Abstract In this preface of the Special Issue on Future Directions of Further Developments in Mathematics, we discuss about mathematics, how it should be, what it was and how we hope it will develop.

Abstract Let G G be a π \pi -separable group for some set π \pi of primes, let χ ∈ B π ( G ) \chi \in {B}_{\pi }\left(G) and let N ◃ G N\hspace{0.33em}<mml:mpadded xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">\triangleleft </mml:mpadded>\hspace{0.33em}G . In this article, we explore how to construct a nucleus for an irreducible constituent of χ N {\chi }_{N} via the given nucleus ( W , γ ) \left(W,\gamma ) for χ \chi .

Abstract In this article, we are interested in uniformly p r pr -ideals with order ≤ 2 \le 2 (which we call 2 r 2r -ideals) introduced by Rabia Üregen in [On uniformly pr-ideals in commutative rings, Turkish J. Math. 43 (2019), no. 4, 18781886]. Several characterizations and properties of these ideals are given. Moreover, the comparison between the (nonzero) 2 r 2r -ideals and certain classes of classical ideals gives rise to characterizations of certain rings based only on the properties of the ideals consisting only of zero-divisors. Namely, among other things, we compare the class of (nonzero) 2 r 2r -ideals with the class of (minimal) prime ideals, the class of minimal prime ideals and their squares, and the class of primary ideals. The study of 2 r 2r -ideal in polynomial rings allows us to give a new characterization of the rings satisfying the famous A A -property.

Abstract We consider the problem of scheduling jobs with equal lengths and arbitrary sizes on uniform parallel batch machines with different capacities. Each machine can only process the jobs whose sizes are not larger than its capacity. Several jobs can be processed as a batch simultaneously on a machine, as long as their total size does not exceed the machine’s capacity. The objective is to minimize makespan. Under a divisibility constraint, we obtain two efficient exact algorithms. For the general problem, we obtain an efficient 2-approximation algorithm. Previous work has shown that the problem cannot be approximated to within an approximation ratio better than 2, unless P = NP, even when all machines have identical speeds and capacities.

Abstract In this article, the general Liénard system d x d t = ϕ ( y ) − F ( x ) , d y d t = − g ( x ) \left\{\begin{array}{l}\frac{{\rm{d}}x}{{\rm{d}}t}=\phi (y)-F\left(x),\\ \frac{{\rm{d}}y}{{\rm{d}}t}=-g\left(x)\end{array}\right. is studied. By using the Filippov transformation, combined with the careful estimation of divergence along the closed orbit, we prove the sufficient conditions for the uniqueness of limit cycles in this system. Our results extend almost all the related existing studies on the Liénard system.

Abstract The aim of this article is to design a new iteration process for solving certain fixed-point problems. In particular, we prove weak and strong convergence theorems for generalized nonexpansive mappings in the framework of uniformly convex Banach spaces. In addition, we discuss the stability of the solution under mild conditions. Further, we provide some numerical examples to indicate that the proposed method works properly.

Abstract Let X X be a compact Riemann surface of genus g ≥ 2 g\ge 2 and ℳ ( E 6 ) {\mathcal{ {\mathcal M} }}\left({E}_{6}) be the moduli space of E 6 {E}_{6} -Higgs bundles over X X . We consider the automorphisms σ + {\sigma }_{+} of ℳ ( E 6 ) {\mathcal{ {\mathcal M} }}\left({E}_{6}) defined by σ + ( E , φ ) = ( E ∗ , − φ t ) {\sigma }_{+}\left(E,\varphi )=\left({E}^{\ast },-{\varphi }^{t}) , induced by the action of the outer involution of E 6 {E}_{6} in ℳ ( E 6 ) {\mathcal{ {\mathcal M} }}\left({E}_{6}) , and σ − {\sigma }_{-} defined by σ − ( E , φ ) = ( E ∗ , φ t ) {\sigma }_{-}\left(E,\varphi )=\left({E}^{\ast },{\varphi }^{t}) , which results from the combination of σ + {\sigma }_{+} with the involution of ℳ ( E 6 ) {\mathcal{ {\mathcal M} }}\left({E}_{6}) , which consists on a change of sign in the Higgs field. In this work, we describe the fixed points of σ + {\sigma }_{+} and σ − {\sigma }_{-} , as F 4 {F}_{4} -Higgs bundles, F 4 {F}_{4} -Higgs pairs associated with the fundamental irreducible representation of F 4 {F}_{4} , and PSp ( 8 , C ) {\rm{PSp}}\left(8,{\mathbb{C}}) -Higgs pairs associated with the second symmetric power or the second wedge power of the fundamental representation of Sp ( 8 , C ) {\rm{Sp}}\left(8,{\mathbb{C}}) . Finally, we describe the reduced notions of semistability and polystability for these objects.

Abstract The main aim of this article is to study admissible congruences on a type B semigroup. First, we give characterizations of the minimum admissible congruence whose trace is a normal congruence on a type B semigroup. After obtaining some properties of admissible congruences with the same trace on a type B semigroup, we introduce the notion of a normal subsemigroup and give characterizations of the minimum and maximum admissible congruences whose kernels are normal subsemigroups. Finally, the concept of a congruence pair of a type B semigroup is given, and two congruences associated with the congruence pair are obtained.

Abstract Let f n {f}_{n} be the n n th Fibonacci number with f 1 = f 2 = 1 {f}_{1}={f}_{2}=1 . Recently, the exact values of ∑ k = n ∞ 1 f k s − 1 ⌊{\left({\sum }_{k=n}^{\infty }\frac{1}{{f}_{k}^{s}}\right)}^{-1}⌋ have been obtained only for s = 1 , 2 s=1,2 , where ⌊ x ⌋ \lfloor x\rfloor is the floor function. It has been an open problem for s ≥ 3 s\ge 3 . In this article, we consider the case of s = 4 s=4 and show that ∑ k = n ∞ 1 f k 4 − 1 = f n 4 − f n − 1 4 + 2 ( − 1 ) n 5 f 2 n − 1 − n + 2 5 , ⌊{\left(\mathop{\sum }\limits_{k=n}^{\infty }\frac{1}{{f}_{k}^{4}}\right)}^{-1}⌋={f}_{n}^{4}-{f}_{n-1}^{4}+\frac{2{\left(-1)}^{n}}{5}{f}_{2n-1}-\left\{\frac{n+2}{5}\right\}, where { x } ≔ x − ⌊ x ⌋ \left\{x\right\}:= x-\lfloor x\rfloor .

Abstract The normalizer problem of integral group rings has been studied extensively in recent years due to its connection with the longstanding isomorphism problem of integral group rings. Class-preserving Coleman automorphisms of finite groups occur naturally in the study of the normalizer problem. Let G G be a finite group with a nilpotent subgroup N N . Suppose that G / N G\hspace{-0.0em}\text{/}\hspace{-0.0em}N acts faithfully on the center of each Sylow subgroup of N N . Then it is proved that every class-preserving Coleman automorphism of G G is an inner automorphism. In addition, if G G is the product of a cyclic normal subgroup and an abelian subgroup, then it is also proved that every class-preserving Coleman automorphism of G G is an inner automorphism. Other similar results are also obtained in this article. As direct consequence, the normalizer problem has a positive answer for such groups.