36 resultados para problem complexity

em Helda - Digital Repository of University of Helsinki


Relevância:

20.00% 20.00%

Publicador:

Resumo:

In this study I consider what kind of perspective on the mind body problem is taken and can be taken by a philosophical position called non-reductive physicalism. Many positions fall under this label. The form of non-reductive physicalism which I discuss is in essential respects the position taken by Donald Davidson (1917-2003) and Georg Henrik von Wright (1916-2003). I defend their positions and discuss the unrecognized similarities between their views. Non-reductive physicalism combines two theses: (a) Everything that exists is physical; (b) Mental phenomena cannot be reduced to the states of the brain. This means that according to non-reductive physicalism the mental aspect of humans (be it a soul, mind, or spirit) is an irreducible part of the human condition. Also Davidson and von Wright claim that, in some important sense, the mental aspect of a human being does not reduce to the physical aspect, that there is a gap between these aspects that cannot be closed. I claim that their arguments for this conclusion are convincing. I also argue that whereas von Wright and Davidson give interesting arguments for the irreducibility of the mental, their physicalism is unwarranted. These philosophers do not give good reasons for believing that reality is thoroughly physical. Notwithstanding the materialistic consensus in the contemporary philosophy of mind the ontology of mind is still an uncharted territory where real breakthroughs are not to be expected until a radically new ontological position is developed. The third main claim of this work is that the problem of mental causation cannot be solved from the Davidsonian - von Wrightian perspective. The problem of mental causation is the problem of how mental phenomena like beliefs can cause physical movements of the body. As I see it, the essential point of non-reductive physicalism - the irreducibility of the mental - and the problem of mental causation are closely related. If mental phenomena do not reduce to causally effective states of the brain, then what justifies the belief that mental phenomena have causal powers? If mental causes do not reduce to physical causes, then how to tell when - or whether - the mental causes in terms of which human actions are explained are actually effective? I argue that this - how to decide when mental causes really are effective - is the real problem of mental causation. The motivation to explore and defend a non-reductive position stems from the belief that reductive physicalism leads to serious ethical problems. My claim is that Davidson's and von Wright's ultimate reason to defend a non-reductive view comes back to their belief that a reductive understanding of human nature would be a narrow and possibly harmful perspective. The final conclusion of my thesis is that von Wright's and Davidson's positions provide a starting point from which the current scientistic philosophy of mind can be critically further explored in the future.

Relevância:

20.00% 20.00%

Publicador:

Resumo:

In the last thirty years, primarily feminist scholars have drawn attention to and re-evaluated the philosophy of Simone de Beauvoir (1908 1986). Her philosophical practice has been described as non-systematic, and her literary writing has been viewed as part of her non-systematic mode of philosophising. This dissertation radically deepens the question concerning Beauvoir s philosophical motivations for turning to literature as a mode to express subjectivity. It explicates the central concepts of Beauvoir s philosophy of existence, which are subjectivity, ambiguity, paradox and temporality, and their background in the modern traditions of existential philosophy and phenomenology. It also clarifies Beauvoir s main reason to turn to literature in order to express subjectivity as both singular and universal: as a specific mode of communication, literature is able to make the universality of existence manifest in the concrete, singular and temporal texture of life. In addition, the thesis gives examples of how Beauvoir s literary works contribute to an understanding of the complexity of subjectivity. I use the expression poetics of subjectivity to refer to the systematic relation between Beauvoir s existential and phenomenological notion of subjectivity and her literary works, and to her articulations of a creative mode of using language, especially in the novel. The thesis is divided into five chapters, of which the first three investigate Beauvoir s philosophy of existence at the intersection of the modern traditions of thought that began with René Descartes and Søren Kierkegaard s intuitions about subjectivity. Chapter 1 interprets Beauvoir s notion of ambiguity, as compared to paradox, and argues that both determine her notion of existence. Chapters 2 and 3 investigate the phenomenological side of Beauvoir s philosophy through a study of her response to early French interpretations of transcendental subjectivity, especially in the works of Jean-Paul Sartre and Maurice Merleau-Ponty. My analysis shows that Edmund Husserl s distinction between different levels of subjective experience is central to Beauvoir s understanding of subjectivity and to the different ego concepts she uses. Chapter 4 is a study of Beauvoir s reflections on the expression of subjective thought, and, more specifically, her philosophical conceptions of the metaphysical novel and the autobiography as two modes of indirect communication. Chapter 5, finally, compares two modes of investigating concrete subjectivity; Beauvoir s conceptual study of femininity in Le deuxième sexe and her literary expression of subjectivity in the novel L Invitée. My analysis reveals and explicates Beauvoir s original contribution to a comprehensive understanding of the becoming and paradox of human existence: the fundamental insight that these phenomena are sexed, historically as well as imaginatively.

Relevância:

20.00% 20.00%

Publicador:

Resumo:

The main research problem of this study was to explain how and why background music is used in Finnish department stores and how it is related to their marketing. The problem was investigated through the opinions, attitudes, and conceptions of the managers of Anttila, Sokos and Stockmann department stores. The data of study (N = 31) constituted of a www-survey to which the managers were asked to answer. In the first chapter of the study s theoretical section, the relationship between background music and an enterprise was examined. It was found that background music can serve as an aid in seeking competitive advantage. In the second chapter, the service encounter s environment and atmosphere in relation to marketing was examined and it was found that they are a part of customer s product or service experience. In the third chapter, the interaction process between service encounter atmosphere and consumer behaviour was examined and the essential finding was that atmospheric stimuli affects an individual through emotional, cognitive, and physiological processes, in which individual s personal characteristics are also in a great role. In the fourth chapter, the significance of background music s musical features was examined but the research results were found so contradictory that only the complexity of the studied phenomenon became clear. Findings from the study s empirical section showed that all examined department stores play background music and the usage of music is chain-controlled. The respondents considered background music in department stores as a fundamental element and they understood its significance in enterprise s marketing. The respondents also believed that customers consider background music important and pleasant. Respondents views on background music s effects to purchasing behaviour divided opinions more, but the majority however believed that background music has effects to purchasing behaviour. The main conclusion of the study was that background music is an important marketing tool, at least in a department store type service encounter.

Relevância:

20.00% 20.00%

Publicador:

Resumo:

The study is a philosophical analysis of Israel Scheffler’s philosophy of education, focusing on three crucial conceptions in his philosophy: the conception of rationality, the conception of human nature, and the conception of reality. The interrelations of these three concepts as well as their relations to educational theorizing are analysed and elaborated. A conceptual problem concerning Scheffler’s ideal of rationality derives from Scheffler’s supposition of the strong analogy between science education and moral education in terms of the ideal of rationality. This analogy is argued to be conceptually problematic, since the interconnections of rationality, objectivity, and truth, appear to differ from each other in the realms of ethics and science, given the presuppositions of ontological realism and ethical naturalism, to which Scheffler explicitly subscribes. This study considers two philosophical alternatives for solving this problem. The first alternative relates the analogy to the normative concept of personhood deriving from the teleological understanding of human nature. Nevertheless, this position turns out to be problematic for Scheffler, since he rejects all teleological thinking in his philosophy. The problem can be solved, as it is argued, by limiting Scheffler’s rejection of teleology – in light of his philosophical outlook on the whole – in a manner that allows a modest version of a teleological conception of human nature. The second alternative, based especially on Scheffler’s later contributions, is to suggest that reality is actually more complex and manifold than it appears to be in light of a contemporary naturalist worldview. This idea of plurealism – Scheffler’s synthesis of pluralism and realism – is represented especially in Scheffler’s contributions related to his debate with Nelson Goodman dealing with both constructivism and realism. The idea of plurealism is not only related to the ethics-science-distinction, but is more widely related to the relationship between ontological realism and the incommensurable systems of description in diverse realms of human understanding. The Scheffler-Goodman debate is also analysed in relation to the contemporary constructivism-realism debate in educational philosophy. In terms of educational questions, Scheffler’s plurealism is argued as offering a fruitful perspective. Scheffler’s philosophy of education can be interpreted as searching for solutions to the problems deriving from the tension between the tradition of analytical philosophy and the complexity and multiplicity of educational reality. The complexity of reality combined with the supposition of the limitedness of human knowledge does not lead Scheffler to relativism or particularism, but, in contrast, Schefflerian formulations of rationality and objectivity preserve the possibility for critical inquiry in all realms of educational reality. In light of this study, Scheffler’s philosophy of education provides an exceptional example of combining ontological realism, epistemological fallibilism, and the defence of the ideal of rationality, combined with a wide-ranging understanding of educational reality.

Relevância:

20.00% 20.00%

Publicador:

Resumo:

Failures in industrial organizations dealing with hazardous technologies can have widespread consequences for the safety of the workers and the general population. Psychology can have a major role in contributing to the safe and reliable operation of these technologies. Most current models of safety management in complex sociotechnical systems such as nuclear power plant maintenance are either non-contextual or based on an overly-rational image of an organization. Thus, they fail to grasp either the actual requirements of the work or the socially-constructed nature of the work in question. The general aim of the present study is to develop and test a methodology for contextual assessment of organizational culture in complex sociotechnical systems. This is done by demonstrating the findings that the application of the emerging methodology produces in the domain of maintenance of a nuclear power plant (NPP). The concepts of organizational culture and organizational core task (OCT) are operationalized and tested in the case studies. We argue that when the complexity of the work, technology and social environment is increased, the significance of the most implicit features of organizational culture as a means of coordinating the work and achieving safety and effectiveness of the activities also increases. For this reason a cultural perspective could provide additional insight into the problem of safety management. The present study aims to determine; (1) the elements of the organizational culture in complex sociotechnical systems; (2) the demands the maintenance task sets for the organizational culture; (3) how the current organizational culture at the case organizations supports the perception and fulfilment of the demands of the maintenance work; (4) the similarities and differences between the maintenance cultures at the case organizations, and (5) the necessary assessment of the organizational culture in complex sociotechnical systems. Three in-depth case studies were carried out at the maintenance units of three Nordic NPPs. The case studies employed an iterative and multimethod research strategy. The following methods were used: interviews, CULTURE-survey, seminars, document analysis and group work. Both cultural analysis and task modelling were carried out. The results indicate that organizational culture in complex sociotechnical systems can be characterised according to three qualitatively different elements: structure, internal integration and conceptions. All three of these elements of culture as well as their interrelations have to be considered in organizational assessments or important aspects of the organizational dynamics will be overlooked. On the basis of OCT modelling, the maintenance core task was defined as balancing between three critical demands: anticipating the condition of the plant and conducting preventive maintenance accordingly, reacting to unexpected technical faults and monitoring and reflecting on the effects of maintenance actions and the condition of the plant. The results indicate that safety was highly valued at all three plants, and in that sense they all had strong safety cultures. In other respects the cultural features were quite different, and thus the culturally-accepted means of maintaining high safety also differed. The handicraft nature of maintenance work was emphasised as a source of identity at the NPPs. Overall, the importance of safety was taken for granted, but the cultural norms concerning the appropriate means to guarantee it were little reflected. A sense of control, personal responsibility and organizational changes emerged as challenging issues at all the plants. The study shows that in complex sociotechnical systems it is both necessary and possible to analyse the safety and effectiveness of the organizational culture. Safety in complex sociotechnical systems cannot be understood or managed without understanding the demands of the organizational core task and managing the dynamics between the three elements of the organizational culture.

Relevância:

20.00% 20.00%

Publicador:

Resumo:

Design embraces several disciplines dedicated to the production of artifacts and services. These disciplines are quite independent and only recently has psychological interest focused on them. Nowadays, the psychological theories of design, also called design cognition literature, describe the design process from the information processing viewpoint. These models co-exist with the normative standards of how designs should be crafted. In many places there are concrete discrepancies between these two in a way that resembles the differences between the actual and ideal decision-making. This study aimed to explore the possible difference related to problem decomposition. Decomposition is a standard component of human problem-solving models and is also included in the normative models of design. The idea of decomposition is to focus on a single aspect of the problem at a time. Despite its significance, the nature of decomposition in conceptual design is poorly understood and has only been preliminary investigated. This study addressed the status of decomposition in conceptual design of products using protocol analysis. Previous empirical investigations have argued that there are implicit and explicit decomposition, but have not provided a theoretical basis for these two. Therefore, the current research began by reviewing the problem solving and design literature and then composing a cognitive model of the solution search of conceptual design. The result is a synthetic view which describes recognition and decomposition as the basic schemata for conceptual design. A psychological experiment was conducted to explore decomposition. In the test, sixteen (N=16) senior students of mechanical engineering created concepts for two alternative tasks. The concurrent think-aloud method and protocol analysis were used to study decomposition. The results showed that despite the emphasis on decomposition in the formal education, only few designers (N=3) used decomposition explicitly and spontaneously in the presented tasks, although the designers in general applied a top-down control strategy. Instead, inferring from the use of structured strategies, the designers always relied on implicit decomposition. These results confirm the initial observations found in the literature, but they also suggest that decomposition should be investigated further. In the future, the benefits and possibilities of explicit decomposition should be considered along with the cognitive mechanisms behind decomposition. After that, the current results could be reinterpreted.

Relevância:

20.00% 20.00%

Publicador:

Resumo:

The purpose of the present study was to increase understanding of the interaction of rural people and, specifically, women with the environment in a dry area in Sudan. The study that included both nomadic pastoralists and farmers aimed at answering two main research questions, namely: What kinds of roles have the local people, and the women in particular, had in land degradation in the study area and what kinds of issues would a gender-sensitive, forestry-related environmental rehabilitation intervention need to consider there? The study adopted the definition of land degradation as proposed by the United Nations Convention to Combat Desertification (UNCCD), which describes land degradation as reduction or loss the biological or economic productivity and complexity of land in arid, semi-arid and dry sub-humid areas. The Convention perceives desertification as land degradation. The dry study area in Sudan, South of the Sahara, has been the subject of land degradation or desertification discussions since the 1970s, and other studies have been also conducted to assess the degradation in the area. Nevertheless, the exact occurrence, scale and local significance of land degradation in the area is still unclear. This study explored how the rural population whose livelihood depended on the area, perceived environmental changes occurring there and compared their conceptions with other sources of information of the area such as research reports. The main fieldwork methods included interviews with open-ended questions and observation of people and the environment. The theoretical framework conceptualised the rural population as land users whose choices of environmental activities are affected by multiple factors in the social and biophysical contexts in which they live. It was emphasised that these factors have their own specific characteristics in different contexts, simultaneously recognising that there are also factors that generally affect environmental practices in various areas such as the land users' environmental literacy (conceptions of the environment), gender and livelihood needs. The people studied described that environmental changes, such as reduced vegetation cover and cropland production, had complicated the maintenance of their livelihoods in the study area. Some degraded sites were also identified through observations during the fieldwork. Whether a large-scale reduction of cropland productivity had occurred in the farmers' croplands remained, however, unclear. The study found that the environmental impact of the rural women's activities varied and was normally limited. The women's most significant environmental impact resulted from their cutting of trees, which was likely to contribute, at least in some places, to land degradation, affecting the environment together with climate and livestock. However, when a wider perspective is taken, it becomes questionable whether the women have really played roles in land degradation, since gender, poverty and the need to maintain livelihood had caused them to conduct environmentally harmful activities. The women have had, however, no power to change the causes of their activities. The findings further suggested that an inadequate availability of food was the most critical problem in the study area. Therefore, an environmental programme in the area was suggested to include technical measures to increase the productivity of croplands, opportunities for income generation and readiness to co-operate with other programmes to improve the local people's abilities to maintain their livelihoods. In order to protect the environment and alleviate the women's work burden, the introduction of fuel-saving stoves was also suggested. Furthermore, it was suggested that increased planting of trees on homesteads would be supported by an easy availability of tree seedlings. Planting trees on common property land was, however, perceived as extremely demanding in the study area, due to scarcity of such land. In addition, it became apparent that the local land users, and women in particular, needed to allocate their labour to maintain the immediate livelihood of their families and were not motivated to allocate their labour solely for environmental rehabilitation. Nonetheless, from the point of view of the existing social structures, women's active participation in a community-based environmental programme would be rather natural, particularly among the farmer women who had already formed a women's group and participated in communal decision making. Forming of a women group or groups was suggested to further support both the farmer women's and pastoral women's active participation within an environmental programme and their general empowerment. An Environmental programme would need to acknowledge that improving rural people's well-being and maintaining their livelihood in the study area requires development and co-operation with various sectors in Sudan.

Relevância:

20.00% 20.00%

Publicador:

Resumo:

The problem of recovering information from measurement data has already been studied for a long time. In the beginning, the methods were mostly empirical, but already towards the end of the sixties Backus and Gilbert started the development of mathematical methods for the interpretation of geophysical data. The problem of recovering information about a physical phenomenon from measurement data is an inverse problem. Throughout this work, the statistical inversion method is used to obtain a solution. Assuming that the measurement vector is a realization of fractional Brownian motion, the goal is to retrieve the amplitude and the Hurst parameter. We prove that under some conditions, the solution of the discretized problem coincides with the solution of the corresponding continuous problem as the number of observations tends to infinity. The measurement data is usually noisy, and we assume the data to be the sum of two vectors: the trend and the noise. Both vectors are supposed to be realizations of fractional Brownian motions, and the goal is to retrieve their parameters using the statistical inversion method. We prove a partial uniqueness of the solution. Moreover, with the support of numerical simulations, we show that in certain cases the solution is reliable and the reconstruction of the trend vector is quite accurate.

Relevância:

20.00% 20.00%

Publicador:

Resumo:

The object of this dissertation is to study globally defined bounded p-harmonic functions on Cartan-Hadamard manifolds and Gromov hyperbolic metric measure spaces. Such functions are constructed by solving the so called Dirichlet problem at infinity. This problem is to find a p-harmonic function on the space that extends continuously to the boundary at inifinity and obtains given boundary values there. The dissertation consists of an overview and three published research articles. In the first article the Dirichlet problem at infinity is considered for more general A-harmonic functions on Cartan-Hadamard manifolds. In the special case of two dimensions the Dirichlet problem at infinity is solved by only assuming that the sectional curvature has a certain upper bound. A sharpness result is proved for this upper bound. In the second article the Dirichlet problem at infinity is solved for p-harmonic functions on Cartan-Hadamard manifolds under the assumption that the sectional curvature is bounded outside a compact set from above and from below by functions that depend on the distance to a fixed point. The curvature bounds allow examples of quadratic decay and examples of exponential growth. In the final article a generalization of the Dirichlet problem at infinity for p-harmonic functions is considered on Gromov hyperbolic metric measure spaces. Existence and uniqueness results are proved and Cartan-Hadamard manifolds are considered as an application.

Relevância:

20.00% 20.00%

Publicador:

Resumo:

Metabolism is the cellular subsystem responsible for generation of energy from nutrients and production of building blocks for larger macromolecules. Computational and statistical modeling of metabolism is vital to many disciplines including bioengineering, the study of diseases, drug target identification, and understanding the evolution of metabolism. In this thesis, we propose efficient computational methods for metabolic modeling. The techniques presented are targeted particularly at the analysis of large metabolic models encompassing the whole metabolism of one or several organisms. We concentrate on three major themes of metabolic modeling: metabolic pathway analysis, metabolic reconstruction and the study of evolution of metabolism. In the first part of this thesis, we study metabolic pathway analysis. We propose a novel modeling framework called gapless modeling to study biochemically viable metabolic networks and pathways. In addition, we investigate the utilization of atom-level information on metabolism to improve the quality of pathway analyses. We describe efficient algorithms for discovering both gapless and atom-level metabolic pathways, and conduct experiments with large-scale metabolic networks. The presented gapless approach offers a compromise in terms of complexity and feasibility between the previous graph-theoretic and stoichiometric approaches to metabolic modeling. Gapless pathway analysis shows that microbial metabolic networks are not as robust to random damage as suggested by previous studies. Furthermore the amino acid biosynthesis pathways of the fungal species Trichoderma reesei discovered from atom-level data are shown to closely correspond to those of Saccharomyces cerevisiae. In the second part, we propose computational methods for metabolic reconstruction in the gapless modeling framework. We study the task of reconstructing a metabolic network that does not suffer from connectivity problems. Such problems often limit the usability of reconstructed models, and typically require a significant amount of manual postprocessing. We formulate gapless metabolic reconstruction as an optimization problem and propose an efficient divide-and-conquer strategy to solve it with real-world instances. We also describe computational techniques for solving problems stemming from ambiguities in metabolite naming. These techniques have been implemented in a web-based sofware ReMatch intended for reconstruction of models for 13C metabolic flux analysis. In the third part, we extend our scope from single to multiple metabolic networks and propose an algorithm for inferring gapless metabolic networks of ancestral species from phylogenetic data. Experimenting with 16 fungal species, we show that the method is able to generate results that are easily interpretable and that provide hypotheses about the evolution of metabolism.

Relevância:

20.00% 20.00%

Publicador:

Resumo:

Reuse of existing carefully designed and tested software improves the quality of new software systems and reduces their development costs. Object-oriented frameworks provide an established means for software reuse on the levels of both architectural design and concrete implementation. Unfortunately, due to frame-works complexity that typically results from their flexibility and overall abstract nature, there are severe problems in using frameworks. Patterns are generally accepted as a convenient way of documenting frameworks and their reuse interfaces. In this thesis it is argued, however, that mere static documentation is not enough to solve the problems related to framework usage. Instead, proper interactive assistance tools are needed in order to enable system-atic framework-based software production. This thesis shows how patterns that document a framework s reuse interface can be represented as dependency graphs, and how dynamic lists of programming tasks can be generated from those graphs to assist the process of using a framework to build an application. This approach to framework specialization combines the ideas of framework cookbooks and task-oriented user interfaces. Tasks provide assistance in (1) cre-ating new code that complies with the framework reuse interface specification, (2) assuring the consistency between existing code and the specification, and (3) adjusting existing code to meet the terms of the specification. Besides illustrating how task-orientation can be applied in the context of using frameworks, this thesis describes a systematic methodology for modeling any framework reuse interface in terms of software patterns based on dependency graphs. The methodology shows how framework-specific reuse interface specifi-cations can be derived from a library of existing reusable pattern hierarchies. Since the methodology focuses on reusing patterns, it also alleviates the recog-nized problem of framework reuse interface specification becoming complicated and unmanageable for frameworks of realistic size. The ideas and methods proposed in this thesis have been tested through imple-menting a framework specialization tool called JavaFrames. JavaFrames uses role-based patterns that specify a reuse interface of a framework to guide frame-work specialization in a task-oriented manner. This thesis reports the results of cases studies in which JavaFrames and the hierarchical framework reuse inter-face modeling methodology were applied to the Struts web application frame-work and the JHotDraw drawing editor framework.

Relevância:

20.00% 20.00%

Publicador:

Resumo:

In this thesis we study a series of multi-user resource-sharing problems for the Internet, which involve distribution of a common resource among participants of multi-user systems (servers or networks). We study concurrently accessible resources, which for end-users may be exclusively accessible or non-exclusively. For all kinds we suggest a separate algorithm or a modification of common reputation scheme. Every algorithm or method is studied from different perspectives: optimality of protocols, selfishness of end users, fairness of the protocol for end users. On the one hand the multifaceted analysis allows us to select the most suited protocols among a set of various available ones based on trade-offs of optima criteria. On the other hand, the future Internet predictions dictate new rules for the optimality we should take into account and new properties of the networks that cannot be neglected anymore. In this thesis we have studied new protocols for such resource-sharing problems as the backoff protocol, defense mechanisms against Denial-of-Service, fairness and confidentiality for users in overlay networks. For backoff protocol we present analysis of a general backoff scheme, where an optimization is applied to a general-view backoff function. It leads to an optimality condition for backoff protocols in both slot times and continuous time models. Additionally we present an extension for the backoff scheme in order to achieve fairness for the participants in an unfair environment, such as wireless signal strengths. Finally, for the backoff algorithm we suggest a reputation scheme that deals with misbehaving nodes. For the next problem -- denial-of-service attacks, we suggest two schemes that deal with the malicious behavior for two conditions: forged identities and unspoofed identities. For the first one we suggest a novel most-knocked-first-served algorithm, while for the latter we apply a reputation mechanism in order to restrict resource access for misbehaving nodes. Finally, we study the reputation scheme for the overlays and peer-to-peer networks, where resource is not placed on a common station, but spread across the network. The theoretical analysis suggests what behavior will be selected by the end station under such a reputation mechanism.

Relevância:

20.00% 20.00%

Publicador:

Resumo:

Minimum Description Length (MDL) is an information-theoretic principle that can be used for model selection and other statistical inference tasks. There are various ways to use the principle in practice. One theoretically valid way is to use the normalized maximum likelihood (NML) criterion. Due to computational difficulties, this approach has not been used very often. This thesis presents efficient floating-point algorithms that make it possible to compute the NML for multinomial, Naive Bayes and Bayesian forest models. None of the presented algorithms rely on asymptotic analysis and with the first two model classes we also discuss how to compute exact rational number solutions.