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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; Şengil, Uluç; Master Student; Department of Mathematics; 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 chain-binomial model for pull and push-based information diffusion(IEEE, 2006) Department of Mathematics; Department of Computer Engineering; Çağlar, Mine; Özkasap, Öznur; Faculty Member; Faculty Member; Department of Mathematics; Department of Computer Engineering; College of Sciences; College of Engineering; 105131; 113507We compare pull and push-based epidemic paradigms for information diffusion in large scale networks. Key benefits of these approaches are that they are fully distributed, utilize local information only via pair-wise interactions, and provide eventual consistency, scalability and communication topology-independence, which make them suitable for peer-to-peer distributed systems. We develop a chain-Binomial epidemic probability model for these algorithms. Our main contribution is the exact computation of message delivery latency observed by each peer, which corresponds to a first passage time of the underlying Markov chain. Such an analytical tool facilitates the comparison of pull and push-based spread for different group sizes, initial number of infectious peers and fan-out values which are also accomplished in this study. Via our analytical stochastic model, we show that push-based approach is expected to facilitate faster information spread both for the whole group and as experienced by each member.Publication Open Access A generalization of the Hardy-Littlewood conjecture(Colgate University, 2022) Department of Mathematics; Alkan, Emre; Faculty Member; Department of Mathematics; 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 Open Access A note on weakly compact homomorphisms between uniform algebras(Polish Academy of Sciences, 1997) Department of Mathematics; Ülger, Ali; Faculty Member; Department of Mathematics; College of SciencesPublication Open Access A polynomial embedding of pair of partial orthogonal latin squares(Elsevier, 2014) Donovan, Diane M.; Department of Mathematics; Yazıcı, Emine Şule; Faculty Member; Department of Mathematics; College of Sciences; 27432We show that a pair of orthogonal partial Latin squares of order n can be embedded in a pair of orthogonal Latin squares of order at most 16n(4) and all orders greater than or equal to 48n(4). This paper provides the first direct polynomial order embedding construction for pairs of orthogonal partial Latin squares.Publication Open Access A statistical subgrid scale model for large eddy simulations(American Institute of Physics (AIP) Publishing, 2013) Kara, Rukiye; Department of Mathematics; Çağlar, Mine; Faculty Member; Department of Mathematics; College of Sciences; 105131Çinlar velocity is a promising subgrid velocity model for large eddy simulation. The energy spectrum plays a central role for modeling the subgrid stress term in filtered Navier-Stokes equations. Considering a truncated Gamma distribution for radius of eddies, the subgrid scale energy spectrum has been computed analytically. In this study, we develop a new subgrid stress model for representing the small scale effects in LES by defining the parameters of the energy spectrum.Publication Metadata only A survey of the additive dilogarithm(Birkhauser, 2021) Department of Mathematics; Ünver, Sinan; Faculty Member; Department of Mathematics; College of Sciences; 177871Borel’s construction of the regulator gives an injective map from the algebraic K–groups of a number field to its Deligne–Beilinson cohomology groups. This has many interesting arithmetic and geometric consequences. The formula for the regulator is expressed in terms of the classical polyogarithm functions. In this paper, we give a survey of the additive dilogarithm and the several different versions of the weight two regulator in the infinitesimal setting. We follow a historical approach which we hope will provide motivation for the definitions and the constructions.Publication Open Access Addendum to 'Unidirectionally invisible potentials as local building blocks of all scattering potentials'(American Physical Society (APS), 2014) Department of Mathematics; Department of Physics; Mostafazadeh, Ali; Faculty Member; Department of Mathematics; Department of Physics; College of Sciences; 4231In [Phys. Rev. A 90, 023833 (2014)], we offer a solution to the problem of constructing a scattering potential v(x) which possesses scattering properties of one's choice at an arbitrarily prescribed wave number. This solution involves expressing v(x) as the sum of n <= 6 finite-range unidirectionally invisible potentials. We improve this result by reducing the upper bound on n from 6 to 4. In particular, we show that we can construct v(x) as the sum of up to n = 3 finite-range unidirectionally invisible potentials, unless if it is required to be bidirectionally reflectionless.Publication Open Access Adiabatic approximation, semiclassical scattering, and unidirectional invisibility(Institute of Physics (IOP) Publishing, 2014) Department of Mathematics; Mostafazadeh, Ali; Faculty Member; Department of Mathematics; College of Sciences; 4231The transfer matrix of a possibly complex and energy-dependent scattering potential can be identified with the S-matrix of a two-level time-dependent non-Hermitian Hamiltonian H(tau). We show that the application of the adiabatic approximation to H(tau) corresponds to the semiclassical description of the original scattering problem. In particular, the geometric part of the phase of the evolving eigenvectors of H(tau) gives the pre-exponential factor of the WKB wave functions. We use these observations to give an explicit semiclassical expression for the transfer matrix. This allows for a detailed study of the semiclassical unidirectional reflectionlessness and invisibility. We examine concrete realizations of the latter in the realm of optics.Publication Open Access Adiabatic series expansion and higher-order semiclassical approximations in scattering theory(Institute of Physics (IOP) Publishing, 2014) Department of Mathematics; Mostafazadeh, Ali; Faculty Member; Department of Mathematics; College of Sciences; 4231The scattering properties of any complex scattering potential, nu: R -> C, can be obtained from the dynamics of a particular non-unitary two-level quantum system. S-nu. The application of the adiabatic approximation to S-nu yields a semiclassical treatment of the scattering problem. We examine the adiabatic series expansion for the evolution operator of S-v and use it to obtain corrections of arbitrary order to the semiclassical formula for the transfer matrix of S-nu. This results in a high-energy approximation scheme that unlike the semiclassical approximation can be applied for potentials with large derivatives.