Transforms Without Tears

The mathematician Alexander Borovik speaks of the importance of the `vertical unity' of mathematics. By this he means to draw our attention to the fact that many sophisticated mathematical concepts, even those introduced at the cutting-edge of research, have their roots in our most basic conceptualisations of the world. If this is so, we might expect any truly fundamental mathematical language to detect such structural commonalities. It is reasonable to suppose then that the lack of philosophical interest in such vertical unity is related to the prominence given by philosophers to languages which do not express well such relations. In this chapter, I suggest that we look beyond set theory to the newly emerging homotopy type theory, which makes plain what there is in common between very simple aspects of logic, arithmetic and geometry and much more sophisticated concepts. With this language in mind, new light is thrown on thenature of mathematical concepts with clear benefits for educationalists.

This review article aims to investigate the applicability of mathematical concepts in traditional fishing practices, with an insight into geometrical shapes embedded in fishing instruments through the lens of the Indian Knowledge System (IKS). Ubiratan D’Ambrosio, the father of ethnomathematics, emphasizes the cultural aspect of mathematical knowledge. This descriptive review article uncovers how traditional communities intuitively apply complex mathematical ideas like the isoperimetric inequality and geometric optimization by analysing the various fishing tools such as nets and traps. For instance, the circular design of cast nets demonstrates an understanding of maximizing area with a fixed perimeter. Similarly, other geometrical shapes like a cone, cylinder, and a rectangle can also be identified in various fishing instruments. This evidence indicates how various sophisticated mathematical concepts are embedded in traditional fishing activities, and we are using these concepts without having any formal theoretical knowledge. This study bridges the gap between culture and mathematics. As advocated by NEP-2020, this type of culturally based mathematics will profoundly impact the teaching and learning process.

Social network analysis involves delicate and sophisticated mathematical concepts which are abstract and challenging to acquire by traditional methods. Many studies show that female students perform poorly in computer science-related courses compared to male students. To address these issues, this research investigates the impact of employing a web-based interactive programming tool, Jupyter notebooks, on supporting deeper conceptual understanding and, therefore, better attainment levels of the course learning outcomes in a female setting. The work also highlights the overall experience and enjoyment this tool brought to the classroom. Document analysis and questionnaire were used as data collection methods. A mixed approach was applied, mid-term exam documents were investigated qualitatively, and the questionnaire was analyzed quantitatively. Our results showed that most students correctly perceived the learning outcomes and knowledge introduced within the Jupyter environment. Moreover, the interactive nature of Jupyter enhanced engagement and brought enjoyment to the learning experience.

Bio-inspired computing has demonstrated to be useful in several application areas. Over the last decade, new bio-inspired algorithms have emerged with applications for detection, optimisation and classification for use in computer vision tasks. On the other hand, automatic circle detection in digital images is considered an important and complex task for the computer vision community. Consequently, a tremendous amount of research has been devoted to find an optimal circle detector. This article presents an algorithm for the automatic detection of circular shapes from complicated and noisy images with no consideration of the conventional Hough transform principles. The proposed algorithm is based on newly developed Artificial Immune Optimisation (AIO) technique, known as the Clonal Selection Algorithm (CSA). The CSA is an effective method for searching and optimising following the Clonal Selection Principle (CSP) in the human immune system which generates a response according to the relationship between antigens (Ags), i.e. patterns to be recognised and antibodies (Abs), i.e. possible solutions. The algorithm uses the encoding of three points as candidate circles (x,y,r) over the edge image. An objective function evaluates if such candidate circles (Ab) are actually present in the edge image (Ag). Guided by the values of this objective function, the set of encoded candidate circles are evolved using the CSA so that they can fit to the actual circles on the edge map of the image. Experimental results over several synthetic as well as natural images with varying range of complexity validate the efficiency of the proposed technique with regard to accuracy, speed and robustness.

Automatic circle detection in digital images has received considerable attention over the last years in computer vision as several novel efforts aim for an optimal circle detector. This paper presents an algorithm for automatic detection of circular shapes considering the overall process as an optimization problem. The approach is based on the Harmony Search Algorithm (HSA), a derivative free meta-heuristic optimization algorithm inspired by musicians improvising new harmonies while playing. The algorithm uses the encoding of three points as candidate circles (harmonies) over the edge-only image. An objective function evaluates (harmony quality) if such candidate circles are actually present in the edge image. Guided by the values of this objective function, the set of encoded candidate circles are evolved using the HSA so that they can fit into the actual circles on the edge map of the image (optimal harmony). Experimental results from several tests on synthetic and natural images with a varying complexity range have been included to validate the efficiency of the proposed technique regarding accuracy, speed and robustness.

The pixel-wise Markov image models introduced in the previous chapters have a limited ability in dealing with texture images. In order to describe large scale behavior in texture images, it is necessary to increase the size of the neighborhood. However, this dramatically increases the number of parameters and makes the computation for the parameter estimation quite difficult. Considering that the scales of the texture elements can be diverse, one possible way of dealing with various scales of texture elements is to express image data hierarchically using a multiscale transform. Laplacian pyramids [27] and wavelet multiresolution representations [124] are techniques that can decompose an image into multiple scales. Successive filtering and decimation operations in multiscale transforms provide a pyramid structure for image data. As a result, we have a sequence of image data for various scales and frequency bands. Specifically, coarse image components with smaller image resolution are located at the higher level and fine image components capturing detail can be found at the lower levels of the image pyramid. This multiscale image structure can provide useful information for the analysis and representation of the image data, especially for texture images. Processing successively from coarse to fine levels of the pyramid, image features at different scales and frequency bands are captured and class labels are assigned accordingly to large regions with low frequency and then refined to small image blocks and eventually to pixels with higher frequency information. This multiscale processing also allows us to save computations. That is, coarsening the image data for a multiscale image representation may smooth out local minima and result in faster convergence with a reduced sensitivity to an initial class label. Then, the optimal class labels at the current level of the pyramid to the next finer level can be obtained by searching in the vicinity of the class labels obtained at the previous coarser level. Since there exist strong inter scale and intra scale dependencies among vertically and horizontally connected neighboring data in the image pyramid, the multiscale images can be described more efficiently via Markov modeling. Adopting Markov models for the multiscale image data, the following issues arise: (i) how to describe various inter scale and intra scale interactions in terms of the Markov models, (ii) how to formulate the optimization problem for the class labels in the image pyramid, (iii) how to obtain the optimal class labels defined in (ii). These issues are treated in this chapter.

Digital image processing, post-processing, and data integration techniques as applied to airborne and satellite remotely sensed data for the purpose of extracting useful Earth resources information will be discussed in this chapter. Image preprocessing and data reduction tools are described in the previous chapter. The concepts discussed in this chapter include: • Image processing techniques such as unsupervised image classifications, supervised image classifications, neural network classifiers, simulated annealing classifiers, and fuzzy logic classification systems • The most widely accepted indices and land use/land cover classification schemes • Post-processing techniques such as filtering and change detection • Accuracy assessment and validation of results • Data integration and spatial modeling including examples of integration of remotely sensed data with other conventional survey and map form data for Earth observation purposes

The previous chapters have shown how diverse are the patterns of household food production in Russia; most rural households grow a mixture of vegetables, roots, and fruit and might keep some poultry and small and large livestock for personal consumption, but many also have developed one or more branches geared to the market. Specialization is a feature of the post-Soviet ‘personal subsidiary economy’ and it has been associated with a deepening geographical division of labour in the sector. In this concluding chapter, we examine the extent to which the degree of market engagement of households maps onto the patterns of specialization and diversification we have described in the previous pages. The aim is to shed some light on the questions posed at the beginning of this book about the ‘nature’ of household production and its place in Russia’s evolving agri-food economy. It seems to us self-evident that the degree and character of household production’s subsumption to the market is the key to understanding the different directions in which it might be taken in the future. There is still little consensus among theorists of the peasantry about capitalism’s impact on small producers, but few would disagree that the market is the dominant transformative process whether, as in the case of households located in the suburbs of the major cities, it is to bind them ever more tightly into the market nexus or, as in the northern peripheries, its very absence reproduces their marginality. In the villages in which we conducted our surveys we encountered households positioned at different points along a ‘market’ spectrum. At one end there were those that were producing food exclusively to meet their own and their family’s consumption needs. These households sell surpluses, if they have them, in order to earn a money income to buy the staples they cannot produce themselves, but this is not their principal motivation. The noted Russian rural sociologist Vinogradskii (1999), among others, has identified the existence of spontaneous non-monetary networks cooperation developing among rural households aimed at achieving food security, but even in the most remote places we visited, we found that primitive markets did exist.

In previous chapters we have seen how system-level virtual prototypes get created and how they are often partitioned into pieces intended to be implemented as either software or hardware. This partitioning was done to manage costs and performance and to optimize many other attributes of the system. For the parts of the system intended to be hardware, several choices still remain. It is possible that those functionalities can be implemented using off-the-shelf components, such as processors or DSPs, or as components that require small extensions in order to meet the necessary requirements, such as configurable and extensible processors. It is likely that there will be a few components for which dedicated custom hardware is required and we now need to refine those models in ways that make them more amenable to the transformation process that is often called behavioral synthesis or high-level synthesis.

<strong class="journal-contentHeaderColor">Abstract.</strong> Wake meandering studies require knowledge of the instantaneous wake evolution. Scanning lidar data are used to identify the wind flow behind offshore wind turbines but do not immediately reveal the wake edges and centerline. The precise wake identification helps to build models predicting wake behavior. The conventional Gaussian fit methods are reliable in the near-wake area but lose precision with distance from the rotor and require good data resolution for an accurate fit. The thresholding methods, i.e., selection of a threshold that splits the data into background flow and wake, usually imply a fixed value or manual estimation, which hinders the wake identification on a large data set. We propose an automatic thresholding method for the wake shape and centerline detection, which is less dependent on the data resolution and quality and can also be applied to the image data. We show that the method performs reasonably well on large-eddy simulation data and apply it to the data set containing lidar measurements of the two wakes. Along with the wake identification, we use image processing statistics, such as entropy analysis, to filter and classify lidar scans. The automatic thresholding method and the subsequent centerline search algorithm are developed to reduce dependency on the supplementary data such as free-flow wind speed and direction. We focus on the technical aspect of the method and show that the wake shape and centerline found from the thresholded data are in a good agreement with the manually detected centerline and the Gaussian fit method. We also briefly discuss a potential application of the method to separate the near and far wakes and to estimate the wake direction.

The emphasis will be to demonstrate the fundamental mathematical concepts in calculating rates of reactions and the meaning of chemical reaction mechanisms. Reactions as mentioned in a previous chapter can be frequently classified as homogeneous or heterogeneous. Homogeneous reactions occur in a single phase for example, the gas phase reaction involving NO and Oxygen

Our general argument about strategy can now be linked to the previous chapters. We have seen how, during the 1960s and 1970s, the organizational field which characterized the financial services in the UK began to break up. Actors within firms, aided and pushed by key outside interests such as shareholders, the government and management consultancies, began to search for ways of understanding what was happening and, in the process, of exerting some control over their futures. In this process, the 'need for strategy' began to loom as a central part of their transformation process. They began to articulate this new language and to work out its effects. In the process, old practices and languages began to fall by the wayside or to be reconstructed from the point of view of strategy. These processes were by no means painless. They involved the ejection of certain organizations and individuals from the industry, the repositioning of firms (which had implications for the status, rewards and careers of individuals within them) and the reconstruction of relations between the firms, the state and consumers (which had effects on the financial well-being of millions of people). These processes of change were neither smooth nor unidirectionalKeywordsFinancial InstitutionFinancial ServiceBusiness StrategyManagement ConsultantConsultancy FirmThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The previous chapter showed how the unfolding of complexity is a useful tool to describe and design the way an organization groups (or should structure) its primary activities. In this chapter the unfolding of complexity is used to discuss the distribution of resources and discretion from the organization’s global level to the local level of the most basic primary activities. For this purpose it uses the Recursion/Functions Table. This is a tool to discuss the centralization and decentralization of organizational resources and decision-making. Some resources may be centralized but at the same time may be functionally decentralized. Supported by multiple examples of particular transformation processes we discuss in this chapter criteria to decentralize or otherwise the organization’s resources. The Table is used to give systemic meaning to business functions; are these functions regulating the inside and now of the organization or are they providing capacity to deal with the outside and then? For a primary activity to be autonomous, and viable in its own right, it needs resources and discretion to make decisions and develop its own identity. In the end this chapter offers a model for the distribution of resources, relations and information throughout the organization to support the design of its structural mechanisms. As such it is a powerful tool to map its complexity.

In the previous chapter we learned about corner detectors which find particularly distinctive points in a scene. These points can be reliably detected in different views of the same scene irrespective of viewpoint or lighting conditions. Such points are characterized by high image gradients in orthogonal directions and typically occur on the corners of objects. However the 3-dimensional coordinate of the corresponding world point was lost in the perspective projection process which we discussed in Chap. 6 – we mapped a 3-dimensional world point to a 2-dimensional image coordinate. All we know is that the world point lies along some ray in space corresponding to the pixel coordinate, as shown in Fig. 6.6. To recover the missing third dimension we need additional information. In Sect. 6.2.3 the additional information was camera calibration parameters plus a geometric object model, and this allowed us to estimate the object’s 3-dimensional pose from 2-dimensional image data.

Machine recognition has drawn considerable interest and attention from researches in intelligent system design and computer vision communities over the recent past. Understandably there are a large number of commercial, law enforcement, control and forensic applications to this. We human beings have natural ability to recognize persons at a glance. Motivated by our remarkable ability, a series of attempts [1–4] have been made to simulate this ability in machines. The development of human recognition system in machines is quite difficult because the natural objects are complex, multidimensional, and corresponds to environmental changes [3, 5–7]. There are two important issues that need to be addressed in machine recognition: (1) how the features are adopted to represent an object under environmental changes and (2) how we classify an object image based on a chosen representation. Over the years, researches have developed a number of methods for feature extraction and classification. All of these, however, have their own merits and demerits. Most of the work is related to the real domain. The outperformance of complex-valued neuron over conventional neuron has been well established in previous chapters. Few researchers have recently tried multivariate statistical techniques in the complex domain, like complex principal component analysis (PCA) for 2D vector field analysis [8] and complex independent component analysis (ICA) for performing source separation on functional magnetic resonance imaging data [9, 10]. But, no attempts have been made to develop techniques for feature extraction using their concepts. This chapter presents formal procedures for feature extraction using unsupervised learning techniques in complex domain. Efficient learning and better precision in result offered by feature extractor and classifier, considering simulations in complex domain, figure out their technical benefits over conventional methods. Notably, the success of machine recognition is limited by variations in features resulting from the natural environment. These may be due to instrument distortion, acquisition in an outdoor environment, different noises, complex background, occlusion and illumination. A solid set of examples presented in this chapter demonstrate the superiority of feature representation and classification in complex domain.KeywordsFeature ExtractionIndependent Component AnalysisIndependent Component AnalysisComplex DomainFalse Acceptance RateThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.