FEF - Makale Koleksiyonu | Matematik Bölümü / Department of Mathematics
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Öğe Tulczyjew's triplet for Lie groups III: higher order dynamics and reductions for iterated bundles(Serbian Society of Mechanics, 2021) Esen, Oğul; Gümral, Hasan; Sütlü, SerkanGiven a Lie group G, we elaborate the dynamics on T*T*G and T*TG, which is given by a Hamiltonian, as well as the dynamics on the Tul-czyjew symplectic space TT * G, which may be defined by a Lagrangian or a Hamiltonian function. As the trivializations we adapted respect the group structures of the iterated bundles, we exploit all possible subgroup reductions (Poisson, symplectic or both) of higher order dynamics.Öğe On extensions, Lie-Poisson systems, and dissipation(Heldermann Verlag, 2022-07-06) Esen, Oğul; Özcan, Gökhan; Sütlü, SerkanLie-Poisson systems on the dual spaces of unified products are studied. Having been equipped with a twisted 2-cocycle term, the extending structure framework allows not only to study the dynamics on 2-cocycle extensions, but also to (de)couple mutually interacting Lie-Poisson systems. On the other hand, symmetric brackets; such as the double bracket, the Cartan-Killing bracket, the Casimir dissipation bracket, and the Hamilton dissipation bracket are worked out in detail. Accordingly, the collective motion of two mutually interacting irreversible dynamics, as well as the mutually interacting metriplectic flows, are obtained. The theoretical results are illustrated in three examples. As an infinite-dimensional physical model, decompositions of the BBGKY hierarchy are presented. As for the finite-dimensional examples, the coupling of two Heisenberg algebras, and the coupling of two copies of 3D dynamics are studied.Öğe Inverse solution of thermoacoustic wave equation for cylindrical layered media(Frontiers Media S.A., 2022-03-30) Elmas, Demet; Ünalmış Uzun, BanuThermoacoustic imaging is a crossbred approach taking advantages of electromagnetic and ultrasound disciplines, together. A significant number of current medical imaging strategies are based on reconstruction of source distribution from information collected by sensors over a surface covering the region to be imaged. Reconstruction in thermoacoustic imaging depends on the inverse solution of thermoacoustic wave equation. Homogeneous assumption of tissue to be imaged results in degradation of image quality. In our paper, inverse solution of the thermoacoustic wave equation using layered tissue model consisting of concentric annular layers on a cylindrical cross-section is investigated for cross-sectional thermoacustic imaging of breast and brain. By using Green’s functions and surface integral methods we derive an exact analytic inverse solution of thermoacoustic wave equation in frequency domain. Our inverse solution is an extension of conventional solution to layered cylindrical structures. By carrying out simulations, using numerical test phantoms consisting of thermoacoustic sources distributed in the layered model, our layered medium assumption solution was tested and benchmarked with conventional solutions based on homogeneous medium assumption in frequency domain. In thermoacoustic image reconstruction, where the medium is assumed as homogeneous medium, the solution of nonhomogeneous thermoacoustic wave equation results in geometrical distortions, artifacts and reduced image resolution due to inconvenient medium assumptions.Öğe Slant curves in the Lorentzian warped product manifold - I× fE²(Birkhauser, 2022-03-15) Dursun, UğurIn this work, we study slant curves in the 3-dimensional Lorentzian warped product - I× fE², where E² is a 2-dimensional Euclidean plane, I? R is an open interval equipped with the metric dt², and f is a positive smooth function on I. First we give a characterization of slant curves, and then we obtain a classification of all slant curves in - I× fE². We also compute their curvature and torsion, and we obtaine some results on slant curves and helices in the de Sitter space S13(1) and in the Minkowski space E13. Moreover we determined some biharmonic slant curves in S13(1).Öğe Sönümlemeli sistemlerin eşlenmesi üzerine(Afyon Kocatepe Üniversitesi, 2021) Esen, Oğul; Özcan, Gökhan; Sütlü, SerkanBu makalede karşılıklı etki tepki içindeki iki sönümlemeli sistemin beraber (kollektif- eşlenmiş) hareketinin analizi üzerine bir yöntem öneriyoruz. Aşikardır ki; eşlenmiş hareketi kontrol eden diferansiyel denklemler iki sistemin denklemlerini bir arada yazmak dışında karşılıklı etki tepkinin doğurduğu fazladan terimler içerecektir. Karşılıklı etkiyi belirleyen ilave terimler, Lie cebirlerinin karşılıklı etkisi ile üretilecektir ve bu şekilde pür geometrik/cebirsel bir inşa gerçekleştirilecektir. Sonrasında elde ettiğimiz sonuçları 3 ve 4 boyutlu örneklerde göstereceğiz.Öğe A new form of Kakutani fixed point theorem and intersection theorem with applications(Ministry Communications & High Technologies, 2021) Farajzadeh, Ali P.; Zanganehmehr, Parastoo; Hasanoğlu, ElmanThe first aim of this paper is to extend the Kakutani's fixed point theorem from locally convex topological vector spaces to linear topological spaces. The second goal is to present sufficient conditions under which the intersection of a family of sets is nonempty. The third step is to provide an existence theorem for a set valued mapping. Finally, as an application, an existence result of a solution for quasi-equilibrium problem is given.Öğe Epidemiyolojideki kompartman modellerinin eşlenmiş Hamilton analizi(Marmara Üniversitesi Fen Bilimleri Enstitüsü, 2021-01-13) Ateşli, Begüm; Esen, Oğul; Sütlü, SerkanEpidemiyolojideki SIR, SEIR, 2-SIR ve 2-SEIR modellerinin Hamilton formülasyonu verildi. Eşlenmiş Lie-Poisson sistemleri yazıldı. SIR ve SEIR modellerinin eşlenmiş Lie-Poisson sistemi oldukları gösterildi. Bir Lie cebiri için bükülmüş eşçevrim genişlemesi çalışıldı. Bu genişlemenin dual uzayı üzerinde eşlenmiş Lie-Poisson denklemleri elde edildi. SIR ve SEIR kompartman modellerinin iki popülasyon karşılığı olan 2-SIR ve 2-SEIR modellerinin bükülmüş eşçevrim genişlemesiyle elde edilmiş Lie-Poisson sistemi olarak ifade edilebilecekleri gösterildi.Öğe Eşlenmiş Lie grupları üzerindeki Lagrange fark denklemleri(Marmara Üniversitesi Fen Bilimleri Enstitüsü, 2021-03-09) Esen, Oğul; Kudeyt, Mahmut; Sütlü, SerkanBu makalede, kesikli (discrete) dinamiğin Lagrange formülasyonu eşlenmiş (matched pair) Lie grupları üzerinde çalışılmıştır. Sonuç olarak, karşılıklı etkileşim altındaki iki sistemin kolektif davranışını belirleyen eşlenmiş (Lagrange) fark denklemleri elde edilmiştir. İki örnek verilmiştir. İlki, bir Lie grubunun tanjant grubu üzerindeki fark denklemleri, ikincisi ise Heisenberg grubu üzerindeki fark denklemleridir.Öğe Lie cebiroidleri üzerindeki Lagrange dinamiğinin eşlenmesi problemi üzerine(Süleyman Demirel Üniversitesi, 2021-08-15) Esen, Oğul; Kaya, Hanife Kübra; Sütlü, SerkanLie cebiroidleri, bir anlamda tanjant demetini ve Lie cebiri yapısını beraber ihtiva eden ve fakat daha genel olan geometrik inşaalardır. Lagrange dinamiğinin en genel ifadesi Lie cebiroidleri üzerinde mümkündür. Bu makalede, karşılıklı (Lie cebiroidi üzerinde tanımlı) etki içindeki iki Lagrange dinamiğinin beraber davranışı, geometrik ve cebirsel bir yol ile elde edilecektir. Bu bakış açısı ile etkileşim, Lie cebiroidlerinin birbirleri üzerine olan lineer temsilleri (etkileri) ifade edilecektir. Böylece, belirli uyumluluk şartını sağlayan karşılıklı etki içindeki iki Lie cebiroidinin eşlenmesi, diğer bir ifade ile tek bir Lie cebiroidi olarak yazılması sağlanacaktır. Sonrasında ise eşlenmiş Lie cebiroidi üzerinde Lagrange dinamiği yazılacaktır. Elde edilecek kollektif (eşlenmiş) hareket denklemleri, bireysel davranışların gözlemlenmesinin yanı sıra karşılıklı etki terimlerinin de belirlenmesine olanak verecektir. Çalışmamız esnasında bir çok örnek sunularak teorik tanımların daha net anlatımı yakalanmaya çalışılacaktır.Öğe Matched pair analysis of the Vlasov plasma(American Institute of Mathematical Sciences-AIMS, 2021-06) Esen, Oğul; Sütlü, SerkanWe present the Hamiltonian (Lie-Poisson) analysis of the Vlasov plasma, and the dynamics of its kinetic moments, from the matched pair decomposition point of view. We express these (Lie-Poisson) systems as couplings of mutually interacting (Lie-Poisson) subdynamics. The mutual interaction is beyond the well-known semi-direct product theory. Accordingly, as the geometric framework of the present discussion, we address the matched pair Lie-Poisson formulation allowing mutual interactions. Moreover, both for the kinetic moments and the Vlasov plasma cases, we observe that one of the constitutive subdynamics is the compressible isentropic fluid flow, and the other is the dynamics of the kinetic moments of order >= 2. In this regard, the algebraic/geometric (matched pair) decomposition that we offer, is in perfect harmony with the physical intuition. To complete the discussion, we present a momentum formulation of the Vlasov plasma, along with its matched pair decomposition.Öğe Constant angle surfaces in the Lorentzian warped product manifold – I × fE²(Birkhauser, 2021-06) Dursun, Uğur; Turgay, Nurettin CenkIn this work, we study constant angle space-like and time-like surfaces in the 3-dimensional Lorentzian warped product manifold - I× fE² with the metric g~ = - d t²+ f²(t) (d x²+ d y²) , where I is an open interval, f is a strictly positive function on I, and E² is the Euclidean plane. We obtain a classification of all constant angle space-like and time-like surfaces in - I× fE². In this classification, we determine space-like and time-like surfaces with zero mean curvature, rotational surfaces, and surfaces with constant Gaussian curvature. Also, we obtain some results on constant angle space-like and time-like surfaces of the de Sitter space S13(1).Öğe Discrete dynamical systems over double cross-product Lie groupoids(World Scientific, 2021-03) Esen, Oğul; Sütlü, SerkanDiscrete Euler-Lagrange equations are studied over double cross product Lie groupoids. As such, a geometric framework for the local analysis of a discrete dynamical system is established. The arguments are elucidated on the local discrete dynamics of a gauge groupoid. The discrete Elroy's beanie is studied as a physical example.Öğe Some boundary Harnack principles with uniform constants(Springer Science and Business Media B.V., 2022-10) Barlow, Martin T.; Karlı, DenizWe prove two versions of a boundary Harnack principle in which the constants do not depend on the domain by using probabilistic methods.Öğe Homology of quantum linear groups(Int Press Boston, 2021-03-24) Kaygun, Atabey; Sütlü, SerkanFor every n >= 1, we calculate the Hochschild homology of the quantum monoids M-q(n), and the quantum groups GL(q)(n) and SLq(n) with coefficients in a 1-dimensional module coming from a modular pair in involution.Öğe Adaptive convolution kernel for artificial neural networks(Academic Press Inc., 2021-02) Tek, Faik Boray; Çam, İlker; Karlı, DenizMany deep neural networks are built by using stacked convolutional layers of fixed and single size (often 3 × 3) kernels. This paper describes a method for learning the size of convolutional kernels to provide varying size kernels in a single layer. The method utilizes a differentiable, and therefore backpropagation-trainable Gaussian envelope which can grow or shrink in a base grid. Our experiments compared the proposed adaptive layers to ordinary convolution layers in a simple two-layer network, a deeper residual network, and a U-Net architecture. The results in the popular image classification datasets such as MNIST, MNIST-CLUTTERED, CIFAR-10, Fashion, and ‘‘Faces in the Wild’’ showed that the adaptive kernels can provide statistically significant improvements on ordinary convolution kernels. A segmentation experiment in the Oxford-Pets dataset demonstrated that replacing ordinary convolution layers in a U-shaped network with 7 × 7 adaptive layers can improve its learning performance and ability to generalize.Öğe Öğe Second order Lagrangian dynamics on double cross product groups(Elsevier B.V., 2021-02) Oğul, Esen; Kudeyt, Mahmut; Sütlü, Serkan SelçukWe observe that the iterated tangent group of a Lie group may be realized as a double cross product of the 2nd order tangent group, with the Lie algebra of the base Lie group. Based on this observation, we derive the 2nd order Euler–Lagrange equations on the 2nd order tangent group from the 1st order Euler–Lagrange equations on the iterated tangent group. We also present in detail the 2nd order Lagrangian dynamics on the 2nd order tangent group of a double cross product group.Öğe Some results on a subclass of harmonic mappings of order alpha(Işık University Press, 2014) Varol, Dürdane; Aydoğan, Seher Melike; Owa, ShigeyoshiLet SH be the class of harmonic mappings defined by SH = { f = h(z) + g(z)| h(z) = z + ?? n=2 anz?, g(z) = b1z + ?? n=2 bnz?, b1 < 1 } where h(z) and g(z) are analytic. Additionally f(z) ? SH(?) ? | zh? (z) ? zg?(z) h(z) + g(z) ? 1 ? b1 1 + b1| < | 1 ? b1 1 + b1| ? ?, z ? U, 0 ? ? < 1 ? b1 1 + b1 In the present work, by considering the analyticity of the functions defined by R. M. Robinson [7], we discuss the applications to the harmonic mappings.Öğe Harmonic mappings related to starlike function of complex order ?(Işık University Press, 2014) Aydoğan, Seher MelikeLet SH be the class of harmonic mappings defined by SH = { f = h(z) + g(z) | h(z) = z + ?? n=2 anz?, g(z) = ?? n=1 bnz?} The purpose of this talk is to present some results about harmonic mappings which was introduced by R. M. Robinson [8].
