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If (S, ◦,≤) is an ordered hypersemigroup, an equivalence relation ρ on S is called congruence if (a, b) ∈ ρ implies (a ◦ x, b ◦ x) ∈ ρ and (x ◦ a, x ◦ b) ∈ ρ for every x ∈ S ; in the sense that for every u ∈ a ◦ x there exists v ∈ b ◦ x such that (u, v) ∈ ρ and for every u ∈ x◦a there exists v ∈ x◦b such that (u, v) ∈ ρ. It has been proved in Turk J Math 2021(5) [On the paper “A study on (strong) order-congruences in ordered semihypergroups”] that if S is an ordered hypersemigroup, then there exists a congruence ρ on S such that S/ρ is an ordered hypersemigroup. This result, is the main result for an involution ordered hypersemigroup by Xinyang Feng, Jian Tang and Yanfeng Luo in U.P.B. Sci. Bull, Series A, 2018, but its proof is wrong; the correct proof is given in the present paper. Examples illustrate the results. [ABSTRACT FROM AUTHOR]

This note is written to show that the definition of the LA-semihypergroup by V. Amjad, K. Hila and F. Yousafzai "Generalized hyperideals in locally associative left almost semihypergroups, New York J. Math. 2014" should be corrected and that it is not enough to replace the multiplication "·" of an LA-semigroup by the hyperoperation "ο" to pass from an LA-semigroup to an LA-semihypergroup. The two examples of the paper based on the definition of the LA-semihypergroup are wrong that is a further indication that this definition needs correction. According to the last section of the paper, the paper generalizes the results of an LA-semigroup by M. Akram, N. Yaqoob and M. Khan "On (m, n) -ideals of left almost semigroups, Appl. Math. Sci. (Ruse) 2013" while the paper duplicates, without citation, the section 4 of the paper by W. Khan, F. Yousafzai, W. Guo and M. Khan "On (m, n) -ideals of left almost semigroups, J. Semigroup Theory Appl. 2014" with the usual change of "·" to "ο". [ABSTRACT FROM AUTHOR]

As a continuation of the paper "Adjunction Identity to Hypersemigroup" in Turk J Math 2022; 46 (7): 2834-2853, it has been proved here that the adjunction of a greatest element to an ordered hypersemigroup is actually an embedding problem. The concept of pseudoideal has been introduced and has been proved that for each ordered hypersemigroup S an ordered hypersemigroup V having a greatest element (poe-hypersemigroup) can be constructed in such a way that there exists a pseudoideal T of S such that S is isomorphic to T. If S does not have a greatest element, then this can be regarded as the embedding of an ordered hypersemigroup in an ordered semigroup with greatest element. [ABSTRACT FROM AUTHOR]

Inequalities are frequently used in various fields of mathematics to prove theorems. The existence of inequalities contributes significantly to the foundations of such branches. In this paper, we study the properties of order relations in the system of dual numbers, which is inspired by order relations defined on real numbers. Besides, some special inequalities that are used in various fields of mathematics, such as Cauchy-Schwarz, Minkowski, and Chebyshev are studied in this framework. An example is also provided to validate our research findings. [ABSTRACT FROM AUTHOR]

This is from the paper "Hypergroupes canoniques valués et hypervalués" by J. Mittas in Mathematica Balkanica 1971: "The concept of hypergroup introduced by Fr. MARTY in 1934 [Actes du Congrès des Math. Scand. Stocholm 1935, p. 45] is as follows: "A hypergroup is a nonempty set H endowed with a multiplication xy such that, for every x, y, z ? H, the following hold: (1) xy ? H; (2) x(yz) = (xy)z and (3) xH = Hx = H. The first condition expresses that the multiplication is an hyperoperation on H, in other words, the composition of two elements x, y of H is a subset of H. It is very easy to prove that for any x, y ? H, we have xy ?= Ø." Although according to Mittas "it is very easy to prove that xy ?= Ø", this is not possible. The notation x(yz) has a meaning of course if we identify the x by {x} and define an operation between sets. The authors working on hypersemigroups added in the definition by Mittas, the following: x(yz) = (xy)z means that ? u?yz xu = ? v?xy vz. But we never use this last equality in the papers on hypersemigroups in which we always use the x(yz) = (xy)z. As a result, most of the results of ordered hypersemigroups are copies from corresponding results on ordered semigroups in which the multiplication "·" has been replaced by "?". [ABSTRACT FROM AUTHOR]

The weak L1 -space meets in many areas of mathematics. For example, the conjugate functions of Lebesgue integrable functions belong to the weak L1 -space. The difficulty of working with the weak L1 -space is that the weak L1 -space is not a normed space. Moreover, infinitely differentiable (even continuous) functions are not dense in this space. Due to this, the theory of approximation was not produced in this space. In the present paper, we introduced the concept of the modulus of continuity of the functions from the weak L1 -space, studied its properties, found a criterion for convergence to zero of the modulus of continuity of the function from the weak L1 -space, and proved in this space an analogue of the Jackson-type theorem. [ABSTRACT FROM AUTHOR]

In a recent paper, Yu et al. obtained convergence and stability results of the K* iterative method for quasistrictly contractive mappings [An iteration process for a general class of contractive-like operators: Convergence, stability and polynomiography. AIMS Mathematics 2021; 6 (7): 6699-6714.]. To guarantee these convergence and stability results, the authors imposed some strong conditions on parametric control sequences which are used in the K* iterative method. The aim of the presented work is twofold: (a) to recapture the aforementioned results without any restrictions imposed on the mentioned parametric control sequences (b) to complete the work of Yu et al. by adding a result regarding the data dependency of the fixed points of quasi-strictly contractive mappings. We also furnish some illustrative examples to support our results. Our work can be considered an important refinement and complement of the work of Yu et al. [ABSTRACT FROM AUTHOR]

In this paper, we have characterized the nature and form of solutions of the following nonlinear delaydifferential equation where bi C, L(z, f) are a linear delay-differential polynomial of f; n is positive integers; q, Q and P respectively are nonzero, nonconstant and any polynomials. Different special cases of our result will accommodate all the results of [J. Math. Anal. Appl., 452(2017), 1128-1144; Mediterr. J. Math., 13(2016), 3015-3027; Open Math., 18(2020), 1292-1301]. Thus our result can be considered as an improvement of all of them. We have also illustrated a handful number of examples to show that all the cases as demonstrated in our theorem actually occur and consequently the same are automatically applicable to the previous results. [ABSTRACT FROM AUTHOR]

The concept of Γ-hypersemigroup has been introduced in Turk J Math 2020; 44 (5): 1835-1851 in which it has in which it has been shown that various results on Γ-hypersemigroups can be obtained directly as corollaries of more general results from the theory of le-semigroups (i.e. lattice ordered semigroups having a greatest element) or poe-semigroups. As a continuation of the paper mentioned above, in the present paper, the concept of ordered Γ-hypersemigroups has been introduced, and their relation to lattice ordered semigroups is given. It has been shown that although the results on ordered Γ-hypersemigroups cannot be obtained as corollaries to the corresponding results of le or poe-semigroups, still the main idea comes from the le-semigroups or poe-semigroups, and the proofs go along the lines of the le or poe-semigroups. [ABSTRACT FROM AUTHOR]

The mathematics of DNA molecules is often studied as a double helix model. This paper focuses on the modelling of infinitesimal bending of DNA helices. The paper checks the flexibility of DNA molecule, i.e. the flexibility of double helix in infinitesimal bending theory. Actually, first, we find infinitesimal bending field of an arbitrary curve on the helicoid, that leaves bent curves on the helicoid, and show that helix bending on the helicoid is not possible. Finally, we deal with the infinitesimal bending of helicoid, using PDEs. Visualization of infinitesimal bending was done in Mathematica. [ABSTRACT FROM AUTHOR]

In a very recent work, Şeker and Sümer Eker [On subclasses of bi-close-to-convex functions related to the odd-starlike functions. Palestine Journal of Mathematics 2017; 6: 215-221] defined two subclasses of analytic bi-closeto- convex functions related to the odd-starlike functions in the open unit disk U. The main purpose of this paper is to generalize and improve the results of Şeker and Sümer Eker (in the aforementioned study) defining a comprehensive subclass of bi-close-to-convex functions. Also, we investigate the Fekete-Szegö type coefficient bounds for functions belonging to this new class. [ABSTRACT FROM AUTHOR]

In a very recent work, Şeker [Seker B. On a new subclass of bi-univalent functions defined by using Salagean operator. Turkish Journal of Mathematics 2018; 42: 2891-2896] defined two subclasses of analytic bi-univalent functions by means of Sălăgean differential operator and he obtained the initial Taylor-Maclaurin coefficient estimates for functions belonging to these classes. The main purpose of this paper is to improve the results obtained by Şeker in the aforementioned study. For this purpose, we define a general subclass of bi-univalent functions. [ABSTRACT FROM AUTHOR]

Let A(n) be the class of functions f (z) = anzn + an+1zn+1 + ... (n ∈ ℕ), which are analytic in the open unit disk 핌, where an ≠ 0. For f (z) ∈ A(n), Miller and Mocanu in 1978 showed a very interesting result for f (z). Applying the result due to Miller and Mocanu, we would like to consider some new results for such functions. Our results in this paper are generalizations for results by Nunokawa in 1992. [ABSTRACT FROM AUTHOR]

A basic goal of this paper is to investigate the tubular surface constructed by the spherical indicatrices of any spatial curve in the Euclidean 3-space. This kind of tubular surface is designed for the alternative moving frame {N, C, W} in conjunction with finding a relationship between the tubular surfaces and their special curves, such as geodesic curves, asymptotic curves, and minimal curves. The minimal curve γ on a surface is defined by the property that its fundamental coefficients satisfy Eq. (3.7) along the curve γ. At the end of this article, we exemplify these curves on the tubular surfaces with their figures using the program Mathematica. [ABSTRACT FROM AUTHOR]

In this paper, an integral representation is given for special bounded solutions of pseudoparabolic equations of the form ... by means of a generating pair of the corresponding class of the generalized Q-holomorphic functions in Lp,2(C), for p > 2, where a, b, c, d are functions of z alone. [ABSTRACT FROM AUTHOR]

In this paper, we define the digital Lusternik-Schnirelmann category catκ, introduce some of its properties, and discuss how the adjacency relation affects the digital Lusternik-Schnirelmann category. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with nonzero identity and n be a positive integer. In this paper, we study the concepts of n-absorbing δ-primary ideals and weakly n-absorbing δ-primary ideals, which are the generalizations of δ-primary ideals and weakly δ-primary ideals, respectively. We introduce the concepts of n-absorbing δ-primary ideals and weakly n-absorbing δ-primary ideals. Moreover, we give many properties of these new types of ideals and investigate the relations between these structures. [ABSTRACT FROM AUTHOR]

The computation of the distance between any two points of the Sierpinski gasket with respect to the intrinsic metric has already been investigated by several authors. However, to the best of our knowledge, in the literature there is not an explicit formula obtained by using the code set of the Sierpinski gasket. In this paper, we obtain an explicit formula for the intrinsic metric on the Sierpinski gasket via the code representations of its points. We finally give an important geometrical property of the Sierpinski gasket with regard to the intrinsic metric by using its code representation. [ABSTRACT FROM AUTHOR]

Let q ≥ 3 be a prime number and let H(λq) be the eλtended Hecke group associated with q. In this paper, we determine the presentation of the commutator subgroup (H (λq)a)' of the normal subgroup H (λq)α, where H (λq)a is a subgroup of index 2 in H (λq). Next we discuss the commutator subgroup (H2)' (λq) of the principal congruence subgroup H2 (λq) of H (λq). Then we show that some quotient groups of H (λq) are generalized M* -- groups. Finally, we prove some results related to some normal subgroups of H (λq), especially in the case q = 5. [ABSTRACT FROM AUTHOR]

In this paper, the notion of the pseudofield of values of matrices is introduced and studied. The relationship between quantum states and the field of values of matrices is mentioned. The notion of the pseudopolynomial numerical hull, as a generalization of the pseudofield of values, of matrices is introduced and some properties of this notion are investigated. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the lower bound for the dcsl index δκ of a κ-uniform dcsl of even cycle C2n, n ≥ 2, in terms of the dimension of a poset and prove that dim(F) ≤ δκ(C2n), where F is the range of any κ-uniform dcsl f of C2n, n ≥ 2. [ABSTRACT FROM AUTHOR]

In this paper, we develop an efficient technique in the framework of multiplicative calculus and suggest a new class of numerical methods for solving multiplicative nonlinear equations g (x) = 1. We also develop the convergence criteria of the proposed methods. We solve the population growth model and minimization problem, which demonstrate the implementation and efficiency of the new techniques. We also show that these techniques perform much better as compared to the similar ordinary methods for solving ordinary nonlinear equations f (x) = 0. [ABSTRACT FROM AUTHOR]

The aim of this paper is to estimate the shape of an unknown part of the boundary of a geometrical domain. The identification technique used to estimate this part is the observation of the solution of a diffusion problem on the known part of this boundary. This technique is based on the sentinels theory. [ABSTRACT FROM AUTHOR]

In the paper we discuss two functionals of the Fekete-Szegö type: Φf(µ) = a2a4 -- µ,a3² and ©/(µ) = a4 -- µa2a3 for an analytic function f (z) = z + a2z² + a3z³ + . . ., z € Δ, (Δ = {z € C : |z| < 1}) and a real number µ. We focus our research on the estimation of (µ)| and \©f (µ)|, while f is either in S* (the class of starlike functions) or in Κ (the class of convex functions). [ABSTRACT FROM AUTHOR]

In this paper, we define the delta derivative in the integer group ring and we show that the delta derivative is well defined on the free groups. We also define a polynomial invariant of oriented knot and link by carrying the delta derivative to the link group. Since the delta derivative is a generalization of the free derivative, this polynomial invariant called the delta polynomial is a generalization of the Alexander polynomial. in addition, we present a new polynomial called the difference polynomial of oriented knot and link, which is similar to the Alexander polynomial and is a special case of the delta polynomial. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a class of unitary operators defined on the Bergman space L²a (C+) of the right half plane C+ and study certain algebraic properties of these operators. Using these results, we then show that a bounded linear operator S from L²a(C+) into itself commutes with all the weighted composition operators Wa,a £ D if and only if S(w) = {Sbw,bw),w £ C+ satisfies a certain averaging condition. Here for a = c + id £ D, f £ L2a(C+),Waf = ... Some applications of these results are also discussed. [ABSTRACT FROM AUTHOR]

The paper is devoted to the study of metallic Riemannian structures. An integrability condition and curvature properties for these structures by means of a Φ-operator applied to pure tensor fields are presented. Examples of these structures are also given. [ABSTRACT FROM AUTHOR]

This paper deals with the global attractivity of positive solutions of the second-order nonlinear difference equationxn+1 = axkn + bk-1Σj=1xjnxk-jn-1 + cxkn-1/Axkn + Bk-1Σj=1xjnxk-jn-1 + Cxkn-1, k = 3, 4; ..., n = 0, 1, ..., where the parameters a, b, c, A, B, C and the initial values x0, x-1 are arbitrary positive real numbers. [ABSTRACT FROM AUTHOR]

An almost null ring is a ring R in which for all a, b ∊ R, a³ = 0, Ma² = 0 for some square-free integer M that depends on a and ab = ka² = lb² for some integers k, l. This paper is devoted to the classification of the almost null rings. [ABSTRACT FROM AUTHOR]

In this paper we prove that spacelike and timelike magnetic trajectories corresponding to the para-Kähler 2-form on a para-Kähler manifold (M, p, g) are circles on M. We then classify all para-Kähler magnetic curves in pseudo-Euclidean spaces E2nn. [ABSTRACT FROM AUTHOR]

Golchin and Rezaei (Commun Algebra 2009; 37: 1995-2007) introduced the weak version of Condition (PWP) for S -posets, called Condition (PWP)w. In this paper, we continue to study this condition. We first present a necessary and sufficient condition under which the S -poset A(I) satisfies Condition (PWP)w. Furthermore, we characterize pomonoids S over which all cyclic (Rees factor) S -posets satisfy Condition (PWP)w, and pomonoids S over which all Rees factor S -posets satisfying Condition (PWP)w have a certain property. Finally, we consider direct products of S -posets satisfying Condition (PWP)w. [ABSTRACT FROM AUTHOR]