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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]

Throughout the paper in the title by Jian Tang, Yanfeng Luo and Xiangyun Xie in Turk J Math 42 (2018) the following lemma has been used. Lemma: Let (S, *) be a semihypergroup and ρ an equivalence relation on S. Then (i) If ρ is a congruence, then (S/ρ, ⊗) is a semihypergroup with respect to the hyperoperation (a)ρ ⊗ (b)ρ = ... (c)ρ . (ii) If ρ is a strong congruence, then (S/ρ, ⊗) is a semigroup with respect to the operation (a)ρ ⊗ (b)ρ = (c)ρ for all c ∈ a * b. The property (i) of the paper is certainly wrong as ... (c)ρ is a subset of S and not a nonempty subset of S/ρ as it should be. Property (ii) has no sense in the way is written. In addition, according to the authors, as an application of the results of this paper they solved the open problem on ordered semihypergroups given by Davvaz, Corsini and Changphas in European J Combin 44 (2015). The problem is that the above mentioned problem has not been solved in the above mentioned article; we point out the reason, and we solve it in the present paper. Some further related results; also results necessarily for the completeness of the paper are given. 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]

The Randić index (R) and the geometric-arithmetic index (GA) are found to be useful tools in QSPR and QSAR studies. In the Journal of Inequalities and Applications 180, 1-7, Lokesha, Shwetha Shetty, Ranjini, Cangul, and Cevik gave "New bounds for Randić and GA indices." In the paper, we first point out that Theorems 1, 2, and 4 are incorrect and in this short note we present the correct inequalities for Randić and GA indices. In the same paper, we provide the equality cases for Theorems 3, 5, and 6. [ABSTRACT FROM AUTHOR]

In this study M1 (⌈0 (3) (χ-3), M2 (⌈0 (5), χ5) and M3 (Γ0 (7), χ - 7) have been examined and the formulas for the Fourier Coefficients of theta series and the representation number of positive integers by some quadratic forms 3x1² + 3x1x2 + x2², 5(x1² + x1x2 + x1x3 + x14 + x1² + x2x3 + x2x4 + 3² + x3x4) + 2x4², and 7(x1² + x1x2 + x1x3 + x1x4 + x1x5 + x2² + x2x3 + x2x4 + x2x5 + x3² + x3x4 + x3x5 + x4² + x4x5 + x5² + 7 (x1x6 + x2x6 + x3x6 + x4x6) + 3x6², are determined. This work is a correction to a paper of the same title by Ahmet Tekcan [5]. [ABSTRACT FROM AUTHOR]

In this paper, Caputo boundary value problems of order 3 < ζ ≤ 4 are investigated on the interval [0, 1]. By Guo-Krasnoselskii fixed point theorem, some criteria of existence and multiplicity of positive and decreasing solutions are established. The main novelty of the paper lies in its capability to achieve positive solutions while the corresponding Green’s function changes sign. Finally, two examples are provided to illustrate the application of these results. [ABSTRACT FROM AUTHOR]

Let G be a finite, weighted graph, and let T be a time-scale with a fixed point t0 such that sup T = ∞. In this paper, we construct the heat kernel on G in time-scale T in terms of a certain convolution series involving the heat operator acting on a parametrix, which is a fairly general function depending on the vertex set of G and the time variable t ∈ T. We develop some applications by choosing different parametrices and various time-scales. The results we obtain here extend, in part, aspects of the recent articles in that the time-scale considered in this paper is arbitrary. [ABSTRACT FROM AUTHOR]

In this paper, we study the permutation properties of the class of trinomials of the form f(x) = x4q+1 + λ1xq+4 + λ2x2q+3 ∈ Fq²[x] where λ1, λ2 ∈ Fq and they are not simultaneously zero. We find all necessary and sufficient conditions on λ1 and λ2 such that f(x) permutes Fq², where q is odd and q = 22k+1, k ∈ N. [ABSTRACT FROM AUTHOR]

This paper is concerned with the oscillation of solutions to a class of first-order differential equations with variable coefficients and a general delay argument. New oscillation criteria are established, which improve and extend many known results reported in the literature. A couple of illustrative examples are given to show the efficiency of the newly obtained results. In particular, it is shown that our criteria partially fulfill a remaining gap in a recent sharp result by Pituk et al. [31]. [ABSTRACT FROM AUTHOR]

In this paper, we prove an identity for differentiable convex functions related to conformable fractional integrals. Moreover, some parameterized inequalities are established by using conformable fractional integrals. To be more precise, parameterized inequalities are obtained by taking advantage of the convexity, the Hölder inequality, and the power mean inequality. Furthermore, previous and new results are presented by using special cases of the obtained theorems. [ABSTRACT FROM AUTHOR]

The present paper deals with the approximation properties of the linear positive operators, including Δh -Gould-Hopper Appell polynomials. We investigate some theorems for convergence of the operators and their approximation degrees with the help of the classical approach, Peetre's K-functional, Lipschitz class and Voronovskajatype theorem. In the last section of the paper, we introduce the Kantorovich form of the operators and examine the approximation degree. [ABSTRACT FROM AUTHOR]

In this paper, we establish the uniqueness and existence of the classical solution to higher-order boundary value problems. Then, we give a new Lyapunov-type inequalities for higher order boundary value problems. Our study is based on Green’s functions corresponding to the five different types of two-point boundary value problems. In addition, some applications of the obtained inequalities are given. [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]

Ever since Lotfi A. Zadeh published the paper "Fuzzy Sets" in 1965 setting the basis of a new theory named fuzzy sets theory, many scientists have developed this theory and its applications. Mathematicians were especially interested in extending classical mathematical results in the fuzzy context. Such an extension was also done relating fuzzy sets theory and geometric theory of analytic functions. The study begun in 2011 has many interesting published outcomes and the present paper follows the line of the previous research in the field. The aim of the paper is to give some references related to the connections already made between fuzzy sets theory and geometric theory of analytic functions and to present some new results that might prove interesting for mathematicians willing to enlarge their views on certain aspects of the merge between the two theories. Using the notions of fuzzy differential subordination and the classical notion of differential subordination for analytic functions, two criteria for the univalence of the analytic functions are stated in this work. [ABSTRACT FROM AUTHOR]

In this paper, we apply the criterion of Mikhalev and Umirbaev for the invertibility of an endomorphism of a finitely generated free Leibniz algebra via its Jacobian matrix to determine whether a given endomorphism is an automorphism. Moreover, it is shown that the invertibility of the determinant of the Jacobian matrix of an automorphism implies its wildness. [ABSTRACT FROM AUTHOR]

In this paper, transmission eigenvalues of a Schrödinger equation have been studied by constructing a new inner product and using Weyl theory. Necessary conditions for these eigenvalues to be negative, real, and finite have been examined. This method has provided a new framework related to transmission eigenvalue problems and the investigation of their properties. The conclusions have been verified for the special case of the problem. [ABSTRACT FROM AUTHOR]

In this paper, the connection between the ϵ -pseudospectrum of the tensor product operator A1 ⊗ A2 and the ϵ -pseudospectrums of operators A1 and A2 has been investigated and some results are given about this connection under certain conditions. [ABSTRACT FROM AUTHOR]

This paper explores the ideals and their structural properties in two generalizations of the partial transformation semigroup. Furthermore, principal, maximal, and minimal ideals within these semigroups are elucidated. [ABSTRACT FROM AUTHOR]

The Euclidean hull of a linear code C is defined as C ∩ C⊥, where C⊥ denotes the dual of C under the Euclidean inner product. A linear code with the trivial hull is called a linear complementary dual (LCD) code. A pair (C, D) of linear codes of length n over the finite field Fq is called a linear complementary pair (LCP) of codes if C ⊕ D = Fnq . More generally, a pair (C, D) of linear codes of the same length over Fq is called a linear ℓ -intersection pair of codes if C ∩ D has dimension ℓ as a vector space over Fq . In this paper, we give characterizations of LCD, LCP of cyclic codes and one-dimensional hull cyclic codes of length qm− 1, m ≥ 1, over Fq in terms of their basic dual zero sets and their trace representations. We also formulate the hull dimension of a cyclic code of arbitrary length over Fq with respect to its basic dual zero set. Moreover, we provide a general formula for the dimension ℓ of the intersection of two cyclic codes of arbitrary length over Fq based on their basic dual zero sets. [ABSTRACT FROM AUTHOR]

In this paper, we examine stacky structures in certain Einstein gravity theories. In brief, using the classical formulation of (vacuum) gravity, with vanishing cosmological constant, we first construct the stack of solutions to Einstein field equations on any given fixed manifold. Using a similar approach and setup, we also study Einstein’s gravity on families of manifolds and define another stack encoding this situation. Later on, we focus on the gauge theoretical interpretation of 3D gravity and provide a natural stack associated with that interpretation. Finally, in a particular setup, we give a natural morphism between the two stacks arising from different descriptions of 3D gravity. [ABSTRACT FROM AUTHOR]

This paper addresses the second-order nonlocal dynamic thermistor problem with two-point boundary conditions on time scales. Utilizing the fixed point theorems by Schaefer and Rus, we establish some sufficient conditions for the existence and uniqueness of solutions. Furthermore, we discuss the continuous dependence of solutions and four types of Ulam stability. We provide examples to support the applicability of our results. [ABSTRACT FROM AUTHOR]

In this paper, we show that in an almost product manifold there exist curves that are slant submanifolds. We characterize these curves and study them in two and three-dimensional locally product manifolds. Finally, we construct curves in a hypersurface of a Kaehler manifold. [ABSTRACT FROM AUTHOR]

In this paper, we study Bernstein-Nikol’skii-Markov type inequalities for arbitrary algebraic polynomials with respect to a weighted Lebesgue space, where the weight functions have some singularities on a given contour. We consider curves which can contain a finite number of exterior and interior corners with power law tangency of the boundary arcs at those points where the weight functions have both zeros and poles of finite order. The estimates are given for the growth of the module of derivatives for algebraic polynomials on the closure of a region bounded by a given curve, depending on the behavior of weight functions, on the property of curve, and on the degree of contact of the boundary arcs, which form zero angles on the boundary. [ABSTRACT FROM AUTHOR]

The onset of magnetoconvection (known as Rayleigh-Bénard-Chandrasekhar convection) in two relaxation time viscoelastic liquids is studied here without seeking explicit recourse to a normal stress formulation as is usually done in these studies. Magnetoconvection refers to the flow of fluid in the presence of both thermal gradients (Rayleigh-Bénard convection) and a magnetic field. When these two effects are combined, they can lead to interesting and complex patterns of fluid motion. Understanding magnetoconvection in viscoelastic liquids is crucial for various industrial and scientific applications. The hyperbolic-type of linear momentum equation is decomposed into two first-order equations in time by cleverly separating the viscoelastic effect from the other effects in a clever manner as reported in a recent paper. The results of Maxwell, Rivlin-Ericksen, Walters' liquid B, and Newtonian liquids are obtained as limiting cases of the present study. This research contributes to the understanding of magnetoconvection in viscoelastic liquids by using a novel approach that decouples the viscoelastic effect from other influences. The results obtained shed light on the behaviour of various types of viscoelastic materials and provide valuable insights for practical applications in fields such as materials science, engineering, and geophysics. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of positive periodic solutions of a third-order nonlinear integrodifferential equation with distributed delays, by using the Green function and the Krasnosel'skii fixed point theorem in cones of Banach spaces, providing new results on this field. Three examples are analyzed to illustrate the effectiveness of the abstract results. [ABSTRACT FROM AUTHOR]

Several recent papers were devoted to various modifications of limited, Grothendieck, and Dunford-Pettis operators, etc., through involving the Banach lattice structure. In the present paper, it is shown that many of these operators appear as operators affiliated to well-known properties of Banach lattices, like the disjoint (dual) Schur property, the disjoint Grothendieck property, the property (d), the sequential w* -continuity of the lattice operations, etc. We also introduce new classes of operators such as the s-GPP-operators, s-BDP-operators, and bi-sP-operators. It is proved that the spaces consisting of regular versions of the above-mentioned operators are all the Banach spaces. The domination problem for these operators is investigated. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to establish the eigenvalues and the eigenfunctions of both the q -Durrmeyer operators Dn,q and the limit q -Durrmeyer operators D∞,q introduced by V. Gupta in the case 0 < q < 1. All moments for Dn,q and D∞,q are provided. The coefficients for the eigenfunctions of the operators are explicitly derived and the eigenfunctions of these operators are illustrated by graphical examples. [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]

In this paper, we define the ruled surface in terms of the Darboux vector field of an isotropic curve in a complex 3-space, and we study the ruled surface as an elastic strip along the isotropic curve. First of all, we show that elastic strips along isotropic curves that serve critical points of the modified Sadowsky functional are characterized by three Euler-Lagrange equations. As a result, we give two conservation laws to characterize elastic strips of isotropic curves in complex 3-space and explain an isotropic helix in terms of the force vector. Finally, we give some examples to illustrate elastic strips with isotropic curves in a complex 3-space. [ABSTRACT FROM AUTHOR]

This paper is devoted to studying the existence and uniqueness of continuous solutions of the following iterative functional differential equation ... By using of Boyd-Wong's fixed point theorem and under suitable conditions, we establish the existence and uniqueness of a continuous solution. [ABSTRACT FROM AUTHOR]

For n = 2, let Pn, In, Tn, and Sn be the partial transformation semigroup, symmetric inverse semigroup, (full) transformation semigroup, and symmetric group on the set Xn = {1, . . ., n}, respectively. In this paper, we find the ranks of certain subsemigroups of Pn, In, and Tn consisting of transformations with idempotent complement whose restrictions to the set Xm belong to the (possible) semigroup Sm, Im, Tm, or Pm for 1 = m = n - 1. [ABSTRACT FROM AUTHOR]

In a number of applied problems (immunology, epidemiology, virology, etc.), there is a need to study the dynamics of the trajectories of the compositions of the Lotka-Volterra mappings. In this paper, some variants of these compositions are considered, in particular, the dynamics of the trajectories of the compositions of the Lotka-Volterra mappings acting in a two-dimensional simplex, with transitive tournaments, which can be applied in the study of these processes. Cards of fixed points are constructed for compositions and the characters of fixed points are studied. [ABSTRACT FROM AUTHOR]

In this paper, we define a mixed-base number system over a Weyl group Dn, the group of even-signed permutations. We introduce one-to-one correspondence between the positive integers of the set {1, · · ·, 2n-1n!} and elements of this group, after constructing the subexceedant function associated with the group. Thus, the integer representations of all the classical Weyl groups are now completed. Furthermore, we present an inversion statistic on the group Dn by using a decomposition of a positive root system of this reflection group. This inversion statistic is compatible with the length function on the group Dn. Then we derive some combinatorial properties for the inversion statistic. In addition, we prove that the D-major index is equi-distributed with this inversion statistic on Dn. Finally, we propose a public-key cryptosystem based on both the generalized hidden discrete logarithm problem and the integer representation over the group Dn. [ABSTRACT FROM AUTHOR]

In the present paper, we study timelike surfaces with parallel normalized mean curvature vector field in the four-dimensional Minkowski space. We introduce special isotropic parameters on each such surface, which we call canonical parameters, and prove a fundamental existence and uniqueness theorem stating that each timelike surface with parallel normalized mean curvature vector field is determined up to a rigid motion in the Minkowski space by three geometric functions satisfying a system of three partial differential equations. In this way, we minimize the number of functions and the number of partial differential equations determining the surface, thus solving the Lund-Regge problem for this class of surfaces. [ABSTRACT FROM AUTHOR]

For a nonlightlike nondegenerate regular curve, we have the arc-length parameter and the Frenet-Serret type formula by using a moving frame like a regular space curve in the Euclidean space. If a point of the curve moves between spacelike and timelike regions, then there is a lightlike point. In this paper, we consider mixed types of not only regular curves but also curves with singular points. In order to consider mixed type of curves with singular points, we introduce a frame, so-called the lightcone frame, and lightcone framed curves. We investigate differential geometric properties of lightcone framed curves. [ABSTRACT FROM AUTHOR]

Gelfand and Shilov give the definition of the composition δ(g(x)) for an infinitely differentiable function g(x) having any number of simple roots. In the paper, we consider their definition for an infinitely differentiable function having any number of multiple roots by using the method of the discarding of unwanted infinite quantities from asymptotic expansions and give some examples. Further, we define the compositions δ(g+) and δ(g-) for a locally summable function g(x). [ABSTRACT FROM AUTHOR]

In this paper, we introduce in a braided setting the notion of left module for an invertible 1-cocycle and we prove some categorical equivalences between categories of modules associated to an invertible 1-cocycle and categories of modules associated to Hopf braces. [ABSTRACT FROM AUTHOR]

This paper is concerned with the oscillation and asymptotic behavior of solutions of third-order nonlinear neutral differential equations with a middle term and mixed nonlinear neutral terms in the case of the canonical operator. We establish several oscillation criteria that guarantee that all solutions are oscillatory or converge to zero. The given results are obtained by applying the comparison method, the Riccati transformation and the integral averaging technique. The results improve significantly and extend existing ones in the literature. Finally, illustrative examples are given. [ABSTRACT FROM AUTHOR]

In this paper, we prove a generalization of a discreteness criteria for a large class of subgroups of PSL2(C). In particular, given a finitely generated purely loxodromic free Kleinian group Γ = ⟨ξ1, ξ2,. . ., ξn⟩ for n ≥ 2, we show that |trace2(ξi) - 4| + |trace(ξiξjξ-1i ξ-1j) - 2| ≥ 2 sinh2 (14 log αn) for some ξi and ξj for i ̸= j in Γ provided that certain conditions on the hyperbolic displacements given by ξi, ξj and their length 3 conjugates formed by the generators are satisfied. Above, the constant αn turns out to be the real root strictly larger than (2n-1)2 of a fourth degree integer coefficient polynomial obtained by solving a family of optimization problems via the Karush-Kuhn-Tucker theory. The use of this theory in the context of hyperbolic geometry is another novelty of this work. [ABSTRACT FROM AUTHOR]

This paper is devoted to defining the delayed analogue of the Mittag-Leffler type function with three parameters and investigating a representation of a solution to Langevin delayed equations with Prabhakar derivatives involving two generalized fractional distinct orders, which are first introduced and investigated, by means of the Laplace integral transform. It is verified by showing the solution satisfies the introduced system. Special cases which are also novel are presented as examples. The findings are illustrated with the help of the RLC circuits. [ABSTRACT FROM AUTHOR]

The paper deals with the twisted Sasaki metric on the unit tangent bundle of n-dimensional Riemannian manifold Mn. The main purpose of the research is to find deformations that preserve the existence harmonic left-invariant unit vector fields on 3-dimensional unimodular Lie groups G with the left invariant metric and harmonic maps G → T1G in case of twisted Sasaki metric on the unit tangent bundle. The necessary and sufficient conditions for harmonicity of left-invariant unit vector field and map Mn → T1Mn are obtained. The necessary and sufficient conditions for harmonicity of left-invariant unit vector field and map M² → T1M² with respect to some orthonormal frame are obtained. Left-invariant harmonic unit vector fields and harmonic maps G → T1G, where G is a three-dimensional unimodular Lie group with left-invariant metric, using some orthonormal frame are described. Left-invariant harmonic unit vector fields which determine harmonic maps G → T1G, where G is a three-dimensional unimodular Lie group with left-invariant metric in the particular case of twisted Sasaki metric, namely the vertical rescaled metric are classified. [ABSTRACT FROM AUTHOR]

Let Xn denote the chain {1, 2, ..., n} under its natural order. We denote the semigroups consisting of all order-preserving transformations and all orientation-preserving transformations on Xn by On and OPn, respectively. We denote by E(U) the set of all idempotents of a subset U of a semigroup S. In this paper, we first determine the cardinalities of ... (1 ≤ r ≤ n) and then, by using these results, we determine the numbers of idempotents in On and OPn by a new method. Let OP-n denote the semigroup of all orientation-preserving and order-decreasing transformations on Xn. Moreover, we determine the cardinalities of OP-n, OP-n,Y = {α ∈ OP-n: fix(α) = Y } for any nonempty subset Y of Xn and OP-n,r = {α ∈ OP-n: |fix(α)| = r} for 1 ≤ r ≤ n. Also, we determine the number of idempotents in OP- n and the number of nilpotents in OP- n. [ABSTRACT FROM AUTHOR]