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Publication Open Access 2D hybrid meshes for direct simulation Monte Carlo solvers(Institute of Physics (IOP) Publishing, 2013) Şengil, Nevsan; Department of Mathematics; Department of Mathematics; Şengil, Uluç; Master Student; College of SciencesThe efficiency of the direct simulation Monte Carlo (DSMC) method decreases considerably if gas is not rarefied. In order to extend the application range of the DSMC method towards non-rarefied gas regimes, the computational efficiency of the DSMC method should be increased further. One of the most time consuming parts of the DSMC method is to determine which DSMC molecules are in close proximity. If this information is calculated quickly, the efficiency of the DSMC method will be increased. Although some meshless methods are proposed, mostly structured or non-structured meshes are used to obtain this information. The simplest DSMC solvers are limited with the structured meshes. In these types of solvers, molecule indexing according to the positions can be handled very fast using simple arithmetic operations. But structured meshes are geometry dependent. Complicated geometries require the use of unstructured meshes. In this case, DSMC molecules are traced cell-by-cell. Different cell-by-cell tracing techniques exist. But, these techniques require complicated trigonometric operations or search algorithms. Both techniques are computationally expensive. In this study, a hybrid mesh structure is proposed. Hybrid meshes are both less dependent on the geometry like unstructured meshes and computationally efficient like structured meshes.
Publication Metadata only A characterization of heaviness in terms of relative symplectic cohomology(Wiley, 2024) Mak, Cheuk Yu; Sun, Yuhan; Department of Mathematics; Department of Mathematics; Varolgüneş, Umut; College of SciencesFor a compact subset K$K$ of a closed symplectic manifold (M,omega)$(M, \omega)$, we prove that K$K$ is heavy if and only if its relative symplectic cohomology over the Novikov field is nonzero. As an application, we show that if two compact sets are not heavy and Poisson commuting, then their union is also not heavy. A discussion on superheaviness together with some partial results is also included.Item Metadata only A characterization of quasipositive two-bridge knots(World Scientific Publ Co Pte Ltd, 2024) 0000-0002-9758-1045; Orevkov, Stepan; Department of Mathematics; Özbağcı, Burak; Faculty Member; College of Sciences; 29746We prove a simple necessary and sufficient condition for a two-bridge knot K(p,q) to be quasipositive, based on the continued fraction expansion of p/q. As an application, coupled with some classification results in contact and symplectic topology, we give a new proof of the fact that smoothly slice two-bridge knots are non-quasipositive. Another proof of this fact using methods within the scope of knot theory is presented in Appendix A, by Stepan Orevkov.Publication Metadata only A differential integrability condition for two-dimensional Hamiltonian systems(Czech Technical Univ Prague, 2014) N/A; Department of Mathematics; Department of Mathematics; Mostafazadeh, Ali; Faculty Member; College of Sciences; 4231We review, restate, and prove a result due to Kaushal and Korsch [Phys. Lett. A 276, 47 (2000)] on the complete integrability of two-dimensional Hamiltonian systems whose Hamiltonian satisfies a set of four linear second order partial differential equations. In particular, we show that a two- dimensional Hamiltonian system is completely integrable, if the Hamiltonian has the form H = T + V where V and T are respectively harmonic functions of the generalized coordinates and the associated momenta.Publication Metadata only A dynamical formulation of one-dimensional scattering theory and its applications in optics(Academic Press Inc Elsevier Science, 2014) NA; Department of Mathematics; Department of Mathematics; Mostafazadeh, Ali; Faculty Member; College of Sciences; 4231We develop a dynamical formulation of one-dimensional scattering theory where the reflection and transmission amplitudes for a general, possibly complex and energy-dependent, scattering potential are given as solutions of a set of dynamical equations. By decoupling and partially integrating these equations, we reduce the scattering problem to a second order linear differential equation with universal initial conditions that is equivalent to an initial-value time-independent Schrodinger equation. We give explicit formulas for the reflection and transmission amplitudes in terms of the solution of either of these equations and employ them to outline an inverse-scattering method for constructing finite-range potentials with desirable scattering properties at any prescribed wavelength. In particular, we construct optical potentials displaying threshold lasing, antilasing, and unidirectional invisibility.Publication Metadata only A fredholm alternative-like result on power bounded operators(Scientific Technical Research Council Turkey-Tubitak, 2011) Yavuz, Onur; Department of Mathematics; Department of Mathematics; Ülger, Ali; Faculty Member; College of Sciences; N/ALet X be a complex Banach space and T:X\rightarrow X be a power bounded operator, i.e., \sup_{n \geq 0}\ T^n\Publication Open Access A generalization of the Hardy-Littlewood conjecture(Colgate University, 2022) Department of Mathematics; Department of Mathematics; Alkan, Emre; Faculty Member; College of Sciences; 32803A famous conjecture of Hardy and Littlewood claims the subadditivity of the prime counting function, namely that ?(x+y) ? ?(x)+?(y) holds for all integers x, y ? 2, where ?(x) is the number of primes not exceeding x. It is widely believed nowadays that this conjecture is not true since Hensley and Richards stunningly discovered an incompatibility with the prime k-tuples conjecture. Despite this drawback, here we generalize the subadditivity conjecture to subsets of prime numbers possessing a rich collection of preassigned structures. We show that subadditivity holds in this extended manner over certain ranges of the parameters which are wide enough to imply that it holds in an almost all sense. Under the prime k-tuples conjecture, very large values of convex combinations of the prime counting function are obtained infinitely often, thereby indicating a strong deviation of ?(x) from being convex, even in a localized form. Finally, a Tauberian type condition is given for subsets of prime numbers which in turn implies an extension of a classical phenomenon, originally suggested by Legendre, about the asymptotically best fit functions to ?(x) of the shape x/(log x ? A).Publication Metadata only A generalization of the Hardy-Littlewood conjecture(Colgate University, 2022) Department of Mathematics; Department of Mathematics; Alkan, Emre; Faculty Member; College of Sciences; 32803A famous conjecture of Hardy and Littlewood claims the subadditivity of the prime counting function, namely that π(x+y) ≤ π(x)+π(y) holds for all integers x, y ≥ 2, where π(x) is the number of primes not exceeding x. It is widely believed nowadays that this conjecture is not true since Hensley and Richards stunningly discovered an incompatibility with the prime k-tuples conjecture. Despite this drawback, here we generalize the subadditivity conjecture to subsets of prime numbers possessing a rich collection of preassigned structures. We show that subadditivity holds in this extended manner over certain ranges of the parameters which are wide enough to imply that it holds in an almost all sense. Under the prime k-tuples conjecture, very large values of convex combinations of the prime counting function are obtained infinitely often, thereby indicating a strong deviation of π(x) from being convex, even in a localized form. Finally, a Tauberian type condition is given for subsets of prime numbers which in turn implies an extension of a classical phenomenon, originally suggested by Legendre, about the asymptotically best fit functions to π(x) of the shape x/(log x − A). © 2022, Colgate University. All rights reserved.Publication Metadata only A Hamiltonian formulation of the Pais-Uhlenbeck oscillator that yields a stable and unitary quantum system(Elsevier Science Bv, 2010) NA; Department of Mathematics; Department of Mathematics; Mostafazadeh, Ali; Faculty Member; College of Sciences; 105131We offer a new Hamiltonian formulation of the classical Pais-Uhlenbeck oscillator and consider its canonical quantization. We show that for the non-degenerate case where the frequencies differ, the quantum Hamiltonian operator is a Hermitian operator with a positive spectrum, i.e., the quantum system is both stable and unitary. Furthermore it yields the classical Pais-Uhlenbeck oscillator in the classical limit. A consistent description of the degenerate case based on a Hamiltonian that is quadratic in momenta requires its analytic continuation into a complex Hamiltonian system possessing a generalized PT-symmetry (an involutive antilinear symmetry). We devise a real description of this complex system, derive an integral of motion for it, and explore its quantization.Publication Metadata only A new correlation coefficient for bivariate time-series data(Elsevier Science Bv, 2014) Erdem, Orhan; Varlı, Yusuf; Department of Mathematics; Department of Mathematics; Ceyhan, Elvan; Faculty Member; College of SciencesThe correlation in time series has received considerable attention in the literature. Its use has attained an important role in the social sciences and finance. For example, pair trading in finance is concerned with the correlation between stock prices, returns, etc. In general, Pearson's correlation coefficient is employed in these areas although it has many underlying assumptions which restrict its use. Here, we introduce a new correlation coefficient which takes into account the lag difference of data points. We investigate the properties of this new correlation coefficient. We demonstrate that it is more appropriate for showing the direction of the covariation of the two variables overtime. We also compare the performance of the new correlation coefficient with Pearson's correlation coefficient and Detrended Cross-Correlation Analysis (DCCA) via simulated examples. (C) 2014 Elsevier B.V. All rights reserved.