Different Testing Results on SVM with Double Penalty Parameters

The Support Vector Machine proposed by Vapnik is a generalized linear classifier which makes binary classification of data based on the supervised learning. SVM has been rapidly developed and has derived a series of improved and extended algorithms, which have been applied in pattern recognition, image recognition, etc. Among the many improved algorithms, the technique of regulating the ratio of two penalty parameters according to the ratio of the sample quantities of the two classes has been widely accepted. However, the technique has not been verified in the way of rigorous mathematical proof. The experiments based on USPS sets in the study were designed to test the accuracy of the theory. The optimal parameters of the USPS sets were found through the grid-scanning method, which showed that the theory is not accurate in any case because there is absolutely no linear relationship between ratios of penalty parameters and sample sizes.

The Essence of the Variational Iteration Method and the Homotopy Analysis Method Is Different

The recently published paper “The variational iteration method is a special case of the homotopy analysis method” by Robert A. Van Gorder [1], weakly pointed out that the variational iteration method and all of its optimal analogues are specific cases of the more general homotopy analysis method. This assertion was not truly supported by a rigorous mathematical proof, nor by an accessible example from the attributed papers. In this brief, we refute the author's claim by supplementing three simple examples, which do not indicate that the variational iteration method is a special case of the homotopy analysis method. This is justified by a Theorem to compute the rate of convergence of both methods.

Fuzzy portfolio selection with prospect consistency constraint based on possibility theory

Considering that most studies have taken the investors’ preference for risk into account but ignored the investors’ preference for assets, in this paper, we combine the prospect theory and possibility theory to provide investors with a portfolio strategy that meets investors’ preference for assets. Firstly, a novel reference point is proposed to give investors a comprehensive impression of assets. Secondly, the prospect return rate of assets is quantified as trapezoidal fuzzy number, and its possibilistic mean value and variance are regarded as prospect return and risk and then used to define the fuzzy prospect value. This new definition is presented to denote the score of an asset in investors’ subjective cognition. And then, a prospect asset filtering frame is proposed to help investors select assets according to their preference. When assets are selected, another new definition called prospect consistency coefficient is proposed to measure the deviation of a portfolio strategy from investors’ preference. Some properties of the definition are presented by rigorous mathematical proof. Based on the definition and its properties, a possibilistic model is constructed, which can not only provide investors optimal strategies to make profit and reduce risk as much as possible, but also ensure that the deviation between the strategies and investors’ preference is tolerable. Finally, a numerical example is given to validate the proposed method, and the sensitivity analysis of parameters in prospect value function and prospect consistency constraint is conducted to help investors choose appropriate values according to their preferences. The results show that compared with the general M-V model, our model can not only better satisfy investors’ preference for assets, but also disperse risk effectively.

RIGOUR AND PROOF

This paper puts forward a new account of rigorous mathematical proof and its epistemology. One novel feature is a focus on how the skill of reading and writing valid proofs is learnt, as a way of understanding what validity itself amounts to. The account is used to address two current questions in the literature: that of how mathematicians are so good at resolving disputes about validity, and that of whether rigorous proofs are necessarily formalizable.

No hair theorem for massless scalar fields outside asymptotically flat horizonless reflecting compact stars

In a recent paper, Hod started a study on no scalar hair theorem for asymptotically flat spherically symmetric neutral horizonless reflecting compact stars. In fact, Hod’s approach only rules out massive scalar fields. In the present paper, for massless scalar fields outside neutral horizonless reflecting compact stars, we provide a rigorous mathematical proof on no hair theorem. We show that asymptotically flat spherically symmetric neutral horizonless reflecting compact stars cannot support exterior massless scalar field hairs.

Active Contours With Stochastic Fronts and Mechanical Topology Optimization

Active contours with stochastic fronts (ACSF) is developed and investigated in this article to address the dependency of the existing algorithms of topology optimization on the initial guesses. The promising results of ACSF confirms that the use of this approach leads to higher chance of escaping from local solutions compared to the classic level set method. ACSF as a special case of the stochastic active contours (SAC), has a simplified structure that makes its implementation easier, and at the same time it has a rigorous mathematical proof of convergence. Although propitious, there is still a slight chance of trapping scenarios for ACSF that is observed in the presented results.

CONTROLLABLE WORKSPACE OF CABLE-DRIVEN REDUNDANT PARALLEL MANIPULATORS BY FUNDAMENTAL WRENCH ANALYSIS

Workspace analysis is always a crucial issue in robotic manipulator design. This paper introduces a set of newly defined fundamental wrenches that opens new horizons to physical interpretation of controllable workspace of a general cable-driven redundant parallel manipulator. Based on this set of fundamental wrenches, a novel tool is presented to determine configurations of cable-driven redundant parallel manipulator that belong to the controllable workspace. Analytical expressions of such workspace boundaries are obtained in an implicit form and a rigorous mathematical proof is provided for this method. Finally, the proposed method is implemented on a spatial cable-driven manipulator of interest.

On the Convergence of Truncated Processes of Multiserver Retrial Queues

Retrial queues can only be solved in a closed form in very few and simple cases, so researchers must resort to approximate models. However, most of the papers that propose approximate models assume the convergence of the proposed models to their exact counterparts, without providing a rigorous mathematical proof. In this paper we demonstrate the convergence of finite truncated models with two reattempt orbits.