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Yayın Some remarks on uniform boundary Harnack principles(Cornell Univ, 2021-03-18) 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.Yayın Matched pair analysis of the Vlasov plasma(Cornell Univ, 2021-02-09) 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.Yayın Cohomologies and generalized derivation extensions of n-Lie algebras(Cornell Univ, 2021-04-18) Ateşli, Begüm; Esen, Oğul; Sütlü, SerkanA cohomology theory, associated to a n-Lie algebra and a representation space of it, is introduced. It is observed that this cohomology theory is qualified to encode the generalized derivation extensions, and that it coincides, for n = 3, with the known cohomology of n-Lie algebras. The abelian extensions and infinitesimal deformations of n-Lie algebras, on the other hand, are shown to be characterized by the usual cohomology of n-Lie algebras. Furthermore, the Hochschild-Serre spectral sequence of the Lie algebra cohomology is upgraded to the level of n-Lie algebras, and is applied to the cohomology of generalized derivation extensions.Yayın Tulczyjew's triplet for Lie groups III : higher order dynamics and reductions for iterated bundles(Cornell Univ, 2021-02-23) Esen, Oğul; Gümral, Hasan; Sütlü, SerkanGiven a Lie group G, we elaborate the dynamics on T*T*G and T*T G, which is given by a Hamiltonian, as well as the dynamics on the Tulczyjew 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.Yayın Quantum van Est isomorphism(Cornell Univ, 2022-05-07) Kaygun, Atabey; Sütlü, SerkanMotivated by the fact that the Hopf-cyclic (co)homology of the quantized algebras of functions and quantized universal enveloping algebras are the correct analogues of the Lie algebra and Lie group (co)homologies, we hereby construct three van Est type isomorphisms between the Hopf-cyclic (co)homologies of Lie groups and Lie algebras, and their quantum groups and corresponding enveloping algebras, both in h-adic and q-deformation frameworks.Yayın Higher analogues of discrete topological complexity(Cornell Univ, 2024-04-16) Alabay, Hilal; Borat, Ayşe; Cihangirli, Esra; Erdal, Esma DiricanIn this paper, we introduce the n−th discrete topological complexity and study its properties such as its relation with simplicial LusternikSchnirelmann category and how the higher dimensions of discrete topological complexity relate with each other. Moreover, we find a lower bound of n−discrete topological complexity which is given by the n−th 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 twisted torsion of compact 3-manifolds(Cornell Univ, 2024-08-20) Erdal, Esma DiricanLet M be a 3-manifold with connected non-vacuos boundary which is not spherical. Assume that N is another 3-manifold with vacuous boundary and N∗ is the 3-manifold obtained by removing from N the interior of a 3-cell. In the present paper, we find a relationship between the multiplicative property of the twisted Reidemeister torsion and the connected sum operation on these manifolds in order to understand their topology and geometry.Yayın On extensions, Lie-Poisson systems, and dissipation(Cornell Univ, 2021-01-08) Esen, Oğul; Özcan, Gökhan; Sütlü, SerkanOn the dual space of extended structure, equations governing the collective motion of two mutually interacting Lie-Poisson systems are derived. By including a twisted 2-cocycle term, this novel construction is providing the most general realization of (de)coupling of Lie-Poisson systems. A double cross sum (matched pair) of 2-cocycle extensions are constructed. The conditions are determined for this double cross sum to be a 2-cocycle extension by itself. On the dual spaces, Lie-Poisson equations are computed. We complement the discussion by presenting a double cross sum of some symmetric brackets, such as double bracket, Cartan-Killing bracket, Casimir dissipation bracket, and Hamilton dissipation bracket. Accordingly, the collective motion of two mutually interacting irreversible dynamics, as well as 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 finite-dimensional examples, the coupling of two Heisenberg algebras and coupling of two copies of 3D dynamics are studied.Yayın Early Alzheimer's disease detection from retinal OCT images: a UK Biobank study(Cornell Univ, 2025-11-07) Turkan, Yasemin; Tek, Faik Boray; Nazlı, M. Serdar; Eren, ÖyküAlterations in retinal layer thickness, measurable using Optical Coherence Tomography (OCT), have been associated with neurodegenerative diseases such as Alzheimer's disease (AD). While previous studies have mainly focused on segmented layer thickness measurements, this study explored the direct classification of OCT B-scan images for the early detection of AD. To our knowledge, this is the first application of deep learning to raw OCT B-scans for AD prediction in the literature. Unlike conventional medical image classification tasks, early detection is more challenging than diagnosis because imaging precedes clinical diagnosis by several years. We fine-tuned and evaluated multiple pretrained models, including ImageNet-based networks and the OCT-specific RETFound transformer, using subject-level cross-validation datasets matched for age, sex, and imaging instances from the UK Biobank cohort. To reduce overfitting in this small, high-dimensional dataset, both standard and OCT-specific augmentation techniques were applied, along with a year-weighted loss function that prioritized cases diagnosed within four years of imaging. ResNet-34 produced the most stable results, achieving an AUC of 0.62 in the 4-year cohort. Although below the threshold for clinical application, our explainability analyses confirmed localized structural differences in the central macular subfield between the AD and control groups. These findings provide a baseline for OCT-based AD prediction, highlight the challenges of detecting subtle retinal biomarkers years before AD diagnosis, and point to the need for larger datasets and multimodal approaches.
