Mathematics Research Workshttp://hdl.handle.net/1903/15952016-10-26T02:55:18Z2016-10-26T02:55:18ZOn 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.
2008-01-01T00:00:00ZLONG-TIME EXISTENCE OF SMOOTH SOLUTIONS FOR THLong time existence of smooth solutions for the rapidly rotating shallow-water and Euler equationsECHENG, BINTADMOR, EITANhttp://hdl.handle.net/1903/86632016-03-29T07:21:42Z2008-01-01T00:00:00ZLONG-TIME EXISTENCE OF SMOOTH SOLUTIONS FOR THLong time existence of smooth solutions for the rapidly rotating shallow-water and Euler equationsE
CHENG, BIN; TADMOR, EITAN
We study the stabilizing effect of rotational forcing in the nonlinear setting of twodimensional
shallow-water and more general models of compressible Euler equations. In [Phys. D,
188 (2004), pp. 262–276] Liu and Tadmor have shown that the pressureless version of these equations
admit a global smooth solution for a large set of subcritical initial configurations. In the present
work we prove that when rotational force dominates the pressure, it prolongs the lifespan of smooth
solutions for t <∼
ln(δ−1); here δ � 1 is the ratio of the pressure gradient measured by the inverse
squared Froude number, relative to the dominant rotational forces measured by the inverse
Rossby number. Our study reveals a “nearby” periodic-in-time approximate solution in the small δ
regime, upon which hinges the long-time existence of the exact smooth solution. These results are
in agreement with the close-to-periodic dynamics observed in the “near-inertial oscillation” (NIO)
regime which follows oceanic storms. Indeed, our results indicate the existence of a smooth, “approximate
periodic” solution for a time period of days, which is the relevant time period found in
NIO obesrvations.
2008-01-01T00:00:00Z