On the H\'enon-Lane-Emden conjecture

We consider Liouville-type theorems for the following H\'{e}non-Lane-Emden system \hfill -\Delta u&=& |x|^{a}v^p \text{in} \mathbb{R}^N, \hfill -\Delta v&=& |x|^{b}u^q \text{in} \mathbb{R}^N, when $pq>1$, $p,q,a,b\ge0$. The main conjecture states that there is no non-trivial non-negative solution whenever $(p,q)$ is under the critical Sobolev hyperbola, i.e. $\frac{N+a}{p+1}+\frac{N+b}{q+1}>{N-2}$. We show that this is indeed the case in dimension N=3 provided the solution is also assumed to be bounded, extending a result established recently by Phan-Souplet in the scalar case. Assuming stability of the solutions, we could then prove Liouville-type theorems in higher dimensions. For the scalar cases, albeit of second order ($a=b$ and $p=q$) or of fourth order ($a\ge 0=b$ and $p>1=q$), we show that for all dimensions $N\ge 3$ in the first case (resp., $N\ge 5$ in the second case), there is no positive solution with a finite Morse index, whenever $p$ is below the corresponding critical exponent, i.e $1<p<\frac{N+2+2a}{N-2}$ (resp., $1<p<\frac{N+4+2a}{N-4}$). Finally, we show that non-negative stable solutions of the full H\'{e}non-Lane-Emden system are trivial provided \label{sysdim00} N<2+2(\frac{p(b+2)+a+2}{pq-1}) (\sqrt{\frac{pq(q+1)}{p+1}}+ \sqrt{\frac{pq(q+1)}{p+1}-\sqrt\frac{pq(q+1)}{p+1}}).Comment: Theorem 4 has been added in the new version. 23 pages, Comments are welcome. Updated version - if any - can be downloaded at http://www.birs.ca/~nassif/ or http://www.math.ubc.ca/~fazly/research.htm

Maximal codeword lengths in Huffman codes

The following question about Huffman coding, which is an important technique for compressing data from a discrete source, is considered. If p is the smallest source probability, how long, in terms of p, can the longest Huffman codeword be? It is shown that if p is in the range 0 less than p less than or equal to 1/2, and if K is the unique index such that 1/F(sub K+3) less than p less than or equal to 1/F(sub K+2), where F(sub K) denotes the Kth Fibonacci number, then the longest Huffman codeword for a source whose least probability is p is at most K, and no better bound is possible. Asymptotically, this implies the surprising fact that for small values of p, a Huffman code's longest codeword can be as much as 44 percent larger than that of the corresponding Shannon code

Behavioral analysis of anisotropic diffusion in image processing

©1996 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or distribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE. This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyright. In most cases, these works may not be reposted without the explicit permission of the copyright holder.DOI: 10.1109/83.541424In this paper, we analyze the behavior of the anisotropic diffusion model of Perona and Malik (1990). The main idea is to express the anisotropic diffusion equation as coming from a certain optimization problem, so its behavior can be analyzed based on the shape of the corresponding energy surface. We show that anisotropic diffusion is the steepest descent method for solving an energy minimization problem. It is demonstrated that an anisotropic diffusion is well posed when there exists a unique global minimum for the energy functional and that the ill posedness of a certain anisotropic diffusion is caused by the fact that its energy functional has an infinite number of global minima that are dense in the image space. We give a sufficient condition for an anisotropic diffusion to be well posed and a sufficient and necessary condition for it to be ill posed due to the dense global minima. The mechanism of smoothing and edge enhancement of anisotropic diffusion is illustrated through a particular orthogonal decomposition of the diffusion operator into two parts: one that diffuses tangentially to the edges and therefore acts as an anisotropic smoothing operator, and the other that flows normally to the edges and thus acts as an enhancement operator

The Catalyst Effect of Historic Preservation: A Spatial Analysis of the Impact of Historic District Designation on Housing Renovations in New York City

The constraints on property use along with the economic merits of historic districts are critical points in the debate over preservation policies. Improving the existing housing stock is a substantial economic activity and a significant part of the nation’s overall construction industry. The purpose of this study is to present an empirical analysis of the relationship between historic district designation and renovation decisions. The results of the analysis found that historic preservation does not inhibit the renovation activity of single-family homeowners. However, contrary to the claims made by proponents, it does not give a powerful incentive for owners in undesignated neighborhoods