Let G be a graph with no three independent vertices. How many edges of G can be packed with edge-disjoint copies of Kk? More specifically, let fk(n, m) be the largest integer t such that, for any graph with n vertices, m edges, and independence number 2, at least t edges can be packed with edge-disjoint copies of Kk. Turán's theorem together with Wilson's Theorem assert that if . A conjecture of Erdős states that for all plausible m. For any ε > 0, this conjecture was open even if . Generally, f_k(n,m) may be significantly smaller than . Indeed, for k=7 it is easy to show that for m ≈ 0.3n2. Nevertheless, we prove the following result. For every k≥ 3 there exists γ>0 such that if then . In the special case k=3 we obtain the reasonable bound γ ≥ 10−4. In particular, the above conjecture of Erdős holds whenever G has fewer than 0.2501n2 edges.

A family of subsets of an n-set is 2-cancellative if, for every four-tuple {A, B, C, D} of its members, A∪ B∪C=A∪ B∪ D implies C = D. This generalizes the concept of cancellative set families, defined by the property that A∪B ≠A ∪ C for A, B, C all different. The asymptotics of the maximum size of cancellative families of subsets of an n-set is known (Tolhuizen [7]). We provide a new upper bound on the size of 2-cancellative families, improving the previous bound of 20.458n to 20.42n.

Spreadsheets are a widespread tool for a variety of tasks, particularly in business settings. Spreadsheet users employ a form of programming that, although popular, is highly error-prone and has limited expressiveness. A promising approach to overcome these shortcomings is to augment spreadsheets with logic-based knowledge representation and reasoning (KR&R) functionality. In this paper, we present Logic Embedded in SpreadSheets (LESS), a system which integrates PowerLoom, a highly expressive logic-based KR&R system, with Microsoft (MS) Excel. The design of LESS provides different tiers of functionality that explore trade-offs between direct access to the underlying logic engine and user-friendly support for spreadsheets users. A prototype of LESS was implemented as an MS Excel add-in.

We show that a random graph studied by Ioffe and Levit is an example of an inhomogeneous random graph of the type studied by Bollobás, Janson and Riordan, which enables us to give a new, and perhaps more revealing, proof of their result on a phase transition.

We exploit the spreadsheet metaphor to make deductive problem-solving methods available to the vast population of spreadsheet end-users. In particular, we show how the function-based problem-solving capabilities of spreadsheets can be extended to include logical deductive methods in a way that is consistent with the existing spreadsheet ‘look and feel’. The foundation of our approach is the integration of a standard deductive logic system into a successful Commercial-Off-The-Shelf (COTS) spreadsheet. We have demonstrated this by designing and implementing an extension to Excel that manages the integration of Excel and a deductive logic engine based on the World Wide Web Consortium (W3C) standard ontology language OWL + SWRL.

Let d=1≤d1≤ d2≤···.≤ dn be a non-decreasing sequence of n positive integers, whose sum is even. Let denote the set of graphs with vertex set [n]={1,2,. . .., n} in which the degree of vertex i is di. Let Gn,d be chosen uniformly at random from . Let d=(d1+d2+···.+dn)/n be the average degree. We give a condition on d under which we can show that w.h.p. the chromatic number of is Θ(d/ln d). This condition is satisfied by graphs with exponential tails as well those with power law tails.

A novel approach to estimate the real-time moving trajectory of an object is proposed in this paper. The object's position is obtained from the image data of a charge coupled device (CCD) camera, while a state estimator predicts the linear and angular velocities of the moving object. To overcome the uncertainties and noises residing in the input data, a Kalman filter and neural networks are utilized cooperatively. Since the Kalman filter needs to approximate a nonlinear system into a linear model in order to estimate the states, there still exist errors as well as uncertainties. To resolve this problem, in this approach, the Kohonen networks, which have a high adaptability to the memory of the input–output relationship, are utilized for the nonlinear region. In addition to this, the Kohonen network, as a sort of neural network, can effectively adapt to the dynamic variations and become robust against noises. This approach is derived from the observation that the Kohonen network is a type of a self-organized map and is spatially oriented, which makes it suitable for determining the trajectories of moving objects. The superiority of the proposed algorithm compared with the extended Kalman filter is demonstrated through real experiments.

Robotic manipulators that have interacted with uncalibrated environments typically have limited positioning and tracking capabilities, if control tasks cannot be appropriately encoded using available features in the environments. Specifically, to perform 3-D trajectory following operations employing binocular vision, it seems necessary to have a priori knowledge on pointwise correspondence information between two image planes. However, such an assumption cannot be made for any smooth 3-D trajectories. This paper describes how one might enhance autonomous robotic manipulation for 3-D trajectory following tasks using eye-to-hand binocular visual servoing. Based on a novel encoded error, an image-based feedback control law is proposed without assuming pointwise binocular correspondence information. The proposed control approach can guarantee task precision by employing only an approximately calibrated binocular vision system. The goal of the autonomous task is to drive a tool mounted on the end-effector of the robotic manipulator to follow a visually determined smooth 3-D target trajectory in desired speed with precision. The proposed control architecture is suitable for applications that require precise 3-D positioning and tracking in unknown environments. Our approach is successfully validated in a real task environment by performing experiments with an industrial robotic manipulator.

In this paper we prove polynomial versions of the Carlson–Simpson theorem and the Graham–Rothschild theorem on parameter sets. To do so we prove a useful extension of the polynomial Hales–Jewett theorem.

This paper presents the Genetic Algorithm Optimized Fourier Series Formulation (GAOFSF) method for stable gait generation in bipedal locomotion. It uses a Truncated Fourier Series (TFS) formulation with its coefficients determined and optimized by Genetic Algorithm. The GAOFSF method can generate human-like stable gaits for walking on flat terrains as well as on slopes in a uniform way. Through the adjustment of only a single or two parameters, the step length and stride-frequency can easily be adjusted online, and slopes of different gradients are accommodated. Dynamic simulations show the robustness of the GAOFSF, with stable gaits achieved even if the step length and stride frequency are adjusted by significant amounts. With its ease of adjustments to accommodate different gait requirements, the approach lends itself readily for control of walking on a rough terrain and in the presence of external perturbations.

This research presents a new and generic geometric approach that characterizes the kinematic singularity of wheeled mobile robots. First, the kinematic models of all the common wheels are obtained: fixed, centered orientable, castor, and Swedish. Then, a procedure for generating robot kinematic models is presented based on the set of wheel equations and the null space concept. Next, two examples are developed to illustrate the nongeneric singularity characterization. In order to improve that approach, a generic and practical geometric approach is established to characterize the singularity of any kinematic model of a wheeled mobile robot (WMR). Finally, the singular configurations for many types of mobile robots are depicted employing the proposed approach.

In this paper, we present our proposal to Constraint Functional Logic Programming over Finite Domains (CFLP()) with a lazy functional logic programming language which seamlessly embodies finite domain () constraints. This proposal increases the expressiveness and power of constraint logic programming over finite domains (CLP()) by combining functional and relational notation, curried expressions, higher-order functions, patterns, partial applications, non-determinism, lazy evaluation, logical variables, types, domain variables, constraint composition, and finite domain constraints. We describe the syntax of the language, its type discipline, and its declarative and operational semantics. We also describe , an implementation for CFLP(), and a comparison of our approach with respect to CLP() from a programming point of view, showing the new features we introduce. And, finally, we show a performance analysis which demonstrates that our implementation is competitive with respect to existing CLP() systems and that clearly outperforms the closer approach to CFLP().

We give a combinatorial proof of the result of Kahn, Kalai and Linial [16], which states that every balanced boolean function on the n-dimensional boolean cube has a variable with influence of at least . The methods of the proof are then used to recover additional isoperimetric results for the cube, with improved constants.

We also state some conjectures about optimal constants.

In this paper, we investigate the problem of measuring the shape of a continuum robot manipulator using visual information from a fixed camera. Specifically, we capture the motion of a set of fictitious planes, each formed by four or more feature points, defined at various strategic locations along the body of the robot. Then, utilizing expressions for the robot forward kinematics as well as the decomposition of a homography relating a reference image of the robot to the actual robot image, we obtain the three-dimensional shape information continuously. We then use this information to demonstrate the development of a kinematic controller to regulate the manipulator end-effector to a constant desired position and orientation.

In combinatorial optimization, a popular approach to NP-hard problems is the design of approximation algorithms. These algorithms typically run in polynomial time and are guaranteed to produce a solution which is within a known multiplicative factor of optimal. Unfortunately, the known factor is often known to be large in pathological instances. Conventional wisdom holds that, in practice, approximation algorithms will produce solutions closer to optimal than their proven guarantees. In this paper, we use the rigorous-analysis-of-heuristics framework to investigate this conventional wisdom.

We analyse the performance of three related approximation algorithms for the uncapacitated facility location problem (from Jain, Mahdian, Markakis, Saberi and Vazirani (2003) and Mahdian, Ye and Zhang (2002)) when each is applied to an instances created by placing n points uniformly at random in the unit square. We find that, with high probability, these 3 algorithms do not find asymptotically optimal solutions, and, also with high probability, a simple plane partitioning heuristic does find an asymptotically optimal solution.

The design and implementation of an incremental copying heap garbage collector for WAM-based Prolog systems is presented. Its heap layout consists of a number of equal-sized blocks. Other changes to the standard WAM allow these blocks to be garbage collected independently. The independent collection of heap blocks forms the basis of an incremental collecting algorithm which employs copying without marking (contrary to the more frequently used mark© or mark&slide algorithms in the context of Prolog). Compared to standard semi-space copying collectors, this approach to heap garbage collection lowers in many cases the memory usage and reduces pause times. The algorithm also allows for a wide variety of garbage collection policies including generational ones. The algorithm is implemented and evaluated in the context of hProlog.

This paper deals with the stability of null-space velocity control algorithms in extended operational space for redundant robots. We compare the performance of the control algorithm based on the minimal null-space projection and generalized-inverse-based projection into the Jacobian null-space. We show how the null-space projection affects the performance of the null-space tracking algorithm. The results are verified with the simulation and real implementation on a redundant mobile robot composed of 3 degrees of freedom (DOFs) mobile platform and 7-DOF robot arm.

A logical spreadsheet is a spreadsheet in which the formula language is composed of logical expressions. Logical spreadsheets were invented shortly after traditional electronic spreadsheets were introduced, but since then logical spreadsheet research has been somewhat sparse. Recently, however, there has been a resurgence in the interest of logical spreadsheets in the research community. In this article, we summarize logical spreadsheet research up to this point.