AgentGeom: a multiagent system for pedagogical support in geometric proof problems

This paper aims, first, to describe the fundamental characteristics and workings of the AgentGeom artificial tutorial system, which is designed to help students develop knowledge and skills related to problem solving, mathematical proof in geometry, and the use of mathematical language. Following this, we indicate the manner in which a secondary school student can appropriate these abilities through interactions with the system. Our system uses strategic messages of the agent tutor in an argumentative process that collaborates with a student in the construction of a proof.     

The concept of proof in the light of mathematical work

ZDM – Mathematics Education - Our article aims to show how illuminating mathematical work as a concept from didactics of mathematics is useful in understanding issues relating to proving and...     

Existence and instability of steady states for a triangular cross-diffusion system: A computer-assisted proof

In this paper, we present and apply a computer-assisted method to study steady states of a triangular cross-diffusion system. Our approach consist in an a posteriori validation procedure, that is based on using a fixed point argument around a numerically computed solution, in the spirit of the Newton–Kantorovich theorem. It allows to prove the existence of various non homogeneous steady states for different parameter values. In some situations, we obtain as many as 13 coexisting steady states. We also apply the a posteriori validation procedure to study the linear stability of the obtained steady states, proving that many of them are in fact unstable.     

A computer-assisted proof for the existence of horseshoe in a novel chaotic system

The dynamics of a novel chaotic system are studied, and a rigorous computer-assisted proof for existence of horseshoe in this system is given. A Poincaré section is properly chosen to obtain the Poincaré map, which is proved to be semi-conjugate to the 4-shift map by utilizing topological horseshoe theory. This implies the entropy of the system is no less than log 4, and the system definitely exhibits chaos.     

Automatic differentiation: Reduced gradient and reduced Hessian matrix

This paper is concerned with automatic differentiation methods for computing the reduced gradient M ^t G and the reduced Hessian matrix M ^t HM. Hereby G is the gradient and H is the Hessian matrix of a real function F of n variables, and M is a matrix with n rows and k columns where kn. The reduced quantities are of particular interest in constrained optimization with objective function F. Two automatic differentiation methods are described, a standard method that produces G and H as intermediate results, and an economical method that takes a shortcut directly to the reduced quantites. The two methods are compared on the basis of the reqired computing time and storage. It is shown that the costs for the economical method are less than (k ²+3k+2)/(n ²+3n+2) times the expenses for the standard method.     

Logic of agreement: Foundations,semantic system and proof theory

In this paper a multi-valued propositional logic — logic of agreement — in terms of its model theory and inference system is presented. This formal system is the natural consequence of a new way to approach concepts as commonsense knowledge, uncertainty and approximate reasoning — the point of view of agreement. Particularly, it is discussed a possible extension of the Classical Theory of Sets based on the idea that, instead of trying to conceptualize sets as “fuzzy” or “vague” entities, it is more adequate to define membership as the result of a partial agreement among a group of individual agents. Furthermore, it is shown that the concept of agreement provides a framework for the development of a formal and sound explanation for concepts (e.g. fuzzy sets) which lack formal semantics. According to the definition of agreement, an individual agent agrees or not with the fact that an object possesses a certain property. A clear distinction is then established, between an individual agent — to whom deciding whether an element belongs to a set is just a yes or no matter — and a commonsensical agent — the one who interprets the knowledge shared by a certain group of people. Finally, the logic of agreement is presented and discussed. As it is assumed the existence of several individual agents, the semantic system is based on the perspective that each individual agent defines her/his own conceptualization of reality. So the semantics of the logic of agreement can be seen as being similar to a semantics of possible worlds, one for each individual agent. The proof theory is an extension of a natural deduction system, using supported formulas and incorporating only inference rules. Moreover, the soundness and completeness of the logic of agreement are also presented.     

Simplified modelling of joints and beam-like structures for BIW optimization in a concept phase of the vehicle design process

The paper proposes an engineering approach for the replacement of beam-like structures and joints in a vehicle model. The final goal is to provide the designer with an effective methodology for creating a concept model of such automotive components, so that an NVH optimization of the body in white (BIW) can be performed at the earliest phases of the vehicle design process. The proposed replacement methodology is based on the reduced beam and joint modelling approach, which involves a geometric analysis of beam-member cross-sections and a static analysis of joints. The first analysis aims at identifying the beam center nodes and computing the equivalent beam properties. The second analysis produces a simplified model of a joint that connects three or more beam-members through a static reduction of the detailed joint FE model.In order to validate the proposed approach, an industrial case-study is presented, where beams and joints of the upper region of a vehicle's BIW are replaced by simplified models. Two static load-cases are defined to compare the original and the simplified model by evaluating the stiffness of the full vehicle under torsion and bending in accordance with the standards used by automotive original equipment manufacturer (OEM) companies. A dynamic comparison between the two models, based on global frequencies and modal shapes of the full vehicle, is presented as well.     

A proof of a conjecture about a symmetric system of orthogonal polynomials

ABSTRACT

The purpose of this note is to give an affirmative answer to a conjecture appearing in Berg [Open problems. Integral Transforms Spec Funct. 2015;26(2):90–95].     

Lagrange optimality system for a class of nonsmooth convex optimization

In this paper, we revisit the augmented Lagrangian method for a class of nonsmooth convex optimization. We present the Lagrange optimality system of the augmented Lagrangian associated with the problems, and establish its connections with the standard optimality condition and the saddle point condition of the augmented Lagrangian, which provides a powerful tool for developing numerical algorithms: we derive a Lagrange–Newton algorithm for the nonsmooth convex optimization, and establish the nonsingularity of the Newton system and the local convergence of the algorithm.     

An elementary proof of an equivalence theorem relevant in the theory of optimization

A corrigendum in Ref. 1 is given.     

An elementary proof of an equivalence theorem relevant in the theory of optimization

The authors give an elementary proof of an equivalence theorem of analysis which is often used in optimization theory. The theorem asserts that certain conditions are equivalent to weak convergence inL ₁. One is the Dunford-Pettis condition concerning absolute integrability. Two others are expressed in terms of Nagumo functions, and can be thought of as growth properties. The original proofs of the various parts of the theorem are scattered in different and specialized mathematical publications. The authors feel it useful to present here a straightforward proof of the various parts in terms of standard Lebesgue integration theory.     

Analysis and optimization of the rectilinear motion of a two-body system

The rectilinear motion of a system of two interacting bodies when there is a dry friction force acting on both of them is considered. It is assumed that the relative velocity of the bodies can vary practically instantaneously, while the distance between them has upper and lower limits. The periodic motion of the system as a whole is constructed, and the mean velocity of motion and the energy costs per unit of path are determined. The optimum values of the parameters for which the highest mean velocity is reached with the superimposed limitations are obtained.     

Switched system modeling and robust steering control of the tail end phase in a hot strip mill

In this article, a robust steering control for the last phase of the rolling process in a hot strip mill is proposed. This phase, called tail end phase, may be modelled as a linear switched system. The switchings make the system unstable and the task of the tail end steering control consists in guaranteeing the safety of the industrial plant. The system involves a two time scales dynamics. Hence, the singular perturbation method is used in order to design the control law. The controller has to take into account the physical variations of the rolled products and an uncertainty in the switching time. Results concerning the ArcelorMittal hot strip mill of Eisenhüttenstadt are presented.     

A heuristic algorithm for the optimization of a retrial system with Bernoulli vacation

In this study, we consider an M/M/c retrial queue with Bernoulli vacation under a single vacation policy. When an arrived customer finds a free server, the customer receives the service immediately; otherwise the customer would enter into an orbit. After the server completes the service, the server may go on a vacation or become idle (waiting for the next arriving, retrying customer). The retrial system is analysed as a quasi-birth-and-death process. The sufficient and necessary condition of system equilibrium is obtained. The formulae for computing the rate matrix and stationary probabilities are derived. The explicit close forms for system performance measures are developed. A cost model is constructed to determine the optimal values of the number of servers, service rate, and vacation rate for minimizing the total expected cost per unit time. Numerical examples are given to demonstrate this optimization approach. The effects of various parameters in the cost model on system performance are investigated.     

A new convergence proof of the Adomian decomposition method for a mixed hyperbolic elliptic system of conservation laws

In this paper, we propose a new convergence proof of the Adomian’s decomposition method (ADM), applied to the generalized nonlinear system of partial differential equations (PDE’s) based on new formula for Adomian polynomials. The decomposition scheme obtained from the ADM yields an analytical solution in the form of a rapidly convergent series for a system of conservation laws. Systems of conservation laws is presented, we obtain the stability of the approximate solution when the system changes type. We show with an explicit example that the latter property is true for general Cauchy problem satisfying convergence hypothesis. The results indicate that the ADM is effective and promising.