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Yayın Reidemeister-Franz torsion as a homomorphism(Yıldız Teknik Üniversitesi, 2025-05-09) Dirican Erdal, Esma; Zeren, Yusuf; Kirişci, Murat; Çevikel, Adem CengizLet us denote the Diff-isomorphism classes of n-dimensional non-empty, closed, connected, oriented differentiable manifolds by Mn Diff. An n-manifold Mn is called highly connected if π1(Mn)=0 for i=0,…, ⌊n/2⌋-1. So the Diff-isomorphism classes of n-dimensional highly connected differentiable manifolds is given as MnDiff,hc= {Mn𝜖 MDiff | Mn is highly con-nected}. Hence by [1], MnDiff and MnDiff,hc are abelian monoids under the connected sum. By [1]-[3], the monoid M2nDiff,hc is a unique factorisation monoid provided that n ≡ 3,5,7 mod 8 except for n=15 or n=31. Suppose that W2n 𝜖 M2nDiff,hc .Then W2n admits a unique connected sum decomposition into 2n-manifolds that can not be decomposed any further, W2n=M1#M2#⋯#Mj. By using such a connected sum decomposition, we prove that Rei- demeister-Franz torsion can be seen as a monoid homomorphism |T RF − |: M2nDiff,hc → R+given by | T RF (W2n)| = | T RF (M1)| x |T RF (M2)| x ⋯ x | T RF (Mj)|.Yayın Çift manifoldların asiklik olmayan hücresel zincir kompleksleri için torsiyon formülü(Akdeniz Üniversitesi, 2025-09-04) Dirican Erdal, EsmaBir n-manifold Mn, yüksek derecede bağlantılı olarak adlandırılır eğer her i = 0, . . . , bn/2c−1 için πi(Mn) = 0 sağlanırsa. Kabul edelim ki n-boyutlu yüksek derecede bağlantılı kapalı yönlendirilebilir türevlenebilir manifoldların difeomorfizm sınıfları MDiff,hcn ile gösterilsin. Eğer n ≡ 3, 5, 7 (mod 8) ve n, 15 ve 31’ e eşit değil ise MDiff,hc2n manifoldlar üzerinde tek türlü çarpanlara ayırma monoididir [2]. Böylece herhangi bir 2n-manifold W2n ∈ MDiff,hc2n için W2n = M2n1 #M2n2 # . . . #M2nk parçalanışı mevcuttur. Milnor, Reidemeister-Franz torsiyonunun yapıştırmalara göre çarpımsal olarak etki ettiğini bir düzeltici terim farkıyla ispatlamıştır. Ayrıca Milnor, eğer manifoldun hücre parçalanışının zincir kompleksi asiklik ise düzeltici terimin 1 olduğunu ispatlamıştır [1]. Bu çalışmada asiklik varsayımı olmadan, düzeltici terimin 1 olduğu W2n ∈ MDiff,hc2n manifoldlarının torsiyonunu hesaplayan bir formül elde edilmiştir.Yayın A multiplicative gluing formula Reidemeister-Franz torsion of high dimensional closed manifolds(IKSAD Publications, 2024-03-23) Dirican Erdal, Esma; Şahin, HasanLet M be a 2n-dimensional (n≥2), closed, oriented, smooth manifold which is obtained by a connected sum of two closed, oriented, smooth manifolds and . Milnor shows that Reidemeister-Franz torsion acts multiplicatively with respect to such gluings. Namely, the torsion of M is the product of the torsions of , and the torsion of (2n-1)-sphere times a corrective term T(H*) coming from homologies. In this work, by using homological algebra techniques, we obtain a multiplicative gluing formula for the Reidemeister-Franz torsion of M with untwisted ℝ-coefficients so that the corrective term T(H*) becomes 1. Moreover, considering a connected sum decomposition for any 2ndimensional, closed, oriented, smooth manifold W, we develop a useful formula, without a corrective term, to compute the Reidemeister-Franz torsion of W with untwisted ℝ-coefficients in terms of the Reidemeister-Franz torsions of its building blocks in the decomposition.Yayın Kinetic moments of Vlasov dynamics: a matched pair analysis(Marmara Üniversitesi, 2020-10-30) Esen, Oğul; Sütlü, Serkan; Yıldırım, KenanThis talk is based on [1]. We first present a gentle introduction to the Hamiltonian (Lie-Poisson) analysis of dynamical systems. Then we recast the dynamics of kinetic moments of Vlasov equation from the matched pair decomposition point of view. That is, we present the moment dynamics as a coupling of mutually interacting (Lie-Poisson) subdynamics. We observe that one of the constitutive subdynamics is the compressible isentropic fluid flow.Yayın On matched pair Hamiltonian analysis of the compartmental models(Marmara Üniversitesi, 2020-10-30) Ateşli, Begüm; Esen, Oğul; Sütlü, Serkan; Yıldırım, KenanEpidemiological compartmental models predict the spread of an infectious disease that a specific population encounter. The population is divided into compartments representing different stages of the epidemic and the change of these compartments in time is given by nonlinear differential equations. In previous studies, the Hamiltonian analysis of these models is included. In this work, we briefly explain SIR, SEIR, 2-SIR and 2-SEIR models, and their Hamiltonian analysis. We recollect the matched pair Lie-Poisson systems and observe that SIR and SEIR models can be written as matched pair Lie-Poisson systems. We generalize the matched pair Lie-Poisson systems using the twisted cocycle extension. We attain that matched pair Lie-Poisson systems obtained by the twisted cocycle extension is convenient for 2-SIR and 2-SEIR models.Yayın On Lie groupoids, Lie algebroids and equations of motion under mutual actions(Marmara Üniversitesi, 2020-10-30) Esen, Oğul; Kaya, Hanife Kübra; Sütlü, Serkan; Yıldırım, KenanIn this work, we present matching of two mutually interacting Lagrangian systems on Lie algebroid frameworks. Firstly, we mention Lie groupoids and Lie algebroids formally. Due to the mutual actions of two Lie groupoids and two Lie algebroids onto each other, we refer the matched Lie groupoids and Lie algebroids structures. Then, we obtain Euler-Lagrange equations on the matched Lie algebroids which involve individual behaviours and mutual action terms. During this work, we provide many examples.Yayın Eşlenmiş Lie cebiroidlerinin Dual Uzayı üzerindeki Hamilton dinamiği(İstanbul Üniversitesi, 2021-09-03) Esen, Oğul; Kaya, Hanife Kübra; Sütlü, SerkanLie cebiroidleri, hem teğet demetini hem de Lie cebiri yapısını bünyesinde barındıran ancak daha genel olarak ifade edilen yapılardır. Lie cebiroidlerinin dual uzayları üzerindeki Poisson yapısı sayesinde Hamilton dinamiği için uygun bir geometrik altyapı oluşur. Karşılıklı etki içindeki iki Lie cebiroidi, belirli uygunluk koşulları altında eşlenerek etki terimlerinin de gözüktüğü tek bir Lie cebiroidi olarak yazılabilir. Çalışmamızda, karşılıklı etki içindeki iki Hamilton sisteminin beraber hareketini yöneten denklemler eşlenmiş Lie cebiroidlerinin dual uzayı üzerinde tanımlanacak Hamilton formülasyonu ile elde edilecektir.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.Yayın On pseudo-umbilical rotational surfaces with pointwise 1-type gauss map in E-2(4)(Istanbul Technical Univ, 2017-01-01) Bektaş, Burcu; Canfes, Elif Özkara; Dursun, UğurIn this work, we study two families of rotational surfaces in the pseudo Euclidean space 1E4 with profile curves lying in 2 dimensional planes. First, we obtain a classification of pseudo umbilical spacelike surfaces and timelike surfaces in these families. Then, we show that in this classification there exists no a pseudo umbilical rotational surface in 1E4 with pointwise 1 type Gauss map of second kind. Finally, we determine such pseudo umbilical rotational surfaces in 1E4 having pointwise 1 type Gauss map of first kind.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 Nonlinear waves in a stenosed elastic tube filled with viscous fluid: Forced perturbed korteweg-de vries equation(Springer Science and Business Media, LLC, 2008) Tay, Kim Gaik; Demiray, Hilmi; Tiong, Ong CheeIn the present work, treating the artery as a prestressed thin-walled and long circularly cylindrical elastic tube with a mild symmetrical stenosis and the blood as an incompressible Newtonian fluid, we have studied the propagation of weakly nonlinear waves in such a composite medium, in the long wave approximation, by use of the reductive perturbation method. By introducing a set of stretched coordinates suitable for the boundary value type of problems and expanding the field variables into asymptotic series of the smallness parameter of nonlinearity and dispersion, we obtained a set of nonlinear differential equations governing the terms at various order. By solving these nonlinear differential equations, we obtained the forced perturbed Korteweg-de Vries equation with variable coefficient as the nonlinear evolution equation. By use of the coordinate transformation, it is shown that this type of nonlinear evolution equation admits a progressive wave solution with variable wave speed.Yayın “International conference on vibration problems” ICOVP-2007 and a short history(Springer Science and Business Media, LLC, 2008) İnan, Esin[No abstract available]Yayın Nonlinear wave modulation in a prestressed thin elastic tube filled with an inviscid fluid(Wit Press, 2002) Bakırtaş, İlkay; Demiray, HilmiIn the present work, employing the nonlinear equations of an incompressible, isotropic and elastic thin tube and the approximate equations of an incompressible inviscid fluid and then utilizing the reductive perturbation technique, the amplitude modulation of weakly nonlinear waves is examined. It is shown that, the amplitude modulation of these waves is governed by a nonlinear Schrödinger (NLS) equation. The result is compared with some previous works on the same subject. The modulational instability of the monochromatic wave solution is discussed for some elastic materials and initial deformations. It is shown that the amplitude modulation of weakly nonlinear waves near the marginal state is governed by the Generalized Nonlinear Schrödinger equation (GNLS).Yayın On complex solutions of the eikonal equation(IEEE, 2007) Hasanoğlu, ElmanIn this paper a new approach of complex rays in an inhomogeneous medium is presented. Complex rays are complex solutions of the eikonal equation, the main equation of the geometical optics. It is shown that solving the eikonal equation by using the characteristic method naturally leads to the pseudoriemann and Minkowski geometries. In framework of these geometries complex rays , like the real ones, can be drawn in real space and they may have caustics, and caustics also can be drawn in real space.Yayın Beam tracing theory in Minkowski space(IEEE, 2011) Hasanoğlu, ElmanThis paper provides a novel approach to beam theory in homogeneous lossless mediun. The main idea is to interpret the classic eikonal equation in three dimensional Minkowski space.Yayın Variable coefficient Korteweg-deVries equation in fluid-filled elastic tubes(Technical University Liberec, 2011-09-05) Demiray, HilmiIn the present work, treating the arteries as a prestressed thin elastic tube with a stenosis and the blood as an inviscid fluid, we have studied the propagation of weakly nonlinear waves in such a medium by use of the reductive perturbation method and obtained the variable coefficient Korteweg-deVries (KdV) equation as the evolution equation. A progressive wave type of solution to this evolution equation, in the sense of distribution, is presented and the result is discussed.Yayın Cross-sectional thermoacoustic imaging using multi-layer cylindrical media(IEEE, 2017-11-10) Elmas, Demet; Ünalmış Uzun, Banu; İdemen, Mehmet Mithat; Karaman, MustafaFor cross-sectional two-dimensional thermoacustic imaging of breast and brain, we explored solution of the wave equation using layered tissue model consisting of concentric annular layers on a cylindrical cross-section. To obtain the forward and inverse solutions of the thermoacoustic wave equation, we derived the Green's function involving Bessel and Hankel functions by employing the geometrical and acoustic parameters (densities and velocities) of layered media together with temporal initial condition, radiation conditions and continuity conditions on the layers' boundaries. The image reconstruction based on this approach involves the layer parameters as the apriori information which can be estimated from the acquired thermoacoustic data. To test and compare our layered solution with conventional solution based on homogeneous medium assumption, we performed simulations using numerical test phantoms consisting of sources distributed in the layered structure.Yayın Thermoacoustic image reconstruction based on layered tissue model(SPIE-Int Soc Optical Engineering, 2017) Bayıntır, Hazel; İdemen, Mehmet Mithat; Ünalmış Uzun, Banu; Karaman, Mustafa; Elmas, DemetWe derived analytical forward and inverse solution of thermoacoustic wave equation for inhomogeneous multi layered planar and cylindrical mediums with the source distribution existing in all layers. These solutions are applicable for imaging of organs such as breast and brain, whose structures are suitable for multi-layer modelling. For qualitative testing and comparison of the point-spread-functions associated with the homogeneous and layered solutions, we performed numerical simulations. Our simulation results show that the conventional inverse solution based on homogeneous medium assumption, as expected, produces incorrect locations of point sources and significantly increased side lobes, whereas our inverse solution involving the multi-layered medium produces point sources at the correct locations with lower side lobes.Yayın Çok katmanlı silindirik yapılar için termoakustik dalga denkleminin ters çözümü(IEEE, 2017-06-27) Elmas, Demet; Ünalmış Uzun, Banu; İdemen, Mehmet Mithat; Karaman, MustafaTermoakustik görüntüleme, elektromanyetik enerji uyarımı ile ultrason dalgaları oluşumunu sağlayan yeni bir yöntemdir. Bu sistemin görüntüleme işlemi termoakustik dalga denkleminin ters çözümüne dayanmaktadır. Ters çözümde görüntülenecek dokunun homojen yapıda olduğu varsayımı, görüntü kalitesinin azalmasına neden olur. Bu çalışmada, meme ve beyinin görüntülemesinde uygulanabilecek üç boyutlu eş merkezli silindirik çok katmanlı yapılar için termoakustik dalga denkleminin analitik ters çözümü elde edilmiştir. Çözümü sayısal olarak test etmek için, noktasal kaynaklar içeren üç katmanlı fantom kullanılarak numerik simulasyonlar elde edilmiştir.Yayın Çok katmanlı düzlemsel ortam için termoakustik dalga denkleminin çözümü(IEEE, 2017-06-27) Bayıntır, Hazel; Ünalmış Uzun, Banu; İdemen, Mehmet Mithat; Karaman, MustafaBu çalışmada, farklı akustik parametrelere sahip çok katmanlı düzlemsel ortam için tüm katmanlarda kaynak dağılımı olduğu varsayımı altında termoakustik dalga denkleminin analitik olarak düz ve ters çözümü elde edilmiştir. Çok katmanlı düzlemsel ortam için elde edilen analitik çözüm katmanlı düzlemsel green fonksiyonlarına dayanmaktadır. Çok katmanlı düzlemsel modelleme meme, deri ve karın bölgesi görüntülemelerine uygun bir modellemedir. Elde edilen analitik çözüm ile literatürde var olan homojen ortam varsayımına dayanan çözüm her katmanda noktasal kaynak alınarak sayısal olarak karşılaştırılmıştır.
