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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]

In this note, we slightly generalize Theorem 2 in the paper by M. Tunç and point out that the assumption of Theorem 3 is not sufficient. A misuse of the term 'mean' is also discussed. [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]

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]

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]

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]

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]

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]

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]

In this paper we classify the finite groups whose abelian subgroups of equal order (B*-groups) are conjugate. The classification has been achieved by means of a lot of general structure properties of B*-groups, provided in the course of the paper. [ABSTRACT FROM AUTHOR]

There are many old results of integrals involving Bessel functions, currently available in handbooks, but we found no recourse in the well-known references to how they were established. In this paper, we attempt to have a clear way of proving some of these results. In fact, we consider a certain class of Bessel integrals where we prove that such integrals vanish under certain conditions. To this end some theorems regarding this class of integrals with their proofs are put forward. A computer algorithm is provided to implement some of our results. The result in this paper extend the work in [3], and it concludes by indicating the wide range of old and new results that can be obtained. [ABSTRACT FROM AUTHOR]

We define new special curves in Euclidean 3-space which we call slant helices and conical geodesic curves. Those notions are generalizations of the notion of cylindrical helices. One of the results in this paper gives a classification of special developable surfaces under the condition of the existence of such a special curve as a geodesic. As a result, we consider geometric invariants of space curves. By using these invariants, we can estimate the order of contact with those special curves for general space curves. All arguments in this paper are straight forward and classical. However, there have been no papers which have investigated slant helices and conical geodesic curves so far as we know. [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]

The main interest of the present paper is to classify the almost cosymplectic 3-manifolds that satisfy ∥gradλ∥ = const.(≠ 0) and ∇ξh = 2ahΦ. [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]

This paper examines the existence of nontrivial periodic solutions for the nonlinear functional differential system with feedback control: (Multiple line equation(s) cannot be represented in ASCII text) Under certain growth conditions on the nonlinearity f , several sufficient conditions for the existence of nontrivial solution are obtained by using Leray-Schauder nonlinear alternative. [ABSTRACT FROM AUTHOR]

In this paper we first review the definitions of crossed module [10], pullback crossed module and cat¹-object in the category of commutative algebras. We then describe a certain pullback of cat¹-commutative algebras. [ABSTRACT FROM AUTHOR]

The concept of the ‘relation of comparability’ was introduced by Maillet in [7], who showed that if α,β are comparable Liouville numbers then each of the numbers α+β, α-β, αβ and α/β is either a rational or Lioville number. Moreover those which are Liouville numbers are comparable aamong theem and toα and β Maillet's proof uses in an essential way the transitivity of the comparability relation. Unfortunately, as the comparability relation is not transitive, his proof is defective. In this paper, without using the comparability relation, we obtain some uncountable subfields of p-adic numbers field, Qp. In [1] using a different notion of comparability, Alnıaçık was able to define some uncountable subfields of C. In this paper, without using comparability relation, we define irregular semistrong p-adic Um numbers and obtain some uncountable subfields of p-adic numbers field Qp. [ABSTRACT FROM AUTHOR]

In this paper, the geometry of F-invariant submanifolds of a Kaehlerian product manifold is studied. The fundamental properties of these submanifolds are investigated such as pseudo umbilical, curvature invariant, totally geodesic, mixed geodesic submanifold and locally decomposable Riemannian product manifold. [ABSTRACT FROM AUTHOR]

In this paper, we give a new characterization of Möbius transformations. To this end, a new concept of "Apollonius points of pentagons" is used. [ABSTRACT FROM AUTHOR]

In this paper, we present some results concerning two generalized derivations on a semiprime ring. These results are a generalization of results of M. Brešar and J. Vukman in [2], which are related to a theorem of E. Posner for the product of derivations on a prime ring. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of two-sided &alpha-derivation of a near-ring and give some generalizations of [1]. Let N be a near ring. An additive mapping f : N → N is called an (α β)-derivation if there exist functions α, β : N → N such that f(xy) = f(x)α(y) + β(x)f(y) for all x, y ∈ N. An additive mapping d : N → N is called a two-sided α-derivation if d is an (α 1)-derivation as well as a (1, α)-derivation. The purpose of this paper is to prove the following two assertions: (i) Let N be a semiprime near-ring, I be a subset of N such that 0 &sisn; I, IN ⊆ I and d be a two-sided α-derivation of N. If d acts as a homomorphism on I or as an anti-homomorphism on I under certain conditions on α, then d(I) = {0}. (ii) Let N be a prime near-ring, I be a nonzero semigroup ideal of N, and d be a (α, 1)-derivation on N. If d + d is additive on I, then (N,+) is abelian. [ABSTRACT FROM AUTHOR]

In two previous papers, the two first-named authors introduced a notion of contact r-surgery along Legendrian knots in contact 3-manifolds. They also showed how (at least in principle) to convert any contact r-surgery into a sequence of contact (± 1)-surgeries, and used this to prove that any (closed) contact 3-manifold can be obtained from the standard contact structure on S3 by a sequence of such contact (± 1)-surgeries. In the present paper, we give a shorter proof of that result and a more explicit algorithm for turning a contact r-surgery into (± 1)-surgeries. We use this to give explicit surgery diagrams for all contact structures on S3 and S1 x S2, as well as all overtwisted contact structures on arbitrary closed, orientable 3-manifolds. This amounts to a new proof of the Lutz-Martinet theorem that each homotopy class of 2-plane fields on such a manifold is represented by a contact structure. [ABSTRACT FROM AUTHOR]

Let Nn(4c) be the complex space form of constant holomorphic sectional curvature 4c , φ : M → Nn(4c) be an immersion of an n-dimensional Lagrangian manifold M in Nn(4c). Denote by S and H the square of the length of the second fundamental form and the mean curvature of M. Let ρ be the non-negative function on M defined by ρ2 = S - nH2 , Q be the function which assigns to each point of M the infimum of the Ricci curvature at the point. In this paper, we consider the variational problem for non-negative functional U(φ) = ∫∫M ρ2dv = ∫M(S - nH2)dv . We call the critical points of U(φ) the Extremal submanifold in complex space form Nn(4c) . We shall get the new Euler-Lagrange equation of U(φ) and prove some integral inequalities of Simons' type for n-dimensional compact Extremal Lagrangian submanifolds φ : M → Nn(4c) in the complex space form Nn(4c) in terms of ρ2, Q,H and give some rigidity and characterization Theorems. [ABSTRACT FROM AUTHOR]

In this paper, using the concept of A-statistical convergence for double sequences, we investigate a Korovkin-type approximation theorem for sequences of positive linear operator on the space of all continuous real valued functions defined on any compact subset of the real two-dimensional space. Then we display an application which shows that our new result is stronger than its classical version. We also obtain a Voronovskaya-type theorem and some differential properties for sequences of positive linear operators constructed by means of the Bernstein polynomials of two variables. [ABSTRACT FROM AUTHOR]

In the paper we consider the problem of uniqueness of derivatives of meromorphic functions when they share two or three sets and obtained five results which will improve all the existing results. [ABSTRACT FROM AUTHOR]

The concept of differential subordination was introduced in [3] by S.S. Miller and P.T. Mocanu and the concept of strong differential subordination was introduced in [1], [2] by J.A. Antonino and S. Romaguera. In [5] we have studied the strong differential subordinations in the general case and in [6] we have studied the first order linear strong differential subordinations. In this paper we study the second order linear strong differential subordinations. Our results may be applied to deduce sufficient conditions for univalence in the unit disc, such as starlikeness, convexity, alpha-convexity, close-to-convexity respectively. [ABSTRACT FROM AUTHOR]

The present paper has the object of showing some interesting relationship on the maximum modulus, the maximum term, the index of maximum term and the coefficients of entire functions defined by Dirichlet series of slow growth; some properties like Taylor entire functions are obtained. [ABSTRACT FROM AUTHOR]

A criterion is established on the bounded solutions of type higher-order nonlinear neutral differential equations of type oscillatory or tending to zero at infinity [a(t)[χ(t) + r(t)x(κ(t))](n-1)]′ + p(t)F(x(t(t))) + q(t)G(χ(σ(t))) = ϕ(t), where t ≥ t0 , n = 2, a, p are positive, r, q, ϕ are allowed to alternate in sign infinitely many times, F,G are continuous functions, and κ, 㱔, σ are strictly increasing unbounded continuous delay functions. [ABSTRACT FROM AUTHOR]

In this paper we consider a class of polynomials p(z) = a0 + ∑v=tn avzv , t ≥ 1, not vanishing in ∣z∣ < k, k ≥ 1 and investigate the dependence of max ∣z∣=1 ∣p(Rz) - p(z)∣ on max ∣z∣=1 ∣p(z)∣. Our result not only generalizes some polynomial inequalities, but also a variety of interesting results can be deduced from it by a fairly uniform procedure. [ABSTRACT FROM AUTHOR]

In this paper, we study the basic results on the general study of null curves in the Minkowski 4-space R14. A transversal vector bundle of a null curve in R14 is constructed using a frenet Frame consisting of two real null and two space-like vectors. The null curves are characterized by using the Frenet frame. [ABSTRACT FROM AUTHOR]

In this paper we characterize Q-modules and almost Q-modules. Next we establish some equivalent conditions for an almost Q-module to be a Q-module. Using these results, some characterizations are given for Noetherian Q-modules. [ABSTRACT FROM AUTHOR]

The concept of differential subordination was introduced in [4] by S. S. Miller and P. T. Mocanu and the concept of strong differential subordination was introduced in [1] by J. A. Antonino and S. Romaguera. This last concept was applied in the special case of Briot-Bouquet strong differential subordination. In this paper we study the strong differential subordinations in the general case, following the general theory of differential subordinations presented in [4]. [ABSTRACT FROM AUTHOR]

In this paper, we study some of the properties of gr-multiplication ideals in a graded ring R. We first characterize finitely generated gr-multiplication ideals and then give a characterization of gr-multiplication ideals by using the gr-localization of R. Finally we determine the set of gr-P -primary ideals of R when P is a gr-multiplication gr-prime ideal of R. [ABSTRACT FROM AUTHOR]

Let Ap denote the class of functions of the form f (z) = zp+ap+1 zp+1+a p+2zp+2+…which are regular and p-valent in the open unit disc D = {z : |z| < 1}. Let Mp(α) be the subclass of Ap consisting of functions f(z) which satisfy Re (z f′ (z)/ f(z) < α, (z ε D) for some real α (α > 1). The aim of this paper is to give a representation theorem, a distortion theorem and a coefficient inequality for the class Mp(α). [ABSTRACT FROM AUTHOR]

Let γ = 2cos π/5 and let H5 be the Hecke group associated to γ. In this paper, the normalizers of the congruence subgroups H 05 (I) in PSL (2, 핫[γ]) are studied in the case where I = (2)αI′, (2, I′) = 1 and I′ is a prime ideal. [ABSTRACT FROM AUTHOR]

In this paper we define a diagonal lift Dg of Riemannian metric g of manifold Mn to the tangent bundle of order two denoted by T²Mn of Mn, we associate to Dg its Levi-civita connection of T² M and we investigate applications of the diagonal lifts in the killing vectors and geodesics. [ABSTRACT FROM AUTHOR]

In the present paper we shall give the radius of starlikeness for the classes of p-valent analytic functions in the unit disc D = {z ||z| lt; 1}. [ABSTRACT FROM AUTHOR]

In 2004, Al-Hawary and Al-Omari introduced and explored the class of ωº--open sets which is strictly stronger than the class of ω--open sets and weaker than that of open sets. In this paper, we introduce what we call ωº--continuity and ωXº--continuity and we give several characterizations and two decompositions of ωº--continuity. Finally, new decompositions of continuity are provided. [ABSTRACT FROM AUTHOR]

In this paper, we obtain vertical, complete and horizontal lifts of paracomplex geometric structures on paracomplex manifolds to its tangent bundle. Also, we obtain integrability on paracomplex tangent bundle. [ABSTRACT FROM AUTHOR]

In this paper, we introduce some characterizations for augmented graded rings in special cases. [ABSTRACT FROM AUTHOR]

In this paper we aim to extend a result of Hörmander's, that Bp ,kloc (Ω) ⊂ Cm (Ω) if (1+|⋅|)m/k ∈ Lp′, to the setting of vector valued local Hörmander-Beurling spaces, as well as to show that the space ∩j=1∞ Bpj, kjloc (Ω, E) (1 ≤ pj ≤ &infin, kj = ejw, j = 1, 2, . . .) is topologically isomorphic to Ew (Ω, E). Moreover, it is well known that the union of Sobolev spaces Hsloc(Ω) (= B...loc(Ω)) coincides with the space D′F (Ω ) of finite order distributions on Ω. We show that this is also verified in the context of vector valued Beurling ultradistributions. [ABSTRACT FROM AUTHOR]

The main result in this paper is the following: Let G be a torsion-free locally nilpotent group and let F be a finitely generated subgroup of G. If every subgroup of G containing F is subnormal in G, then G is nilpotent. [ABSTRACT FROM AUTHOR]