Progress in the mathematical theory of quantum disordered systems

We review recent progress in the mathematical theory of quantum disordered systems: the Anderson transition, including some joint work with Marchetti, the (quantum and classical) Edwards-Anderson (EA) spin-glass model and return to equilibrium for a class of spin-glass models, which includes the EA model initially in a very large transverse magnetic field. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4770066]

Mathematical models of cognitive processes

The research activity carried out during the PhD course was focused on the development of mathematical models of some cognitive processes and their validation by means of data present in literature, with a double aim: i) to achieve a better interpretation and explanation of the great amount of data obtained on these processes from different methodologies (electrophysiological recordings on animals, neuropsychological, psychophysical and neuroimaging studies in humans), ii) to exploit model predictions and results to guide future research and experiments. In particular, the research activity has been focused on two different projects: 1) the first one concerns the development of neural oscillators networks, in order to investigate the mechanisms of synchronization of the neural oscillatory activity during cognitive processes, such as object recognition, memory, language, attention; 2) the second one concerns the mathematical modelling of multisensory integration processes (e.g. visual-acoustic), which occur in several cortical and subcortical regions (in particular in a subcortical structure named Superior Colliculus (SC)), and which are fundamental for orienting motor and attentive responses to external world stimuli. This activity has been realized in collaboration with the Center for Studies and Researches in Cognitive Neuroscience of the University of Bologna (in Cesena) and the Department of Neurobiology and Anatomy of the Wake Forest University School of Medicine (NC, USA). PART 1. Objects representation in a number of cognitive functions, like perception and recognition, foresees distribute processes in different cortical areas. One of the main neurophysiological question concerns how the correlation between these disparate areas is realized, in order to succeed in grouping together the characteristics of the same object (binding problem) and in maintaining segregated the properties belonging to different objects simultaneously present (segmentation problem). Different theories have been proposed to address these questions (Barlow, 1972). One of the most influential theory is the so called “assembly coding”, postulated by Singer (2003), according to which 1) an object is well described by a few fundamental properties, processing in different and distributed cortical areas; 2) the recognition of the object would be realized by means of the simultaneously activation of the cortical areas representing its different features; 3) groups of properties belonging to different objects would be kept separated in the time domain. In Chapter 1.1 and in Chapter 1.2 we present two neural network models for object recognition, based on the “assembly coding” hypothesis. These models are networks of Wilson-Cowan oscillators which exploit: i) two high-level “Gestalt Rules” (the similarity and previous knowledge rules), to realize the functional link between elements of different cortical areas representing properties of the same object (binding problem); 2) the synchronization of the neural oscillatory activity in the γ-band (30-100Hz), to segregate in time the representations of different objects simultaneously present (segmentation problem). These models are able to recognize and reconstruct multiple simultaneous external objects, even in difficult case (some wrong or lacking features, shared features, superimposed noise). In Chapter 1.3 the previous models are extended to realize a semantic memory, in which sensory-motor representations of objects are linked with words. To this aim, the network, previously developed, devoted to the representation of objects as a collection of sensory-motor features, is reciprocally linked with a second network devoted to the representation of words (lexical network) Synapses linking the two networks are trained via a time-dependent Hebbian rule, during a training period in which individual objects are presented together with the corresponding words. Simulation results demonstrate that, during the retrieval phase, the network can deal with the simultaneous presence of objects (from sensory-motor inputs) and words (from linguistic inputs), can correctly associate objects with words and segment objects even in the presence of incomplete information. Moreover, the network can realize some semantic links among words representing objects with some shared features. These results support the idea that semantic memory can be described as an integrated process, whose content is retrieved by the co-activation of different multimodal regions. In perspective, extended versions of this model may be used to test conceptual theories, and to provide a quantitative assessment of existing data (for instance concerning patients with neural deficits). PART 2. The ability of the brain to integrate information from different sensory channels is fundamental to perception of the external world (Stein et al, 1993). It is well documented that a number of extraprimary areas have neurons capable of such a task; one of the best known of these is the superior colliculus (SC). This midbrain structure receives auditory, visual and somatosensory inputs from different subcortical and cortical areas, and is involved in the control of orientation to external events (Wallace et al, 1993). SC neurons respond to each of these sensory inputs separately, but is also capable of integrating them (Stein et al, 1993) so that the response to the combined multisensory stimuli is greater than that to the individual component stimuli (enhancement). This enhancement is proportionately greater if the modality-specific paired stimuli are weaker (the principle of inverse effectiveness). Several studies have shown that the capability of SC neurons to engage in multisensory integration requires inputs from cortex; primarily the anterior ectosylvian sulcus (AES), but also the rostral lateral suprasylvian sulcus (rLS). If these cortical inputs are deactivated the response of SC neurons to cross-modal stimulation is no different from that evoked by the most effective of its individual component stimuli (Jiang et al 2001). This phenomenon can be better understood through mathematical models. The use of mathematical models and neural networks can place the mass of data that has been accumulated about this phenomenon and its underlying circuitry into a coherent theoretical structure. In Chapter 2.1 a simple neural network model of this structure is presented; this model is able to reproduce a large number of SC behaviours like multisensory enhancement, multisensory and unisensory depression, inverse effectiveness. In Chapter 2.2 this model was improved by incorporating more neurophysiological knowledge about the neural circuitry underlying SC multisensory integration, in order to suggest possible physiological mechanisms through which it is effected. This endeavour was realized in collaboration with Professor B.E. Stein and Doctor B. Rowland during the 6 months-period spent at the Department of Neurobiology and Anatomy of the Wake Forest University School of Medicine (NC, USA), within the Marco Polo Project. The model includes four distinct unisensory areas that are devoted to a topological representation of external stimuli. Two of them represent subregions of the AES (i.e., FAES, an auditory area, and AEV, a visual area) and send descending inputs to the ipsilateral SC; the other two represent subcortical areas (one auditory and one visual) projecting ascending inputs to the same SC. Different competitive mechanisms, realized by means of population of interneurons, are used in the model to reproduce the different behaviour of SC neurons in conditions of cortical activation and deactivation. The model, with a single set of parameters, is able to mimic the behaviour of SC multisensory neurons in response to very different stimulus conditions (multisensory enhancement, inverse effectiveness, within- and cross-modal suppression of spatially disparate stimuli), with cortex functional and cortex deactivated, and with a particular type of membrane receptors (NMDA receptors) active or inhibited. All these results agree with the data reported in Jiang et al. (2001) and in Binns and Salt (1996). The model suggests that non-linearities in neural responses and synaptic (excitatory and inhibitory) connections can explain the fundamental aspects of multisensory integration, and provides a biologically plausible hypothesis about the underlying circuitry.

Modeling the spatial dynamics of economic models

The advances that have been characterizing spatial econometrics in recent years are mostly theoretical and have not found an extensive empirical application yet. In this work we aim at supplying a review of the main tools of spatial econometrics and to show an empirical application for one of the most recently introduced estimators. Despite the numerous alternatives that the econometric theory provides for the treatment of spatial (and spatiotemporal) data, empirical analyses are still limited by the lack of availability of the correspondent routines in statistical and econometric software. Spatiotemporal modeling represents one of the most recent developments in spatial econometric theory and the finite sample properties of the estimators that have been proposed are currently being tested in the literature. We provide a comparison between some estimators (a quasi-maximum likelihood, QML, estimator and some GMM-type estimators) for a fixed effects dynamic panel data model under certain conditions, by means of a Monte Carlo simulation analysis. We focus on different settings, which are characterized either by fully stable or quasi-unit root series. We also investigate the extent of the bias that is caused by a non-spatial estimation of a model when the data are characterized by different degrees of spatial dependence. Finally, we provide an empirical application of a QML estimator for a time-space dynamic model which includes a temporal, a spatial and a spatiotemporal lag of the dependent variable. This is done by choosing a relevant and prolific field of analysis, in which spatial econometrics has only found limited space so far, in order to explore the value-added of considering the spatial dimension of the data. In particular, we study the determinants of cropland value in Midwestern U.S.A. in the years 1971-2009, by taking the present value model (PVM) as the theoretical framework of analysis.

The Devil’s Calculus: Mathematical Models of Civil War

In spite of the movement to turn political science into a real science, various mathematical methods that are now the staples of physics, biology, and even economics are thoroughly uncommon in political science, especially the study of civil war. This study seeks to apply such methods - specifically, ordinary differential equations (ODEs) - to model civil war based on what one might dub the capabilities school of thought, which roughly states that civil wars end only when one side’s ability to make war falls far enough to make peace truly attractive. I construct several different ODE-based models and then test them all to see which best predicts the instantaneous capabilities of both sides of the Sri Lankan civil war in the period from 1990 to 1994 given parameters and initial conditions. The model that the tests declare most accurate gives very accurate predictions of state military capabilities and reasonable short term predictions of cumulative deaths. Analysis of the model reveals the scale of the importance of rebel finances to the sustainability of insurgency, most notably that the number of troops required to put down the Tamil Tigers is reduced by nearly a full order of magnitude when Tiger foreign funding is stopped. The study thus demonstrates that accurate foresight may come of relatively simple dynamical models, and implies the great potential of advanced and currently unconventional non-statistical mathematical methods in political science.

Mathematical models for the arrangement and the cleaning of the river basin and surroundings of the river sandy grounds to its passage by the city of Salta (Argentina)

The province of Salta is located the Northwest of Argentina in the border with Bolivia, Chile and Paraguay. Its Capital is the city of Salta that concentrates half of the inhabitants of the province and has grown to 600000 hab., from a small active Spanish town well founded in 1583. The city is crossed by the Arenales River descending from close mountains at North, source of water and end of sewers. But with actual growing it has become a focus of infection and of remarkable unhealthiness. It is necessary to undertake a plan for the recovery of the river, directed to the attainment of the well-being and to improve the life?s quality of the Community. The fundamental idea of the plan is to obtain an ordering of the river basin and an integral management of the channel and its surroundings, including the cleaning out. The improvement of the water?s quality, the healthiness of the surroundings and the improvement of the environment, must go hand by hand with the development of sport activities, of relaxation, tourism, establishment of breeding grounds, kitchen gardens, micro enterprises with clean production and other actions that contribute to their benefit by the society, that being a basic factor for their care and sustainable use. The present pollution is organic, chemical, industrial, domestic, due to the disposition of sweepings and sewer effluents that affects not only the flora and small fauna, destroying the biodiversity, but also to the health of people living in their margins. Within the plan it will be necessary to consider, besides hydric and environmental cleaning and the prevention of floods, the planning of the extraction of aggregates, the infrastructure and consolidation of margins works and the arrangement of all the river basin. It will be necessary to consider the public intervention at state, provincial and local level, and the private intervention. In the model it has been necessary to include the sub-model corresponding to the election of the entity to be the optimal instrument to reach the proposed objectives, giving an answer to the social, environmental and economic requirements. For that the authors have used multi-criteria decision methods to qualify and select alternatives, and for the programming of their implementation. In the model the authors have contemplated the short, average and long term actions. They conform a Paretooptimal alternative which secures the ordering, integral and suitable management of the basin of the Arenales River, focusing on its passage by the city of Salta.

Development of mathematical models to elaborate strategies, select alternatives and development of plans for adaptation of communities to climate change in different geographical areas including costs to implement it.

There is evidence that the climate changes and that now, the change is influenced and accelerated by the CO2 augmentation in atmosphere due to combustion by humans. Such ?Climate change? is on the policy agenda at the global level, with the aim of understanding and reducing its causes and to mitigate its consequences. In most countries and international organisms UNO (e.g. Rio de Janeiro 1992), OECD, EC, etc . . . the efforts and debates have been directed to know the possible causes, to predict the future evolution of some variable conditioners, and trying to make studies to fight against the effects or to delay the negative evolution of such. The Protocol of Kyoto 1997 set international efforts about CO2 emissions, but it was partial and not followed e.g. by USA and China . . . , and in Durban 2011 the ineffectiveness of humanity on such global real challenges was set as evident. Among all that, the elaboration of a global model was not boarded that can help to choose the best alternative between the feasible ones, to elaborate the strategies and to evaluate the costs, and the authors propose to enter in that frame for study. As in all natural, technological and social changes, the best-prepared countries will have the best bear and the more rapid recover. In all the geographic areas the alternative will not be the same one, but the model must help us to make the appropriated decision. It is essential to know those areas that are more sensitive to the negative effects of climate change, the parameters to take into account for its evaluation, and comprehensive plans to deal with it. The objective of this paper is to elaborate a mathematical model support of decisions, which will allow to develop and to evaluate alternatives of adaptation to the climatic change of different communities in Europe and Latin-America, mainly in especially vulnerable areas to the climatic change, considering in them all the intervening factors. The models will consider criteria of physical type (meteorological, edaphic, water resources), of use of the ground (agriculturist, forest, mining, industrial, urban, tourist, cattle dealer), economic (income, costs, benefits, infrastructures), social (population), politician (implementation, legislation), educative (Educational programs, diffusion) and environmental, at the present moment and the future. The intention is to obtain tools for aiding to get a realistic position for these challenges, which are an important part of the future problems of humanity in next decades.

Development of mathematical models to elaborate strategies, to select alternatives and the development of plans for the adaptation of communities to climate change in different geographical areas including costs to implement it

Climate change is on the policy agenda at the global level, with the aim of understanding and reducing its causes and to mitigate its consequences. In most of the countries and international organisms UNO, OECD, EC, etc … the efforts and debates have been directed to know the possible causes, to predict the future evolution of some variable conditioners, and trying to make studies to fight against the effects or to delay the negative evolution of such. Nevertheless, the elaboration of a global model was not boarded that can help to choose the best alternative between the feasible ones, to elaborate the strategies and to evaluate the costs. As in all natural, technological and social changes, the best-prepared countries will have the best bear and the more rapid recover. In all the geographic areas the alternative will not be the same one, but the model should help us to make the appropriated decision. It is essential to know those areas that are more sensitive to the negative effects of climate change, the parameters to take into account for its evaluation, and comprehensive plans to deal with it. The objective of this paper is to elaborate a mathematical model support of decisions, that will allow to develop and to evaluate alternatives of adaptation to the climatic change of different communities in Europe and Latin-America, mainly, in vulnerable areas to the climatic change, considering in them all the intervening factors. The models will take into consideration criteria of physical type (meteorological, edaphic, water resources), of use of the ground (agriculturist, forest, mining, industrial, urban, tourist, cattle dealer), economic (income, costs, benefits, infrastructures), social (population), politician (implementation, legislation), educative (Educational programs, diffusion), sanitary and environmental, at the present moment and the future.

Introduction to Coding Theory for Flow Equations of Complex Systems Models

The modeling of complex dynamic systems depends on the solution of a differential equations system. Some problems appear because we do not know the mathematical expressions of the said equations. Enough numerical data of the system variables are known. The authors, think that it is very important to establish a code between the different languages to let them codify and decodify information. Coding permits us to reduce the study of some objects to others. Mathematical expressions are used to model certain variables of the system are complex, so it is convenient to define an alphabet code determining the correspondence between these equations and words in the alphabet. In this paper the authors begin with the introduction to the coding and decoding of complex structural systems modeling.

Saint Mathew Law and Bonini Paradox in Textual Theory of Complex Models

The mathematical models of the complex reality are texts belonging to a certain literature that is written in a semi-formal language, denominated L(MT) by the authors whose laws linguistic mathematics have been previously defined. This text possesses linguistic entropy that is the reflection of the physical entropy of the processes of real world that said text describes. Through the temperature of information defined by Mandelbrot, the authors begin a text-reality thermodynamic theory that drives to the existence of information attractors, or highly structured point, settling down a heterogeneity of the space text, the same one that of ontologic space, completing the well-known law of Saint Mathew, of the General Theory of Systems and formulated by Margalef saying: “To the one that has more he will be given, and to the one that doesn't have he will even be removed it little that it possesses.

Műszaki és gazdasági hatékonyság Koopmans termeléselméletében (Technical and economic efficiency in Koopmans' theory of production)

Koopmans gyakorlati problémák megoldása során szerzett tapasztalatait általánosítva fogott hozzá a lineáris tevékenységelemzési modell kidolgozásához. Meglepődve tapasztalta, hogy a korabeli közgazdaságtan nem rendelkezett egységes, kellően egzakt termeléselmélettel és fogalomrendszerrel. Úttörő dolgozatában ezért - mintegy a lineáris tevékenységelemzési modell elméleti kereteként - lerakta a technológiai halmazok fogalmán nyugvó axiomatikus termeléselmélet alapjait is. Nevéhez fűződik a termelési hatékonyság és a hatékonysági árak fogalmának egzakt definíciója, s az egymást kölcsönösen feltételező viszonyuk igazolása a lineáris tevékenységelemzési modell keretében. A hatékonyság manapság használatos, pusztán műszaki szempontból értelmezett definícióját Koopmans csak sajátos esetként tárgyalta, célja a gazdasági hatékonyság fogalmának a bevezetése és elemzése volt. Dolgozatunkban a lineáris programozás dualitási tételei segítségével rekonstruáljuk ez utóbbira vonatkozó eredményeit. Megmutatjuk, hogy egyrészt bizonyításai egyenértékűek a lineáris programozás dualitási tételeinek igazolásával, másrészt a gazdasági hatékonysági árak voltaképpen a mai értelemben vett árnyékárak. Rámutatunk arra is, hogy a gazdasági hatékonyság értelmezéséhez megfogalmazott modellje az Arrow-Debreu-McKenzie-féle általános egyensúlyelméleti modellek közvetlen előzményének tekinthető, tartalmazta azok szinte minden lényeges elemét és fogalmát - az egyensúlyi árak nem mások, mint a Koopmans-féle hatékonysági árak. Végezetül újraértelmezzük Koopmans modelljét a vállalati technológiai mikroökonómiai leírásának lehetséges eszközeként. Journal of Economic Literature (JEL) kód: B23, B41, C61, D20, D50. /===/ Generalizing from his experience in solving practical problems, Koopmans set about devising a linear model for analysing activity. Surprisingly, he found that economics at that time possessed no uniform, sufficiently exact theory of production or system of concepts for it. He set out in a pioneering study to provide a theoretical framework for a linear model for analysing activity by expressing first the axiomatic bases of production theory, which rest on the concept of technological sets. He is associated with exact definition of the concept of production efficiency and efficiency prices, and confirmation of their relation as mutual postulates within the linear model of activity analysis. Koopmans saw the present, purely technical definition of efficiency as a special case; he aimed to introduce and analyse the concept of economic efficiency. The study uses the duality precepts of linear programming to reconstruct the results for the latter. It is shown first that evidence confirming the duality precepts of linear programming is equal in value, and secondly that efficiency prices are really shadow prices in today's sense. Furthermore, the model for the interpretation of economic efficiency can be seen as a direct predecessor of the Arrow–Debreu–McKenzie models of general equilibrium theory, as it contained almost every essential element and concept of them—equilibrium prices are nothing other than Koopmans' efficiency prices. Finally Koopmans' model is reinterpreted as a necessary tool for microeconomic description of enterprise technology.

A gazdaságpolitikai döntéshozatal nemzetek fölötti centralizációja és a közösségi gazdaságtan. Az adóverseny elméletének néhány tanulsága (Supranational centralization of economic policy-making and community economics: some lessons from the theory of tax competition)

Mennyiben képes jelenleg a közösségi gazdaságtan az adópolitikák nemzetek fölötti centralizációjára vonatkozó politikai döntések megalapozására? Válaszunk röviden az lesz, hogy a közösségi gazdaságtan főárama - noha számos releváns gazdasági és politikai tényező hatását sikeresen elemzi - jelenleg nem kínál kielégítőnek tekinthető döntési kritériumokat a döntéshozók számára. Ennek oka, hogy központi szerepet játszik benne egy, a modellek szempontjából exogén és a közgazdasági elmélettől idegen tényező: a kormányzatok jóindulatára, pontosabban annak mértékére vonatkozó premissza. Tanulmányunk az adóverseny fiskális föderalista elméletét vizsgálja, és megpróbál általánosabb szinten is a közszektor gazdaságelméletének jelenlegi állapotára, valamint továbbfejlesztésére vonatkozó tanulságokat levonni. A kiutat az elméleti zsákutcából a kormányzati működés és döntéshozatal, valamint a kívánatos gazdaságpolitikai döntések elméletének összekapcsolása jelentheti. Erre megtörténtek az első kísérletek, de a szisztematikus és átfogó elemzés egyelőre várat magára. / === / How far can community economics provide a basis for political decision-making on supranational centralization of taxation policies? The short answer here will be that although the mainstream of community economics succeeds in analysing many relevant economic and political factors, it fails at present to provide satisfactory criteria for decisionmakers. This is because a central role is played in it by a factor exogenous to the models and alien to economic theory: the premise of the measure of goodwill from governments. The study examines the fiscal federalist theory of tax competition. It tries to draw conclusions, on a more general level, about the present state of the economic theory of the public sector and future development of it. The way out of the theoretical blind alley could be to link the theories of government operation and decision-making and of desirable economic-policy decision-making. The first attempts to do so have been made, but a systematic and comprehensive analysis is still awaited.

Mathematical constraints on a theory of human memory - Response

Colonius suggests that, in using standard set theory as the language in which to express our computational-level theory of human memory, we would need to violate the axiom of foundation in order to express meaningful memory bindings in which a context is identical to an item in the list. We circumvent Colonius's objection by allowing that a list item may serve as a label for a context without being identical to that context. This debate serves to highlight the value of specifying memory operations in set theoretic notation, as it would have been difficult if not impossible to formulate such an objection at the algorithmic level.

Risk Estimates of Dengue in Travelers to Dengue Endemic Areas Using Mathematical Models

Dengue has emerged as a frequent problem in international travelers. The risk depends on destination, duration, and season of travel. However, data to quantify the true risk for travelers to acquire dengue are lacking. We used mathematical models to estimate the risk of nonimmune persons to acquire dengue when traveling to Singapore. From the force of infection, we calculated the risk of dengue dependent on duration of stay and season of arrival. Our data highlight that the risk for nonimmune travelers to acquire dengue in Singapore is substantial but varies greatly with seasons and epidemic cycles. For instance, for a traveler who stays in Singapore for 1 week during the high dengue season in 2005, the risk of acquiring dengue was 0.17%, but it was only 0.00423% during the low season in a nonepidemic year such as 2002. Risk estimates based on mathematical modeling will help the travel medicine provider give better evidence-based advice for travelers to dengue endemic countries.