Decision problems for 3-manifolds and their fundamental groups

We survey the status of some decision problems for 3-manifolds and their fundamental groups. This includes the classical decision problems for finitely presented groups (Word Problem, Conjugacy Problem, Isomorphism Problem), and also the Homeomorphism Problem for 3-manifolds and the Membership Problem for 3-manifold groups.Comment: 31 pages, final versio

Theory of the decision/problem state

A theory of the decision-problem state was introduced and elaborated. Starting with the basic model of a decision-problem condition, an attempt was made to explain how a major decision-problem may consist of subsets of decision-problem conditions composing different condition sequences. In addition, the basic classical decision-tree model was modified to allow for the introduction of a series of characteristics that may be encountered in an analysis of a decision-problem state. The resulting hierarchical model reflects the unique attributes of the decision-problem state. The basic model of a decision-problem condition was used as a base to evolve a more complex model that is more representative of the decision-problem state and may be used to initiate research on decision-problem states

The Best Choice Problem under ambiguity

We model and solve Best Choice Problems in the multiple prior framework: An ambiguity averse decision maker aims to choose the best among a fixed number of applicants that appear sequentially in a random order. The decision faces ambiguity about the probability that a candidate a relatively top applicant is actually best among all applicants. We show that our model covers the classical secretary problem, but also other interesting classes of problems. We provide a closed form solution of the problem for time-consistent priors using minimax backward induction. As in the classical case the derived stopping strategy is simple. Ambiguity can lead to substantial differences to the classical threshold rule.Best Choice Problem

Strategy for quantum algorithm design assisted by machine learning

We propose a method for quantum algorithm design assisted by machine learning. The method uses a quantum-classical hybrid simulator, where a "quantum student" is being taught by a "classical teacher." In other words, in our method, the learning system is supposed to evolve into a quantum algorithm for a given problem assisted by classical main-feedback system. Our method is applicable to design quantum oracle-based algorithm. As a case study, we chose an oracle decision problem, called a Deutsch-Jozsa problem. We showed by using Monte-Carlo simulations that our simulator can faithfully learn quantum algorithm to solve the problem for given oracle. Remarkably, learning time is proportional to the square root of the total number of parameters instead of the exponential dependance found in the classical machine learning based method.Comment: published versio

Exercise Strategies for American Exotic Options under Ambiguity

We analyze several exotic options of American style in a multiple prior setting and study the optimal exercise strategy from the perspective of an ambiguity averse buyer in a discrete time model of Cox–Ross–Rubinstein style. The multiple prior model relaxes the assumption of a known distribution of the stock price process and takes into account decision maker’s inability to completely determine the underlying asset’s price dynamics. In order to evaluate the American option the decision maker needs to solve a stopping problem. Unlike the classical approach ambiguity averse decision maker uses a class of measures to evaluate her expected payoffs instead of a unique prior. Given time-consistency of the set of priors an appropriate version of backward induction leads to the solution as in the classical case. Using a duality result the multiple prior stopping problem can be related to the classical stopping problem for a certain probability measure – the worst-case measure. Therefore, the problem can be reduced to identifying the worst-case measure. We obtain the form of the worstcase measure for different classes of exotic options explicitly exploiting the observation that the options can be decomposed in simpler event-driven claims.American option, optimal stopping, ambiguity, uncertainty aversion