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The cells of the sinoatrial node (SAN) are self-excitable entities that show a coupled electrical pattern that consists of the synchronized activation of action potentials that determine the heart rate. To accurately describe the behavior of cell membrane proteins, theoretical biophysicists have devoted themselves to the study of the electrical activity of individual cells, which involves solving a large number of coupled differential equations. This computational limitation makes the modeling of a large number of cells unattainable, since the intracellular distribution of Ca 2+ must be considered and this fact increases in grand extent the number of differential equations involved. In this work, we explore different parallel architectures (using OpenMP, MPI, and CUDA libraries) to show advances in the computational modeling and simulation of the SAN using a multicellular array in which the cells are endowed of heterogeneous conductances and are electrically coupled, considering a variable connectivity among them.

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We prove the existence of multiple nontrivial solutions for the semilinear elliptic problem −∆u = h (λu + g(u)) in R N , u ∈ D 1,2 , where h ∈ L 1 ∩ L α for α > N/2, N 3, g is a C 1 (R, R) function that has at most linear growth at infinity, g(0) = 0, and λ is an eigenvalue of the corresponding linear problem −∆u = λhu in R N , u ∈ D 1,2 . Existence of multiple solutions, for certain values of g (0), is obtained by imposing a generalized Landesman-Lazer type condition. We use the saddle point theorem of Ambrosetti and Rabinowitz and the mountain pass theorem, as well as a Morse-index result of Ambrosetti [A. Ambrosetti, Differential Equations with Multiple Solutions and Nonlinear Functional Analysis, Equadiff 82, Lecture Notes in Math., vol. 1017, Springer-Verlag, Berlin, 1983] and a Leray-Schauder index theorem for mountain pass type critical points due to Hofer [H. Hofer, A note on the Topological Degree at a critical Point of Mountain Pass Type, Proc. Amer. Math. Soc. 90 (1984) 309-315]. The results of this paper are based upon multiplicity results for resonant problems on bounded domains in [E. Landesman, S. Robinson, A. Rumbos, Multiple solutions of semilinear elliptic problems at resonance, Nonlinear Anal. 24 (1995) 1049-1059] and [S. Robinson, Multiple solutions for semilinear elliptic boundary value problems at resonance, Electron. J. Differential Equations 1995 (1995) 1-14], and complement a previous existence result by the authors in [G. López Garza, A. Rumbos, Resonance and strong resonance for semilinear elliptic equations in R N , Electron.

Resumen. En varios libros de texto la demostración basada en las ideas de Dandelin del teorema atribuido a Apolonio que establece que una cónica resulta de la intersección de un plano con un cono circular recto utiliza geometría sintética y la definición mediante la excentricidad de una cónica. Aquí presentamos una demostración que usa solo la definición elemental de cónica mediante distancias entre puntos con cálculos e imágenes realizadas dentro del contexto de la Geometría Analítica.