Mathematics Research Workshttp://hdl.handle.net/1903/15952017-09-25T02:35:20Z2017-09-25T02:35:20ZOn The Number of Unlabeled Bipartite GraphsAtmaca, AbdullahOruc, Yavuz Ahttp://hdl.handle.net/1903/191862017-06-30T03:35:04Z2016-01-01T00:00:00ZOn The Number of Unlabeled Bipartite Graphs
Atmaca, Abdullah; Oruc, Yavuz A
Let $I$ and $O$ denote two sets of vertices, where $I\cap O =\Phi$, $|I| = n$, $|O| = r$, and $B_u(n,r)$ denote the set of unlabeled graphs whose edges connect vertices in $I$ and $O$. It is shown that the following two-sided equality holds.
$\displaystyle \frac{\binom{r+2^{n}-1}{r}}{n!} \le |B_u(n,r)| \le 2\frac{\binom{r+2^{n}-1}{r}}{n!} $
This paper describes a result that has been obtained in joint work with Abdullah Atmaca of Bilkent University, Ankara, Turkey
2016-01-01T00:00:00ZOn Number Of Partitions Of An Integer Into A Fixed Number Of Positive IntegersOruc, A. Yavuzhttp://hdl.handle.net/1903/163512016-03-29T02:48:54Z2015-04-01T00:00:00ZOn Number Of Partitions Of An Integer Into A Fixed Number Of Positive Integers
Oruc, A. Yavuz
This paper focuses on the number of partitions of a positive integer $n$ into $k$ positive summands, where $k$ is an integer between $1$ and $n$. Recently some upper bounds were reported for this number in [Merca14]. Here, it is shown that these bounds are not as tight as an earlier upper bound proved in [Andrews76-1] for $k\le 0.42n$. A new upper bound for the number of partitions of $n$ into $k$ summands is given, and shown to be tighter than the upper bound in [Merca14] when $k$ is between $O(\frac{\sqrt{n}}{\ln n})$ and $n-O(\frac{\sqrt{n}}{\ln n})$. It is further shown that the new upper bound is also tighter than two other upper bounds previously reported in~[Andrews76-1] and [Colman82]. A generalization of this upper bound to number of partitions of $n$ into at most $k$ summands is also presented.
Submitted to Journal of Number Theory.
2015-04-01T00:00:00ZSPECTRAL METHODS FOR HYPERBOLIC PROBLEMSTadmor, Eitanhttp://hdl.handle.net/1903/86872016-03-29T05:29:08Z1994-01-01T00:00:00ZSPECTRAL METHODS FOR HYPERBOLIC PROBLEMS
Tadmor, Eitan
We review several topics concerning spectral approximations of time-dependent problems,
primarily | the accuracy and stability of Fourier and Chebyshev methods for the
approximate solutions of hyperbolic systems.
To make these notes self contained, we begin with a very brief overview of Cauchy
problems. Thus, the main focus of the �rst part is on hyperbolic systems which are dealt
with two (related) tools: the energy method and Fourier analysis.
The second part deals with spectral approximations. Here we introduce the main ingredients
of spectral accuracy, Fourier and Chebyshev interpolants, aliasing, di�erentiation
matrices ...
The third part is devoted to Fourier method for the approximate solution of periodic
systems. The questions of stability and convergence are answered by combining ideas from
the �rst two sections. In this context we highlight the role of aliasing and smoothing; in
particular, we explain how the lack of resolution might excite small scales weak instability,
which is avoided by high modes smoothing.
The forth and �nal part deals with non-periodic problems. We study the stability of
the Chebyshev method, paying special attention to the intricate issue of the CFL stability
restriction on the permitted time-step.
1994-01-01T00:00:00ZRECOVERY OF EDGES FROM SPECTRAL DATA WITH NOISE—A NEW PERSPECTIVEENGELBERG, SHLOMOTADMOR, EITANhttp://hdl.handle.net/1903/86642016-03-29T07:23:58Z2008-01-01T00:00:00ZRECOVERY OF EDGES FROM SPECTRAL DATA WITH NOISE—A NEW PERSPECTIVE
ENGELBERG, SHLOMO; TADMOR, EITAN
We consider the problem of detecting edges—jump discontinuities in piecewise
smooth functions from their N-degree spectral content, which is assumed to be corrupted by noise.
There are three scales involved: the “smoothness” scale of order 1/N, the noise scale of order √η,
and the O(1) scale of the jump discontinuities. We use concentration factors which are adjusted to
the standard deviation of the noise √η � 1/N in order to detect the underlying O(1)-edges, which
are separated from the noise scale √η � 1.
2008-01-01T00:00:00Z