From linear to non-linear/chaotic pendulum: a computational study

Analytical study on reinforced concrete two dimensional frame with knee bracing under cyclic loading

Recognizing Hamming graphs in linear time and space

We give an algorithm which computes a (1-e)-approximately maximum st-flow in an undirected uncapacitated graph in time O(1/e√m/F⋅ m log2 n) where F is the flow value. By trading this off against the Karger-Levine algorithm for undirected graphs which takes ~O(m+nF) time, we obtain a running time of ~O(m n1/3/e2/3) for uncapacitated graphs, improving the previous best dependence on e by a factor of O(1/e3). Like the algorithm of Christiano, Kelner, Madry, Spielman and Teng, our algorithm reduces the problem to electrical flow computations which are carried out in linear time using fast Laplacian solvers. However, in contrast to previous work, our algorithm does not reweight the edges of the graph in any way, and instead uses local (i.e., non s-t) electrical flows to reroute the flow on congested edges. The algorithm is simple and may be viewed as trying to find a point at the intersection of two convex sets (the affine subspace of st-flows of value F and the l∞ ball) by an accelerated version of the method of alternating projections due to Nesterov.By combining this with Ford and Fulkerson's augmenting paths algorithm, we obtain an exact algorithm with running time ~O(m5/4 F1/4) for uncapacitated undirected graphs, improving the previous best running time of ~O(m+ min(nF,m3/2)).We give a related algorithm with the same running time for approximate minimum cut, based on minimizing a smoothed version of the l1 norm inside the cut space of the input graph. We show that the minimizer of this norm is related to an approximate blocking flow and use this to give an algorithm for computing a length k approximately blocking flow in time ~O(m √k).

현대사회는 보안과 안전이 중요해지면서 감시카메라들이 여러 곳에 설치되어 있다. 하지만 감시영상을 보고 상황을 파악하는 것은 여전히 사람의 몫으로 인력과 시간이 소모된다. 그래서 자동으로 감시영상을 분석하여 주요 사건 중심으로 요약해 주는 연구의 필요성이 커지고 있다. 본 논문에서는 감시영상에서 존재하는 다수의 사람을 추적하고, 추적을 통해 얻은 정보를 이용하여 감시영상을 요약하는 방법을 제안한다. 제안하는 감시영상 요약 시스템은 조명보정을 적용하여 배경제거한 후 다수의 사람을 추출하고, 추출된 사람의 추적 정보를 상태 데이터베이스에 저장한다. 추적을 통해 얻은 정보로 추적 대상들의 추적경로, 움직임 상태, 지체시간, 카메라 안으로의 출입시간 등을 사용한다. 또 사람의 움직임에 따라 6 가지(Enter, Stay, Slow, Normal, Fast and Exit)로 움직임 상태를 분류하였고, 움직임 상태를 시간별, 공간별로 요약 그래프로 나타내 추적대상의 움직임 상태를 빠르게 파악할 수 있다.

Chaos is an active research subject in the fields of science in recent years. It is a complex and an erratic behaviour that is possible in very simple systems. In the present day, the chaotic behaviour can be observed in experiments. Many studies have been made in chaotic dynamics during the past three decades and many simple chaotic systems have been discovered. In this work, the behaviour of some simple dynamical systems is studied by constructing mathematical models. Investigations are made on the periodic orbits for continuous maps and idea of sensitive dependence on initial conditions, which is the hallmark of chaos, is obtained. A small attempt has been made to find out the reasons / unknown conditions for the production of chaos in a system. This is explained through simple dynamical systems. (Why chaos is produced in a forced damped simple pendulum?) Besides, another attempt has been made to identify, algebraically simplest chaotic flow. These are the significance of this study - “Bifurcations and chaos in simple dynamical systems”. Accordingly, an analysis is done on different dynamical systems. The exact solution is obtained by solving the differential equation by using Runge-Kutta method; as a result, it is clear from the analysis that, period multiplication occurring in a forced damped simple pendulum can leads to chaos. This result is proving that it is possible to find out the reasons / unknown conditions under which chaotic behaviour exhibits in various systems. The importance of result is, “why chaos is produced in various systems? - may be identified in future. Key Words: Bifurcations, chaos, dynamical systems.

It was argued before that the Hilbert space of quantum harmonic oscillators can be realized in terms of particular trees. We generalize this approach for the Wigner functions defined on the phase space. Along this way, we derive a new representation for the Laguerre function in terms of canonical partition function of the statistical model defined on particular trees with loops.

<i>Introduction to the Modeling and Analysis of Complex Systems.</i> H. Sayama (Ed.). (2015, Open SUNY Textbooks). Free open access PDF, 498 pp. ISBN 978-1-942341-06-2 (deluxe color edition). ISBN 978-1-942341-08-6 (print edition). ISBN 978-1-942341-09-3 (ebook).

Dynamical systems are commonly used to describe the state of time‐dependent systems. In many engineering and control problems, the state space is high‐dimensional making it difficult to analyze and visualize the behavior of the system for varying input conditions. We present a novel dimensionality reduction technique that is tailored to high‐dimensional dynamical systems. In contrast to standard general purpose dimensionality reduction algorithms, we use energy minimization to preserve properties of the flow in the high‐dimensional space. Once the projection operator is optimized, further high‐dimensional trajectories are projected easily. Our 3D projection maintains a number of useful flow properties, such as critical points and flow maps, and is optimized to match geometric characteristics of the high‐dimensional input, as well as optional user constraints. We apply our method to trajectories traced in the phase spaces of second‐order dynamical systems, including finite‐sized objects in fluids, the circular restricted three‐body problem and a damped double pendulum. We compare the projections with standard visualization techniques, such as PCA, t‐SNE and UMAP, and visualize the dynamical systems with multiple coordinated views interactively, featuring a spatial embedding, projection to subspaces, our dimensionality reduction and a seed point exploration tool.

PropCart is a three wheeled model vehicle with a variable pitch pusher propeller and rubber band motor. This paper presents the construction and testing of this vehicle as a platform for teaching concepts of linear and rotational mechanics, and elementary aerodynamics of airfoils applied in aircraft and wind turbines. Building and testing the vehicle is proposed as a 12th Grade STEM project: SCIENCE component: Physics; Linear and Rotational Mechanics with practical determination of basic and derived quantities in particular uniformly accelerated motion. TECHNOLOGY component: Design and construction of the vehicle. ENGINEERING component (overlapping technology): Measurement of thrust of the propeller at different setting angles using video data of propeller angular velocity and cart linear velocity; enabling determination of relative wind and angle of attack, calculation of Reynolds number and identifying stall angle without the use of a wind tunnel. MATHEMATICS component: Data analysis using Excel. A teaching course based mainly on the Khan academy physics programme is provided to support concepts used in the propeller cart project. PropCart is based on a four wheeled vehicle with fixed pitch propeller, devised by David Newton (1999). A redesigned three wheel vehicle with a variable pitch propeller was used in a practical project in a 12 week introductory engineering course at the University of Canterbury (2002) for 15 Petronas students. 12 students completed the project: Test videos for each student were taken in PAL format at 25fps for time intervals of 1–2 seconds for propeller setting angles of 15, 30 and 45 degrees; with student choice of the number of rubber bands used and the number of windup turns. Readings of linear and angular motion were taken with software which could select frame by frame display. Linear and angular displacement against time were plotted against time and time squared. Separate experiments were conducted on the performance of the propeller with the cart fixed with motor axis normal to an annular scale graduated in degrees. The propeller using 2 rubber bands was wound up by 30 turns for each test. Setting angles from 0 to 90 degrees were used. Graphs of angular displacement against time were compared over one second and plotted on a common time axis for setting angles 0 to 90 degrees. The graphs show that stalling occurs at angles greater than 15 degrees. Cart linear motion and propeller angular velocity within a one second time interval were related. Angular velocity of the propeller against time was a piecewise function: At low setting angles near 15 degrees tending to an acceleration sub-function followed by a constant velocity sub-function the latter giving constant thrust and constant linear acceleration of the cart. The angular displacement against time squared graph showed non uniform decreasing acceleration, with the rate of decrease increasing with time squared due to drag; further supported by the test results of the linear motion of the students’ vehicles. A large number of activities are detailed for the teaching of linear and rotational mechanics. For example, the rotational inertia of the propeller can be determined by mounting it as a compound pendulum; this activity importantly uses the parallel axis theorem. Taking the propeller blade as a flat plate airfoil: The relative air flow at mid span can be determined as falling along the resultant of air relative to the cart and air relative to the reference point on the blade: Enabling the determination of flow angle and angle of attack for a given setting angle. A parallel linkage with adjustable link lengths was developed for measuring setting angle by reflecting a laser beam off mirror foil on the blades, this is briefly discussed.

Using hierarchical definitions one can describe very large graphs in small space. The blow-up from the length of the hierarchical description to the size of the graph can be as large as exponential. If the efficiency of graph algorithms is measured in terms of the length of the hierarchical description rather than in terms of the graph size, algorithms that do not exploit the hierarchy become hopelessly inefficient. Whether the hierarchy can be exploited to speed up the solution of graph problems depends on the hierarchical graph model. In the literature, hierarchical graph models have been described that allow almost no exploitation of the hierarchy [W 84]. We present a hierarchical graph model that permits to exploit the hierarchy. For this model we give algorithms that test planarity of a hierarchically described graph in linear time in the length of the hierarchical description.

Minimal separators in [formula omitted]-sparse graphs

Implicit computation of maximum bipartite matchings by sublinear functional operations

Graph isomorphism is a core problem in graph analysis of various application domains. Given two graphs, the graph isomorphism problem is to determine whether there exists an isomorphism between them. As real-world graphs are getting bigger and bigger, applications demand practically fast algorithms that can run on large-scale graphs. However, existing approaches such as graph canonization and subgraph isomorphism show limited performances on large-scale graphs either in time or space. In this paper, we propose a new approach to graph isomorphism, which is the framework of pairwise color refinement and efficient backtracking. The main features of our approach are: (1) pairwise color refinement and binary cell mapping (2) compressed CS (candidate space), and (3) partial failing set, which together lead to a much faster and scalable algorithm for graph isomorphism. Extensive experiments with real-world datasets show that our approach outperforms state-of-the-art algorithms by up to orders of magnitude in terms of running time.

Kannan and Warnow [Triangulating Three-Colored Graphs, Proc. 2nd SODA, 1991, pp. 337–343 and SIAM J. Discrete Math., 5 (1992), pp. 249–258] describe an algorithm to decide whether a three-colored graph can be triangulated so that all the edges connect vertices of different colors. This problem is motivated by a problem in evolutionary biology. Kannan and Warnow have two implementation strategies for their algorithm: one uses slightly superlinear time, while the other uses linear time but quadratic space. We note that three-colored triangulatable graphs are always planar, and we use this fact to modify Kannan and Warnow's algorithm to obtain an algorithm that uses both linear time and linear space.

A general theory for characterizing and then realizing algorithms in hardware is given. The physical process of computation is interpreted in terms of a graph in physical space and time, and then an embedding into this graph of another graph which characterizes data flow in particular algorithms is given. The types of the special class of computational structures called systolic arrays which can occur physically are completely described, and a technique is developed for mapping the graph of a particular systolic algorithm into a physical array. Examples illustrate the methodology.