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Ítem Analytical solution for transient flow of a generalized bingham fluid with memory in a movable tube using computer algebra(SPRINGER, 2007-01-01) Ospina, Juan; Velez, Mario; Ospina, Juan; Velez, Mario; Universidad EAFIT. Departamento de Ciencias; Lógica y ComputaciónA rheological linear model for a certain generalized Bingham fluid with rheological memory, which flows in a movable tube is proposed and analytically solved. The model is a system of two linear and coupled partial differential equations with integral memory. We apply the Laplace transform method making the inverse transform by means of the Bromwich integral and the theorem of residues and the analytical solution are obtained using computer algebra. We deduce the explicit forms of the velocity and stress profiles for the generalized Bingham fluid in terms of Bessel and Struve functions. Various limit cases are obtained and the standard Hagen-Poiseuille and Buckingham-Reiner equations are recovered from more general equations. This works shows the powerful of Maple to solve complex rheological problems in an analytical form as it is presented here by the first time. © Springer-Verlag Berlin Heidelberg 2007.Ítem Basic reproductive rate of a spatial epidemic model using computer algebra software(2005-01-01) Doracelly Hincapié, P.; Juan Ospina, G.; Doracelly Hincapié, P.; Juan Ospina, G.; Universidad EAFIT. Departamento de Ciencias; Lógica y ComputaciónUsing computer algebra software we obtain the basic reproductive rate corresponding to the propagation of a directly transmitted disease in a circular habitat when the disease is endemic at the boundary. The method used is the Laplace Transform Technique and calculus of residues. The results that were obtained include both the explicit form of the R0 for the boundary condition that was considered, as the explicit symbolic solution of the model equation. The method that was used can be extended to other more complex problems such as indirectly transmitted diseases with one or more intermediary hosts or effects of genetic, immunological, geographical or social heterogeneity in the human population. This application indicates that the computer algebra software for symbolic computation has a very promissory future in mathematical epidemiology.Ítem Biomedical computer vision using computer algebra: Analysis of a case of rhinocerebral mucormycosis in a diabetic boy(Springer Science + Business Media, 2010-01-01) Vélez, M.; Ospina, J.; Vélez, M.; Ospina, J.; Universidad EAFIT. Departamento de Ciencias; Lógica y ComputaciónComputer algebra is applied to biomedical computer vision. Specifically certain biomedical images resulting from a case of rhinocerebral mucormysocis in a diabetic boy are analyzed using techniques in computational geometry and in algebraic-geometric topology. We apply convolution and deblurring via diffusion equation from the side of computational geometry and knot theory, graph theory and singular homology form the side of algebraic-geometric topology. Our strategy consists in to represent the biomedical images using algebraic structures in such way that the peculiarities of the images are represented using algebraic complexities. With our strategy we obtain an automatic procedure for the analysis and the diagnostic based on biomedical images. © 2010 Springer-Verlag Berlin Heidelberg.Ítem CADI- cálculo diferencial asistido por computador(Universidad EAFIT, 1990) López Ortiz, León; Universidad EAFITÍtem Mathematical model for dengue with three states of infection(SPIE-INT SOC OPTICAL ENGINEERING, 2012-01-01) Hincapie, D.; Ospina, J.; Hincapie, D.; Ospina, J.; Universidad EAFIT. Departamento de Ciencias; Lógica y ComputaciónA mathematical model for dengue with three states of infection is proposed and analyzed. The model consists in a system of differential equations. The three states of infection are respectively asymptomatic, partially asymptomatic and fully asymptomatic. The model is analyzed using computer algebra software, specifically Maple, and the corresponding basic reproductive number and the epidemic threshold are computed. The resulting basic reproductive number is an algebraic synthesis of all epidemic parameters and it makes clear the possible control measures. The microscopic structure of the epidemic parameters is established using the quantum theory of the interactions between the atoms and radiation. In such approximation, the human individual is represented by an atom and the mosquitoes are represented by radiation. The force of infection from the mosquitoes to the humans is considered as the transition probability from the fundamental state of atom to excited states. The combination of computer algebra software and quantum theory provides a very complete formula for the basic reproductive number and the possible control measures tending to stop the propagation of the disease. It is claimed that such result may be important in military medicine and the proposed method can be applied to other vector-borne diseases. © 2012 SPIE.Ítem Mathematical modeling of Chikungunya fever control(SPIE-INT SOC OPTICAL ENGINEERING, 2015-01-01) Hincapie-Palacio, Doracelly; Ospina, Juan; Hincapie-Palacio, Doracelly; Ospina, Juan; Universidad EAFIT. Departamento de Ciencias; Lógica y ComputaciónChikungunya fever is a global concern due to the occurrence of large outbreaks, the presence of persistent arthropathy and its rapid expansion throughout various continents. Globalization and climate change have contributed to the expansion of the geographical areas where mosquitoes Aedes aegypti and Aedes albopictus (Stegomyia) remain. It is necessary to improve the techniques of vector control in the presence of large outbreaks in The American Region. We derive measures of disease control, using a mathematical model of mosquito-human interaction, by means of three scenarios: a) a single vector b) two vectors, c) two vectors and human and non-human reservoirs. The basic reproductive number and critical control measures were deduced by using computer algebra with Maple (Maplesoft Inc, Ontario Canada). Control measures were simulated with parameter values obtained from published data. According to the number of households in high risk areas, the goals of effective vector control to reduce the likelihood of mosquito-human transmission would be established. Besides the two vectors, if presence of other non-human reservoirs were reported, the monthly target of effective elimination of the vector would be approximately double compared to the presence of a single vector. The model shows the need to periodically evaluate the effectiveness of vector control measures. © 2015 SPIE.Ítem MultiZ: A Library for Computation of High-order Derivatives Using Multicomplex or Multidual Numbers(Association for Computing Machinery (ACM), 2020-01-01) Aguirre-Mesa A.M.; Garcia M.J.; Millwater H.; Mecánica AplicadaMulticomplex and multidual numbers are two generalizations of complex numbers with multiple imaginary axes, useful for numerical computation of derivatives with machine precision. The similarities between multicomplex and multidual algebras allowed us to create a unified library to use either one for sensitivity analysis. This library can be used to compute arbitrary order derivates of functions of a single variable or multiple variables. The storage of matrix representations of multicomplex and multidual numbers is avoided using a combination of one-dimensional resizable arrays and an indexation method based on binary bitwise operations. To provide high computational efficiency and low memory usage, the multiplication of hypercomplex numbers up to sixth order is carried out using a hard-coded algorithm. For higher hypercomplex orders, the library uses by default a multiplication method based on binary bitwise operations. The computation of algebraic and transcendental functions is achieved using a Taylor series approximation. Fortran and Python versions were developed, and extensions to other languages are self-evident. © 2020 ACM.Ítem Possible quantum algorithm for the Lipshitz-Sarkar-Steenrod square for Khovanov homology(SPIE-INT SOC OPTICAL ENGINEERING, 2013-01-01) Ospina, Juan; Ospina, Juan; Universidad EAFIT. Departamento de Ciencias; Lógica y ComputaciónRecently the celebrated Khovanov Homology was introduced as a target for Topological Quantum Computation given that the Khovanov Homology provides a generalization of the Jones polynomal and then it is possible to think about of a generalization of the Aharonov.-Jones-Landau algorithm. Recently, Lipshitz and Sarkar introduced a space-level refinement of Khovanov homology. which is called Khovanov Homotopy. This refinement induces a Steenrod square operation Sq?2 on Khovanov homology which they describe explicitly and then some computations of Sq?2 were presented. Particularly, examples of links with identical integral Khovanov homology but with distinct Khovanov homotopy types were showed. In the presente work we will introduce possible quantum algorithms for the Lipshitz-Sarkar-Steenrod square for Khovanov Homolog and their possible simulations using computer algebra. © 2013 SPIE.Ítem Una primera lección de Geometría algebraica(Universidad EAFIT, 2005) Cadavid, Carlos; Universidad EAFIT. Departamento de Ingeniería Mecánica; Laboratorio CAD/CAM/CAEEn este artículo se explica cómo aparece la Geometría Algebraica, partiendo del estudio de los conjuntos de soluciones de sistemas algebraicosÍtem Solving stochastic epidemiological models using computer algebra(SPIE-INT SOC OPTICAL ENGINEERING, 2011-01-01) Hincapie, D.; Ospina, J.; Hincapie, D.; Ospina, J.; Universidad EAFIT. Departamento de Ciencias; Lógica y ComputaciónMathematical modeling in Epidemiology is an important tool to understand the ways under which the diseases are transmitted and controlled. The mathematical modeling can be implemented via deterministic or stochastic models. Deterministic models are based on short systems of non-linear ordinary differential equations and the stochastic models are based on very large systems of linear differential equations. Deterministic models admit complete, rigorous and automatic analysis of stability both local and global from which is possible to derive the algebraic expressions for the basic reproductive number and the corresponding epidemic thresholds using computer algebra software. Stochastic models are more difficult to treat and the analysis of their properties requires complicated considerations in statistical mathematics. In this work we propose to use computer algebra software with the aim to solve epidemic stochastic models such as the SIR model and the carrier-borne model. Specifically we use Maple to solve these stochastic models in the case of small groups and we obtain results that do not appear in standard textbooks or in the books updated on stochastic models in epidemiology. From our results we derive expressions which coincide with those obtained in the classical texts using advanced procedures in mathematical statistics. Our algorithms can be extended for other stochastic models in epidemiology and this shows the power of computer algebra software not only for analysis of deterministic models but also for the analysis of stochastic models. We also perform numerical simulations with our algebraic results and we made estimations for the basic parameters as the basic reproductive rate and the stochastic threshold theorem. We claim that our algorithms and results are important tools to control the diseases in a globalized world. © 2011 Copyright Society of Photo-Optical Instrumentation Engineers (SPIE).Ítem Symbolic solution for generalized quantum cylindrical wells using computer algebra(2008-01-01) Pulgarin, E.Y.V.; Pulgarin, E.Y.V.; Universidad EAFIT. Departamento de Ciencias; Lógica y ComputaciónThis paper present how to solve the problem of cylindrical quantum wells with potential energy different from zero and with singularity of the energy on the axis of the cylinder. The solution to the problem was obtained using methods of computer algebra. The results depend of Bessel and Kummer functions. This paper present energy levels and wave functions in some of the cases with an exactly form and in other cases with an approximated form, this form depended on the possibility of integrating the special functions and calculating the zeros of these functions. Here we can see the power of the method in the applications concerning complex problems of quantum mechanics, and the possibility of being able to apply this method in order to solve other problems in science and also in engineering.Ítem Topological and geometrical quantum computation in cohesive Khovanov homotopy type theory(SPIE-INT SOC OPTICAL ENGINEERING, 2015-05-21) Ospina, Juan; Ospina, Juan; Universidad EAFIT. Departamento de Ciencias; Lógica y ComputaciónThe recently proposed Cohesive Homotopy Type Theory is exploited as a formal foundation for central concepts in Topological and Geometrical Quantum Computation. Specifically the Cohesive Homotopy Type Theory provides a formal, logical approach to concepts like smoothness, cohomology and Khovanov homology; and such approach permits to clarify the quantum algorithms in the context of Topological and Geometrical Quantum Computation. In particular we consider the so-called a "open-closed stringy topological quantum computera" which is a theoretical topological quantum computer that employs a system of open-closed strings whose worldsheets are open-closed cobordisms. The open-closed stringy topological computer is able to compute the Khovanov homology for tangles and for hence it is a universal quantum computer given than any quantum computation is reduced to an instance of computation of the Khovanov homology for tangles. The universal algebra in this case is the Frobenius Algebra and the possible open-closed stringy topological quantum computers are forming a symmetric monoidal category which is equivalent to the category of knowledgeable Frobenius algebras. Then the mathematical design of an open-closed stringy topological quantum computer is involved with computations and theorem proving for generalized Frobenius algebras. Such computations and theorem proving can be performed automatically using the Automated Theorem Provers with the TPTP language and the SMT-solver Z3 with the SMT-LIB language. Some examples of application of ATPs and SMT-solvers in the mathematical setup of an open-closed stringy topological quantum computer will be provided. © 2015 SPIE.Ítem Two qutrits universal quantum gates from the nine-dimensional unitary solutions of the Yang-Baxter equation(SPIE-INT SOC OPTICAL ENGINEERING, 2007-04-25) Vélez, M.; Ospina, J.; Vélez, M.; Ospina, J.; Universidad EAFIT. Departamento de Ciencias; Lógica y ComputaciónUsing the Kauffman-Lomonaco method, some two-qutrits universal quantum gates are derived from the nine-dimensional unitary solutions of the Yang-Baxter equations associated with algebraic structures like the partial transpose operator and the dihedral group, which admit three dimensional representations. The Yang-Baxterization method given by Zhang-Kauffman-Ge is continuously used to obtain two-qutrits quantum gates and certain Hamiltonians for the evolution of the quantum gates are obtained, being such Hamiltonians interpreted as physical Hamiltonians of chain of particles of spin 1. Finally, the generalization for systems of two qudits is presented in the case of Yang-Baxterization of representations of braided monoidal algebra like the BH algebra and the bicolored Birman-Wenzl-Muraki algebra For these algebras the corresponding two-qudits quantum gates are constructed jointly with the associated Hamiltonians interpreted like physical chains of particles with spin d . It is conjectured that the derived two-qdits quantum gates and the Hamiltonians may be implemented over bi-dimensional lattice systems like anyons systems or more generally over any physical systems ruled by the Yang-Baxter equations.Ítem Universal quantum gates Via Yang-baxterization of dihedral quantum double(SPRINGER, 2007-01-01) Velez, Mario; Ospina, Juan; Velez, Mario; Ospina, Juan; Universidad EAFIT. Departamento de Ciencias; Lógica y ComputaciónThe recently discovered Yang-Baxterization process for the quantum double of the dihedral group algebra, is presented keeping on mind the quantum computation. The products resultant from Yang-Baxterization process are interpreted as universal quantum gates using the Bryslinski's theorem. Results are obtained for two-qubits and two-qutrits gates. Using the Zhang-Kauffman-Ge method (ZKGM), certain Hamiltonians responsible for the quantum evolution of the quantum gates are obtained. Possible physical systems such as anyons systems are mentioned as referents for practical implementation. © Springer-Verlag Berlin Heidelberg 2007.