The paper considers the intellectual computer mathematics system InparSolver, which is designed to automatically explore and solve basic classes of computational mathematics problems on multi-core computers with graphics accelerators. The problems of results reliability of solving problems with approximate input data are outlined. The features of the use of existing computer mathematics systems are analyzed, their weaknesses are found. The functionality of InparSolver, some innovative approaches to the implementation of effective solutions to problems in a hybrid architecture are (...) described. Examples of applied usage of InparSolver for processes mathematical modeling in various subject areas are given. Nowadays, new more complex objects and phenomena in many subject areas are constantly emerging, which are subject to mathematical research on a computer. This encourages the development of new numerical methods and technologies of mathematical modeling, as well as the creation of more powerful computers for their implementation. With the advent and constant development of supercomputers of various architectures, the problems of their effective use, expansion of tasks range should be solved, ensuring the reliability of computer results and increasing the level of intellectual information support for users ‒ specialists in various fields. Today, the issues of solving these problems are given special attention by many specialists in the fields of information technology and parallel programming. The world's leadingscientists in the field of computer technology see the solution to the problems of efficient usage of modern supercomputers in algorithmic software creation that easily adapts to different computer architectures with different types of memory and coprocessors, supports efficient parallelism on millions of cores etc. In addition, improving the efficiency of high-performance computing on modern supercomputers is provided by their intellectualization, transferring to the computer to perform a significant part of the functions. The industry of development and usage of intelligent computer technologies is one of the main directions of science and technology development in modern society. (shrink)

The article analyses innovative ethno-geographical systems of the first century BC. During Hellenistic times, the science of geography made use of increasingly advanced mathematical and astronomical skills to ensure a scientific basis for the cartographical project; however, this geographical research apparently disregarded the natural and human environments. There is a paradigm change in the referred century. The Stoic Posidonius focuses on the concept of zones found in the early philosophers and finds a compromise between the ‘scientific’ and the ‘descriptive’ (...) geographies. Likewise, Vitruvius conveys a geographical system which associates climatic, somatic, and psychic features. (shrink)

The ARXMLIV corpus is a remarkable collection of text containing scientific mathematical discourse. With more than half a million documents, it is an ambitious target for large scale linguistic and semantic analysis, requiring a generalized and distributed approach. In this paper we implement an architecture which solves and automates the issues of knowledge representation and knowledge management, providing an abstraction layer for distributed development of semantic analysis tools. Furthermore, we enable document interaction and visualization and present current implementations (...) of semantic tools and follow-up applications using this architecture. We identify five different stages, or purposes, which such architecture needs to address, encapsulating each in an independent module. These stages are determined by the different properties of the document formats used, as well as the state of processing and linguistic enrichment introduced so far. We discuss the need of migration between XML representations and the challenges it would pose on our system, revealing the benefits and trade-off of each format we employ. In the heart of the architecture lies the Semantic Blackboard module. The Semantic Blackboard comprises a system based on a centralized RDF database which can facilitate distributed corpus analysis of arbitrary applications, or analysis modules. This is achieved by providing a document abstraction layer and a mechanism for storing, reusing and communicating results via RDF stand-off annotations deposited in the central database. Achieving a properly encapsulated and automated pipeline from the input corpus document to a semantically enriched output in a state-of-the-art representation is the task of the Preprocessing, Semantic Result and Output Generation modules. Each of them addresses the task of format migration and enhances the document for further semantic enrichment or aggregation. The fifth module, targeting Visualization and Feedback, enables user interaction and display of different stages of processing.. (shrink)

We present a search engine for mathematical formulae. The MathWebSearch system harvests the web for content representations (currently MathML and OpenMath) of formulae and indexes them with substitution tree indexing, a technique originally developed for accessing intermediate results in automated theorem provers. For querying, we present a generic language extension approach that allows constructing queries by minimally annotating existing representations. First experiments show that this architecture results in a scalable application.

MKM has been defined as the quest for technologies to manage mathematical knowledge. MKM “in the small” is well-studied, so the real problem is to scale up to large, highly interconnected corpora: “MKM in the large”. We contend that advances in two areas are needed to reach this goal. We need representation languages that support incremental processing of all primitive MKM operations, and we need software architectures and implementations that implement these operations scalably on large knowledge bases. We present (...) instances of both in this paper: the MMT framework for modular theory-graphs that integrates meta-logical foundations, which forms the base of the next OMDOC version; and TNTBase, a versioned storage system for XML- based document formats. TNTBase becomes an MMT database by instantiating it with special MKM operations for MMT. (shrink)

Cover -- Title -- Copyright -- Dedication -- Contents -- List of Figures -- List of Tables -- Acknowledgements -- 1 Physics, Knowledge and Semiosis -- 2 Language, Knowledge and Description -- 3 Mathematical Statements and Expressions -- 4 Mathematical Symbols and the Architecture of the Grammar of Mathematics -- 5 Genres of Language and Mathematics -- 6 Images and the Knowledge of Physics -- 7 Physics and Semiotics -- Appendix A System Network Conventions -- Appendix (...) B Full System Networks for Mathematics -- Appendix C Details of Corpus -- Index. (shrink)

Algorithmic software for mathematical modeling of structural stability is considered, which is reduced to solving a partial generalized eigenvalues problem of sparse matrices, with automatic parallelization of calculations on modern parallel computers with graphics processors. Peculiarities of realization of parallel algorithms for different structures of sparse matrices are presented. The times of solving the problem of stability of composite materialsusing a three-dimensional model of "finite size fibers" on computers of different architectures are given. In mathematical modeling of physical (...) and technical processes in many cases there is a need to solve problems of algebraic problem of eigenvalues with sparse matrices of large volumes. In particular, such problems arise in the analysis of the strength of structures in civil and industrial construction, aircraft construction, electric welding, etc. The solving to these problems is to determine the eigenvalues and eigenvectors of sparse matrices of different structure. The efficiency of solving these problems largely depends on the effectiveness of mathematical modeling of the problem as a whole. Continuous growth of task parameters, calculation of more complete models of objects and processes on computers require an increase in computer productivity. High-performance computing requirements are far ahead of traditional parallel computing, even with multicore processors. High-performance computing requirements are far ahead of traditional parallel computing, even with multicore processors. Today, this problem is solved by using powerful supercomputers of hybrid architecture, such as computers with multicore processors and graphics processors, which combine MIMD and SIMD architectures. But the potential of high-performance computers can be used to the fullest only with algorithmic software that takes into account both the properties of the task and the features of the hybrid architecture. Complicating the architecture of modern high-performance supercomputers of hybrid architecture, which are actively used for mathematical modeling means a significant complication of efficient use of these resources in creating parallel algorithms and programs. here are problems with the creation of algorithmic software with automatic execution of stages of work, which are associated with the efficient use of computing resources, ways to store and process sparse matrices, analysis of the reliability of computer results. This makes it possible to significantly increase the efficiency of mathematical modeling of practical problems on modern high-performance computers, as well as free users from the problems of parallelization of complex problems. he developed algorithmic software automatically implements all stages of parallel computing and processing of sparse matrices on a hybrid computer. It was used at the Institute of Mechanics named after S.P. Tymoshenko NAS of Ukraine in modeling the strength problems of composite material. A significant improvement in the time characteristics of mathematical modeling was obtained. Problems of mathematical modeling of the properties of composite materials has an important role in designing the processes of deformation and destruction of products in various subject areas. Algorithmic software for mathematical modeling of structural stability is considered, which is reduced to solving a partial generalized problem of eigen values of sparse matrices of different structure of large orders, with automatic parallelization of calculations on modern parallel computers with graphics processors. The main methodological principles and features of implementation of parallel algorithms for different structures of sparse matrices are presented, which ensure effective implementation of multilevel parallelism of a hybrid system and reduce data exchange time during the computational process. As an example of these approaches, a hybrid algorithm of the iteration method in subspace for tape and block-diagonal matrices with a frame for computers of hybrid architecture is given. Peculiarities of data decomposition for matrices of profile structure at realization of parallel algorithms are considered. The proposed approach provides automatic determination of the required topology of the hybrid computer and the optimal amount of resources for the organization of an efficient computational process. The results of testing the developed algorithmic software for problems from the collection of the University of Florida, as well as the times of solving the problem of stability of composite materials using a three-dimensional model of "finite size fibers" on computers of different architectures. The results show a significant improvement in the time characteristics of solving problems. (shrink)

The ability to improve in speed and accuracy as a result of repeating some task is an important hallmark of intelligent biological systems. Although gradual behavioral improvements from practice have been modeled in spiking neural networks, few such models have attempted to explain cognitive development of a task as complex as addition. In this work, we model the progression from a counting-based strategy for addition to a recall-based strategy. The model consists of two networks working in parallel: a slower basal (...) ganglia loop and a faster cortical network. The slow network methodically computes the count from one digit given another, corresponding to the addition of two digits, whereas the fast network gradually “memorizes” the output from the slow network. The faster network eventually learns how to add the same digits that initially drove the behavior of the slower network. Performance of this model is demonstrated by simulating a fully spiking neural network that includes basal ganglia, thalamus, and various cortical areas. Consequently, the model incorporates various neuroanatomical data, in terms of brain areas used for calculation and makes psychologically testable predictions related to frequency of rehearsal. Furthermore, the model replicates developmental progression through addition strategies in terms of reaction times and accuracy, and naturally explains observed symptoms of dyscalculia. (shrink)

While many different mechanisms contribute to the generation of spatial order in biological development, the formation of morphogenetic fields which in turn direct cell responses giving rise to pattern and form are of major importance and essential for embryogenesis and regeneration. Most likely the fields represent concentration patterns of substances produced by molecular kinetics. Short range autocatalytic activation in conjunction with longer range “lateral” inhibition or depletion effects is capable of generating such patterns (Gierer and Meinhardt, 1972). Non-linear reactions are (...) required, and mathematical criteria were derived to design molecular models capable of pattern generation. The classical embryological feature of proportion regulation can be incorporated into the models. The conditions are mathematically necessary for the simplest two-factor case, and are likely to be a fair approximation in multi-component systems in which activation and inhibition are systems parameters subsuming the action of several agents. Gradients, symmetric and periodic patterns, in one or two dimensions, stable or pulsing in time, can be generated on this basis. Our basic concept of autocatalysis in conjunction with lateral inhibition accounts for self-regulatory biological features, including the reproducible formation of structures from near-uniform initial conditions as required by the logic of the generation cycle. Real tissue form, for instance that of budding Hydra, may often be traced back to local curvature arising within an initially relatively flat cell sheet, the position of evagination being determined by morphogenetic fields. Shell theory developed for architecture may also be applied to such biological processes. (shrink)

In this contribution we propose an agent architecture for theorem proving which we intend to investigate in depth in the future. The work reported in this paper is in an early state, and by no means finished. We present and discuss our proposal in order to get feedback from the Calculemus community.

We present a search engine for mathematical formulae. The MathWebSearch system harvests the web for content representations of formulae and indexes them with substitution tree indexing. In version 0.4 we have parallelized and distributed the search server and augmented the web interface with a new JavaScript-based visual editor for content math formulae. Furthermore, we have extended the query language by generalization, variants, unification, and text search facilities, which can also be mixed. Our experiments show that this architecture (...) results in a scalable application. (shrink)

Between 1906 and 1911, as a response to Betrand’s Russell’s review of La Science et l’Hypothèse, Henri Poincaré launched an attack on the movement to formalise the foundations of mathematics reducing it to logic. The main point is the following: the universality of logic is based on the idea that their truth is independent of any context including epistemic and cultural contexts. From the free context notion of truth and proof it follows that, given an axiomatic system, nothing new (...) can follow. One of the main strategies of Poincaré’s solution to this dilemma is based on the notions of understanding and of grasping the architecture of the propositions of mathematics. According to this view mathematic rigour does not reduce to “derive blindly” without gaps from axioms,mathematical rigour is, according to Poincaré, closely linked to the ability to grasp the architecture of mathematics and contribute to an extension of the meaning embedded in structures that constitute the architecture of mathematical propositions. The focus of my paper relates precisely to the notion of architecture and to the notion of understanding. According to my reconstruction, Poincaré’s suggestions could be seen as pointing out that understanding is linked to reason not only within a structure but reasoning about the structure. (shrink)

While collaboration has always played an important role in many cases of discovery and creation, recent developments such as the web facilitate and encourage collaboration at scales never seen before, even in areas such as mathematics, where contributions by single individuals have historically been the norm. This new scenario poses a challenge at the theoretical level, as it brings out the importance of various issues which, as of yet, have not been sufficiently central to the study of problem-solving, discovery, and (...) creativity. We analyze the case of collective and web-based proof events in mathematics, which share their temporal and social nature with every case of collective problem-solving. We propose that some ideas from cognitive architectures, in particular, the notion of codelet—understood as an agent engaged in one of a multitude of available tasks—can illuminate our understanding of collective problem-solving and act as a natural bridge from some of the theoretical aspects of collective, web-based discovery to the practical concern of designing cognitively inspired systems to support collective problem-solving. We use the Pythagorean Theorem and its many proofs as a case study to illustrate our approach. (shrink)

Intelligent Guided Learning System is a computer system that uses computers to imitate the experience and methods of teaching experts to assist in teaching; ITS provides learners with personalized learning resources and adaptive teaching methods, thus reducing students’ dependence on teachers and realizing independent learning. After years of research and development, and with the help of artificial intelligence technology, ITS has basically formed a stable structure and unified implementation specifications and has given birth to many excellent products in (...) disciplines such as basic computer, language, medicine, and mathematics. Based on the above background, this paper takes the system structure of universal ITS as the basis, combines the characteristics of music sight-singing subject in teaching contents and teaching methods, researches the intelligent guidance system model of music sight-singing, and completes the design of the overall system architecture. The specific strategies and implementation methods of the teaching methods, resource recommendation, and ability assessment components of the teaching model are studied. The research includes the definition of the difficulty characteristics of the score, the design of the score recommendation algorithm, the design of the sight-singing scoring algorithm, and the experimental analysis of each algorithm. It is proposed that the difficulty feature-based score recommendation algorithm is the core component of resource recommendation in the teacher model, and the sight-singing scoring algorithm is the basis for ability assessment and learner model update. (shrink)

Explorations in Systems Phenomenology in Relation to Ontology, Hermeneutics and the Meta-dialectics of Design -/- SYNOPSIS A Phenomenological Analysis of Emergent Design is performed based on the foundations of General Schemas Theory. The concept of Sign Engineering is explored in terms of Hermeneutics, Dialectics, and Ontology in order to define Emergent Systems and Metasystems Engineering based on the concept of Meta-dialectics. -/- ABSTRACT Phenomenology, Ontology, Hermeneutics, and Dialectics will dominate our inquiry into the nature of the Emergent Design of the (...) System and its inverse dual, the Meta-system. This is an speculative dissertation that attempts to produce a philosophical, mathematical, and theoretical view of the nature of Systems Engineering Design. Emergent System Design, i.e., the design of yet unheard of and/or hitherto non-existent Systems and Metasystems is the focus. This study is a frontal assault on the hard problem of explaining how Engineering produces new things, rather than a repetition or reordering of concepts that already exist. In this work the philosophies of E. Husserl, A. Gurwitsch, M. Heidegger, J. Derrida, G. Deleuze, A. Badiou, G. Hegel, I. Kant and other Continental Philosophers are brought to bear on different aspects of how new technological systems come into existence through the midwifery of Systems Engineering. Sign Engineering is singled out as the most important aspect of Systems Engineering. We will build on the work of Pieter Wisse and extend his theory of Sign Engineering to define Meta-dialectics in the form of Quadralectics and then Pentalectics. Along the way the various ontological levels of Being are explored in conjunction with the discovery that the Quadralectic is related to the possibility of design primarily at the Third Meta-level of Being, called Hyper Being. Design Process is dependent upon the emergent possibilities that appear in Hyper Being. Hyper Being, termed by Heidegger as Being (Being crossed-out) and termed by Derrida as Differance, also appears as the widest space within the Design Field at the third meta-level of Being and therefore provides the most leverage that is needed to produce emergent effects. Hyper Being is where possibilities appear within our worldview. Possibility is necessary for emergent events to occur. Hyper Being possibilities are extended by Wild Being propensities to allow the embodiment of new things. We discuss how this philosophical background relates to meta-methods such as the Gurevich Abstract State Machine and the Wisse Metapattern methods, as well as real-time architectural design methods as described in the Integral Software Engineering Methodology. One aim of this research is to find the foundation for extending the ISEM methodology to become a general purpose Systems Design Methodology. Our purpose is also to bring these philosophical considerations into the practical realm by examining P. Bourdieu’s ideas on the relationship between theoretical and practical reason and M. de Certeau’s ideas on practice. The relationship between design and implementation is seen in terms of the Set/Mass conceptual opposition. General Schemas Theory is used as a way of critiquing the dependence of Set based mathematics as a basis for Design. The dissertation delineates a new foundation for Systems Engineering as Emergent Engineering based on General Schemas Theory, and provides an advanced theory of Design based on the understanding of the meta-levels of Being, particularly focusing upon the relationship between Hyper Being and Wild Being in the context of Pure and Process Being. (shrink)

Contemporary architecture is increasingly grounded in science and mathematics. Architectural discourse has shifted radically from the sometimes disorienting Derridean deconstruction, to engaging scientific terms such as fractals, chaos, complexity, nonlinearity, and evolving systems. That's where the architectural action is -- at least for cutting-edge architects and thinkers -- and every practicing architect and student needs to become conversant with these terms and know what they mean. Unfortunately, the vast majority of architecture faculty are unprepared to explain them to (...) students, not having had a scientific education themselves. Here is an architecture book by an architect/scientist, just in time to help architects in the new millennium. Alexander discusses many of the scientific terms arising in cutting-edge architecture, and explains them to those who don't have scientific training or advanced mathematical knowledge. We find discussions of the evolution of forms; the importance of process in design; iteration; genetic algorithms; sequences of transformations; different levels of scale (i.e. fractals); etc. They are explained here by an architect who is also a scientist, because he wants to change the way architects think and build. Alexander is not merely popularizing other scientists' results and making them accessible to architects: he is in fact presenting new and original scientific work that ties many of these concepts together in a way that will be useful to architects. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Complex Mimetic SystemsHans Weigand (bio)The goal of science is to make the wonderful and complex understandable and simple—but not less wonderful.—Herb Simon, The Sciences of the Artificial11. IntroductionComplex systems theory stands for an approach in the social as well as natural and computational sciences that studies how interactions between parts give rise to collective behaviors of a system, and how the system interacts and forms relationships with (...) its environment. A typical example is the market economy, in which the order is emergent: it is the result of human action, but not the execution of any human design.2 Complex systems theory is currently a popular vehicle for understanding the complexities of the Internet; other examples of its application are as diverse as the stock market, ant colonies, epidemics, and the spread of business innovations. The focus in this article is on modeling social systems rather than physical or biological phenomena. [End Page 63]The mimetic theory developed by René Girard fits well with a systems perspective, as it has specific ideas about the drivers of human action (at the micro level) and cultural processes (at the macro level). The relationship has been explored earlier in several books and articles;3 in particular, in the work of the Centre de Recherche en Epistémologie Appliquée (CREA) in Paris. The systems perspective received a new stimulus in the 1990s in the form of complex adaptive systems (CAS),4 not least because of the agent-based computer simulations adopted at that time. The objective of this article is to introduce a special version of CAS, called complex mimetic systems (CMS), which builds on mimetic theory. The purpose of this article is twofold. On the one hand, the phrasing of mimetic theory in terms of CMS poses an interesting challenge to the theory to formalize its concepts and explicate possible limitations. On the other hand, CMS might be a useful contribution to the CAS field, as it incorporates concepts such as “social influence” that are very basic to but typically not spelled out in current CAS work.The structure of this article is as follows. In section 2, I briefly review the CAS literature and introduce the idea of complex mimetic systems. Section 3 describes a straw-man agent architecture for CMS, whereas in section 4, I move to the macro level and formalize the culture reproduction cycle. To illustrate the applicability of the CMS framework, I discuss in section 5 Ingo Piepers’s recent CAS analysis of the European international system in the period 1490–1945, and comment on it from a CMS perspective.2. Complex Mimetic SystemsThe characteristic features of complex systems are as follows:• Nonlinear effects: a small perturbation may have a large effect.• Feedback loops: the effects of an element’s behavior are fed back to it in some way. The feedback can be positive (amplification, runaway) or negative (damping).• Openness: a typical complex system interacts with its environment.• Memory: the history of a complex system may be important. Prior states may have an influence on present states.• Nesting, hierarchy: a complex system may itself be an element of a larger complex system. For example, clans in a tribal society, nation-states in an international system, or divisions in a company. [End Page 64]• Topology: the interactions of the elements are enabled and constrained by the network, or grid, of relationships between them.Complex Adaptive SystemsComplex adaptive systems (CAS) are a modern variant of complex systems theory. The elements of a complex adaptive system are called agents, as CAS builds on computational research in the field of multiagent systems.5 Although there is not much difference in the overall objectives, the use of agent simulation opens up interesting research opportunities. Let me clarify this by means of an introductory example given by Miller and Page, the standing ovation problem.6Standing ovations, in which waves of audience members stand up to express their particular appreciation of a performance, usually arise and evolve spontaneously. Suppose that we want to model this phenomenon. A starting point could be a simple mathematical model in which we distinguish N people, each receiving a signal... (shrink)

"The Emperor's New Mind" by Roger Penrose has received a great deal of both praise and criticism. This review discusses philosophical aspects of the book that form an attack on the "strong" AI thesis. Eight different versions of this thesis are distinguished, and sources of ambiguity diagnosed, including different requirements for relationships between program and behaviour. Excessively strong versions attacked by Penrose (and Searle) are not worth defending or attacking, whereas weaker versions remain problematic. Penrose (like Searle) regards the notion (...) of an algorithm as central to AI, whereas it is argued here that for the purpose of explaining mental capabilities the architecture of an intelligent system is more important than the concept of an algorithm, using the premise that what makes something intelligent is not what it does but how it does it. What needs to be explained is also unclear: Penrose thinks we all know what consciousness is and claims that the ability to judge Go "del's formula to be true depends on it. He also suggests that quantum phenomena underly consciousness. This is rebutted by arguing that our existing concept of "consciousness" is too vague and muddled to be of use in science. This and related concepts will gradually be replaced by a more powerful theory-based taxonomy of types of mental states and processes. The central argument offered by Penrose against the strong AI thesis depends on a tempting but unjustified interpretation of Goedel's incompleteness theorem. Some critics are shown to have missed the point of his argument. A stronger criticism is mounted, and the relevance of mathematical Platonism analysed. Architectural requirements for intelligence are discussed and differences between serial and parallel implementations analysed. (shrink)

This insight to art came from chess composition concentrating art in a very dense form. To identify and mathematically assess the uniqueness is the key applicable to other areas eg. computer programming. Maximization of uniqueness is minimization of entropy that coincides as well as goes beyond Information Theory (Shannon, 1948). The reusage of logic as a universal principle to minimize entropy, requires simplified architecture and abstraction. Any structures (e.g. plugins) duplicating or dividing functionality increase entropy and so unreliability (eg. (...) British Airways IT system). The ideas here were verified by my chess compositions, art works and complex information system as an author of each co uk, and were presented at conferences in Santorini, Adelaide, Geneva, Daejon and virtually. (shrink)

Neuroscience has illuminated the neural basis of decision-making, providing evidence that supports specific models of decision-processes. These models typically are quite mechanical, the realization of abstract mathematical “diffusion to bound” models. While effective decision-making seems to be essential for sophisticated behavior, central to an account of freedom, and a necessary characteristic of self-governing systems, it is not clear how the simple models neuroscience inspires can underlie the notion of self-governance. Drawing from both philosophy and neuroscience I explore ways in (...) which the proposed decision-making architectures can play a role in systems that can reasonably be thought of as “self-governing”. (shrink)

This book is the integrated presentation of a large body of work on understanding the operation of biological brains as systems.

Epistasis plays an important role in the genetic architecture of common human diseases and can be viewed from two perspectives, biological and statistical, each derived from and leading to different assumptions and research strategies. Biological epistasis is the result of physical interactions among biomolecules within gene regulatory networks and biochemical pathways in an individual such that the effect of a gene on a phenotype is dependent on one or more other genes. In contrast, statistical epistasis is defined as deviation (...) from additivity in a mathematical model summarizing the relationship between multilocus genotypes and phenotypic variation in a population. The goal of this essay is to review definitions and examples of biological and statistical epistasis and to explore the relationship between the two. Specifically, we present and discuss the following two questions in the context of human health and disease. First, when does statistical evidence of epistasis in human populations imply underlying biomolecular interactions in the etiology of disease? Second, when do biomolecular interactions produce patterns of statistical epistasis in human populations?Answers to these two reciprocal questions will provide an important framework for using genetic information to improve our ability to diagnose, prevent and treat common human diseases. We propose that systems biology will provide the necessary information for addressing these questions and that model systems such as bacteria, yeast and digital organisms will be a useful place to start. BioEssays 27:637–646, 2005. © 2005 Wiley Periodicals, Inc. (shrink)

The hybridization of two or more energy sources into a single power station is one of the widely discussed solutions to address the demand and supply havoc generated by renewable production, heating power, and cooling power) and its energy storage issues. Hybrid energy sources work based on the complementary existence of renewable sources. The combined cooling, heating, and power is one of the significant systems and shows a profit from its low environmental impact, high energy efficiency, low economic investment, and (...) sustainability in the industry. This paper presents an economic model of a microgrid system containing the CCHP system and energy storage considering the energy coupling and conversion characteristics, the effective characteristics of each microsource, and energy storage unit is proposed. The random forest regression model was optimized by the gravitational search algorithm. The test results show that the GSA-RFR model improves prediction accuracy and reduces the generalization error. The detail of the MG network and the energy storage architecture connected to the other renewable energy sources is discussed. The mathematical formulation of energy coupling and energy flow of the MG network including wind turbines, photovoltaic, CCHP system, fuel cell, and energy storage devices are presented. The testing system has been analysed under load peak cutting and valley filling of energy utilization index, energy utilization rate, the heat pump, the natural gas consumption of the microgas turbine, and the energy storage unit. The energy efficiency costs were observed as 88.2% and 86.9% with heat pump and energy storage operation comparing with GSA-RFR-based operation costs as 93.2% and 93% in summer and winter season, respectively. The simulation results extended the rationality and economy of the proposed model. (shrink)

We characterize abstraction in computer science by first comparing the fundamental nature of computer science with that of its cousin mathematics. We consider their primary products, use of formalism, and abstraction objectives, and find that the two disciplines are sharply distinguished. Mathematics, being primarily concerned with developing inference structures, has information neglect as its abstraction objective. Computer science, being primarily concerned with developing interaction patterns, has information hiding as its abstraction objective. We show that abstraction through information hiding is a (...) primary factor in computer science progress and success through an examination of the ubiquitous role of information hiding in programming languages, operating systems, network architecture, and design patterns. (shrink)

In setting up a formal system to specify a grammar formalism, the conventional (mathematical) wisdom is to start with primitives (basic primitive structures) as simple as possible, and then introduce various operations for constructing more complex structures. An alternate approach is to start with complex (more complicated) primitives, which directly capture some crucial linguistic properties and then introduce some general operations for composing these complex structures. These two approaches provide different domains of locality, i.e., domains over which various (...) types of linguistic dependencies can be specified. The latter approach, characterized as complicate locally, simplify globally (CLSG), pushes non‐local dependencies to become local, i.e., they arise in the basic primitive structures to start with.The CLSG approach has led to some new insights into syntactic description, semantic composition, language generation, statistical processing, and psycholinguistic phenomena, all these with possible relevance to the cognitive architecture of language. In this paper, we will describe these results in an introductory manner making use of the framework of lexicalized tree‐adjoining grammar (LTAG), a key example of the CLSG approach, thereby describing the interplay between formal analysis on the one hand and linguistic and processing issues on the other hand. (shrink)

Modern building practices rely on Victorian construction methods founded on industrial technologies. This article asks how it may be possible to develop an alternative approach to the construction of our homes and cities that is more environmentally responsive, works with the natural energy flows within matter and which is connected to natural systems, not insulated from them. The approach of plectic systems architecture suggests that it is possible to create living buildings by re-examining the dynamics of terrestrial matter through (...) a new approach to materials science and computation to engage in a fundamental re-visioning of what constitutes an architectural practice for the twenty-first century. (shrink)

Architecture of Mathematics describes the logical structure of Mathematics from its foundations to its real-world applications. It describes the many interweaving relationships between different areas of mathematics and its practical applications, and as such provides unique reading for professional mathematicians and nonmathematicians alike. This book can be a very important resource both for the teaching of mathematics and as a means to outline the research links between different subjects within and beyond the subject. Features All notions and properties are (...) introduced logically and sequentially, to help the reader gradually build understanding. Focusses on illustrative examples that explain the meaning of mathematical objects and their properties. Suitable as a supplementary resource for teaching undergraduate mathematics, and as an aid to interdisciplinary research. Forming the reader's understanding of Mathematics as a unified science, the book helps to increase his general mathematical culture. (shrink)

The article explores the evolution of the idea of correlating numbers and meanings, from ancient numerological systems to modern models of natural language processing based on vector representations and neural networks. The authors demonstrate that the aspiration to uncover hidden properties of objects by associating them with numbers and performing operations on these numbers has been a common thread across various cultures for millennia. The article traces the stages in the formation of the concept of mathesis universalis (universal mathematics), starting (...) from Aristotle and Proclus’s reflections on general mathematics and culminating in R. Descartes’s ideas about the existence of a universally applicable science of order and measure. Particular attention is devoted to G.W. Leibniz’s ambitious project of creating a universal characteristic – a formalized language in which all knowledge could be precisely expressed and logical inference could be carried out through calculations. It is noted that, despite inherent limitations, some of Leibniz’s intuitions have found embodiment in natural language processing technologies based on vector representations of words. However, the simple correlation of concepts with vectors is insufficient for the full realization of the idea of a calculus of reasoning, as it lacks a mechanism for deriving true statements from others. The development of the transformer architecture, which employs the mechanism of self-attention to construct context-dependent vector representations, has been a significant stride towards the automation of reasoning. Nevertheless, modern language models are based on the principle of maximizing likelihood rather than rigorous logical rules. The authors analyze conceptual solutions proposed for creating a comprehensive calculus of knowledge. In conclusion, it is emphasized that, while the idea of reducing reasoning to mathematical operations is undeniably attractive, one should be cautious about its maximalist interpretations. Cognition is constructive in nature, and even the most perfect model of formalization of reasoning cannot substitute for the practical wisdom and freedom of human thought. (shrink)

Science, technology, engineering, and mathematics (STEM) fields are central to any educational system. The term started with the National Science Foundation as “SMET” and was changed to STEM at a later date due to phonetic reasons. The term was not widely used until Virginia Tech University began offering a “STEM education” degree in 2005 (Friedman 2005). The term STEM covers a broad spectrum of different disciplines. While, in general, STEM is used as an umbrella term for the natural sciences, (...) engineering, mathematics, and technology-related fields, there are some variations in its usage. For example, the US National Science Foundation (NSF) characterizes STEM as including chemistry, computers and information technology, engineering, geoscience, life sciences, mathematical sciences, physics and astronomy, psychology, social sciences, educational research, and STEM education (Gonzalez and Kuezi 2012). Whereas this approach considers the social sciences to be of central importance to STEM and the STEM workforce (Spalter-Roth 2004), in the United Kingdom, STEM is seen to include fields such as medicine, dentistry, and architecture but exclude the social sciences and education (United Kingdom, House of Lords, Science, and Technology, Committee 2012). (shrink)

This edited volume, aimed at both students and researchers in philosophy, mathematics and history of science, highlights leading developments in the overlapping areas of philosophy and the history of modern mathematics. It is a coherent, wide ranging account of how a number of topics in the philosophy of mathematics must be reconsidered in the light of the latest historical research and how a number of historical accounts can be deepened by embracing philosophical questions.

This paper aims to show that in ancient Greek architecture, it is possible to find a genesis of the geometric modeling of visual perception present in propositions of Euclid's Optics, considering mathematical knowledge as a human wisdom expression. Let us start by emphasizing that mathematical thinking is not exclusively rooted in mathematical disciplines, but also includes the broad spectrum of human activities, including activities that come from everyday life. Based on this, we present a socio-cultural characterization (...) of human experience as the source and sustainer of mathematical knowledge. Thereafter, on the basis of a content and contextual analysis of _Euclid's Optics_, we explain the use and development of geometry in the study of various optical effects of visual perception in the architectural art embodied in the Acropolis of Athens; and we focus our analysis on one of the most important of these effects, the Acropolis of Athens. We focus our analysis on one of the most emblematic architectural works of ancient Greek culture: the Parthenon of Athens. (shrink)

The journal welcomes theorists and practitioners, working at the leading edge of their field, to guest-edit issues of the journal. In this present case, we are delighted to have secured the collaboration of Professor Neil Spiller and Doctor Rachel Armstrong of the Bartlett. I wish to thank them both, and their selected authors, for the originality and scope of the articles here assembled.

Formal verification involves the use of logical and computational methods to establish claims that are expressed in precise mathematical terms. These can include ordinary mathematical theorems, as well as claims that pieces of hardware or software, network protocols, and mechanical and hybrid systems meet their specifications. In practice, there is not a sharp distinction between verifying a piece of mathematics and verifying the correctness of a system: formal verification requires describing hardware and software systems in mathematical (...) terms, at which point establishing claims as to their correctness becomes a form of theorem proving. Conversely, the proof of a mathematical theorem may require a lengthy computation, in which case verifying the truth of the theorem requires verifying that the computation does what it is supposed to do. The gold standard for supporting a mathematical claim is to provide a proof, and twentieth-century developments in logic show most if not all conventional proof methods can be reduced to a small set of axioms and rules in any of a number of foundational systems. With this reduction, there are two ways that a computer can help establish a claim: it can help find a proof in the first place, and it can help verify that a purported proof is correct. Automated theorem proving focuses on the “finding” aspect. Resolution theorem provers, tableau theorem provers, fast satisfiability solvers, and so on provide means of establishing the validity of formulas in propositional and first-order logic. Other systems provide search procedures and decision procedures for specific languages and domains, such as linear or nonlinear expressions over the integers or the real numbers. Architectures like SMT combine domain-general search methods with domain-specific procedures. Computer algebra systems and specialized mathematical software packages provide means of carrying out mathematical computations, establishing mathematical bounds, or finding mathematical objects. A calculation can be viewed as a proof as well, and these systems, too, help establish mathematical claims. Automated reasoning systems strive for power and efficiency, often at the expense of guaranteed soundness. Such systems can have bugs, and it can be difficult to ensure that the results they deliver are correct. In contrast, interactive theorem proving focuses on the “verification” aspect of theorem proving, requiring that every claim is supported by a proof in a suitable axiomatic foundation. This sets a very high standard: every rule of inference and every step of a calculation has to be justified by appealing to prior definitions and theorems, all the way down to basic axioms and rules. In fact, most such systems provide fully elaborated “proof objects” that can be communicated to other systems and checked independently. Constructing such proofs typically requires much more input and interaction from users, but it allows us to obtain deeper and more complex proofs. The Lean Theorem Prover aims to bridge the gap between interactive and automated theorem proving, by situating automated tools and methods in a framework that supports user interaction and the construction of fully specified axiomatic proofs. The goal is to support both mathematical reasoning and reasoning about complex systems, and to verify claims in both domains. (shrink)

"_The Moment of Complexity_ is a profoundly original work. In remarkable and insightful ways, Mark Taylor traces an entirely new way to view the evolution of our culture, detailing how information theory and the scientific concept of complexity can be used to understand recent developments in the arts and humanities. This book will ultimately be seen as a classic."-John L. Casti, Santa Fe Institute, author of _Gödel: A Life of Logic, the Mind, and Mathematics_ The science of complexity accounts for (...) that inscrutable mix of chaos and order that governs our natural world. Complexity explains how networks emerge and function, how species organize into ecosystems, how stars form into galaxies, and how just a few sequences of DNA can account for so many different life forms. Recently, the idea of complexity has taken the worlds of business and politics by storm. The concept is used to account for phenomena as varied as the behavior of the stock market, the response of voting populations, and the effects of risk management. Even Disney has used complexity theory to manage crowd control at its theme parks. Given the startling development of new information technologies, we now live in a moment of unprecedented complexity, an era in which change occurs faster than our ability to comprehend it. With _The Moment of Complexity_, Mark C. Taylor offers a timely map for this unfamiliar terrain opening in our midst, unfolding an original philosophy through a remarkable synthesis of science and culture. According to Taylor, complexity is not just a breakthrough scientific concept, but the defining quality of the post-Cold War era. The flux of digital currents swirling around us, he argues, has created a new network culture with its own distinctive logic and dynamic. Drawing on resources from information theory and evolutionary biology, Taylor explains the operation of complex adaptive systems in social and cultural processes and captures a whole new zeitgeist in the making. To appreciate the significance of our emerging network culture, he claims, we need not only to understand contemporary scientific and technological transformations, but also to explore the subtle influences of art, architecture, philosophy, religion, and higher education. _The Moment of Complexity_, then, is a remarkable work of cultural analysis on a scale rarely seen today. To follow its trajectory is to learn how we arrived at this critical moment in our culture, and to know where we might head in the twenty-first century. (shrink)

Reasoning with quantifier expressions in natural language combines logical and arithmetical features, transcending strict divides between qualitative and quantitative. Our topic is this cooperation of styles as it occurs in common linguistic usage and its extension into the broader practice of natural language plus ‘grassroots mathematics’.We begin with a brief review of by changing the semantics of counting in natural ways. A first approach replaces cardinalities by abstract but well-motivated values of ‘mass’ or other mereological aggregating notions. A second approach (...) keeps the cardinalities but generalizes the meaning of counting to work in models that allow dependencies between variables.Finally, we return to our starting point in natural language, confronting the architecture of our formal systems with linguistic quantifier vocabulary and syntax, as well as with natural reasoning modules such as the monotonicity calculus. In addition to these encounters with formal semantics, we discuss the role of counting in semantic evaluation procedures for quantifier expressions and determine, for instance, which binary quantifiers are computable by finite ‘semantic automata’. We conclude with some general thoughts on yet further entanglements of logic and counting in formal systems, on rethinking the qualitative/quantitative divide, and on connecting our analysis to empirical findings in cognitive science. (shrink)

Because humans cannot know one another’s minds directly, every form of communication is a solution to the same basic problem: how can privately held information be made publicly accessible through manipulations of the physical environment? Language is by far the best studied response to this challenge. But there are a diversity of non-linguistic strategies for representation with external signs as well, from facial expressions and fog horns to chronological graphs and architectural renderings. The general thesis of this dissertation is that (...) there is an impressively wide spectrum of conventional systems of representation, corresponding to the many ways that the problem of communication can be solved, and that these systems can be described and explained using the tools of contemporary mathematical semantics. As a partial corrective to the countervailing norm, this work concentrates on the class of systems arguably most different from language— those governing the interpretation of pictorial images. Such representations dominate practical communication: witness the proliferation of maps, road signs, newspaper photographs, scientific illustrations, television shows, engineering drawings, and even the fleeting imagery of manual gesture. I argue that systems of depiction and languages embody a parallel technologies of communication. Both are based on semantics: systematic and conventional mappings from signs to representational content. But I also provide evidence that these semantics are profoundly divergent. Whereas the semantics of languages are based on arbitrary associations of signs and denotations, the semantics of systems of depiction are based on rules of geometrical transformation. Drawing on recent research in computer graphics and computational vision, I go on to develop a precise theory of pictorial semantics. This in turn facilitates a detailed comparison of iconic, image-based representation, and symbolic, language-based representation. A consequence of these conclusions is that the traditional, language-centric conception of semantics must be overhauled to allow for a more general semantic theory, one which countenances the wide variety of interpretive mechanisms actually at work in human communication. (shrink)

Development of decision-support and intelligent agent systems necessitates mathematical descriptions of uncertainty and fuzziness in order to model vagueness. This paper seeks to present an outline of Peirce’s triadic logic as a practical new way to model vagueness in the context of artificial intelligence. Charles Sanders Peirce was an American scientist–philosopher and a great logician whose triadic logic is a culmination of the study of semiotics and the mathematical study of anti-Cantorean model of continuity and infinitesimals. After presenting (...) Peircean semiotics within AI perspective, a mathematical formulation of a Peircean triadic set is given in relationship with classical and fuzzy sets. Using basic logical operators, all possible respective implication operators, bi-equivalence operators, valid rules of inference, and associative, distributive and commutative logical properties are derived and verified through the truth function approach. In order to suggest practical directions, aggregation operators for Peirce’s triadic logic have been formulated. A mathematical formulation of a medical diagnostic problem and ER diagram of a library management system using Peirce’s triadic relation show potential for further applications of the proposed triadic set and triadic logic. Alongside, a classical AI game—The Wumpus World—is implemented to show practical efficacy in comparison with binary implementation. Besides giving some preliminary formulations for trichotomous set theory and definition of finite automaton, development of hybrid architectures for intelligent agents and evolutionary computations are discussed as potential practical avenues for Peirce’s triadic logic. (shrink)

This article addresses the ability of Parallel Distributed Processing (PDP) networks to generate stagewise cognitive development in accordance with Piaget's theory of cognitive epigenesis. We carried out a replication study of the simulation experiments by McClelland (1989) and McClelland and Jenkins (1991) in which a PDP network learns to solve balance scale problems. In objective tests motivated from catastrophe theory, a mathematical theory of transitions in epigenetical systems, no evidence for stage transitions in network performance was found. It is (...) concluded that PDP networks lack the ability to recover the positive outcomes of analogous catastrophe analyses of real cognitive developmental data. In an attempt to further characterize the learning behaviour of PDP networks, we carried out a second simulation study using the discrimination‐shift paradigm. The results thus obtained indicate that PDP learning is compatible with the learning of stimulus‐response relationships, not with the acquisition of mediating rules such as conceived in (neo‐)Piagetian theory. In closing, we speculate about the feasibility of simulating stagewise development with alternative network architectures. (shrink)

The crux of the dispute between formalism and intuitionism, it is held, is not whether certain entities exist or not, but how the term function shall be used in mathematics. The identification of effective definition with general recursion fails because an undefined function lies concealed beneath the requirement of a finite number of substitutions, and a fresh characterization of effective definition is sought in terms of a hierarchy of ordinal recursions.A correspondence exists between primitive recursive properties and direct proofs, of (...) irrationality and transcendence for instance, and between general recursive properties and indirect proofs.Mathematics is a concept creating activity and the distinction between a formal mathematics devoid of meaning, at one level, and a meaningful metamathematics at the next is considered to be untenable.RésuméLa controverse entre les formalistes et les intuitionistes ne repose pas, considére‐t‐on, sur l'existence ou la non‐existence de certaines entités, mail sur l'usage du terme fonction en mathématiques. L'identification de la définition effective avec la récursion générate n'est pas valable parce qu'une fonction indéfinié est dissimulée sous la nécessité d'un nombre fini de substitutions, et une nouvelle caractérisation de définition effective est recherchée sous forme de hiérarchie de récursions ordinates.II existe, d'une part, une correspondance entre les propriétés primitives récursives et les preuves directes, d'irrationalité et de transcendance par exemple, et, d'autre part, entre les propriétés récursives générates et les preuves par l'absurde.Les mathématiques sont une activité créatrice de concepts, et la distinction entre les mathématiques formelles dépourvues de sens et, sur le plan supérieur, une méta‐mathématique chargée de sens, ne semble pas justifiée. (shrink)

FAR FROM BEING A PURELY ESOTERIC CONCERN of theoretical mathematicians, the examination of the ontological status of mathematical entities, I submit, has far-reaching implications for a very practical area of knowledge, namely, the method of science in general, and of physics in particular. Although physics and mathematics have since Newton's second derivative been inextricably wedded, modern physics has a particularly mathematical dependence. Physics has moved and continues to move further away from the possibility of direct empirical verification, primarily (...) because of the increasingly complex logistical problems of experimentation within the parameters of the very large and of the very small. As certain areas become more and more theoretical, with developments of this century in astrophysics, cosmology, and quantum mechanics, and more specifically, with the postulation of new hypothetical elementary particles based almost exclusively upon mathematical data, physics is forced to depend increasingly upon mathematics as a method for verifying physical possibility. Typically, a mathematical formulation descriptive of an empirically established phenomenon x is manipulated and made subject to derivation on the assumption that the new formulation will continue to correspond with physical reality, and may even yield new information about the phenomenon's behavior. Why, however, should a coherence between the empirically-defined world and mathematical processes be assumed? This coherence is, above all, dependent upon a hidden metaphysically strong presupposition about the ontological status of mathematical entities and their systems. (shrink)

John von Neumann (1903-1957) was undoubtedly one of the scientific geniuses of the 20th century. The main fields to which he contributed include various disciplines of pure and applied mathematics, mathematical and theoretical physics, logic, theoretical computer science, and computer architecture. Von Neumann was also actively involved in politics and science management and he had a major impact on US government decisions during, and especially after, the Second World War. There exist several popular books on his personality and (...) various collections focusing on his achievements in mathematics, computer science, and economy. Strangely enough, to date no detailed appraisal of his seminal contributions to the mathematical foundations of quantum physics has appeared. Von Neumann's theory of measurement and his critique of hidden variables became the touchstone of most debates in the foundations of quantum mechanics. Today, his name also figures most prominently in the mathematically rigorous branches of contemporary quantum mechanics of large systems and quantum field theory. And finally - as one of his last lectures, published in this volume for the first time, shows - he considered the relation of quantum logic and quantum mechanical probability as his most important problem for the second half of the twentieth century. The present volume embraces both historical and systematic analyses of his methodology of mathematical physics, and of the various aspects of his work in the foundations of quantum physics, such as theory of measurement, quantum logic, and quantum mechanical entropy. The volume is rounded off by previously unpublished letters and lectures documenting von Neumann's thinking about quantum theory after his 1932 Mathematical Foundations of Quantum Mechanics. The general part of the Yearbook contains papers emerging from the Institute's annual lecture series and reviews of important publications of philosophy of science and its history. (shrink)