Multidisciplinary Learning, Another way of Teaching Maths in Engineering

Multidisciplinary training is widely appreciated in industry and business, and nevertheless usually is not addressed in the early stages of most undergraduate programs. We outline here a multidisciplinary course for undergraduates studying engineering in which mathematics would be the common language, the transverse tool. The goal is motivating students to learn more mathematics and as a result, improve the quality of engineering education. The course would be structured around projects in four branches in engineering: mechanical, electrical, civil and bio-tech. The projects would be chosen among a wide variety of topics in engineering practice selected with the guidance of professional engineers. In these projects mathematics should interact with at least two other basic areas of knowledge in engineering: chemistry, computers science, economics, design or physics.

First steps in the discovery of patterns in the academic results of telecommunication engineering students in the subjects Analysis of Circuits and Mathematics

In this paper, the results of six years of research in engineering education, in the application of the European Higher Education Area (EHEA) to improve the performance of the students in the subject Analysis of Circuits of Telecommunication Engineering, are analysed taking into consideration the fact that there would be hidden variables that both separate students into subgroups and show the connection among several basic subjects such as Analysis of Circuits (AC) and Mathematics (Math). The discovery of these variables would help us to explain the characteristics of the students through the teaching and learning methodology, and would show that there are some characteristics that instructors do not take into account but that are of paramount importance

An example of Learning based on competences: Use of Maxima in Linear Algebra for Engineers

This paper analyzes the role of Computer Algebra Systems (CAS) in a model of learning based on competences. The proposal is an e-learning model Linear Algebra course for Engineering, which includes the use of a CAS (Maxima) and focuses on problem solving. A reference model has been taken from the Spanish Open University. The proper use of CAS is defined as an indicator of the generic ompetence: Use of Technology. Additionally, we show that using CAS could help to enhance the following generic competences: Self Learning, Planning and Organization, Communication and Writing, Mathematical and Technical Writing, Information Management and Critical Thinking.

Regularization for sparsity in statistical analysis and machine learning

Pragmatism is the leading motivation of regularization. We can understand regularization as a modification of the maximum-likelihood estimator so that a reasonable answer could be given in an unstable or ill-posed situation. To mention some typical examples, this happens when fitting parametric or non-parametric models with more parameters than data or when estimating large covariance matrices. Regularization is usually used, in addition, to improve the bias-variance tradeoff of an estimation. Then, the definition of regularization is quite general, and, although the introduction of a penalty is probably the most popular type, it is just one out of multiple forms of regularization. In this dissertation, we focus on the applications of regularization for obtaining sparse or parsimonious representations, where only a subset of the inputs is used. A particular form of regularization, L1-regularization, plays a key role for reaching sparsity. Most of the contributions presented here revolve around L1-regularization, although other forms of regularization are explored (also pursuing sparsity in some sense). In addition to present a compact review of L1-regularization and its applications in statistical and machine learning, we devise methodology for regression, supervised classification and structure induction of graphical models. Within the regression paradigm, we focus on kernel smoothing learning, proposing techniques for kernel design that are suitable for high dimensional settings and sparse regression functions. We also present an application of regularized regression techniques for modeling the response of biological neurons. Supervised classification advances deal, on the one hand, with the application of regularization for obtaining a na¨ıve Bayes classifier and, on the other hand, with a novel algorithm for brain-computer interface design that uses group regularization in an efficient manner. Finally, we present a heuristic for inducing structures of Gaussian Bayesian networks using L1-regularization as a filter. El pragmatismo es la principal motivación de la regularización. Podemos entender la regularización como una modificación del estimador de máxima verosimilitud, de tal manera que se pueda dar una respuesta cuando la configuración del problema es inestable. A modo de ejemplo, podemos mencionar el ajuste de modelos paramétricos o no paramétricos cuando hay más parámetros que casos en el conjunto de datos, o la estimación de grandes matrices de covarianzas. Se suele recurrir a la regularización, además, para mejorar el compromiso sesgo-varianza en una estimación. Por tanto, la definición de regularización es muy general y, aunque la introducción de una función de penalización es probablemente el método más popular, éste es sólo uno de entre varias posibilidades. En esta tesis se ha trabajado en aplicaciones de regularización para obtener representaciones dispersas, donde sólo se usa un subconjunto de las entradas. En particular, la regularización L1 juega un papel clave en la búsqueda de dicha dispersión. La mayor parte de las contribuciones presentadas en la tesis giran alrededor de la regularización L1, aunque también se exploran otras formas de regularización (que igualmente persiguen un modelo disperso). Además de presentar una revisión de la regularización L1 y sus aplicaciones en estadística y aprendizaje de máquina, se ha desarrollado metodología para regresión, clasificación supervisada y aprendizaje de estructura en modelos gráficos. Dentro de la regresión, se ha trabajado principalmente en métodos de regresión local, proponiendo técnicas de diseño del kernel que sean adecuadas a configuraciones de alta dimensionalidad y funciones de regresión dispersas. También se presenta una aplicación de las técnicas de regresión regularizada para modelar la respuesta de neuronas reales. Los avances en clasificación supervisada tratan, por una parte, con el uso de regularización para obtener un clasificador naive Bayes y, por otra parte, con el desarrollo de un algoritmo que usa regularización por grupos de una manera eficiente y que se ha aplicado al diseño de interfaces cerebromáquina. Finalmente, se presenta una heurística para inducir la estructura de redes Bayesianas Gaussianas usando regularización L1 a modo de filtro.

Response threshold models, stochastic learning automata and ant colony optimization-based decentralized self-coordination algorithms for heterogeneous multi-tasks distribution in multi-robot systems

In recent decades, there has been an increasing interest in systems comprised of several autonomous mobile robots, and as a result, there has been a substantial amount of development in the eld of Articial Intelligence, especially in Robotics. There are several studies in the literature by some researchers from the scientic community that focus on the creation of intelligent machines and devices capable to imitate the functions and movements of living beings. Multi-Robot Systems (MRS) can often deal with tasks that are dicult, if not impossible, to be accomplished by a single robot. In the context of MRS, one of the main challenges is the need to control, coordinate and synchronize the operation of multiple robots to perform a specic task. This requires the development of new strategies and methods which allow us to obtain the desired system behavior in a formal and concise way. This PhD thesis aims to study the coordination of multi-robot systems, in particular, addresses the problem of the distribution of heterogeneous multi-tasks. The main interest in these systems is to understand how from simple rules inspired by the division of labor in social insects, a group of robots can perform tasks in an organized and coordinated way. We are mainly interested on truly distributed or decentralized solutions in which the robots themselves, autonomously and in an individual manner, select a particular task so that all tasks are optimally distributed. In general, to perform the multi-tasks distribution among a team of robots, they have to synchronize their actions and exchange information. Under this approach we can speak of multi-tasks selection instead of multi-tasks assignment, which means, that the agents or robots select the tasks instead of being assigned a task by a central controller. The key element in these algorithms is the estimation ix of the stimuli and the adaptive update of the thresholds. This means that each robot performs this estimate locally depending on the load or the number of pending tasks to be performed. In addition, it is very interesting the evaluation of the results in function in each approach, comparing the results obtained by the introducing noise in the number of pending loads, with the purpose of simulate the robot's error in estimating the real number of pending tasks. The main contribution of this thesis can be found in the approach based on self-organization and division of labor in social insects. An experimental scenario for the coordination problem among multiple robots, the robustness of the approaches and the generation of dynamic tasks have been presented and discussed. The particular issues studied are: Threshold models: It presents the experiments conducted to test the response threshold model with the objective to analyze the system performance index, for the problem of the distribution of heterogeneous multitasks in multi-robot systems; also has been introduced additive noise in the number of pending loads and has been generated dynamic tasks over time. Learning automata methods: It describes the experiments to test the learning automata-based probabilistic algorithms. The approach was tested to evaluate the system performance index with additive noise and with dynamic tasks generation for the same problem of the distribution of heterogeneous multi-tasks in multi-robot systems. Ant colony optimization: The goal of the experiments presented is to test the ant colony optimization-based deterministic algorithms, to achieve the distribution of heterogeneous multi-tasks in multi-robot systems. In the experiments performed, the system performance index is evaluated by introducing additive noise and dynamic tasks generation over time.

Cryptographic properties of Boolean functions defining elementary cellular automata

In this work, the algebraic properties of the local transition functions of elementary cellular automata (ECA) were analysed. Specifically, a classification of such cellular automata was done according to their algebraic degree, the balancedness, the resiliency, nonlinearity, the propagation criterion and the existence of non-zero linear structures. It is shown that there is not any ECA satisfying all properties at the same time.

The Learning of the Mathematical Models to Aid Decision Making in the High Level Engineering Schools?

At present, in the University curricula in most countries, the decision theory and the mathematical models to aid decision making is not included, as in the graduate program like in Doctored and Master´s programs. In the Technical School of High Level Agronomic Engineers of the Technical University of Madrid (ETSIA-UPM), the need to offer to the future engineers training in a subject that could help them to take decisions in their profession was felt. Along the life, they will have to take a lot of decisions. Ones, will be important and others no. In the personal level, they will have to take several very important decisions, like the election of a career, professional work, or a couple, but in the professional field, the decision making is the main role of the Managers, Politicians and Leaders. They should be decision makers and will be paid for it. Therefore, nobody can understand that such a professional that is called to practice management responsibilities in the companies, does not take training in such an important matter. For it, in the year 2000, it was requested to the University Board to introduce in the curricula an optional qualified subject of the second cycle with 4,5 credits titled " Mathematical Methods for Making Decisions ". A program was elaborated, the didactic material prepared and programs as Maple, Lingo, Math Cad, etc. installed in several IT classrooms, where the course will be taught. In the course 2000-2001 this subject was offered with a great acceptance that exceeded the forecasts of capacity and had to be prepared more classrooms. This course in graduate program took place in the Department of Applied Mathematics to the Agronomic Engineering, as an extension of the credits dedicated to Mathematics in the career of Engineering.

Small Projects: A Method for improving Learning.

Ideas concerning problem-based learning (PBL) developed after running different experiences in different Spanish Universities, are discussed. The driver for introducing PBL has been the requirement for studying Mathematics by the Engineering students. A methodology hybrid of problem-based learning for Mathematics in Engineering studies is proposed. The model is a combination of formal lectures, practical and laboratory sessions with autonomous small projects.

Online games for the teaching of mathematics

In this article we present a didactic experience developed by the GIE (Group of Educational Innovation) “Pensamiento Matemático” of the Polytechnics University of Madrid (UPM), in order to bring secondary students and university students closer to Mathematics. It deals with the development of a virtual board game called Mate-trivial. The mechanics of the game is to win points by going around the board which consists of four types of squares identified by colours: “Statistics and Probability”, “Calculus and Analysis”, “Algebra and Geometry” and “Arithmetic and Number Theory ”. When landing on a square, a question of its category is set out: a correct answer wins 200 points, if wrong it loses 100 points, and not answering causes no effect on the points, but all the same, two minutes out of the 20 minutes that each game lasts are lost. For the game to be over it is necessary, before those 20 minutes run out, to reach the central square and succeed in the final task: four chained questions, one of each type, which must be all answered correctly. It is possible to choose between two levels to play: Level 1, for pre-university students and Level 2 for university students. A prototype of the game is available at the website “Aula de Pensamiento Matemático” developed by the GIE: http://innovacioneducativa.upm.es/pensamientomatematico/. This activity lies within a set of didactic actions which the GIE is developing in the framework of the project “Collaborative Strategies between University and Secondary School Education for the teaching and learning of Mathematics: An Application to solve problems while playing”, a transversal project financed by the UPM.

Contributions to Bayesian network learning with applications to neuroscience

Neuronal morphology is a key feature in the study of brain circuits, as it is highly related to information processing and functional identification. Neuronal morphology affects the process of integration of inputs from other neurons and determines the neurons which receive the output of the neurons. Different parts of the neurons can operate semi-independently according to the spatial location of the synaptic connections. As a result, there is considerable interest in the analysis of the microanatomy of nervous cells since it constitutes an excellent tool for better understanding cortical function. However, the morphologies, molecular features and electrophysiological properties of neuronal cells are extremely variable. Except for some special cases, this variability makes it hard to find a set of features that unambiguously define a neuronal type. In addition, there are distinct types of neurons in particular regions of the brain. This morphological variability makes the analysis and modeling of neuronal morphology a challenge. Uncertainty is a key feature in many complex real-world problems. Probability theory provides a framework for modeling and reasoning with uncertainty. Probabilistic graphical models combine statistical theory and graph theory to provide a tool for managing domains with uncertainty. In particular, we focus on Bayesian networks, the most commonly used probabilistic graphical model. In this dissertation, we design new methods for learning Bayesian networks and apply them to the problem of modeling and analyzing morphological data from neurons. The morphology of a neuron can be quantified using a number of measurements, e.g., the length of the dendrites and the axon, the number of bifurcations, the direction of the dendrites and the axon, etc. These measurements can be modeled as discrete or continuous data. The continuous data can be linear (e.g., the length or the width of a dendrite) or directional (e.g., the direction of the axon). These data may follow complex probability distributions and may not fit any known parametric distribution. Modeling this kind of problems using hybrid Bayesian networks with discrete, linear and directional variables poses a number of challenges regarding learning from data, inference, etc. In this dissertation, we propose a method for modeling and simulating basal dendritic trees from pyramidal neurons using Bayesian networks to capture the interactions between the variables in the problem domain. A complete set of variables is measured from the dendrites, and a learning algorithm is applied to find the structure and estimate the parameters of the probability distributions included in the Bayesian networks. Then, a simulation algorithm is used to build the virtual dendrites by sampling values from the Bayesian networks, and a thorough evaluation is performed to show the model’s ability to generate realistic dendrites. In this first approach, the variables are discretized so that discrete Bayesian networks can be learned and simulated. Then, we address the problem of learning hybrid Bayesian networks with different kinds of variables. Mixtures of polynomials have been proposed as a way of representing probability densities in hybrid Bayesian networks. We present a method for learning mixtures of polynomials approximations of one-dimensional, multidimensional and conditional probability densities from data. The method is based on basis spline interpolation, where a density is approximated as a linear combination of basis splines. The proposed algorithms are evaluated using artificial datasets. We also use the proposed methods as a non-parametric density estimation technique in Bayesian network classifiers. Next, we address the problem of including directional data in Bayesian networks. These data have some special properties that rule out the use of classical statistics. Therefore, different distributions and statistics, such as the univariate von Mises and the multivariate von Mises–Fisher distributions, should be used to deal with this kind of information. In particular, we extend the naive Bayes classifier to the case where the conditional probability distributions of the predictive variables given the class follow either of these distributions. We consider the simple scenario, where only directional predictive variables are used, and the hybrid case, where discrete, Gaussian and directional distributions are mixed. The classifier decision functions and their decision surfaces are studied at length. Artificial examples are used to illustrate the behavior of the classifiers. The proposed classifiers are empirically evaluated over real datasets. We also study the problem of interneuron classification. An extensive group of experts is asked to classify a set of neurons according to their most prominent anatomical features. A web application is developed to retrieve the experts’ classifications. We compute agreement measures to analyze the consensus between the experts when classifying the neurons. Using Bayesian networks and clustering algorithms on the resulting data, we investigate the suitability of the anatomical terms and neuron types commonly used in the literature. Additionally, we apply supervised learning approaches to automatically classify interneurons using the values of their morphological measurements. Then, a methodology for building a model which captures the opinions of all the experts is presented. First, one Bayesian network is learned for each expert, and we propose an algorithm for clustering Bayesian networks corresponding to experts with similar behaviors. Then, a Bayesian network which represents the opinions of each group of experts is induced. Finally, a consensus Bayesian multinet which models the opinions of the whole group of experts is built. A thorough analysis of the consensus model identifies different behaviors between the experts when classifying the interneurons in the experiment. A set of characterizing morphological traits for the neuronal types can be defined by performing inference in the Bayesian multinet. These findings are used to validate the model and to gain some insights into neuron morphology. Finally, we study a classification problem where the true class label of the training instances is not known. Instead, a set of class labels is available for each instance. This is inspired by the neuron classification problem, where a group of experts is asked to individually provide a class label for each instance. We propose a novel approach for learning Bayesian networks using count vectors which represent the number of experts who selected each class label for each instance. These Bayesian networks are evaluated using artificial datasets from supervised learning problems. Resumen La morfología neuronal es una característica clave en el estudio de los circuitos cerebrales, ya que está altamente relacionada con el procesado de información y con los roles funcionales. La morfología neuronal afecta al proceso de integración de las señales de entrada y determina las neuronas que reciben las salidas de otras neuronas. Las diferentes partes de la neurona pueden operar de forma semi-independiente de acuerdo a la localización espacial de las conexiones sinápticas. Por tanto, existe un interés considerable en el análisis de la microanatomía de las células nerviosas, ya que constituye una excelente herramienta para comprender mejor el funcionamiento de la corteza cerebral. Sin embargo, las propiedades morfológicas, moleculares y electrofisiológicas de las células neuronales son extremadamente variables. Excepto en algunos casos especiales, esta variabilidad morfológica dificulta la definición de un conjunto de características que distingan claramente un tipo neuronal. Además, existen diferentes tipos de neuronas en regiones particulares del cerebro. La variabilidad neuronal hace que el análisis y el modelado de la morfología neuronal sean un importante reto científico. La incertidumbre es una propiedad clave en muchos problemas reales. La teoría de la probabilidad proporciona un marco para modelar y razonar bajo incertidumbre. Los modelos gráficos probabilísticos combinan la teoría estadística y la teoría de grafos con el objetivo de proporcionar una herramienta con la que trabajar bajo incertidumbre. En particular, nos centraremos en las redes bayesianas, el modelo más utilizado dentro de los modelos gráficos probabilísticos. En esta tesis hemos diseñado nuevos métodos para aprender redes bayesianas, inspirados por y aplicados al problema del modelado y análisis de datos morfológicos de neuronas. La morfología de una neurona puede ser cuantificada usando una serie de medidas, por ejemplo, la longitud de las dendritas y el axón, el número de bifurcaciones, la dirección de las dendritas y el axón, etc. Estas medidas pueden ser modeladas como datos continuos o discretos. A su vez, los datos continuos pueden ser lineales (por ejemplo, la longitud o la anchura de una dendrita) o direccionales (por ejemplo, la dirección del axón). Estos datos pueden llegar a seguir distribuciones de probabilidad muy complejas y pueden no ajustarse a ninguna distribución paramétrica conocida. El modelado de este tipo de problemas con redes bayesianas híbridas incluyendo variables discretas, lineales y direccionales presenta una serie de retos en relación al aprendizaje a partir de datos, la inferencia, etc. En esta tesis se propone un método para modelar y simular árboles dendríticos basales de neuronas piramidales usando redes bayesianas para capturar las interacciones entre las variables del problema. Para ello, se mide un amplio conjunto de variables de las dendritas y se aplica un algoritmo de aprendizaje con el que se aprende la estructura y se estiman los parámetros de las distribuciones de probabilidad que constituyen las redes bayesianas. Después, se usa un algoritmo de simulación para construir dendritas virtuales mediante el muestreo de valores de las redes bayesianas. Finalmente, se lleva a cabo una profunda evaluaci ón para verificar la capacidad del modelo a la hora de generar dendritas realistas. En esta primera aproximación, las variables fueron discretizadas para poder aprender y muestrear las redes bayesianas. A continuación, se aborda el problema del aprendizaje de redes bayesianas con diferentes tipos de variables. Las mixturas de polinomios constituyen un método para representar densidades de probabilidad en redes bayesianas híbridas. Presentamos un método para aprender aproximaciones de densidades unidimensionales, multidimensionales y condicionales a partir de datos utilizando mixturas de polinomios. El método se basa en interpolación con splines, que aproxima una densidad como una combinación lineal de splines. Los algoritmos propuestos se evalúan utilizando bases de datos artificiales. Además, las mixturas de polinomios son utilizadas como un método no paramétrico de estimación de densidades para clasificadores basados en redes bayesianas. Después, se estudia el problema de incluir información direccional en redes bayesianas. Este tipo de datos presenta una serie de características especiales que impiden el uso de las técnicas estadísticas clásicas. Por ello, para manejar este tipo de información se deben usar estadísticos y distribuciones de probabilidad específicos, como la distribución univariante von Mises y la distribución multivariante von Mises–Fisher. En concreto, en esta tesis extendemos el clasificador naive Bayes al caso en el que las distribuciones de probabilidad condicionada de las variables predictoras dada la clase siguen alguna de estas distribuciones. Se estudia el caso base, en el que sólo se utilizan variables direccionales, y el caso híbrido, en el que variables discretas, lineales y direccionales aparecen mezcladas. También se estudian los clasificadores desde un punto de vista teórico, derivando sus funciones de decisión y las superficies de decisión asociadas. El comportamiento de los clasificadores se ilustra utilizando bases de datos artificiales. Además, los clasificadores son evaluados empíricamente utilizando bases de datos reales. También se estudia el problema de la clasificación de interneuronas. Desarrollamos una aplicación web que permite a un grupo de expertos clasificar un conjunto de neuronas de acuerdo a sus características morfológicas más destacadas. Se utilizan medidas de concordancia para analizar el consenso entre los expertos a la hora de clasificar las neuronas. Se investiga la idoneidad de los términos anatómicos y de los tipos neuronales utilizados frecuentemente en la literatura a través del análisis de redes bayesianas y la aplicación de algoritmos de clustering. Además, se aplican técnicas de aprendizaje supervisado con el objetivo de clasificar de forma automática las interneuronas a partir de sus valores morfológicos. A continuación, se presenta una metodología para construir un modelo que captura las opiniones de todos los expertos. Primero, se genera una red bayesiana para cada experto y se propone un algoritmo para agrupar las redes bayesianas que se corresponden con expertos con comportamientos similares. Después, se induce una red bayesiana que modela la opinión de cada grupo de expertos. Por último, se construye una multired bayesiana que modela las opiniones del conjunto completo de expertos. El análisis del modelo consensuado permite identificar diferentes comportamientos entre los expertos a la hora de clasificar las neuronas. Además, permite extraer un conjunto de características morfológicas relevantes para cada uno de los tipos neuronales mediante inferencia con la multired bayesiana. Estos descubrimientos se utilizan para validar el modelo y constituyen información relevante acerca de la morfología neuronal. Por último, se estudia un problema de clasificación en el que la etiqueta de clase de los datos de entrenamiento es incierta. En cambio, disponemos de un conjunto de etiquetas para cada instancia. Este problema está inspirado en el problema de la clasificación de neuronas, en el que un grupo de expertos proporciona una etiqueta de clase para cada instancia de manera individual. Se propone un método para aprender redes bayesianas utilizando vectores de cuentas, que representan el número de expertos que seleccionan cada etiqueta de clase para cada instancia. Estas redes bayesianas se evalúan utilizando bases de datos artificiales de problemas de aprendizaje supervisado.

Establishing a new paradigm for teaching mathematics at engineering schools

This paper analyzes an ideal model of teaching, thinking after 5-10 years in Universities in the world. We propose the collaborative work for a fruitful learning. According with that, we expose some of our previous projects in this area and some ideas for the ?global education?, focused on the teaching and learning of mathematics to engineering students. Furthermore we explain some of our initiatives for implementing the "Bologna process?. Aspects related to the learning and assessments will be analyzed. The establishment of the new teaching paradigm has to change the learning process and we will suggest some possible initiatives for adapting the learning to the new model. The paper ends by collecting some conclusions.

Generation of human computational models with machine learning

Services in smart environments pursue to increase the quality of people?s lives. The most important issues when developing this kind of environments is testing and validating such services. These tasks usually imply high costs and annoying or unfeasible real-world testing. In such cases, artificial societies may be used to simulate the smart environment (i.e. physical environment, equipment and humans). With this aim, the CHROMUBE methodology guides test engineers when modeling human beings. Such models reproduce behaviors which are highly similar to the real ones. Originally, these models are based on automata whose transitions are governed by random variables. Automaton?s structure and the probability distribution functions of each random variable are determined by a manual test and error process. In this paper, it is presented an alternative extension of this methodology which avoids the said manual process. It is based on learning human behavior patterns automatically from sensor data by using machine learning techniques. The presented approach has been tested on a real scenario, where this extension has given highly accurate human behavior models,

Segregating the functions of human hippocampus

It is now accepted that hippocampal lesions impair episodic memory. However, the precise functional role of the hippocampus in episodic memory remains elusive. Recent functional imaging data implicate the hippocampus in processing novelty, a finding supported by human in vivo recordings and event-related potential studies. Here we measure hippocampal responses to novelty, using functional MRI (fMRI), during an item-learning paradigm generated from an artificial grammar system. During learning, two distinct types of novelty were periodically introduced: perceptual novelty, pertaining to the physical characteristics of stimuli (in this case visual characteristics), and exemplar novelty, reflecting semantic characteristics of stimuli (in this case grammatical status within a rule system). We demonstrate a left anterior hippocampal response to both types of novelty and adaptation of these responses with stimulus familiarity. By contrast to these novelty effects, we also show bilateral posterior hippocampal responses with increasing exemplar familiarity. These results suggest a functional dissociation within the hippocampus with respect to the relative familiarity of study items. Neural responses in anterior hippocampus index generic novelty, whereas posterior hippocampal responses index familiarity to stimuli that have behavioral relevance (i.e., only exemplar familiarity). These findings add to recent evidence for functional segregation within the human hippocampus during learning.

New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan’s tau function

In this paper, we give two infinite families of explicit exact formulas that generalize Jacobi’s (1829) 4 and 8 squares identities to 4n2 or 4n(n + 1) squares, respectively, without using cusp forms. Our 24 squares identity leads to a different formula for Ramanujan’s tau function τ(n), when n is odd. These results arise in the setting of Jacobi elliptic functions, Jacobi continued fractions, Hankel or Turánian determinants, Fourier series, Lambert series, inclusion/exclusion, Laplace expansion formula for determinants, and Schur functions. We have also obtained many additional infinite families of identities in this same setting that are analogous to the η-function identities in appendix I of Macdonald’s work [Macdonald, I. G. (1972) Invent. Math. 15, 91–143]. A special case of our methods yields a proof of the two conjectured [Kac, V. G. and Wakimoto, M. (1994) in Progress in Mathematics, eds. Brylinski, J.-L., Brylinski, R., Guillemin, V. & Kac, V. (Birkhäuser Boston, Boston, MA), Vol. 123, pp. 415–456] identities involving representing a positive integer by sums of 4n2 or 4n(n + 1) triangular numbers, respectively. Our 16 and 24 squares identities were originally obtained via multiple basic hypergeometric series, Gustafson’s Cℓ nonterminating 6φ5 summation theorem, and Andrews’ basic hypergeometric series proof of Jacobi’s 4 and 8 squares identities. We have (elsewhere) applied symmetry and Schur function techniques to this original approach to prove the existence of similar infinite families of sums of squares identities for n2 or n(n + 1) squares, respectively. Our sums of more than 8 squares identities are not the same as the formulas of Mathews (1895), Glaisher (1907), Ramanujan (1916), Mordell (1917, 1919), Hardy (1918, 1920), Kac and Wakimoto, and many others.

Lack of tissue glucocorticoid reactivation in 11β-hydroxysteroid dehydrogenase type 1 knockout mice ameliorates age-related learning impairments

11β-hydroxysteroid dehydrogenase type 1 (11β-HSD-1) intracellularly regenerates active corticosterone from circulating inert 11-dehydrocorticosterone (11-DHC) in specific tissues. The hippocampus is a brain structure particularly vulnerable to glucocorticoid neurotoxicity with aging. In intact hippocampal cells in culture, 11β-HSD-1 acts as a functional 11β-reductase reactivating inert 11-DHC to corticosterone, thereby potentiating kainate neurotoxicity. We examined the functional significance of 11β-HSD-1 in the central nervous system by using knockout mice. Aged wild-type mice developed elevated plasma corticosterone levels that correlated with learning deficits in the watermaze. In contrast, despite elevated plasma corticosterone levels throughout life, this glucocorticoid-associated learning deficit was ameliorated in aged 11β-HSD-1 knockout mice, implicating lower intraneuronal corticosterone levels through lack of 11-DHC reactivation. Indeed, aged knockout mice showed significantly lower hippocampal tissue corticosterone levels than wild-type controls. These findings demonstrate that tissue corticosterone levels do not merely reflect plasma levels and appear to play a more important role in hippocampal functions than circulating blood levels. The data emphasize the crucial importance of local enzymes in determining intracellular glucocorticoid activity. Selective 11β-HSD-1 inhibitors may protect against hippocampal function decline with age.