Integrable deformations of strings on symmetric spaces

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Multiseries Lie groups and asymptotic modules for characterizing and solving integrable models

A multiseries integrable model (MSIM) is defined as a family of compatible flows on an infinite-dimensional Lie group of N-tuples of formal series around N given poles on the Riemann sphere. Broad classes of solutions to a MSIM are characterized through modules over rings of rational functions, called asymptotic modules. Possible ways for constructing asymptotic modules are Riemann-Hilbert and ∂̄ problems. When MSIM's are written in terms of the group coordinates, some of them can be contracted into standard integrable models involving a small number of scalar functions only. Simple contractible MSIM's corresponding to one pole, yield the Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy. Two-pole contractible MSIM's are exhibited, which lead to a hierarchy of solvable systems of nonlinear differential equations consisting of (2 + 1) -dimensional evolution equations and of quite strong differential constraints. © 1989 American Institute of Physics.

Gauge and integrable theories in loop spaces

We propose an integral formulation of the equations of motion of a large class of field theories which leads in a quite natural and direct way to the construction of conservation laws. The approach is based on generalized non-abelian Stokes theorems for p-form connections, and its appropriate mathematical language is that of loop spaces. The equations of motion are written as the equality of a hyper-volume ordered integral to a hyper-surface ordered integral on the border of that hyper-volume. The approach applies to integrable field theories in (1 + 1) dimensions, Chern-Simons theories in (2 + 1) dimensions, and non-abelian gauge theories in (2 + 1) and (3 + 1) dimensions. The results presented in this paper are relevant for the understanding of global properties of those theories. As a special byproduct we solve a long standing problem in (3 + 1)-dimensional Yang-Mills theory, namely the construction of conserved charges, valid for any solution, which are invariant under arbitrary gauge transformations. (C) 2012 Elsevier B.V. All rights reserved.

Selection and hydroponic growth of bread wheat cultivars for bioregenerative life support systems

As part of the ESA-funded MELiSSA program, the suitability, the growth and the development of four bread wheat cultivars were investigated in hydroponic culture with the aim to incorporate such a cultivation system in an Environmental Control and Life Support System (ECLSS). Wheat plants can fulfill three major functions in space: (a) fixation of CO2 and production of O2, (b) production of grains for human nutrition and (c) production of cleaned water after condensation of the water vapor released from the plants by transpiration. Four spring wheat cultivars (Aletsch, Fiorina, Greina and CH Rubli) were grown hydroponically and compared with respect to growth and grain maturation properties. The height of the plants, the culture duration from germination to harvest, the quantity of water used, the number of fertile and non-fertile tillers as well as the quantity and quality of the grains harvested were considered. Mature grains could be harvested after around 160 days depending on the varieties. It became evident that the nutrient supply is crucial in this context and strongly affects leaf senescence and grain maturation. After a first experiment, the culture conditions were improved for the second experiment (stepwise decrease of EC after flowering, pH adjusted twice a week, less plants per m2) leading to a more favorable harvest (higher grain yield and harvest index). Considerably less green tillers without mature grains were present at harvest time in experiment 2 than in experiment 1. The harvest index for dry matter (including roots) ranged from 0.13 to 0.35 in experiment 1 and from 0.23 to 0.41 in experiment 2 with modified culture conditions. The thousand-grain weight for the four varieties ranged from 30.4 to 36.7 g in experiment 1 and from 33.2 to 39.1 g in experiment 2, while market samples were in the range of 39.4–46.9 g. Calcium levels in grains of the hydroponically grown wheat were similar to those from field-grown wheat, while potassium, magnesium, phosphorus, iron, zinc, copper, manganese and nickel levels tended to be higher in the grains of experimental plants. It remains a challenge for future experiments to further adapt the nutrient supply in order to improve senescence of vegetative plant parts, harvest index and the composition of bread wheat grains.

Atomic Quantum Simulation of U(N) and SU(N) Abelian Lattice Gauge Theories

Using ultracold alkaline-earth atoms in optical lattices, we construct a quantum simulator for U(N) and SU(N) lattice gauge theories with fermionic matter based on quantum link models. These systems share qualitative features with QCD, including chiral symmetry breaking and restoration at nonzero temperature or baryon density. Unlike classical simulations, a quantum simulator does not suffer from sign problems and can address the corresponding chiral dynamics in real time.

Ultracold Quantum Gases and Lattice Systems: Quantum Simulation of Lattice Gauge Theories

Abelian and non-Abelian gauge theories are of central importance in many areas of physics. In condensed matter physics, AbelianU(1) lattice gauge theories arise in the description of certain quantum spin liquids. In quantum information theory, Kitaev’s toric code is a Z(2) lattice gauge theory. In particle physics, Quantum Chromodynamics (QCD), the non-Abelian SU(3) gauge theory of the strong interactions between quarks and gluons, is nonperturbatively regularized on a lattice. Quantum link models extend the concept of lattice gauge theories beyond the Wilson formulation, and are well suited for both digital and analog quantum simulation using ultracold atomic gases in optical lattices. Since quantum simulators do not suffer from the notorious sign problem, they open the door to studies of the real-time evolution of strongly coupled quantum systems, which are impossible with classical simulation methods. A plethora of interesting lattice gauge theories suggests itself for quantum simulation, which should allow us to address very challenging problems, ranging from confinement and deconfinement, or chiral symmetry breaking and its restoration at finite baryon density, to color superconductivity and the real-time evolution of heavy-ion collisions, first in simpler model gauge theories and ultimately in QCD.

Quantum Simulation of Non-Abelian Lattice Gauge Theories

We use quantum link models to construct a quantum simulator for U(N) and SU(N) lattice gauge theories. These models replace Wilson’s classical link variables by quantum link operators, reducing the link Hilbert space to a finite number of dimensions. We show how to embody these quantum link models with fermionic matter with ultracold alkaline-earth atoms using optical lattices. Unlike classical simulations, a quantum simulator does not suffer from sign problems and can thus address the corresponding dynamics in real time. Using exact diagonalization results we show that these systems share qualitative features with QCD, including chiral symmetry breaking and we study the expansion of a chirally restored region in space in real time.

The anisotropic λ -deformed SU(2) model is integrable

The all-loop anisotropic Thirring model interpolates between the WZW model and the non-Abelian T-dual of the anisotropic principal chiral model. We focus on the SU(2) case and we prove that it is classically integrable by providing its Lax pair formulation. We derive its underlying symmetry current algebra and use it to show that the Poisson brackets of the spatial part of the Lax pair, assume the Maillet form. In this way we procure the corresponding r and s matrices which provide non-trivial solutions to the modified Yang–Baxter equation.

Generalised integrable λ- and η-deformations and their relation

We construct two-parameter families of integrable λ -deformations of two-dimensional field theories. These interpolate between a CFT (a WZW/gauged WZW model) and the non-Abelian T-dual of a principal chiral model on a group/symmetric coset space. In examples based on the SU(2) WZW model and the SU(2)/U(1) exact coset CFT, we show that these deformations are related to bi-Yang–Baxter generalisations of η-deformations via Poisson–Lie T-duality and analytic continuation. We illustrate the quantum behaviour of our models under RG flow. As a byproduct we demonstrate that the bi-Yang–Baxter σ-model for a general group is one-loop renormalisable.

Classification of discrete systems on a square lattice

We consider the classification up to a Möbius transformation of real linearizable and integrable partial difference equations with dispersion defined on a square lattice by the multiscale reduction around their harmonic solution. We show that the A1, A2, and A3 linearizability and integrability conditions constrain the number of parameters in the equation, but these conditions are insufficient for a complete characterization of the subclass of multilinear equations on a square lattice.

Climate variability effects on summer cropping systems on the Iberian Peninsula

Climate variability and changes in the frequency of extremes events have a direct impact on crop yield and damages. Climate anomalies projections at monthly and yearly timescale allows us for adapting a cropping system (crops, varieties and management) to take advantage of favorable conditions or reduce the effect of adverse conditions. The objective of this work is to develop indices to evaluate the effect of climatic variability in summer cropping systems of Iberian Peninsula, in an attempt of relating yield variability to climate variability, extending the work of Rodríguez-Puebla (2004).

Quality assurance systems in the production and marketing of fresh peaches.

Due to the fast rate of peach post-harvest ripening, damage due to mechanical handling, externally appreciated as bruises and soft areas, is a real problem that leads to an early harvesting and poor quality of the fruits, as perceived by the consumers. More and more, the European consumer asks for good taste and freshness of fruits and vegetables, and these quality factors are not included in standards, nor in most of the producers' practices. Fruit processing and marketing centres (co-operatives) are increasingly interested in adopting quality controls in their processes. ISO 9000 procedures are being applied in some food areas, primarily milk and meat processors, but no generalised procedures have been developed until the present time to be applied to fresh product processes. All different peach and nectarine varieties that are harvested and handled in Murcia cooperatives and sold in a large supermarket in Madrid were analysed during the whole 1997 season (early May to late August). A total number of 78 samples of 25 fruits (co-operative) or 10 fruits (market), were tested in the laboratory for mechanical, optical, chemical and tasting quality. The variability and relationships between all these quality parameters are presented and discussed, and sampling unit sizes which would be advisable for quality control are calculated.

Improving the management of water and nitrogen resources in cropping systems by using simulation models

Los modelos de simulación de cultivos permiten analizar varias combinaciones de laboreo-rotación y explorar escenarios de manejo. El modelo DSSAT fue evaluado bajo condiciones de secano en un experimento de campo de 16 años en la semiárida España central. Se evaluó el efecto del sistema de laboreo y las rotaciones basadas en cereales de invierno, en el rendimiento del cultivo y la calidad del suelo. Los modelos CERES y CROPGRO se utilizaron para simular el crecimiento y rendimiento del cultivo, mientras que el modelo DSSAT CENTURY se utilizó en las simulaciones de SOC y SN. Tanto las observaciones de campo como las simulaciones con CERES-Barley, mostraron que el rendimiento en grano de la cebada era mas bajo para el cereal continuo (BB) que para las rotaciones de veza (VB) y barbecho (FB) en ambos sistemas de laboreo. El modelo predijo más nitrógeno disponible en el laboreo convencional (CT) que en el no laboreo (NT) conduciendo a un mayor rendimiento en el CT. El SOC y el SN en la capa superficial del suelo, fueron mayores en NT que en CT, y disminuyeron con la profundidad en los valores tanto observados como simulados. Las mejores combinaciones para las condiciones de secano estudiadas fueron CT-VB y CT-FB, pero CT presentó menor contenido en SN y SOC que NT. El efecto beneficioso del NT en SOC y SN bajo condiciones Mediterráneas semiáridas puede ser identificado por observaciones de campo y por simulaciones de modelos de cultivos. La simulación del balance de agua en sistemas de cultivo es una herramienta útil para estudiar como el agua puede ser utilizado eficientemente. La comparación del balance de agua de DSSAT , con una simple aproximación “tipping bucket”, con el modelo WAVE más mecanicista, el cual integra la ecuación de Richard , es un potente método para valorar el funcionamiento del modelo. Los parámetros de suelo fueron calibrados usando el método de optimización global Simulated Annealing (SA). Un lisímetro continuo de pesada en suelo desnudo suministró los valores observados de drenaje y evapotranspiración (ET) mientras que el contenido de agua en el suelo (SW) fue suministrado por sensores de capacitancia. Ambos modelos funcionaron bien después de la optimización de los parámetros de suelo con SA, simulando el balance de agua en el suelo para el período de calibración. Para el período de validación, los modelos optimizados predijeron bien el contenido de agua en el suelo y la evaporación del suelo a lo largo del tiempo. Sin embargo, el drenaje fue predicho mejor con WAVE que con DSSAT, el cual presentó mayores errores en los valores acumulados. Esto podría ser debido a la naturaleza mecanicista de WAVE frente a la naturaleza más funcional de DSSAT. Los buenos resultados de WAVE indican que, después de la calibración, este puede ser utilizado como "benchmark" para otros modelos para periodos en los que no haya medidas de campo del drenaje. El funcionamiento de DSSAT-CENTURY en la simulación de SOC y N depende fuertemente del proceso de inicialización. Se propuso como método alternativo (Met.2) la inicialización de las fracciones de SOC a partir de medidas de mineralización aparente del suelo (Napmin). El Met.2 se comparó con el método de inicialización de Basso et al. (2011) (Met.1), aplicando ambos métodos a un experimento de campo de 4 años en un área en regadío de España central. Nmin y Napmin fueron sobreestimados con el Met.1, ya que la fracción estable obtenida (SOC3) en las capas superficiales del suelo fue más baja que con Met.2. El N lixiviado simulado fue similar en los dos métodos, con buenos resultados en los tratamientos de barbecho y cebada. El Met.1 subestimó el SOC en la capa superficial del suelo cuando se comparó con una serie observada de 12 años. El crecimiento y rendimiento del cultivo fueron adecuadamente simulados con ambos métodos, pero el N en la parte aérea de la planta y en el grano fueron sobreestimados con el Met.1. Los resultados variaron significativamente con las fracciones iniciales de SOC, resaltando la importancia del método de inicialización. El Met.2 ofrece una alternativa para la inicialización del modelo CENTURY, mejorando la simulación de procesos de N en el suelo. La continua emergencia de nuevas variedades de híbridos modernos de maíz limita la aplicación de modelos de simulación de cultivos, ya que estos nuevos híbridos necesitan ser calibrados en el campo para ser adecuados para su uso en los modelos. El desarrollo de relaciones basadas en la duración del ciclo, simplificaría los requerimientos de calibración facilitando la rápida incorporación de nuevos cultivares en DSSAT. Seis híbridos de maiz (FAO 300 hasta FAO 700) fueron cultivados en un experimento de campo de dos años en un área semiárida de regadío en España central. Los coeficientes genéticos fueron obtenidos secuencialmente, comenzando con los parámetros de desarrollo fenológico (P1, P2, P5 and PHINT), seguido de los parámetros de crecimiento del cultivo (G2 and G3). Se continuó el procedimiento hasta que la salida de las simulaciones estuvo en concordancia con las observaciones fenológicas de campo. Después de la calibración, los parámetros simulados se ajustaron bien a los parámetros observados, con bajos RMSE en todos los casos. Los P1 y P5 calibrados, incrementaron con la duración del ciclo. P1 fue una función lineal del tiempo térmico (TT) desde emergencia hasta floración y P5 estuvo linealmente relacionada con el TT desde floración a madurez. No hubo diferencias significativas en PHINT entre híbridos de FAO-500 a 700 , ya que tuvieron un número de hojas similar. Como los coeficientes fenológicos estuvieron directamente relacionados con la duración del ciclo, sería posible desarrollar rangos y correlaciones que permitan estimar dichos coeficientes a partir de la clasificación del ciclo. ABSTRACT Crop simulation models allow analyzing various tillage-rotation combinations and exploring management scenarios. DSSAT model was tested under rainfed conditions in a 16-year field experiment in semiarid central Spain. The effect of tillage system and winter cereal-based rotations on the crop yield and soil quality was evaluated. The CERES and CROPGRO models were used to simulate crop growth and yield, while the DSSAT CENTURY was used in the SOC and SN simulations. Both field observations and CERES-Barley simulations, showed that barley grain yield was lower for continuous cereal (BB) than for vetch (VB) and fallow (FB) rotations for both tillage systems. The model predicted higher nitrogen availability in the conventional tillage (CT) than in the no tillage (NT) leading to a higher yield in the CT. The SOC and SN in the top layer, were higher in NT than in CT, and decreased with depth in both simulated and observed values. The best combinations for the dry land conditions studied were CT-VB and CT-FB, but CT presented lower SN and SOC content than NT. The beneficial effect of NT on SOC and SN under semiarid Mediterranean conditions can be identified by field observations and by crop model simulations. The simulation of the water balance in cropping systems is a useful tool to study how water can be used efficiently. The comparison of DSSAT soil water balance, with a simpler “tipping bucket” approach, with the more mechanistic WAVE model, which integrates Richard’s equation, is a powerful method to assess model performance. The soil parameters were calibrated by using the Simulated Annealing (SA) global optimizing method. A continuous weighing lysimeter in a bare fallow provided the observed values of drainage and evapotranspiration (ET) while soil water content (SW) was supplied by capacitance sensors. Both models performed well after optimizing soil parameters with SA, simulating the soil water balance components for the calibrated period. For the validation period, the optimized models predicted well soil water content and soil evaporation over time. However, drainage was predicted better by WAVE than by DSSAT, which presented larger errors in the cumulative values. That could be due to the mechanistic nature of WAVE against the more functional nature of DSSAT. The good results from WAVE indicate that, after calibration, it could be used as benchmark for other models for periods when no drainage field measurements are available. The performance of DSSAT-CENTURY when simulating SOC and N strongly depends on the initialization process. Initialization of the SOC pools from apparent soil N mineralization (Napmin) measurements was proposed as alternative method (Met.2). Method 2 was compared to the Basso et al. (2011) initialization method (Met.1), by applying both methods to a 4-year field experiment in a irrigated area of central Spain. Nmin and Napmin were overestimated by Met.1, since the obtained stable pool (SOC3) in the upper layers was lower than from Met.2. Simulated N leaching was similar for both methods, with good results in fallow and barley treatments. Method 1 underestimated topsoil SOC when compared with a 12-year observed serial. Crop growth and yield were properly simulated by both methods, but N in shoots and grain were overestimated by Met.1. Results varied significantly with the initial SOC pools, highlighting the importance of the initialization procedure. Method 2 offers an alternative to initialize the CENTURY model, enhancing the simulation of soil N processes. The continuous emergence of new varieties of modern maize hybrids limits the application of crop simulation models, since these new hybrids should be calibrated in the field to be suitable for model use. The development of relationships based on the cycle duration, would simplify the calibration requirements facilitating the rapid incorporation of new cultivars into DSSAT. Six maize hybrids (FAO 300 through FAO 700) were grown in a 2-year field experiment in a semiarid irrigated area of central Spain. Genetic coefficients were obtained sequentially, starting with the phenological development parameters (P1, P2, P5 and PHINT), followed by the crop growth parameters (G2 and G3). The procedure was continued until the simulated outputs were in good agreement with the field phenological observations. After calibration, simulated parameters matched observed parameters well, with low RMSE in most cases. The calibrated P1 and P5 increased with the duration of the cycle. P1 was a linear function of the thermal time (TT) from emergence to silking and P5 was linearly related with the TT from silking to maturity . There were no significant differences in PHINT between hybrids from FAO-500 to 700 , as they had similar leaf number. Since phenological coefficients were directly related with the cycle duration, it would be possible to develop ranges and correlations which allow to estimate such coefficients from the cycle classification.

Evaluating sustainability and food security of peasant farming systems in Mesoamerican highlands

En las últimas décadas, la agricultura sostenible ha sido objeto de gran interés y debate académico, no sólo en términos conceptuales, sino también en términos metodológicos. La persistencia de la inseguridad alimentaria y el deterioro de los recursos naturales en muchas regiones del mundo, ha provocado el surgimiento de numerosas iniciativas centradas en revitalizar la agricultura campesina así como renovadas discusiones sobre el rol que juega la agricultura como motor de desarrollo y principal actividad para alivio de la pobreza. Por ello, cuando hablamos de evaluar sistemas campesinos de montaña, debemos considerar tanto la dimensión alimentaria como las especificidades propias de los sistemas montañosos como base fundamental de la sostenibilidad. Al evaluar la contribución que han hecho alternativas tecnológicas y de manejo en la mejora de la sostenibilidad y la seguridad alimentaria de los sistemas campesinos de montaña en Mesoamérica, surgen tres preguntas de investigación: • ¿Se está evaluando la sostenibilidad de los sistemas campesinos teniendo en cuenta la variabilidad climática, la participación de los agricultores y las dinámicas temporales? • ¿Podemos rescatar tendencias comunes en estos sistemas y extrapolar los resultados a otras zonas? • ¿Son inequívocamente positivas las alternativas propuestas que se han llevado a cabo? En este trabajo se presentan tres evaluaciones de sostenibilidad que tratan de poner de manifiesto cuáles son los retos y oportunidades que enfrentan actualmente los sistemas campesinos de montaña. En primer lugar, se evalúan tres sistemas de manejo agrícola bajo dos años meteorológicamente contrastantes. Se determinó que durante el año que experimentó lluvias abundantes y temperaturas moderadas, los sistemas de bajos insumos, basados en el uso de abonos orgánicos y rotación de cultivos, obtuvieron los mejores resultados en indicadores ecológicos y similares resultados en los económicos y sociales que el sistema de altos insumos químicos. En el segundo año, con heladas tempranas y sequía invernal, la productividad se redujo para todos los sistemas pero los sistemas más diversificados (en variedades de maíz y/o siembra de otros cultivos) pudieron resistir mejor los contratiempos climáticos. En segundo lugar, se evalúa el grado de conocimiento (percepción) campesino para determinar los factores claves que determinan la sostenibilidad de sus sistemas y su seguridad alimentaria. Se determinó que los principales puntos críticos identificados por los campesinos (tamaño de parcela y pendiente del terreno) afectan de forma significativa a cuestiones de índole económica, pero no son capaces de explicar los desequilibrios alimenticios existentes. Realizando un análisis comparativo entre comunidades que presentaban buenos y malos resultados en cuanto a aporte energético y proteico, se determinó que la seguridad alimentaria estaba relacionada con la sostenibilidad de los sistemas y que concretamente estaba ligada a los atributos de equidad y autonomía. Otro resultado destacable fue que las comunidades más marginales y con mayor dificultad de acceso mostraron mayores niveles de inseguridad alimentaria, pero la variabilidad intergrupal fue muy alta. Eso demuestra que la seguridad alimentaria y nutricional forma parte de un complejo sistema de estrategias de autoabastecimiento ligada a la idiosincrasia misma de cada uno de los hogares. En tercer lugar, se evaluó el desempeño de las escuelas de campo de agricultores (ECAs) en la mejora de la sostenibilidad y la seguridad alimentaria de un sistema campesino de montaña. Para ver el efecto del impacto de estas metodologías a largo plazo, se estudiaron tres comunidades donde se habían implementado ECAs hace 8, 5 y 3 años. Encontramos que el impacto fue progresivo ya que fue la comunidad más antigua la que mejores valores obtuvo. El impacto de las ECAs fue rápido y persistente en los indicadores relacionados con la participación, el acceso a servicios básicos y la conservación de los recursos naturales. El estudio demostró un claro potencial de las ECAs en la mejora general de la sostenibilidad y la seguridad alimentaria de estos sistemas, sin embargo se observó una relación directa entre el aumento de producción agrícola y el uso de insumos externos, lo que puede suponer un punto crítico para los ideales sostenibles. ABSTRACT During the last decades, sustainable agriculture has been the subject of considerable academic interest and debate, not only in conceptual terms, but also in methodological ones. The persistence of high levels of environmental degradation and food insecurity in many regions has led to new initiatives focused on revitalizing peasant agriculture and renewed discussions of the role of sustainable agriculture as an engine for development, environmental conservation and poverty alleviation. Therefore, to assess mountain farming systems, we must consider food dimension and taking into account the specificities of the mountain systems as the foundation of sustainability. When evaluating contribution of technological and management alternative proposals in achieving sustainability and food security for peasant farming systems in Mesoamerican highlands, three research questions arise: • Is sustainability of peasant-farming systems being evaluated taking into account climate variability, participation of farmers and temporal dynamics? • Can we rescue common trends in these systems and extrapolate the results to other areas? • What alternative proposals that have been conducted are unequivocally positives? In this document, we present three evaluations of sustainability that try to highlight the challenges and opportunities that currently face mountain farming systems in Mesoamerica. First, we evaluate the sustainability of three agricultural management systems in two contrasting weather years. We determined that during the first year that exposed heavy rains and moderate temperatures, low-input systems, which are based on the use of organic fertilizers and crop rotation, provided better results in terms of ecological indicators and equal results in terms of economic and social indicators than those achieved using a high chemical input system. In the second year, which featured early frosts and a winter drought, productivity declined in all systems; however, the most diversified systems (in terms of the maize varieties grown and the sowing of other crops) more successfully resisted these climatic adversities. Second, we evaluate the farmers’ perception to determine the key drivers for achieving their sustainability and food and nutritional security. We determined that the key factors identified by farmers (landholding size and slope of cropland) exerted significant impacts on economic disparities but did not explain the malnutrition levels. We compared two contrasting hamlets according to their energy and protein supply, one namely Limón Timoté (LT), which did not present food problems and Limón Peña Blanca (LP), which did exhibit food insecurity. The results showed that FNS is linked to sustainability, and it is primarily related to the sustainability attributes of self-reliance and equity. Although the more marginated and inaccessible community exhibited more food insecurity, food and nutritional security depend upon a complex array of self-sufficiency strategies that remain linked to individual household idiosyncrasies. Third, we evaluated the impact of farmer field schools for improving the sustainability and food security of peasant mountain systems. In order to appreciate the long-term impact, we studied three communities where FFSs were implemented eight, five and three years ago, respectively. We found that FFSs have a gradual impact, as the community that first implemented FFSs scores highest. The impact of FFSs was broad and long-lasting for indicators related to participation, access to basic services and conservation of natural resources. This study demonstrates the potential of FFSs, but more attention will have to be paid to critical indicators in order to scale up their potential in the future. We observed a direct relationship between the increase in agricultural production and the use of external inputs, which is a critical point for sustainable ideals.

Topological structures of complex belief systems

The concepts of substantive beliefs and derived beliefs are defined, a set of substantive beliefs S like open set and the neighborhood of an element substantive belief. A semantic operation of conjunction is defined with a structure of an Abelian group. Mathematical structures exist such as poset beliefs and join-semilattttice beliefs. A metric space of beliefs and the distance of belief depending on the believer are defined. The concepts of closed and opened ball are defined. S′ is defined as subgroup of the metric space of beliefs Σ and S′ is a totally limited set. The term s is defined (substantive belief) in terms of closing of S′. It is deduced that Σ is paracompact due to Stone's Theorem. The pseudometric space of beliefs is defined to show how the metric of the nonbelieving subject has a topological space like a nonmaterial abstract ideal space formed in the mind of the believing subject, fulfilling the conditions of Kuratowski axioms of closure. To establish patterns of materialization of beliefs we are going to consider that these have defined mathematical structures. This will allow us to understand better cultural processes of text, architecture, norms, and education that are forms or the materialization of an ideology. This materialization is the conversion by means of certain mathematical correspondences, of an abstract set whose elements are beliefs or ideas, in an impure set whose elements are material or energetic. Text is a materialization of ideology.