A.M. Simões1,2,3 and L.P. Castro3,4
1CMA-UBI Centro de Matemática e Aplicações da Universidade da Beira Interior,
Universidade da Beira Interior, Portugal
2Departamento de Matemática, Universidade da Beira Interior, Portugal 3CIDMA Centro de Investigação e Desenvolvimento em Matemática e Aplicações,
Universidade de Aveiro, Portugal
4Departamento de Matemática, Universidade de Aveiro, Portugal
E-mail: [email protected]
Abstract
This talk is devoted to present the σ-semi-Hyers-Ulam stability for higher order integro-dierential equations within appropriate metric spaces. We will show that σ-semi-Hyers-Ulam stability is a new kind of stability somehow be- tween the Hyers-Ulam and the Hyers-Ulam-Rassias stabilities. Su cient con- ditions are obtained in view to guarantee Hyers-Ulam, σ-semi-Hyers-Ulam and Hyers-Ulam-Rassias stabilities for such a class of higher order integro- dierential equations. We will be considering nite intervals as integration domains to de ne the higher order integro-dierential equations. Among the used techniques, we have xed point arguments and generalizations of the Bielecki metric. Some examples of the application of the proposed theory will be included.
Keywords: Hyers-Ulam stability, σ-semi-Hyers-Ulam stability, Hyers-Ulam- Rassias stability, Banach xed point theorem, Bielecki metric, higher order integro-dierential equations, nonlinear integral equation.
Acknowledgements
This work was supported in part by FCTPortuguese Foundation for Sci- ence and Technology through the Center for Research and Development in Mathematics and Applications (CIDMA) of University of Aveiro, within UID- MAT-04106-2013, and through the Center of Mathematics and Applications of University of Beira Interior (CMA-UBI), within project UID-MAT-00212- 2013.
22 Contributed Talks
References
[1] Castro, L.P. and Guerra, R.C., (2013) Hyers-Ulam-Rassias stability of Volterra inte- gral equations within weighted spaces, Lib. Math. (N.S.), 33(2), pp. 2135.
[2] Castro, L.P. and Ramos, A., (2010) Hyers-Ulam and Hyers-Ulam-Rassias stability of Volterra integral equations with a delay, Integral Methods in Science and Engineering, 1, pp. 8594.
[3] Castro, L.P. and Ramos, A., (2009) Hyers-Ulam-Rassias stability for a class of non- linear Volterra integral equations, Banach J. Math. Anal., 3(1), pp. 3643.
[4] Castro, L.P. and Ramos, A., (2011) Hyers-Ulam stability for a class of Fredholm inte- gral equations, Mathematical Problems in Engineering Aerospace and Sciences ICN- PAA 2010, Proceedings of the 8th International Conference of Mathematical Problems in Engineering, Aerospace and Science, pp. 171176.
[5] Castro, L.P. and Simões, A.M., (2017) A New Type of Stability: semi-Hyers-Ulam- Rassias Stability, Book of Abstracts: IV Workshop on Computational Data Analysis and Numerical Methods, pp. 42.
[6] Castro, L.P. and Simões, A.M., (2017) Dierent types of Hyers-Ulam-Rassias Stabil- ities for a class of integro-dierential equations, Filomat, 31(17), pp. 53795390. [7] Castro, L.P. and Simões, A.M., (2018) Hyers-Ulam and Hyers-Ulam-Rassias stabil-
ity for a class of integro-dierential equations, Mathematical Methods in Engineer- ing: Theoretical Aspects, Tas, K. and Baleanu, D. and Tenreiro Machado, J.A. Eds, Springer (to appear).
[8] Castro, L.P. and Simões, A.M., (2017) Hyers-Ulam and Hyers-Ulam-Rassias sta- bility of a class of Hammerstein integral equations, AIP Conference Proceedings, 1798:020036-1/10.
[9] Castro, L.P. and Simões, A.M., (2017) Hyers-Ulam and Hyers-Ulam-Rassias stability of a class of integral equations on nite intervals, CMMSE'17: Proceedings of the 17th International Conference on Computational and Mathematical Methods in Science and Engineering, vol. I-IV, pp. 507515.
[10] Castro, L.P. and Simões, A.M., (2018) Hyers-Ulam-Rassias Stability of Nonlinear Integral Equations Through the Bielecki Metric, Mathematical Methods in the Applied Sciences, (to appear).
Contributed Talks 23
An e cient software for packing boxes in pallets
Ana Moura1, Isabel Cristina Lopes2, Stella Abreu1 and Manuel Cruz3 1LEMA, ISEP, Politécnico do Porto, and CMUP, Portugal
2LEMA, CEOS.PP, ISCAP, Politécnico do Porto, Portugal 3LEMA, ISEP, Politécnico do Porto, Portugal
E-mail: [email protected]
Abstract
In this work, we will describe a software developed to solve an industrial packing problem related to the single container loading problem and the pallet loading problem ([1]): given a set of 3D rectangular boxes of dierent types, the objective is to store the boxes into containers or pallets, as eec- tively as possible, minimizing the number of necessary containers or pallets. Our scenario considers boxes with fragile content, usually marked with the label This side up, to be packed in pallets. Some of the boxes allow rota- tion, others do not. The cargo is weakly heterogeneous, i.e., the assortment of boxes is small, usually less than half a dozen dierent references. The stability of the boxes is also considered. Given the speci cities of our sce- nario, we proposed a tailor-made greedy heuristic, and we implemented it in Matlab. The software developed nds packing solutions, which are easy to understand and execute by a non-expert, for example a worker of some company. The graphical user interface gives a 3D visualization of the boxes that allows rotation of the 3D pallet or container, to see it from all angles. In practice, workers can usually place boxes exceeding the limits of the pal- lets for some centimetres. Our software compares the solutions, considering a chosen tolerance. The heuristic works best for packing into pallets, but it also works for containers of any size. We will present the results of our heuristic and will compare them with other results from the literature [2-5], using a benchmark test set.
Keywords: three dimensional packing, heuristics, container loading, pallet loading, industrial mathematics.
Acknowledgements
This work was partially funded by LEMA (Engineering Mathematical Lab- oratory).
24 Contributed Talks
References
[1] G. Wäscher, H. Hauÿner, H. Schumann, (2007) An improved typology of cutting and packing problems, European Journal of Operational Research, 183, pp. 11091130. [2] E.E. Bischo and M.S.W. Ratcli, (1995) Issues in the development of approaches to
container loading, OMEGA, International Journal of Management Science, 23(4), pp. 377390.
[3] C.H. Che, W. Huang, A. Lim and W. Zhu, (2011) The multiple container loading cost minimization problem, European Journal of Operational Research, 214(3), pp. 501511.
[4] N. Ivancic, K. Mathur and B.B. Mohanty, (1989) An integer-programming based heuristic approach to the three-dimensional packing problem, Journal of Manufac- turing and Operations Management, 2, pp. 268298.
[5] R. Morabito and M. Arenales, (1994) An AND/OR-graph Approach to the Container Loading Problem, International Transactions in Operational Research, 1, pp. 59-73.
Contributed Talks 25
Inference for structured family
Cristina Dias1, Carla Santos2 and João T. Mexia3
1Escola Superior de Tecnologia e Gestão do Instituto Politécnico de Portalegre e Centro
de Matemática e Aplicações da Universidade Nova de Lisboa (CMA), Portugal
2Departamento de Matemática e Ciências Físicas do Instituto Politécnico de Beja e
Centro de Matemática e Aplicações da Universidade Nova de Lisboa (CMA), Portugal
3Departamento de Matemática da Faculdade de Ciências e Tecnologia e Centro de
Matemática e Aplicações da Universidade Nova de Lisboa (CMA), Portugal
E-mail: [email protected]
Abstract
The matrices of a structured family of stochastic symmetric matrices are all of the same order k and correspond to the treatments of a base design. The most interesting case is when the matrices in the family have a dominant rst eigenvalue λ1 see [2]. We then study the action of the factors in the base
design, on the components of the rst structure vector λ1α1with α1 the rst
eigenvector. When the matrices in such families correspond to the treatments of a base design we can carry out ANOVA like analysis of the action of the treatments in the model on the structured vectors see [1] and [3]. This analysis can be transversal when we worked width homologous components and longitudinal when we consider contrast on the components of each structure vector. In this work we consider the models for these matrices and show how to carry out inference for structured family.
Keywords: symmetric matrices, ANOVA like analysis, longitudinal and transversal analysis.
Acknowledgements
This work was partially supported by the Fundação para a Ciência e a Tec- nologia (Portuguese Foundation for Science and Technology) through the project UID-MAT-00297-2013 (Centro de Matemática e Aplicações).
References
[1] Ito, P. K., (1980) Robustness of Anova and Macanova Test Procedures, P. R. Krish- naiah (ed), Handbook of Statistics, Amsterdam, North Holland, V1, pp. 199236. [2] Dias, C., (2013) Modelos e Famílias de Modelos para Matrizes Estocásticas Simétricas,
Ph.D. Thesis, Évora University.
26 Contributed Talks