Are there elimination algorithms for the permanent?

We define the class of elimination algorithms. There are algebraic algorithms for evaluating multivariate polynomials, and include as a special case Gaussian elimination for evaluating the determinant. We show how to find the linear symmetries of a polynomial, defined appropriately, and use these methods to find the linear symmetries of the permanent and determinant. We show that in contrast to the Gaussian elimination algorithm for the determinant, there is no elimination algorithm for the permanent.     

When Are All Functions from a Groupoid into Itself Endomorphisms or Translations?

This paper classifies all groupoids that have the property that every map from the groupoid to itself is either an endomorphism or a multiplication map.1991 Mathematics Subject Classification: 20M15     

Are quantile risk measures suitable for risk-transfer decisions?

Although controversial from the theoretical point of view, quantile risk measures are widely used by institutions and regulators.In this paper, we use a unified approach to find the optimal treaties for an agent who seeks to minimize one of these measures, assuming premium calculation principles of various types.We show that the use of measures like Value at Risk or Conditional Tail Expectation as optimization criteria for insurance or reinsurance leads to treaties that are not enforceable and/or are clearly bad for the cedent. We argue that this is one further argument against the use of quantile risk measures, at least for the purpose of risk-transfer decisions.     

Are dualities appropriate for duality theories in optimization?

We raise some questions about duality theories in global optimization. The main one concerns the possibility to extend the use of conjugacies to general dualities for studying dual optimization problems. In fact, we examine whether dualities are the most general concepts to get duality results. We also consider the passage from a Lagrangian approach to a perturbational approach and the reverse passage in the framework of general dualities. Since a notion of subdifferential can be defined for any duality, it is natural to examine whether the familiar interpretation of multipliers as generalized derivatives of the performance function associated with a dualizing parameterization of the given problem is still valid in the general framework of dualities.     

Are external perturbations responsible for chaotic motion in galaxies?

We study the nature of motion in a logarithmic galactic dynamical model, with an additional external perturbation. Two different cases are investigated. In the first case the external perturbation is fixed, while in the second case it is varying with the time. Numerical experiments suggest, that responsible for the chaotic phenomena is the external perturbation, combined with the dense nucleus. Linear relationships are found to exist, between the critical value of the angular momentum and the dynamical parameters of the galactic system that is, the strength of the external perturbation, the flattening parameter and the radius of the nucleus. Moreover, the extent of the chaotic regions in the phase plane, increases linearly as the strength of the external perturbation and the flattening parameter increases. On the contrary, we observe that the percentage covered by chaotic orbits in the phase plane, decreases linearly, as the scale length of the nucleus increases, becoming less dense. Theoretical arguments are used to support and explain the numerically obtained outcomes. A comparison of the present outcomes with earlier results is also presented.     

When can we improve on sample average approximation for stochastic optimization?

We explore the performance of sample average approximation in comparison with several other methods for stochastic optimization. The methods we evaluate are (a) bagging; (b) kernel density estimation; (c) maximum likelihood estimation; and (d) a Bayesian approach. We use two test sets: first a set of quadratic objective functions allowing different types of interaction between the random component and the univariate decision variable; and second a set of portfolio optimization problems. We make recommendations for effective approaches.     

Are sufficient conditions for conservativity of minimal quantum semigroups necessary?

We show by a counter-example that the two basic operator inequalities in the typical sufficient conditions for conservativity of minimal quantum dynamical semigroups are, in fact, both necessary.     

Are unbounded linear operators computable on the average for Gaussian measures?

Traub and Werschulz [Complexity and Information, Cambridge University Press, New York, 1999] ask whether every linear operator S:⊆X→Y$S:\subseteq X\to Y$ is “computable on the average” w.r.t. a Gaussian measure on X. The question is inspired by an analogous result in information-based complexity on the average-case solvability of linear approximation problems. We give several interpretations of Traub and Werschulz’ question within the framework of type-2 theory of effectivity. We have negative answers to all of these interpretations but the one with minimal requirements on the algorithm's uniformness. On our way to these results, we give an effective version of the Mourier–Prokhorov characterization of Gaussian measures on separable Hilbert spaces.     

How do path generation methods affect the accuracy of quasi-Monte Carlo methods for problems in finance?

Quasi-Monte Carlo (QMC) methods are important numerical tools in computational finance. Path generation methods (PGMs), such as Brownian bridge and principal component analysis, play a crucial role in QMC methods. Their effectiveness, however, is problem-dependent. This paper attempts to understand how a PGM interacts with the underlying function and affects the accuracy of QMC methods. To achieve this objective, we develop efficient methods to assess the impact of PGMs. The first method is to exploit a quadratic approximation of the underlying function and to analyze the effective dimension and dimension distribution (which can be done analytically). The second method is to carry out a QMC error analysis on the quadratic approximation, establishing an explicit relationship between the QMC error and the PGM. Equalities and bounds on the QMC errors are established, in which the effect of the PGM is separated from the effect of the point set (in a similar way to the Koksma–Hlawka inequality). New measures for quantifying the accuracy of QMC methods combining with PGMs are introduced. The usefulness of the proposed methods is demonstrated on two typical high-dimensional finance problems, namely, the pricing of mortgage-backed securities and Asian options (with zero strike price). It is found that the success or failure of PGMs that do not take into account the underlying functions (such as the standard method, Brownian bridge and principal component analysis) strongly depends on the problem and the model parameters. On the other hand, the PGMs that take into account the underlying function are robust and powerful. The investigation presents new insight on PGMs and provides constructive guidance on the implementation and the design of new PGMs and new QMC rules.     

When does inferring reputation probability countervail temptation in cooperative behaviors for the prisoners’ dilemma game?

In evolutionary games, the temptation mechanism reduces cooperation percentage while the reputation mechanism promotes it. Inferring reputation theory proposes that agent's imitating neighbors with the highest reputation takes place with a probability. Although reputation promotes cooperation, when and how it enhances cooperation is still a question. This paper investigates the condition where the inferring reputation probability promotes cooperation. Hence, the effects of reputation and temptation on cooperation are explored under the spatial prisoners’ dilemma game, utilizing the methods of simulation and statistical analysis. Results show that temptation reduces cooperation unconditionally while reputation promotes it conditionally, i.e. reputation countervails temptation conditionally. When the inferring reputation probability is less than 0.5, reputation promotes cooperation substantially and thus countervails temptation. However, when the inferring reputation probability is larger than 0.5, its contribution to cooperation is relatively weak and cannot prevent temptation from undermining cooperation. Reputation even decreases cooperation together with temptation when the probability is higher than 0.8. It should be noticed that inferring reputation does not always succeed to countervail temptation and there is a specific interval for it to promote cooperation.