Stochastic Approximate Algorithms for Uncertain Constrained K-Means Problem

The k-means problem has been paid much attention for many applications. In this paper, we define the uncertain constrained k-means problem and propose a (1+ϵ)-approximate algorithm for the problem. First, a general mathematical model of the uncertain constrained k-means problem is proposed. Second, the random sampling properties of the uncertain constrained k-means problem are studied. This paper mainly studies the gap between the center of random sampling and the real center, which should be controlled within a given range with a large probability, so as to obtain the important sampling properties to solve this kind of problem. Finally, using mathematical induction, we assume that the first j−1 cluster centers are obtained, so we only need to solve the j-th center. The algorithm has the elapsed time O((1891ekϵ2)8k/ϵnd), and outputs a collection of size O((1891ekϵ2)8k/ϵn) of candidate sets including approximation centers.

Ulam Stability of n-th Order Delay Integro-Differential Equations

In this paper, the Ulam stability of an n-th order delay integro-differential equation is given. Firstly, the existence and uniqueness theorem of a solution for the delay integro-differential equation is obtained using a Lipschitz condition and the Banach contraction principle. Then, the expression of the solution for delay integro-differential equation is derived by mathematical induction. On this basis, we obtain the Ulam stability of the delay integro-differential equation via Gronwall–Bellman inequality. Finally, two examples of delay integro-differential equations are given to explain our main results.

Global Exponential Stability of a Discrete-Time Rayleigh System with Delays

This paper deals with the problem of global exponential stability for a discrete-time Rayleigh system with delays. By using the mathematical induction method, some sufficient conditions are proposed for the global exponential stability of the discrete-time Rayleigh system. Finally, a numerical example is given to illustrate the effectiveness and application of the obtained results.

Is the Algorithm of Artificial Neural Network a Deduction or Induction? Discussion between Natural Sciences, Mathematics and Philosophy

The algorithm of artificial neural network (ANN) has been defined as a supervised learning and heuristic algorithms. In training an ANN model, big data is necessary to use as training data to obtain perfectly accurate predicted data. However, big data really have no clear definition. Therefore, adding new training data to re-train an ANN model, by which can improve the predicted accuracy. This action of re-training this ANN model with added new training data is repeated to approach the laws of physics that is accessed to the principle of induction e.g., empirical formulas. However, accessing the principle of induction is limited. If the deduction is found using an ANN model, then approach of this ANN model with added new training data is also performed repeatedly to access the principle of deduction e.g., theory formulas. However, accessing the principle of deduction is also limited. It means the law cannot be easily deduced for an ANN model. Therefore, the algorithm of an ANN is not the canonical classical methods. On the other hand, the algorithm of an ANN does not belong to mathematical induction and deduction.

Bi-Directional Drone Design and its Path Planning

There are two aspects to my project, one is optimized bi-directional drone designing and other is its path planning. The optimization in design lies in the fact that my drone can carry load both in +z and –z axis if its direction of motion is in x-y plane. This design optimization helps the drone to carry more payload than drones of same frame(basic chassis) weight category. In other words, my drone has greater payload to its own weight ratio(almost 0.8) than other drones with almost similar or same functionalities. Coming to path planning algorithm, I have taken a mathematical induction approach to solve the problem statement by clearly defining our conditions to follow to remain in the specified path along with constraints lying in the path. My goal as a path planner has to ensure that the drone follows the conditions specified without grappling into obstacles. Also, to achieve the desired goal in least possible time. The paths traversed by the drone would be stored into the memory processing system of the drone for future development of algorithm.

Reaching a Group-Agreement in Finite Time Under Acyclic Interaction Topology

This paper discusses the finite time agreement problem of networks with acyclic partition topology. In view of the structural characteristics of such network topology, mathematical induction is particularly suitable to prove the main conclusions in the paper. In addition, for the consideration of the finite time consensus problem, in addition to using basic matrix theory to verify the solution of the problem, this brief also has a more detailed analysis of the time required to reach consensus. Based on these two points, it is observed that the solution of this problem is due to the features of acyclic partition interactions and the continuity of the related finite time protocol and contributes to the research on the grouping consensus of multiagent system. Furthermore, simulation examples are presented to verify the theoretical results.

THEOREM ON THE EXCLUSION OF HIT TRIPLES AND ABC CONJECTURE

The proof of Theorem on the exclusion of hit triples and conjecture is given by the method of mathematical induction.

Rational Possibility of Generating Power Laws in the Synthesis of Cam Mechanisms

Introduction.The generation of polynomial power laws of motion for the synthesis of cam mechanisms is complicated by the need to determine the coefficients of power polynomials. The study objective is to discover a rational capability of generating рower law swith arbitrary terms number under s with an rbitrary number of terms under the synthesis of cam mechanisms.Materials and Methods.A unified formula for determining the values of coefficients of power polynomials with any number of integers and/or non-integer exponents is derived through the so-called transfinite mathematical induction. Results.A unified formula for determining the values of coefficients, which gives correct results for any number of even and/or odd exponents, is presented. The correctness of the derived formula is validated by the results on the multiple checks for different numbers, even and odd values of the exponents of quinquinomial and hexanomial power functions. Discussion and Conclusions. A unified formula for determining the values of coefficients of power polynomials makes it possible to rationally define the laws of motion without finite and infinite spikes in the synthesis of elastic cam-lever systems. This provides a rational determination of the laws of motion without finite and infinite spikes in the synthesis of elastic cam-lever systems, and simple verification of the accuracy of the results obtained. The functions are particularly suitable for the synthesis of polydyne cams, as well as cams, since one polynomial can be used throughout the entire geometric mechanism cycle.

Gaussian integral by Taylor series and applications

In this paper, we present a solution for a specific Gaussian integral. Introducing a parameter that depends on a n index, we found out a general solution inspired by the Taylor series of a simple function. We demonstrated that this parameter represents the expansion coefficients of this function, a very interesting and new result. We also introduced some Theorems that are proved by mathematical induction. As a test for the solution presented here, we investigated a non-extensive version for the particle number density in Tsallis framework, which enabled us to evaluate the functionality of the method. Besides, solutions for a certain class of the gamma and factorial functions are derived. Moreover, we presented a simple application in fractional calculus. In conclusion, we believe in the relevance of this work because it presents a solution for the Gaussian integral from an unprecedented perspective.

Thinking Process of Secondary Level Students in Constructing Proof by Mathematical Induction in Terms of Their Attitude toward Mathematics

This research aims to describe secondary-level students' thinking processes in terms of their attitude toward mathematics in constructing proof by mathematical induction. This qualitative research involved two students who were selected from 30 students of second grade of senior high school that categorized into two groups, namely the students who have a positive and negative attitude toward mathematics by using Attitude Toward Mathematics Inventory (ATMI) questionnaire. One student from each category was selected to be given a proving test and interviewed. Students’ process of proving test and interview recording were collected and analyzed to identify their thinking process. This research points out that both students who have positive and negative attitudes towards mathematics can recall the information in their memory regarding the characteristics and steps to prove using mathematical induction. However, the student who has negative attitudes towards mathematics tends to experience some difficulties in processing information, especially in the induction step, caused by panic, depression, and insecurity. The student who has a positive attitude towards mathematics also experiences some difficulties in the proving process, namely understanding what to prove and compiling the induction steps. However, she keeps trying and believes that she would be able to solve it. Hence, she can solve the proof completely.