446,493 research outputs found
Smooth Parametrizations in Dynamics, Analysis, Diophantine and Computational Geometry
Smooth parametrization consists in a subdivision of the mathematical objects
under consideration into simple pieces, and then parametric representation of
each piece, while keeping control of high order derivatives. The main goal of
the present paper is to provide a short overview of some results and open
problems on smooth parametrization and its applications in several apparently
rather separated domains: Smooth Dynamics, Diophantine Geometry, Approximation
Theory, and Computational Geometry.
The structure of the results, open problems, and conjectures in each of these
domains shows in many cases a remarkable similarity, which we try to stress.
Sometimes this similarity can be easily explained, sometimes the reasons remain
somewhat obscure, and it motivates some natural questions discussed in the
paper. We present also some new results, stressing interconnection between
various types and various applications of smooth parametrization
I Don't Want to Think About it Now:Decision Theory With Costly Computation
Computation plays a major role in decision making. Even if an agent is
willing to ascribe a probability to all states and a utility to all outcomes,
and maximize expected utility, doing so might present serious computational
problems. Moreover, computing the outcome of a given act might be difficult. In
a companion paper we develop a framework for game theory with costly
computation, where the objects of choice are Turing machines. Here we apply
that framework to decision theory. We show how well-known phenomena like
first-impression-matters biases (i.e., people tend to put more weight on
evidence they hear early on), belief polarization (two people with different
prior beliefs, hearing the same evidence, can end up with diametrically opposed
conclusions), and the status quo bias (people are much more likely to stick
with what they already have) can be easily captured in that framework. Finally,
we use the framework to define some new notions: value of computational
information (a computational variant of value of information) and and
computational value of conversation.Comment: In Conference on Knowledge Representation and Reasoning (KR '10
A Uniform Mathematical Representation of Logic and Computation.
The current models of computation share varying levels of correspondence with actual implementation schemes. They can be arranged in a hierarchical structure depending upon their level of abstraction. In classical computing, the circuit model shares closest correspondence with physical implementation, followed by finite automata techniques. The highest level in the abstraction hierarchy is that of the theory of computation.Likewise, there exist computing paradigms based upon a different set of defining principles. The classical paradigm involves computing as has been applied traditionally, and is characterized by Boolean circuits that are irreversible in nature. The reversible paradigm requires invertible primitives in order to perform computation. The paradigm of quantum computing applies the theory of quantum mechanics to perform computational tasks.Our analysis concludes that descriptions at lowest level in the abstraction hierarchy should be uniform across the three paradigms, but the same is not true in case of current descriptions. We propose a mathematical representation of logic and computation that successfully explains computing models in all three paradigms, while making a seamless transition to higher levels of the abstraction hierarchy. This representation is based upon the theory of linear spaces and, hence, is referred to as the linear representation. The representation is first developed in the classical context, followed by an extension to the reversible paradigm by exploiting the well-developed theory on invertible mappings. The quantum paradigm is reconciled with this representation through correspondence that unitary operators share with the proposed linear representation. In this manner, the representation is shown to account for all three paradigms. The correspondence shared with finite automata models is also shown to hold implicitly during the development of this representation. Most importantly, the linear representation accounts for the Hamiltonians that define the dynamics of a computational process, thereby resolving the correspondence shared with underlying physical principles.The consistency of the linear representation is checked against a current existing application in VLSI CAD that exploits the linearity of logic functions for symbolic representation of circuits. Some possible applications and applicability of the linear representation to some open problems are also discussed
Some Remarks about the Complexity of Epidemics Management
Recent outbreaks of Ebola, H1N1 and other infectious diseases have shown that
the assumptions underlying the established theory of epidemics management are
too idealistic. For an improvement of procedures and organizations involved in
fighting epidemics, extended models of epidemics management are required. The
necessary extensions consist in a representation of the management loop and the
potential frictions influencing the loop. The effects of the non-deterministic
frictions can be taken into account by including the measures of robustness and
risk in the assessment of management options. Thus, besides of the increased
structural complexity resulting from the model extensions, the computational
complexity of the task of epidemics management - interpreted as an optimization
problem - is increased as well. This is a serious obstacle for analyzing the
model and may require an additional pre-processing enabling a simplification of
the analysis process. The paper closes with an outlook discussing some
forthcoming problems
Computational Social Choice: Prospects and Challenges
AbstractHow should we aggregate the individual views of the members of a group so as to arrive at an adequate representation of the collective view of that group? This is a fundamental question of deep philosophical, economic, and political significance that, around the middle of 20th century, has given rise to the field of Social Choice Theory. More recently, a research trend known as Computational Social Choice has emerged, which studies this question from the perspective of Computer Science. This “computational turn” is fuelled both by the fact that questions of social choice have turned out to be central to a range of application areas, notably in the domain of Information and Communication Technologies, and by the insight that many concepts and techniques originating in Computer Science can be used to solve (or provide a new angle on) problems in Social Choice Theory. In this paper, I give a brief introduction to Computational Social Choice and discuss some of the prospects and challenges for this fast growing area of research
- …