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Max-Sum Diversification, Monotone Submodular Functions and Semi-metric Spaces  [PDF]
Sepehr Abbasi Zadeh,Mehrdad Ghadiri
Computer Science , 2015,
Abstract: In many applications such as web-based search, document summarization, facility location and other applications, the results are preferable to be both representative and diversified subsets of documents. The goal of this study is to select a good "quality", bounded-size subset of a given set of items, while maintaining their diversity relative to a semi-metric distance function. This problem was first studied by Borodin et al\cite{borodin}, but a crucial property used throughout their proof is the triangle inequality. In this modified proof, we want to relax the triangle inequality and relate the approximation ratio of max-sum diversification problem to the parameter of the relaxed triangle inequality in the normal form of the problem (i.e., a uniform matroid) and also in an arbitrary matroid.
Geometric Ruzsa triangle inequality in metric spaces with dilations  [PDF]
Marius Buliga
Mathematics , 2013,
Abstract: This is a short note related to the article arXiv:1212.5056 [math.CO] "On growth in an abstract plane" by Nick Gill, H. A. Helfgott, Misha Rudnev. It contains a general "geometric Ruzsa triangle inequality" which is then applied to get the same kind of inequality in a metric space with dilations.
Reverses of the Triangle Inequality in Inner Product Spaces  [PDF]
Sever Silvestru Dragomir
Mathematics , 2004,
Abstract: Some new reverses for the generalised triangle inequality in inner product spaces and applications are given. Applications in connection to the Schwarz inequality are provided as well.
Additive Reverses of the Generalised Triangle Inequality in Normed Spaces  [PDF]
Sever Silvestru Dragomir
Mathematics , 2004,
Abstract: Some additive reverses of the generalised triangle inequality in normed linear spaces are given. Applications for complex numbers are provided as well.
Quadratic Reverses of the Triangle Inequality in Inner Product Spaces  [PDF]
Sever Silvestru Dragomir
Mathematics , 2004,
Abstract: Some sharp quadratic reverses for the generalised triangle inequality in inner product spaces and applications are given.
Caffarelli-Kohn-Nirenberg inequality on metric measure spaces with applications  [PDF]
Alexandru Kristály,Shin-ichi Ohta
Mathematics , 2012,
Abstract: We prove that if a metric measure space satisfies the volume doubling condition and the Caffarelli-Kohn-Nirenberg inequality with the same exponent $n \ge 3$, then it has exactly the $n$-dimensional volume growth. As an application, if an $n$-dimensional Finsler manifold of non-negative $n$-Ricci curvature satisfies the Caffarelli-Kohn-Nirenberg inequality with the sharp constant, then its flag curvature is identically zero. In the particular case of Berwald spaces, such a space is necessarily isometric to a Minkowski space.
Compressible fluids driven by stochastic forcing: The relative energy inequality and applications  [PDF]
Dominic Breit,Eduard Feireisl,Martina Hofmanova
Mathematics , 2015,
Abstract: We show the relative energy inequality for the compressible Navier-Stokes system driven by a stochastic forcing. As a corollary, we prove the weak-strong uniqueness property (pathwise and in law) and convergence of weak solutions in the inviscid-incompressible limit. In particular, we establish a Yamada-Watanabe type result in the context of the compressible Navier-Stokes system, that is, pathwise weak--strong uniqueness implies weak--strong uniqueness in law.
Some Reverses of the Generalised Triangle Inequality in Complex Inner Product Spaces  [PDF]
Sever Silvestru Dragomir
Mathematics , 2004,
Abstract: Some reverses for the generalised triangle inequality in complex inner product spaces that improve the classical Diaz-Metcalf results and applications are given.
INEQUALITIES IN TRIANGLE  [PDF]
OVIDIU T. POP,NICU?OR MINCULETE,MIHáLY BENCZE
Journal of Science and Arts , 2011,
Abstract: In this paper we will demonstrate a new inequality between sine and a function of first degree. Applications of this inequality in triangle are given.
ON AN INEQUALITY FOR THE MEDIANS OF A TRIANGLE  [PDF]
JIAN LIU
Journal of Science and Arts , 2012,
Abstract: In this paper, we give two new simpler proofs of a sharp inequality for the medians of a triangle. We also establish two new inequalities by using this sharp inequality. Some related conjectures checked by the computer are put forward, which include two conjectures related to the famous Erd s-Mordell inequality.
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