Bivariant and extended Sylvester rank functions

In this paper we consider the algebraic crossed product ${\mathcal{A}}:=C_{K}(X)\rtimes _{T}\mathbb{Z}$ induced by a homeomorphism $T$ on the Cantor set $X$, where $K$ is an arbitrary field with involution and $C_{K}(X)$ denotes the $K$-algebra of locally constant $K$-valued functions on $X$. We investigate the possible Sylvester matrix rank functions that one can construct on ${\mathcal{A}}$ by means of full ergodic $T$-invariant probability measures $\unicode[STIX]{x1D707}$ on $X$. To do so, we present a general construction of an approximating sequence of $\ast$-subalgebras ${\mathcal{A}}_{n}$ which are embeddable into a (possibly infinite) product of matrix algebras over $K$. This enables us to obtain a specific embedding of the whole $\ast$-algebra ${\mathcal{A}}$ into ${\mathcal{M}}_{K}$, the well-known von Neumann continuous factor over $K$, thus obtaining a Sylvester matrix rank function on ${\mathcal{A}}$ by restricting the unique one defined on ${\mathcal{M}}_{K}$. This process gives a way to obtain a Sylvester matrix rank function on ${\mathcal{A}}$, unique with respect to a certain compatibility property concerning the measure $\unicode[STIX]{x1D707}$, namely that the rank of a characteristic function of a clopen subset $U\subseteq X$ must equal the measure of $U$.

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Analysing the Woo-Lam protocol using CSP and rank functions

Proceedings of the 3rd International Workshop on Security in Information Systems ◽

10.5220/0002557000030012 ◽

2005 ◽

Keyword(s):

Rank Functions

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ON THE PIPAGE ROUNDING ALGORITHM FOR SUBMODULAR FUNCTION MAXIMIZATION — A VIEW FROM DISCRETE CONVEX ANALYSIS

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830909000063 ◽

2009 ◽

Vol 01 (01) ◽

pp. 1-23 ◽

Cited By ~ 7

Author(s):

AKIYOSHI SHIOURA

Keyword(s):

Approximation Algorithm ◽

Convex Analysis ◽

Submodular Function ◽

Submodular Functions ◽

Concave Functions ◽

Discrete Convex Analysis ◽

Discrete Convex ◽

Weighted Rank ◽

Rounding Algorithm ◽

Rank Functions

We consider the problem of maximizing a nondecreasing submodular set function under a matroid constraint. Recently, Calinescu et al. (2007) proposed an elegant framework for the approximation of this problem, which is based on the pipage rounding technique by Ageev and Sviridenko (2004), and showed that this framework indeed yields a (1 - 1/e)-approximation algorithm for the class of submodular functions which are represented as the sum of weighted rank functions of matroids. This paper sheds a new light on this result from the viewpoint of discrete convex analysis by extending it to the class of submodular functions which are the sum of M ♮-concave functions. M ♮-concave functions are a class of discrete concave functions introduced by Murota and Shioura (1999), and contain the class of the sum of weighted rank functions as a proper subclass. Our result provides a better understanding for why the pipage rounding algorithm works for the sum of weighted rank functions. Based on the new observation, we further extend the approximation algorithm to the maximization of a nondecreasing submodular function over an integral polymatroid. This extension has an application in multi-unit combinatorial auctions.

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Additive rank functions in noetherian rings

Journal of Algebra ◽

10.1016/0021-8693(90)90092-3 ◽

1990 ◽

Vol 130 (2) ◽

pp. 451-461 ◽

Cited By ~ 9

Author(s):

Günter Krause

Keyword(s):

Noetherian Rings ◽

Rank Functions

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