications in all areas of Pure and Applied Mathematics. *ACM *also publishes timely and thorough survey articles on current trends, new theoretical techniques, novel ideas, and new Mathematical tools in different branches of Mathematics.

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**Editor-in-Chief:** *G. Muhiuddin, University of Tabuk, KSA*

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## Invariant Smooth Extensions

Annals of Communications in Mathematics 2024

, 7 (2)

, 91-94

DOI: https://doi.org/10.62072/acm.2024.070201

**Abstract**We study continuous extension operators for smooth functions from [0, ∞) to R. If A is a topological vector space of smooth functions in R, let us denote by A[0, ∞) the space of restrictions of functions of A to [0, ∞). We show that when A is any of the standard test function spaces D, S, or K then there is a continuous linear operator E from A[0, ∞) to A that satisfies that E (φ) (t) = φ (t) for t ≥ 0 and that satisfies the invariant condition E {ϕ (λx) ;t} = E {ϕ (x) ; λt} , for λ ≥ 0 . However, we show that when A is E, the space of all smooth functions, then such an operator E does not exist.

## Non-homogeneous Quinary Cubic Equation

Annals of Communications in Mathematics 2024

, 7 (2)

, 95-99

DOI: https://doi.org/10.62072/acm.2024.070202

**Abstract**This article is discussed for finding non-zero different solutions in integers to the non-homogeneous cubic equation with five unknowns represented by (x 3 − y ) = (z 3 −w3 )+ 72t 2. Various choices of integer solutions to the above equation are obtained through employing linear transformations and simplification. Some special results based on the solutions are also discussed.

## Almost Bi-ideals and Fuzzy Almost Bi-ideals of Ternary Semigroups

Annals of Communications in Mathematics 2024

, 7 (2)

, 100-107

DOI: https://doi.org/10.62072/acm.2024.070203

**Abstract**In this paper, we present the concepts of almost bi-ideals and fuzzy almost bi-ideals in ternary semigroups. The aim is to study their characterizations and establish the relation between different types of ideals in a ternary semigroup with various examples.

## Dirichlet Problem With Rough Boundary Values

Annals of Communications in Mathematics 2024

, 7 (2)

, 108-113

DOI: https://doi.org/10.62072/acm.2024.070204

**Abstract**Let D be a connected bounded domain in Rn, n ≥ 2, S be its boundary, which is closed and smooth. Consider the Dirichlet problem ∆u = 0 in D, u|S = f, where f ∈ L1 (S) or f ∈ H−ℓ , where H−ℓ is the dual space to the Sobolev space Hℓ := Hℓ (S), ℓ ≥ 0 is arbitrary. The aim of this paper is to prove that the above problem has a solution for an arbitrary f ∈ L1 (S) and this solution is unique and to prove similar result for rough (distributional) boundary values. These results are new. The method of its proof, based on the potential theory, is also new. Definition of the L1 (S)-boundary value and a distributional boundary value of a harmonic in D function is given. For f ∈ L1 (S) the difficulty comes from the fact that the product of an L1 (S) function times the kernel of the potential on S is not absolutely integrable. We prove that an arbitrary f ∈ H−ℓ , ℓ > 0, can be the boundary value of a harmonic function in D.

## Fuzzy Filters in Ordered Semirings

Annals of Communications in Mathematics 2024

, 7 (2)

, 114-127

DOI: https://doi.org/10.62072/acm.2024.070205

**Abstract**We introduce the notion of ideal, prime ideal, filter, fuzzy ideal, fuzzy prime ideal, fuzzy filter of an ordered semiring and study their properties and relations between them. We characterize the prime ideals and filters of an ordered semiring with respect to fuzzy ideals and fuzzy filters respectively. We proved a fuzzy subset µ is a fuzzy filter of an ordered semiring M if and only if µMT β, : X → [0, 1] is a fuzzy filter of an ordered semiring M. M and N be ordered semirings and ϕ : M → N be an onto homomorphism. If f is a ϕ homomorphism invariant fuzzy filter of M then ϕ(f) is a fuzzy filter of N.

## Multivariate Approximation by Parametrized Logistic Activated Multidimensional Convolution Type Operators

Annals of Communications in Mathematics 2024

, 7 (2)

, 128-159

DOI: https://doi.org/10.62072/acm.2024.070206

**Abstract**In this work we introduce for the first time the multivariate parametrized logistic activated convolution type operators in three kinds. We present their approximation properties, that is the quantitative convergence to the unit operator via the multivariate modulus of continuity. We continue with the multivariate global smoothness preservation of these operators. We present extensively the related multivariate iterated approximation, as well as, the multivariate simultaneous approximation and their combinations. Using differentiability into our research, we are producing higher speeds of approximation, multivariate simultaneous global smoothness preservation is also studied.

## Generalizations of Pseudo eBE-algebras

Annals of Communications in Mathematics 2024

, 7 (2)

, 160-175

DOI: https://doi.org/10.62072/acm.2024.070207

**Abstract**After BE-algebra was introduced by Kim et al., its generalization was attempted by several scholars. As a follow-up, Borzooei et al., Rezaei et al. and Sayyad et al. introduced pseudo BE-algebra, eBE-algebra and pseudo eBE-algebra, respectively. The aim of this article is to consider more extended version of pseudo eBE-algebra. The concepts of generalized pseudo eBE-algebra, (generalized) pseudo sub-eBE-algebra, eBEsubalgebra, eBE-filter, right (left) f-section and eBE-upper set are introduced and related properties, interrelationships and characterizations are studied.

## Ordered Nearness Semigroups

Annals of Communications in Mathematics 2024

, 7 (2)

, 176-185

DOI: https://doi.org/10.62072/acm.2024.070208

**Abstract**A semigroup is an algebraic structure that consists of a set and a binary operation that is associative, meaning that the order in which the operations are performed does not affect the outcome. For example, addition and multiplication are associative operations. The term “semigroup” was first used in its modern sense by Harold Hilton in his book on finite groups in 1908 [6]. Semigroups have since then been studied extensively in mathematics, and they have numerous applications in different fields, such as computer science, physics, and economics. Nearenness semigroup is a generalization of semigroups. Near set theory, which is a generalization of rough set theory, is based on the determination of universal sets according to the available information about the objects. Nearenness semigroups extend the concept of nearness from set theory to semigroups. ˙Inan and Öztürk applied the notion of near sets defined by J. F. Peters to the semigroups [10]. Our objective in this paper is to establish the definition of ordered semigroups on weak near approximation spaces. In addition, we investigated certain characteristics of these ordered nearness semigroups.

## Some Results on Multidimensional Fixed Point Theorems in Partially Ordered Generalized Intuitionistic Fuzzy Metric Spaces

Annals of Communications in Mathematics 2024

, 7 (2)

, 186-204

DOI: https://doi.org/10.62072/acm.2024.070209

**Abstract**In this paper, using the idea of a coincidence point for nonlinear mappings in any number of variables, we study a fuzzy contractivity condition to ensure the existence of coincidence points in the framework of generalized intuitionistic fuzzy metric spaces. Recently, many authors have conducted in-depth research on coupling, triple and quadruple fixed point theorems in the context of partially ordered complete metric spaces with different contractive conditions. In partially ordered generalized intuitionistic fuzzy metric spaces, we demonstrate several theorems regarding multidimensional co-incidence points and common fixed points for ϕ -compatible systems.

## q-Deformed and L-parametrized hyperbolic tangent function relied complex valued multivariate trigonometric and hyperbolic neural network approximations

Annals of Communications in Mathematics 2023

, 6 (3)

, Pages: 141-164

DOI: https://doi.org/10.62072/acm.2023.060301

**Abstract**Here we study the multivariate quantitative approximation of complex valued continuous functions on a box of RN , N ∈ N, by the multivariate normalized type neural network operators. We investigate also the case of approximation by iterated multilayer neural network operators. These approximations are achieved by establishing multidimen-sional Jackson type inequalities involving the multivariate moduli of continuity of the en- gaged function and its partial derivatives. Our multivariate operators are defined by using a multidimensional density function induced by a q-deformed and λ-parametrized hyper-bolic tangent function, which is a sigmoid function. The approximations are pointwise and uniform. The related feed-forward neural network are with one or multi hidden layers. The basis of our theory are the introduced multivariate Taylor formulae of trigonometric and hyperbolic type.

## Quotient quasi-ordered residuated systems induced by quasi-valuation maps

Annals of Communications in Mathematics 2023

, 6 (3)

, Pages: 199-208

DOI: https://doi.org/10.62072/acm.2023.060305

**Abstract**The concept of quasi-ordered residuated systems was introduced in 2018 by Bonzio and Chajda as a generalization both of commutative residuated lattices and hoopalgebras. Then this author investigated the substructures of ideals and filters in these algebraic structures. As a continuation of these research, in this article we design the concept of quotient quasi-ordered residuated systems induced by a quasi-valuation on it. Additionally, we prove some important properties of the thus constructed quotient structure.

## Trigonometric generated Lp degree of approximation

Annals of Communications in Mathematics 2023

, 6 (4)

, Pages: 209-219

DOI: https://doi.org/10.62072/acm2023060401

**Abstract**In this article we continue the study of smooth Picard singular integral operators that started in [3], see there chapters 10-14. This time the foundation of our research is a trigonometric Taylor’s formula. We establish the Lp convergence of our operators to the unit operator with rates via Jackson type inequalities engaging the first Lp modulus of continuity. Of interest here is a residual appearing term. Note that our operators are not positive.

## New definition of a singular integral operator

Annals of Communications in Mathematics 2023

, 6 (4)

, Pages: 220-224

DOI: https://doi.org/10.62072/acm.2023.060402

**Abstract**Let D be a connected bounded domain in R^2, S be its boundary which is closed, connected, and smooth, or S=(-∞,∞). Let Φ(z) be the function defined as Φ(z)=1/(2πi) ∫S(f(s)ds)/(s-z), where f∈L^1(S) and z=x+iy. The singular integral operator Af is defined as Af: =1/(iπ) ∫S(f(s)ds)/(s-t), where t∈S. This new definition simplifies the proof of the existence of Φ(t). Necessary and sufficient conditions are given for f∈L^1(S) to be the boundary value of an analytic function in D. The Sokhotsky-Plemelj formulas are derived for f∈L^1(S). Our new definition allows one to treat singular boundary values of analytic functions.

## Fuzzy soft tri-ideals over Gamma-semirings

Annals of Communications in Mathematics 2023

, 6 (4)

, Pages: 225-237

DOI: https://doi.org/10.62072/acm.2023.060403

**Abstract**In this paper, we introduce the notion of a fuzzy soft tri-ideal over Γ−semiring. We characterize the regular Γ−semiring in terms of fuzzy soft tri-ideals, and study some of the properties. M is a regular Γ−semiring, E be a parameters set and A ⊆ E. If (µ, A) is a fuzzy soft left tri-ideal over M, then (µ, A) is a fuzzy soft right ideal over M.

## q-Deformed and L-parametrized hyperbolic tangent function relied complex valued multivariate trigonometric and hyperbolic neural network approximations

Annals of Communications in Mathematics 2023

, 6 (3)

, Pages: 141-164

DOI: https://doi.org/10.62072/acm.2023.060301

**Abstract**Here we study the multivariate quantitative approximation of complex valued continuous functions on a box of RN , N ∈ N, by the multivariate normalized type neural network operators. We investigate also the case of approximation by iterated multilayer neural network operators. These approximations are achieved by establishing multidimen-sional Jackson type inequalities involving the multivariate moduli of continuity of the en- gaged function and its partial derivatives. Our multivariate operators are defined by using a multidimensional density function induced by a q-deformed and λ-parametrized hyper-bolic tangent function, which is a sigmoid function. The approximations are pointwise and uniform. The related feed-forward neural network are with one or multi hidden layers. The basis of our theory are the introduced multivariate Taylor formulae of trigonometric and hyperbolic type.

## Quotient quasi-ordered residuated systems induced by quasi-valuation maps

Annals of Communications in Mathematics 2023

, 6 (3)

, Pages: 199-208

DOI: https://doi.org/10.62072/acm.2023.060305

**Abstract**The concept of quasi-ordered residuated systems was introduced in 2018 by Bonzio and Chajda as a generalization both of commutative residuated lattices and hoopalgebras. Then this author investigated the substructures of ideals and filters in these algebraic structures. As a continuation of these research, in this article we design the concept of quotient quasi-ordered residuated systems induced by a quasi-valuation on it. Additionally, we prove some important properties of the thus constructed quotient structure.