1,252 research outputs found
Shifts of finite type with nearly full entropy
For any fixed alphabet A, the maximum topological entropy of a Z^d subshift
with alphabet A is obviously log |A|. We study the class of nearest neighbor
Z^d shifts of finite type which have topological entropy very close to this
maximum, and show that they have many useful properties. Specifically, we prove
that for any d, there exists beta_d such that for any nearest neighbor Z^d
shift of finite type X with alphabet A for which log |A| - h(X) < beta_d, X has
a unique measure of maximal entropy. Our values of beta_d decay polynomially
(like O(d^(-17))), and we prove that the sequence must decay at least
polynomially (like d^(-0.25+o(1))). We also show some other desirable
properties for such X, for instance that the topological entropy of X is
computable and that the unique m.m.e. is isomorphic to a Bernoulli measure.
Though there are other sufficient conditions in the literature which guarantee
a unique measure of maximal entropy for Z^d shifts of finite type, this is (to
our knowledge) the first such condition which makes no reference to the
specific adjacency rules of individual letters of the alphabet.Comment: 33 pages, accepted by Proceedings of the London Mathematical Societ
Around multivariate Schmidt-Spitzer theorem
Given an arbitrary complex-valued infinite matrix A and a positive integer n
we introduce a naturally associated polynomial basis B_A of C[x0...xn]. We
discuss some properties of the locus of common zeros of all polynomials in B_A
having a given degree m; the latter locus can be interpreted as the spectrum of
the m*(m+n)-submatrix of A formed by its m first rows and m+n first columns. We
initiate the study of the asymptotics of these spectra when m goes to infinity
in the case when A is a banded Toeplitz matrix. In particular, we present and
partially prove a conjectural multivariate analog of the well-known
Schmidt-Spitzer theorem which describes the spectral asymptotics for the
sequence of principal minors of an arbitrary banded Toeplitz matrix. Finally,
we discuss relations between polynomial bases B_A and multivariate orthogonal
polynomials
Welfare Maximization and Truthfulness in Mechanism Design with Ordinal Preferences
We study mechanism design problems in the {\em ordinal setting} wherein the
preferences of agents are described by orderings over outcomes, as opposed to
specific numerical values associated with them. This setting is relevant when
agents can compare outcomes, but aren't able to evaluate precise utilities for
them. Such a situation arises in diverse contexts including voting and matching
markets.
Our paper addresses two issues that arise in ordinal mechanism design. To
design social welfare maximizing mechanisms, one needs to be able to
quantitatively measure the welfare of an outcome which is not clear in the
ordinal setting. Second, since the impossibility results of Gibbard and
Satterthwaite~\cite{Gibbard73,Satterthwaite75} force one to move to randomized
mechanisms, one needs a more nuanced notion of truthfulness.
We propose {\em rank approximation} as a metric for measuring the quality of
an outcome, which allows us to evaluate mechanisms based on worst-case
performance, and {\em lex-truthfulness} as a notion of truthfulness for
randomized ordinal mechanisms. Lex-truthfulness is stronger than notions
studied in the literature, and yet flexible enough to admit a rich class of
mechanisms {\em circumventing classical impossibility results}. We demonstrate
the usefulness of the above notions by devising lex-truthful mechanisms
achieving good rank-approximation factors, both in the general ordinal setting,
as well as structured settings such as {\em (one-sided) matching markets}, and
its generalizations, {\em matroid} and {\em scheduling} markets.Comment: Some typos correcte
Fair Allocation of Utilities in Multirate Multicast Networks: A Framework for Unifying Diverse Fairness Objectives
We study fairness in a multicast network. We assume that different receivers of the same session can receive information at different rates. We study fair allocation of utilities, where utility of a bandwidth is an arbitrary function of the bandwidth. The utility function is not strictly increasing, nor continuous in general. We discuss fairness issues in this general context. Fair allocation of utilities can be modeled as a nonlinear optimization problem. However, nonlinear optimization techniques do not terminate in a finite number of iterations in general. We present an algorithm for computing a fair utility allocation. Using specific fairness properties, we show that this algorithm attains global convergence and yields a fair allocation in polynomial number of iterations
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