Observaciones sobre la matemática aplicada

Nombrado Doctor Honoris Causa en el acto de Apertura del Curso 95/9

Multiple eigenvalues

The dimensions of sets of matrices of various types, with specified eigenvalue multiplicities, are determined. The dimensions of the sets of matrices with given Jordan form and with given singular value multiplicities are also found. Each corresponding codimension is the number of conditions which a matrix of the given type must satisfy in order to have the specified multiplicities.Comment: 19 pages, 1 figure, 2 table

Iterative solution of elliptic problems by approximate factorization

AbstractAn iterative method for the numerical solution of singularly perturbed second-order linear elliptic problems is presented. It is a defect correction iteration in which the approximate operator is the product of two first-order operators, which is readily inverted numerically. The approximate operator is generated by formal asymptotic factorization of the original operator. Hence this is a QUasi Analytic Defect correction iteration (QUAD). Both its continuous and discrete versions are analyzed in one dimension. The scheme is extended to a variety of two dimensional operators and it is analyzed for a model advection-diffusion equation. Numerical calculations show the effectiveness of the scheme over a wide range of values of the small parameter

Which Way Was I Going? Contextual Retrieval Supports the Disambiguation of Well Learned Overlapping Navigational Routes

Groundbreaking research in animals has demonstrated that the hippocampus contains neurons that distinguish betweenoverlapping navigational trajectories. These hippocampal neurons respond selectively to the context of specific episodes despite interference from overlapping memory representations. The present study used functional magnetic resonanceimaging in humans to examine the role of the hippocampus and related structures when participants need to retrievecontextual information to navigate well learned spatial sequences that share common elements. Participants were trained outside the scanner to navigate through 12 virtual mazes from a ground-level first-person perspective. Six of the 12 mazes shared overlapping components. Overlapping mazes began and ended at distinct locations, but converged in the middle to share some hallways with another maze. Non-overlapping mazes did not share any hallways with any other maze. Successful navigation through the overlapping hallways required the retrieval of contextual information relevant to thecurrent navigational episode. Results revealed greater activation during the successful navigation of the overlapping mazes compared with the non-overlapping mazes in regions typically associated with spatial and episodic memory, including thehippocampus, parahippocampal cortex, and orbitofrontal cortex. When combined with previous research, the current findings suggest that an anatomically integrated system including the hippocampus, parahippocampal cortex, and orbitofrontal cortexis critical for the contextually dependent retrieval of well learned overlapping navigational routes