We study a nonlinear system of partial differential equations arising in macroeconomics which utilizes a mean field approximation. This system together with the corresponding data, subject to two moment constraints, is a model for debt and wealth across a large number of similar households, and was introduced in a recent paper of Achdou et al. (Philos Trans R Soc Lond Ser A 372(2028):20130397, 2014). We introduce a relaxation of their problem, generalizing one of the moment constraints; any solution of the original model is a solution of this relaxed problem. We prove existence and uniqueness of strong solutions to the relaxed problem, under the assumption that the time horizon is small. Since these solutions are unique and since solutions of the original problem are also solutions of the relaxed problem, we conclude that if the original problem does have solutions, then such solutions must be the solutions we prove to exist. Furthermore, for some data and for sufficiently small time horizons, we are able to show that solutions of the relaxed problem are in fact not solutions of the original problem. In this way we demonstrate nonexistence of solutions for the original problem in certain cases.
more »
« less
On Incorporating Search-Based Heuristics into Real-World Systems
While massive strides have been made in the field of search-based software testing (SBST) in recent years, there yet remains the problem of transitioning such techniques to reality. This paper discusses this problem in terms of cyber-physical systems, presents research challenges for applying SBST to this domain, and outlines the state-of-the-art achievements of the SBST community in this regard.
more »
« less
- Award ID(s):
- 1657061
- NSF-PAR ID:
- 10092142
- Date Published:
- Journal Name:
- 12th Intl. Workshop on Search-Based Software Testing
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Problem solvers vary their approaches to solving problems depending on the context of the problem, the requirements of the solution, and the ways in which the problems and material to solve the problem are represented, or representations. Representations take many forms (i.e. tables, graphs, figures, images, formulas, visualizations, and other similar contexts) and are used to communicate information to a problem solver. Engagement with certain representations varies between problem solvers and can influence design and solution quality. A problem solver’s evaluation of representations and the reasons for using a representation can be considered factors in problem-solving heuristics. These factors describe unique problem-solving behaviors that can help understand problem solvers. These behaviors may lead to important relationships between a problem solver’s decisions and their ability to solve a problem and overall quality of the solution. Therefore, we pose the following research question: How do factors of problem-solving heuristics describe the unique behaviors of engineering students as they solve multiple problems? To answer this question, we interviewed 16 undergraduate engineering students studying civil engineering. The interviews consisted of a problem-solving portion that was followed immediately by a semi-structured retrospective interview with probing questions created based on the real time monitoring of the problem-solving interview using eye tracking techniques. The problem-solving portion consisted of solving three problems related to the concept of headloss in fluid flow through pipes. Each of the three problems included the same four representations that were used by the students as approaches to solving the problem. The representations are common ways to present the concept of headloss in pipe flow and included two formulas, a set of tables, and a graph. This paper presents a set of common reasons for why decisions were made during the problem-solving process that help to understand more about the problem-solving behavior of engineering students.more » « less
-
Mulzer, Wolfgang ; Phillips, Jeff M (Ed.)We study a fundamental problem in Computational Geometry, the planar two-center problem. In this problem, the input is a set S of n points in the plane and the goal is to find two smallest congruent disks whose union contains all points of S. A longstanding open problem has been to obtain an O(nlog n)-time algorithm for planar two-center, matching the Ω(nlog n) lower bound given by Eppstein [SODA'97]. Towards this, researchers have made a lot of efforts over decades. The previous best algorithm, given by Wang [SoCG'20], solves the problem in O(nlog² n) time. In this paper, we present an O(nlog n)-time (deterministic) algorithm for planar two-center, which completely resolves this open problem.more » « less
-
Problem posing engages students in generating new problems based on given situations (including mathematical expressions or diagrams) or changing (i.e., reformulating) existing problems. Problem posing has been at the forefront of discussion over the past few decades. One of the important topics studied is the process of problem posing as experienced by students and teachers. This paper focuses on problem-posing processes and models thereof. We first provide an overview of previous research and then present the results of a scoping review regarding recent research on problem-posing processes. This review covers 75 papers published between 2017 and 2022 in top mathematics education research journals. We found that some of the prior research directly attempted to examine problem-posing processes, whereas others examined task variables related to problem-posing processes. We conclude this paper by proposing a model for problem-posing processes that encompasses four phases: orientation, connection, generation, and reflection. We also provide descriptions of the four phases of the model. The paper ends with suggestions for future research related to problem-posing processes in general and the problem-posing model proposed in particular.more » « less
-
In this paper we establish the convergence of a numerical scheme based, on the Finite Element Method, for a time-independent problem modelling the deformation of a linearly elastic elliptic membrane shell subjected to remaining confined in a half space. Instead of approximating the original variational inequalities governing this obstacle problem, we approximate the penalized version of the problem under consideration. A suitable coupling between the penalty parameter and the mesh size will then lead us to establish the convergence of the solution of the discrete penalized problem to the solution of the original variational inequalities. We also establish the convergence of the Brezis-Sibony scheme for the problem under consideration. Thanks to this iterative method, we can approximate the solution of the discrete penalized problem without having to resort to nonlinear optimization tools. Finally, we present numerical simulations validating our new theoretical results.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Conference Paper:
- The DOI is not currently available.
