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Yayın Tulczyjew's triplet with an Ehresmann connection I: Trivialization and reduction(World Scientific, 2023-03-30) Esen, Oğul; Kudeyt, Mahmut; Sütlü, SerkanWe study the trivialization and the reduction of Tulczyjew's triplet, in the presence of a symmetry and an Ehresmann connection associated to it. We thus establish a geometric pathway for the Legendre transformations on singular dynamical systems.Yayın Modeling repair demand in existence of a nonstationary installed base(Elsevier B.V., 2023-09) Hekimoğlu, Mustafa; Karlı, DenizLife cycles of products consist of 3 phases, namely growth, maturity, and decline phases. Modeling repair demand is particularly difficult in the growth and decline stages due to nonstationarity. In this study, we suggest respective stochastic models that capture the dynamics of repair demand in these two phases. We apply our theory to two different operations management problems. First, using the moments of spare parts demand, we suggest an algorithm that selects a parametric distribution from the hypergeometric family (Ord, 1967) for each period in time. We utilize the algorithm in a single echelon inventory control problem. Second, we focus on investment decisions of Original Equipment Manufacturers (OEMs) to extend economic lifetimes of products with technology upgrades. Our results indicate that the second moment is sufficient for growing customer bases, whereas using the third moment doubles the approximation quality of theoretical distributions for a declining customer base. From a cost minimization perspective, using higher moments of demand leads to savings up to 13.6% compared to the single-moment approach. Also, we characterize the optimal investment policy for lifetime extension decisions from risk-neutral and risk-averse perspectives. We find that there exists a critical level of investment cost and installed base size for profitability of lifetime extension for OEMs. From a managerial point of view, we find that a risk-neutral decision maker finds the lifetime extension problem profitable. In contrast, even a slight risk aversion can make the lifetime extension decision economically undesirable.Yayın On Caputo fractional Bertrand curves in E3 and E31(Univ Nis, Fac Sci Math, 2024) Taşdemir, Mert; Canfes, Elif Özkara; Uzun, BanuIn this article, we examine Bertrand curves in E3 and E31 by using the Caputo fractional derivative which we call alpha-Bertrand Curves. First, we consider alpha-Bertrand curves in E3 and we give a characterization of them. Then, we study alpha-Bertrand curves in E31 and we prove the necessary and sufficient condition for a alpha-Bertrand curves in E31 by considering time like, space like and null curves. We also give the related examples by using Python.Yayın Bicocycle double cross constructions(World Scientific, 2023-12-01) Esen, Oğul; Guha, Partha; Sütlü, SerkanWe introduce the notion of a bicocycle double cross product (sum) Lie group (algebra), and a bicocycle double cross product bialgebra, generalizing the unified products. On the level of Lie groups the construction yields a Lie group on the product space of two pointed manifolds, none of which being necessarily a subgroup. On the level of Lie algebras, a Lie algebra is obtained on the direct sum of two vector spaces, which are not required to be subalgebras. Finally, on the quantum level a bialgebra is obtained on the tensor product of two (co)algebras that are not necessarily sub-bialgebras.Yayın Constant angle surfaces in the Lorentzian warped product manifold I ×f E²1(Tübitak, 2022-09-14) Dursun, UğurLet I ×f E²1 be a 3-dimensional Lorentzian warped product manifold with the metric g˜ = dt² + f² (t)(dx² ? dy² ), where I is an open interval, f is a strictly positive smooth function on I, and E²1 is the Minkowski 2-plane. In this work, we give a classification of all space-like and time-like constant angle surfaces in I ×f E²1 with nonnull principal direction when the surface is time-like. In this classification, we obtain space-like and time-like surfaces with zero mean curvature, rotational surfaces, and surfaces with constant Gaussian curvature. Also, we have some results on constant angle surfaces of the anti-de Sitter space H³1(?1).Yayın Optimization of wastewater treatment systems for growing industrial parks(Elsevier B.V., 2023-12-20) Savun Hekimoğlu, Başak; İşler, Zülal; Hekimoğlu, Mustafa; Burak, Selmin; Karlı, Deniz; Yücekaya, Ahmet; Akpınar, Ersin; Ediger, Volkan Ş.Wastewater treatment is one of the crucial functions of industrial parks as wastewater from industrial facilities usually contains toxic compounds that can cause damage to the environment. To control their environmental loads, industrial parks make investment decisions for wastewater treatment plants. For this, they need to consider technical and economic factors as well as future growth projections as substantial construction and operational costs of wastewater treatment plants have to be shared by all companies in an industrial park. In this paper, we consider the long-term capacity planning problem for wastewater treatment facilities of a stochastically growing industrial park. By explicitly modeling randomness in the arrival of new tenants and their random wastewater discharges, our model calculates the future mean and variance of wastewater flow in the industrial park. Mean and variance are used in a Mixed Integer Programming Model to optimize wastewater treatment plant selection over a long planning horizon (30 years). By fitting our first model to empirical data from an industrial park in Turkey, we find that considering the variance of wastewater load is critical for long-term planning. Also, we quantify the economic significance of lowering wastewater discharges which can be achieved by water recycling or interplant water exchange.Yayın Matching of cocycle extensions for second tangent groups(American Institute of Physics Inc., 2022-11-07) Uçgun, Filiz Çağatay; Esen, Oğul; Sütlü, SerkanWe present the second-order tangent group of a Lie group as a cocycle extension of the first-order tangent group. We exhibit matching of the second-order tangent groups of two mutually interacting Lie groups. We examine the cocycle extension character of the matched second-order group and arrive at that matched pair of cocycle extensions is a cocycle extension by itself.Yayın Gluing formulas for volume forms on representation varieties of surfaces(Springer Nature, 2025-08-06) Erdal, Esma DiricanLet Σg,n be a compact oriented surface of genus g≥4 with n boundary components. Due to Witten, the twisted Reidemeister torsion coincides with a power of the Atiyah–Bott–Goldman–Narasimhan symplectic form on the space of representations of π1(Σg,0) in any semi-simple Lie group. In the present paper, we first obtain a multiplicative gluing formula for the twisted Reidemeister torsion of Σg,0 in terms of torsions of Σg1,1,Σg2,1, and boundary circle S1, where g=g1+g2 and g1,g2≥2. Then, by using Heusener and Porti’s results on Σg,n, we show that the symplectic volume form on the representation variety of Σg,0 can be expressed as a product of the holomorphic symplectic volume forms on the relative representation varieties of surfaces Σg1,1 and Σg2,1.Yayın Higher analogues of discrete topological complexity(Springer-Verlag Italia S.R.L., 2024-06-13) Alabay, Hilal; Borat, Ayşe; Cihangirli, Esra; Erdal, Esma DiricanIn this paper, we introduce the nth discrete topological complexity and study its properties such as its relation with simplicial Lusternik–Schnirelmann category and how the higher dimensions of discrete topological complexity relate with each other. Moreover, we find a lower bound of n-th discrete topological complexity which is given by the nth usual topological complexity of the geometric realisation of that complex. Furthermore, we give an example for the strict case of that lower bound.Yayın On the theory of single aspheric lenses(Institute of Electrical and Electronics Engineers Inc., 2024) Hasanoğlu, ElmanAspheric lenses play a pivotal role in a myriad of applications, including antenna theory, communication systems, sophisticated optical systems etc. These lenses, unlike their spherical counterparts, possess an infinite number of free parameters and by utilizing these parameters it is possible to achieve optical and antenna systems with very high and sophisticated performances. However, the lack of a rigorous mathematical theory essentially limits possibility to utilize their full potential in the related fields. In our days, the design process relies heavily on elementary mathematical principles and empirical methods [1-2]. Even a single thick aspheric lens lacks a comprehensive mathematical theory. In this paper a new mathematical approach to the theory of single lenses is proposed. Surprisingly, even in this seemingly straightforward case, the serious mathematical theory of a single aspherical lens relies on rather nontrivial mathematical principles as partial differential equations, differential geometry, Riemann geometry.
