Singular values, quasiconformal maps and the Schottky upper bound

1998 ◽  
Vol 41 (12) ◽  
pp. 1241-1247 ◽  
Author(s):  
Songliang Qiu
Keyword(s):  
Upper Bound ◽  
Singular Values ◽  
Quasiconformal Maps

scholarly journals Bounds on condition number of singular matrix

Filomat ◽  
2014 ◽  
Vol 28 (8) ◽  
pp. 1653-1660
Author(s):  
Zhiping Xiong
Keyword(s):  
Condition Number ◽  
Upper Bound ◽  
Singular Values ◽  
Operator Norm ◽  
The Other ◽  
Singular Matrix ◽  
Vector Norm ◽  
Other Hand

For each vector norm ||x||v, a matrix A ? Cmxn has its operator norm ||A||?v = maxx?O ||Ax||?/||x||v. If A is nonsingular, we can define the condition number of A ? Cnxn as P(A) = ||A||vv ||A-1||vv. If A is singular, the condition number of matrix A ? Cmxn may be defined as P+(A)=||A||?v ||A+||v?. Let U be the set of the whole self-dual norms. It is shown that for a singular matrix A ? Cmxn, there is no finite upper bound of P+(A), while ||.|| varies on U. On the other hand, it is shown that inf ||.||? U ||A||?v ||A+||v? = ?1(A)/?r(A), where ?1(A) and ?r(A) are the largest and smallest nonzero singular values of A, respectively.

scholarly journals Bounds for Incidence Energy of Some Graphs

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Weizhong Wang ◽  
Dong Yang
Keyword(s):  
Regular Graph ◽  
Upper Bound ◽  
Maximum Degree ◽  
Incidence Matrix ◽  
Line Graph ◽  
Singular Values ◽  
Simple Graph

LetGbe a simple graph. The incidence energy (IEfor short) ofGis defined as the sum of the singular values of the incidence matrix. In this paper, a new upper bound forIEof graphs in terms of the maximum degree is given. Meanwhile, bounds forIEof the line graph of a semiregular graph and the paraline graph of a regular graph are obtained.

Teichmuller Geometry

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Uniqueness Theorem ◽  
Existence Theorems ◽  
Quadratic Differentials ◽  
Complex Structures ◽  
Metric Geometry ◽  
Quasiconformal Maps ◽  
Hyperbolic Structures ◽  
Teichmüller Metric ◽  
Uniqueness And Existence ◽  
Teichmüller Mapping

This chapter focuses on the metric geometry of Teichmüller space. It first explains how one can think of Teich(Sɡ) as the space of complex structures on Sɡ. To this end, the chapter defines quasiconformal maps between surfaces and presents a solution to the resulting Teichmüller's extremal problem. It also considers the correspondence between complex structures and hyperbolic structures, along with the Teichmüller mapping, Teichmüller metric, and the proof of Teichmüller's uniqueness and existence theorems. The fundamental connection between Teichmüller's theorems, holomorphic quadratic differentials, and measured foliations is discussed as well. Finally, the chapter describes the Grötzsch's problem, whose solution is tied to the proof of Teichmüller's uniqueness theorem.

A SVD and Modified Firefly Optimization Based Robust Digital Image Watermarking Technique

2017 ◽  
Vol 7 (7) ◽  
pp. 268
Author(s):  
Chauhan Usha ◽  
Singh Rajeev Kumar
Keyword(s):  
Digital Media ◽  
Firefly Algorithm ◽  
Image Watermarking ◽  
Singular Values ◽  
Robust Watermarking ◽  
Scaling Factors ◽  
Geometric Attacks ◽  
Host Image ◽  
Firefly Optimization ◽  
Value Decomposition

Digital Watermarking is a technology, to facilitate the authentication, copyright protection and Security of digital media. The objective of developing a robust watermarking technique is to incorporate the maximum possible robustness without compromising with the transparency. Singular Value Decomposition (SVD) using Firefly Algorithm provides this objective of an optimal robust watermarking technique. Multiple scaling factors are used to embed the watermark image into the host by multiplying these scaling factors with the Singular Values (SV) of the host image. Firefly Algorithm is used to optimize the modified host image to achieve the highest possible robustness and transparency. This approach can significantly increase the quality of watermarked image and provide more robustness to the embedded watermark against various attacks such as noise, geometric attacks, filtering attacks etc.

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