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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 3D convective Cahn-Hilliard equation(Amer Inst Mathematical Sciences-Aims, 2007) Eden, Alp; Department of Mathematics; Kalantarov, Varga; Faculty Member; Department of Mathematics; College of Sciences; 117655We consider the initial boundary value problem for the 3D convective Cahn - Hilliard equation with periodic boundary conditions. This gives rise to a continuous dynamical system on L-2(ohm). Absorbing balls in L-2(ohm), H-per(1)(ohm) and H-per(2)(ohm) are shown to exist. Combining with the compactness property of the solution semigroup we conclude the existence of the global attractor. Restricting the dynamical system on the absorbing ball in H-per(2)(ohm) and using the general framework in Eden et. all. [] the existence of an exponential attractor is guaranteed. This approach also gives an explicit upper estimate of the dimension of the exponential attractor, albeit of the global attractor.Publication Metadata only A bias phenomenon on the behavior of Dedekind sums(Int Press Boston, Inc, 2008) Xiong, Maosheng; Zaharescu, Alexandru; Department of Mathematics; Alkan, Emre; Faculty Member; Department of Mathematics; College of Sciences; 32803In this paper we present a bias phenomenon on the behavior of Dedekind sums at visible points in a dilated region. Our results indicate that in more than three quarters of the time the Dedekind sum increases as one moves from one visible point to the next.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 Metadata only A characterization of heaviness in terms of relative symplectic cohomology(Wiley, 2024) Mak, Cheuk Yu; Sun, Yuhan; Department of Mathematics; Varolgüneş, Umut; Department of Mathematics; 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.Publication Metadata only A characterization of the closed unital ideals of the Fourier-Stieltjes algebra B(G) of a locally compact amenable group G(Elsevier, 2003) Department of Mathematics; Ülger, Ali; Faculty Member; Department of Mathematics; College of Sciences; N/ALet G be a locally compact amenable group, B(G) its Fourier–Stieltjes algebra and I be a closed ideal of it. In this paper we prove the following result: The ideal I has a unit element iff it is principal. This is the noncommutative version of the Glicksberg–Host–Parreau Theorem. The paper also contains an abstract version of this theorem.Publication Metadata only A characterization of the invertible measures(Polish Acad Sciences Inst Mathematics, 2007) Department of Mathematics; Ülger, Ali; Faculty Member; Department of Mathematics; College of Sciences; N/ALet G be a locally compact abelian group and M(G) its measure algebra. Two measures mu and lambda are said to be equivalent if there exists an invertible measure pi such that pi * mu = lambda. The main result of this note is the following: A measure mu is invertible iff vertical bar(mu) over cap vertical bar >= epsilon on (G) over cap for some epsilon > 0 and mu is equivalent to a measure lambda of the form lambda = a + theta, where a is an element of L-1(G) and theta is an element of M(G) is an idempotent measure.Publication Open Access A class of Banach algebras whose duals have the Schur property(TÜBİTAK, 1999) Mustafayev, H.; Department of Mathematics; Ülger, Ali; Faculty Member; Department of Mathematics; College of SciencesCall a commutative Banach algebra A a γ-algebra if it contains a bounded group Λ such that aco(Λ) contains a multiple of the unit ball of A. In this paper, first by exhibiting several concrete examples, we show that the class of γ-algebras is quite rich. Then, for a γ-algebra A, we prove that A* has the Schur property iff the Gelfand spectrum Σ of A is scattered iff A* = ap(A) iff A* = Span(Σ).Publication Metadata only A class of banach algebras whose duals have the schur property(Scientific and Technical research Council of Turkey - TUBITAK/Türkiye Bilimsel ve Teknik Araştırma Kurumu, 1999) Mustafayev, Heybetkulu; Department of Mathematics; Ülger, Ali; Faculty Member; Department of Mathematics; College of Sciences; N/ACall a commutative Banach algebra A a γ-algebra if it contains a bounded group Λ such that aco(Λ) contains a multiple of the unit ball of A. In this paper, first by exhibiting several concrete examples, we show that the class of γ-algebras is quite rich. Then, for a γ-algebra A, we prove that A* has the Schur property iff the Gelfand spectrum Σ of A is scattered iff A* = ap(A) iff A* = Span(Σ).Publication Metadata only A comparison of analysis of covariate-adjusted residuals and analysis of covariance(Taylor & Francis Inc, 2009) Goad, Carla L.; Department of Mathematics; Ceyhan, Elvan; Faculty Member; Department of Mathematics; College of Sciences; N/AVarious methods to control the influence of a covariate on a response variable are compared. These methods are ANOVA with or without homogeneity of variances (HOV) of errors and Kruskal-Wallis (K-W) tests on (covariate-adjusted) residuals and analysis of covariance (ANCOVA). Covariate-adjusted residuals are obtained from the overall regression line fit to the entire data set ignoring the treatment levels or factors. It is demonstrated that the methods on covariate-adjusted residuals are only appropriate when the regression lines are parallel and covariate means are equal for all treatments. Empirical size and power performance of the methods are compared by extensive Monte Carlo simulations. We manipulated the conditions such as assumptions of normality and HOV, sample size, and clustering of the covariates. The parametric methods on residuals and ANCOVA exhibited similar size and power when error terms have symmetric distributions with variances having the same functional form for each treatment, and covariates have uniform distributions within the same interval for each treatment. In such cases, parametric tests have higher power compared to the K-W test on residuals. When error terms have asymmetric distributions or have variances that are heterogeneous with different functional forms for each treatment, the tests are liberal with K-W test having higher power than others. The methods on covariate-adjusted residuals are severely affected by the clustering of the covariates relative to the treatment factors when covariate means are very different for treatments. For data clusters, ANCOVA method exhibits the appropriate level. However, such a clustering might suggest dependence between the covariates and the treatment factors, so makes ANCOVA less reliable as well.