Remarks on two fixed-point theorems involving the sum and the product of two operators

In this article, we focus our attention on two fixed-point theorems of Krasnoselskii [1] and Dhage [2]. It is shown that some of the hypotheses of these fixed-point theorems are redundent. Our claim is also illustrated with an example.     

Design and analysis of evolutionary bit-length optimization algorithms for floating to fixed-point conversion

Hardware designs need to obey constraints of resource utilization, minimum clock frequency, power consumption, computation precision and data range, which are all affected by the data type representation. Floating and fixed-point representations are the most common data types to work with real numbers where arithmetic hardware units for fixed-point format can improve performance and reduce energy consumption when compared to floating point solution. However, the right bit-lengths estimation for fixed-point is a time-consuming task since it is a combinatorial optimization problem of minimizing the accumulative arithmetic computation error. This work proposes two evolutionary approaches to accelerate the process of converting algorithms from floating to fixed-point format. The first is based on a classic evolutionary algorithm and the second one introduces a compact genetic algorithm, with theoretical evidence that a near-optimal performance, to find a solution, has been reached. To validate the proposed approaches, they are applied to three computing intensive algorithms from the mobile robotic scenario, where data error accumulated during execution is influenced by sensor noise and navigation environment characteristics. The proposed compact genetic algorithm accelerates the conversion process up to 10.2× against the state of art methods reaching similar bit precision and robustness.     

A fixed-point approach for nonlinear discrete boundary value problems

Existence results are established for second-order discrete boundary value problems.     

Design of fixed-point smoother algorithm for linear discrete time-delay systems

This paper investigates the fixed-point smoothing problems for linear discrete-time systems with multiple time-delays in the observations.The linear discrete-time systems considered have l + 1 output channels.One is instantaneous observation and the others are delayed.The fixed-point smoothers involving recursive algorithm and non-recursive algorithm are designed by using innovation analysis theory without relying on the system augmentation approach.Also, it is further shown that the design of fixed-point smoother comes down to solving l + 1 Riccati equations with the same dimensions as the original systems.     

A new convergence result for fixed-point iteration in bounded interval of

A fixed-point theorem in bounded interval of ⁿ and a new fixed-point iteration on Mann's iteration scheme are established to compute a fixed-point of real Lipschitz functions. Two numerical examples are presented to show that the method is feasible and that it produces reasonable results.     

关于复合函数求导的一个注记

关于复合函数的求导问题是函数导数部分的重点和难点。链式求导法则是一种基本的有效的求导方法。应用链式法则求导,首先要准确地对复合函数的结构进行分层。为了方便地应用该法则,给出将函数复合结构准确分层的几个结论。     

An evaluation of self-adjusting binary search tree techniques

Much has been said in praise of self-adjusting data structures, particularly self-adjusting binary search trees. Self-adjusting trees are most suited to skewed key-access distributions as the techniques attempt to place the most commonly accessed keys near the root of the tree. Theoretical bounds on worst-case and amortized performance (i.e. performance over a sequence of operations) have been derived which compare well with those for optimal binary search trees. In this paper, we compare the performance of three different techniques for self-adjusting trees with that of AVL and random binary search trees. Comparisons are made for various tree sizes, levels of key-access-frequency skewness and ratios of insertions and deletions to searches. The results show that, because of the high cost of maintaining self-adjusting trees, in almost all cases the AVL tree outperforms all the self-adjusting trees and in many cases even a random binary search tree has better performance, in terms of CPU time, than any of the self-adjusting trees. Self-adjusting trees seem to perform best in a highly dynamic environment, contrary to intuition.     

Neural network for optimal steiner tree computation

Hopfield neural network model for finding the shortest path between two nodes in a graph was proposed recently in some literatures. In this paper, we present a modified version of Hopfield model to a more general problem of searching an optimal tree (least total cost tree) from a source node to a number of destination nodes in a graph. This problem is called Steiner tree in graph theory, where it is proved to be a NP-complete. Through computer simulations, it is shown that the proposed model could always find an optimal or near-optimal valid solution in various graphs.     

On finite-state approximants for probabilistic computation tree logic

We generalize the familiar semantics for probabilistic computation tree logic from finite-state to infinite-state labelled Markov chains such that formulas are interpreted as measurable sets. Then we show how to synthesize finite-state abstractions which are sound for full probabilistic computation tree logic and in which measures are approximated by monotone set functions. This synthesis of sound finite-state approximants also applies to finite-state systems and is a probabilistic analogue of predicate abstraction. Sufficient and always realizable conditions are identified for obtaining optimal such abstractions for probabilistic propositional modal logic.     

基于关联度函数的决策树分类算法

为了克服决策树算法中普遍存在的多值偏向问题,提出了一种新的基于关联度函数的决策树算法--AF算法,并从理论上分析了它克服多值偏向的原理。通过实验发现,与ID3算法比较,AF算法不仅克服了多值偏向问题,而且保持了较高的分类正确率。     

Semiring induced valuation algebras: Exact and approximate local computation algorithms

Local computation in join trees or acyclic hypertrees has been shown to be linked to a particular algebraic structure, called valuation algebra. There are many models of this algebraic structure ranging from probability theory to numerical analysis, relational databases and various classical and non-classical logics. It turns out that many interesting models of valuation algebras may be derived from semiring valued mappings. In this paper we study how valuation algebras are induced by semirings and how the structure of the valuation algebra is related to the algebraic structure of the semiring. In particular, c-semirings with idempotent multiplication induce idempotent valuation algebras and therefore permit particularly efficient architectures for local computation. Also important are semirings whose multiplicative semigroup is embedded in a union of groups. They induce valuation algebras with a partially defined division. For these valuation algebras, the well-known architectures for Bayesian networks apply. We also extend the general computational framework to allow derivation of bounds and approximations, for when exact computation is not feasible.     

Polytime model checking for timed probabilistic computation tree logic

We consider the model checking problem for Timed Probabilistic Computation Tree Logic (TPCTL) introduced by H.A. Hansson and D. Jonsson, and studied in a recent book by H.A. Hansson [Han94]. The semantics of TPCTL is defined in terms of probabilistic transition systems. It is shown in [Han94] that model checking for TPCTL has exponential time complexity, the latter being measured in terms of the size of formula, the size of transition system and the value of explicit time that appears in the formula. Besides that, [Han94] describes some polytime decidable classes, the proofs being rather voluminous. We show, by a short proof, that this model checking problem is polytime decidable in the general case. The proof is essentially based on results on the complexity of Markov decision processes. Received: 2 May 1996 / 5 January 1998     

Updating a balanced search tree in O(1) rotations

Olivié has recently introduced the class of ‘half-balanced’ binary search trees, which have O(log n) access time but require only a constant number of single rotations for rebalancing after an insertion or a deletion. In this paper we show that a well-known class of balanced binary trees, the ‘symmetric binary B-trees’ of Bayer, have the same properties. This is not surprising, for Bayer's class and Oliviés class contain exactly the same binary trees.     

An EXCMG accelerated multiscale multigrid computation for 3D Poisson equation

Multiscale multigrid (MSMG) method is an effective computational framework for efficiently computing high accuracy solutions for elliptic partial differential equations. In the current MSMG method, compared to the CPU cost on computing sixth-order solutions by applying extrapolation and other techniques on two fourth-order solutions from different scales grids, much more CPU time is spent on computing fourth-order solutions themselves on coarse and fine grids, particularly for high-dimensional problems. Here we propose to embed extrapolation cascadic multigrid (EXCMG) method into the MSMG framework to accelerate the whole process. Numerical results on 3D Poisson equations show that the new EXCMG–MSMG method is more efficient than the existing MSMG method and the EXCMG method for sixth-order solution computation.     

Learning decision tree for ranking

Decision tree is one of the most effective and widely used methods for classification. However, many real-world applications require instances to be ranked by the probability of class membership. The area under the receiver operating characteristics curve, simply AUC, has been recently used as a measure for ranking performance of learning algorithms. In this paper, we present two novel class probability estimation algorithms to improve the ranking performance of decision tree. Instead of estimating the probability of class membership using simple voting at the leaf where the test instance falls into, our algorithms use similarity-weighted voting and naive Bayes. We design empirical experiments to verify that our new algorithms significantly outperform the recent decision tree ranking algorithm C4.4 in terms of AUC.

Development of a structural equation modeling-based decision tree methodology for the analysis of lung transplantations

Lung transplantation has a vital role among all organ transplant procedures since it is the only accepted treatment for the end-stage pulmonary failure. There have been several research attempts to model the performance of lung transplants. Yet, these early studies either lack model predictive capability by relying on strong statistical assumptions or provide adequate predictive capability but suffer from less interpretability to the medical professionals. The proposed method described in this paper is focused on overcoming these limitations by providing a structural equation modeling-based decision tree construction procedure for lung transplant performance evaluation. Specifically, partial least squares-based path modeling algorithm is used for the structural equation modeling part. The proposed method is validated through a US nation-wide dataset obtained from United Network for Organ Sharing (UNOS). The results are promising in terms of both prediction and interpretation capabilities, and are superior to the existing techniques. Hence, we assert that a decision support system, which is based on the proposed method, can bridge the knowledge-gap between the large amount of available data and betterment of the lung transplantation procedures.     

Faster algorithms for guided tree edit distance

The guided tree edit distance problem is to find a minimum cost series of edit operations that transforms two input forests F and G into isomorphic forests F^′ and G^′ such that a third input forest H is included in F^′ (and G^′). The edit operations are relabeling a vertex and deleting a vertex. We show efficient algorithms for this problem that are faster than the previous algorithm for this problem of Peng and Ting [Z. Peng, H. Ting, Guided forest edit distance: Better structure comparisons by using domain-knowledge, in: Proc. 18th Symposium on Combinatorial Pattern Matching (CPM), 2007, pp. 28-39].     

Parallel multischeme computation with the switch of computation history

A parallel multischeme computation in the solutions of differential equation initial-value problems has been studied. The mathematical switch of computation history is used successfully in the identification of the best approximation among all available ones at a computing step. A solution correction factor is also developed to achieve an extra four digits in solution accuracy. Based on our results, if the truncation error of computation history as we defined it can be properly utilized, then a computation engaging high-order schemes or using fine grids may be unnecessary.