Mathematics / Matematik
Permanent URI for this collectionhttps://hdl.handle.net/11147/8
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Article Citation - WoS: 1Citation - Scopus: 1Unique decompositions into w-ideals for strong Mori domains(World Scientific Publishing, 2022) Hamdi, Haleh; Ay Saylam, Başak; Gürbüz, EzgiA commutative ring R has the unique decomposition into ideals (UDI) property if, for any R-module that decomposes into a finite direct sum of indecomposable ideals, this decomposition is unique up to the order and isomorphism classes of the indecomposable ideals. In [P. Goeters and B. Olberding, Unique decomposition into ideals for Noetherian domains, J. Pure Appl. Algebra 165 (2001) 169-182], the UDI property has been characterized for Noetherian integral domains. In this paper, we aim to study the UDI-like property for strong Mori domains; domains satisfying the ascending chain condition on w-ideals.Article Citation - WoS: 2Citation - Scopus: 2On simple-injective modules(World Scientific Publishing, 2022) Alagöz, Yusuf; Benli Göral, Sinem; Büyükaşık, EnginFor a right module M, we prove that M is simple-injective if and only if M is min-N-injective for every cyclic right module N. The rings whose simple-injective right modules are injective are exactly the right Artinian rings. A right Noetherian ring is right Artinian if and only if every cyclic simple-injective right module is injective. The ring is QF if and only if simple-injective right modules are projective. For a commutative Noetherian ring R, we prove that every finitely generated simple-injective R-module is projective if and only if R = A × B, where A is QF and B is hereditary. An abelian group is simple-injective if and only if its torsion part is injective. We show that the notions of simple-injective, strongly simple-injective, soc-injective and strongly soc-injective coincide over the ring of integers.Article Citation - WoS: 17Citation - Scopus: 17Quantum Calculus of Fibonacci Divisors and Infinite Hierarchy of Bosonic-Fermionic Golden Quantum Oscillators(World Scientific Publishing, 2021) Pashaev, OktayStarting from divisibility problem for Fibonacci numbers, we introduce Fibonacci divisors, related hierarchy of Golden derivatives in powers of the Golden Ratio and develop corresponding quantum calculus. By this calculus, the infinite hierarchy of Golden quantum oscillators with integer spectrum determined by Fibonacci divisors, the hierarchy of Golden coherent states and related Fock-Bargman representations in space of complex analytic functions are derived. It is shown that Fibonacci divisors with even and odd kappa describe Golden deformed bosonic and fermionic quantum oscillators, correspondingly. By the set of translation operators we find the hierarchy of Golden binomials and related Golden analytic functions, conjugate to Fibonacci number F-kappa. In the limit. kappa -> 0, Golden analytic functions reduce to classical holomorphic functions and quantum calculus of Fibonacci divisors to the usual one. Several applications of the calculus to quantum deformation of bosonic and fermionic oscillator algebras, R-matrices, geometry of hydrodynamic images and quantum computations are discussed.Article Locally Isomorphic Torsionless Modules Over Domains of Finite Character(World Scientific Publishing, 2019) Saylam, Başak Ay; Klingler, LeeIn a 2002 paper, P. Goeters and B. Olberding compare local, near, and stable isomorphisms of torsionless modules over h-local domains. In this paper, we compare these weaker forms of isomorphisms of torsionless modules over domains of finite character.Article Citation - WoS: 8Citation - Scopus: 8On the Structure of Modules Defined by Subinjectivity(World Scientific Publishing, 2019) Altınay, Ferhat; Büyükaşık, Engin; Durgun, YılmazThe aim of this paper is to present new results and generalize some results about indigent modules. The commutative rings whose simple modules are indigent or injective are fully determined. The rings whose cyclic right modules are indigent are shown to be semisimple Artinian. We give a complete characterization of indigent modules over commutative hereditary Noetherian rings. We show that a reduced module is indigent if and only if it is a Whitehead test module for injectivity over commutative hereditary noetherian rings. Furthermore, Dedekind domains are characterized by test modules for injectivity by subinjectivity.Article Citation - WoS: 1Citation - Scopus: 1Fully Three-Dimensional Analysis of a Photonic Crystal Assisted Silicon on Insulator Waveguide Bend(World Scientific Publishing, 2018) Eti, Neslihan; Çetin, Zebih; Sözüer, Hüseyin SamiA detailed numerical study of low-loss silicon on insulator (SOI) waveguide bend is presented using the fully three-dimensional (3D) finite-difference time-domain (FDTD) method. The geometrical parameters are optimized to minimize the bending loss over a range of frequencies. Transmission results for the conventional single bend and photonic crystal assisted SOI waveguide bend are compared. Calculations are performed for the transmission values of TE-like modes where the electric field is strongly transverse to the direction of propagation. The best obtained transmission is over 95% for TE-like modes.Article Citation - WoS: 8Citation - Scopus: 9Max-Projective Modules(World Scientific Publishing, 2020) Alagöz, Yusuf; Büyükaşık, EnginWeakening the notion of R-projectivity, a right R-module M is called max-projective provided that each homomorphism f: M ? R/I, where I is any maximal right ideal, factors through the canonical projection : R ? R/I. We study and investigate properties of max-projective modules. Several classes of rings whose injective modules are R-projective (respectively, max-projective) are characterized. For a commutative Noetherian ring R, we prove that injective modules are R-projective if and only if R = A × B, where A is QF and B is a small ring. If R is right hereditary and right Noetherian then, injective right modules are max-projective if and only if R = S × T, where S is a semisimple Artinian and T is a right small ring. If R is right hereditary then, injective right modules are max-projective if and only if each injective simple right module is projective. Over a right perfect ring max-projective modules are projective. We discuss the existence of non-perfect rings whose max-projective right modules are projective. © 2020 World Scientific Publishing Company.