This book presents a comprehensive overview of the sum rule approach to spectral analysis of orthogonal polynomials, which derives from Gábor Szego's classic 1915 theorem and its 1920 extension. Barry Simon emphasizes necessary and sufficient conditions, and provides mathematical background that until now has been available only in journals. Topics include background from the theory of meromorphic functions on hyperelliptic surfaces and the study of covering maps of the Riemann sphere with a finite number of slits removed. This allows (...) for the first book-length treatment of orthogonal polynomials for measures supported on a finite number of intervals on the real line. In addition to the Szego and Killip-Simon theorems for orthogonal polynomials on the unit circle and orthogonal polynomials on the real line, Simon covers Toda lattices, the moment problem, and Jacobi operators on the Bethe lattice. Recent work on applications of universality of the CD kernel to obtain detailed asymptotics on the fine structure of the zeros is also included. The book places special emphasis on OPRL, which makes it the essential companion volume to the author's earlier books on OPUC. (shrink)

The Western cultural archetype of Narcissus experiences a significant turning point in the 1400s, thanks to Leon Battista Alberti’s work. Indeed, the myth evolves from being a subject embodying a taboo in the Antiquity to become the glance that generates the image; in so doing this myth assumes the rank of science and philosophy. Alberti does not follow Pliny’s reading of Ovid’s Metamorphosis, but handles Philostratus’s version; with his visual description he represents in the “Eikones” the darting glance towards the (...) water. The rendering of this image anticipates the notion of centric ray, on which Alberti establishes his perspective’s theory based on the diaphanous nature of “velum”. The art of painting is compared to an “open vitreous and translucent window”. Narcissus does not see the water, so to say, but the image, as well as the painting’s observer instantly grasps the iconic value of it, rather than the material support. Because of this immediacy, the art of painting becomes the terrain of revelation: it makes clear the emerging progressive visibility of “something”, thus concurring with philosophy; better, the art of painting overtakes philosophy with the simple representation of polysemous complexities. This is the new art of painting, in which “there is nothing that is not philosophy”. Furthermore, in this synthetic role of representation in which the icon becomes all one with the spectator – in addition to the centric ray and to the orthogonality of the vision, topics already analyzed by the author in previous studies – consists an outstanding resemblance between Alberti and Cusanus. The two philosophers show their synergy in exploring a new anthropology. From this point of view can be read Alberti’s explanation, often misinterpreted, of Narcissus’ transformation into a flower: a synthesis between “imaginatio” and “executio”, or inactiveness versus activeness. Among all Narcissus’ portrayals of that time, it would be useful to consider the Pseudo Boltraffio’s version of London; this rendering applies thematically Alberti’s theory in its essential aspects, introducing it within Leonardo Da Vinci’s circle of artists. (shrink)

This manuscript explores the orthogonality constraints on configurations and orbitals subject to the property that states are mutually orthogonal. The orthogonality constraints lead to properties that affect the description of chemical systems. When states are described as linear combinations of configurations, the coefficient matrix diagonalises S−1H. Therefore, single-configuration states are only possible in one-electron systems: non-orthogonal configurations yield single-configuration states only if S−1H is diagonal, but this would violate the orthonormalisation constraint. Further, the coefficient matrix is not constrained (...) to be square. Similarly, the orbitals used to construct configurations may also be orthogonal or non-orthogonal; orbitals are only required to be mutually orthogonal at the one-electron limit. Orthogonal orbitals are generally preferred due to their mathematical and conceptual simplicity, leading to fictitious unoccupied orbitals. Since the Fock operator is orthogonality agnostic, non-orthogonal orbitals can be generated by solving the Fock equation independently for each electron; the virtual orbitals produced by this conception are true excitation orbitals as they are eigensolutions of the Fock operator. Additionally, we show that the number of molecular orbitals generated is not restricted to the number of atomic orbitals employed in the computation. This manuscript explores the mathematical relationships that need to be satisfied under the various orthogonality regimes. We also present mathematical relationships that provide results that are independent of the orthogonality approximation within a particular computational method. (shrink)

In recent years, neural networks have attracted much attention in the machine learning and the deep learning technologies. Bio-inspired functions and intelligence are also expected to process efficiently and improve existing technologies. In the visual pathway, the prominent features consist of nonlinear characteristics of squaring and rectification functions observed in the retinal and visual cortex networks, respectively. Further, adaptation is an important feature to activate the biological systems, efficiently. Recently, to overcome short-comings of the deep learning techniques, orthogonality for the (...) weights in the networks has been developed for the signal propagation and the efficient optimization of the learning. In this paper, bio-inspired asymmetric networks with nonlinear characteristics are proposed, which are derived from the retinal networks in the biological visual pathway. The asymmetric network proposed here was verified to detect the movement of the object, efficiently in our previous studies. This paper shows a new characteristic of the adaptive orthogonality in the asymmetric networks. First, it is shown that the asymmetric network with nonlinear characteristics is effective for generating orthogonality. Second, the proposed asymmetric network with Gabor filters is compared with the conventional energy model from the point of the orthogonality characteristics. Finally, the asymmetric networks with nonlinear characteristics can generate the extended orthogonal bases in independent subspaces, which are useful for classification and efficient learning. (shrink)

This paper presents arguments from empirical research and from philosophical considerations to the effect that phenomenality and content are two distinct and independent features of mental representations, which are both relational. Thus, it is argued, classical arguments that infer phenomenality from content have to be rejected. Likewise, theories that try to explain the phenomenal character of experiences by appeal to specific types of content cannot succeed. Instead, a dynamic view of consciousness has to be adopted that seeks to explain consciousness (...) by certain ways of processing representations. Therefore, only empirical methods that are able to investigate the dynamics of the mind can be used for the “quest for consciousness” proper. Moreover, the central intuitions about consciousness are best explained when phenomenality and content are clearly distinguished. (shrink)

Let F be a signature and R a strictly orthogonal rewrite system on ground terms of F . We give an effective proof of a bounding condition for R , based on a detailed analysis of how terms are transformed during the rewrite process, which allows us to give recursive bounds on the derivation lengths of terms. We give a syntactic characterisation of the Grzegorczyk hierarchy and a rewriting schema for calculating its functions. As a consequence of this, using (...) results of Elias Tahhan–Bittar, it can be shown that, for n⩾3 , the derivation length functions for functions in Grzegorczyk class E n belong to Grzegorczyk class E n+1 . We also give recursive bounds for the derivation lengths of functions defined by parameter recursion. (shrink)

Orthogonality in model theory captures the idea of absence of non-trivial interactions between definable sets. We introduce a somewhat opposite notion of cohesiveness, capturing the idea of interaction among all parts of a given definable set. A cohesive set is indecomposable, in the sense that if it is internal to the product of two orthogonal sets, then it is internal to one of the two. We prove that a definable group in an o-minimal structure is a product of cohesive (...) orthogonal subsets. If the group has dimension one, or it is definably simple, then it is itself cohesive. As an application, we show that an abelian group definable in the disjoint union of finitely many o-minimal structures is a quotient, by a discrete normal subgroup, of a direct product of locally definable groups in the single structures. (shrink)

We present a mathematically precise derivation of some generalized orthogonality relations for the discrete series representations of SU(1,1). These orthogonality relations are applied to derive tomographical reconstruction formulas. Their physical interpretation is also discussed.

Modal interpretations constitute a particular approach to associating dynamical variables with physical systems in quantum mechanics. Given the “quantum logical” constraints that are typically adopted by such interpretations, only certain sets of variables can be taken to be simultaneously definite-valued, and only certain sets of values can be ascribed to these variables at a given time. Moreover, each allowable set of variables and values can be uniquely specified by a single “core” projector in the Hilbert space associated with the system. (...) In general, the core projector can be one of several possibilities at a given time. In most previous modal interpretations, the different possible core projectors have formed an orthogonal set. This paper investigates the possibility of adopting a non-orthogonal set. It is demonstrated that such non-orthogonality is required if measurements for which the outcome can be predicted with probability 1 are to reveal the pre-existing value of the variable measured, an assumption which has traditionally constituted a strong motivation for the modal approach. The existing framework for modal interpretations is generalized to explicitly accommodate non-orthogonal core projectors. (shrink)

The primary aim of this original work is to provide a more in-depth insight into the relations between control parameters adjustments, learning techniques, inner swarm dynamics and possible hybridization strategies for popular swarm metaheuristic Firefly Algorithm. In this paper, a proven method, orthogonal learning, is fused with FA, specifically with its hybrid modification Firefly Particle Swarm Optimization. The parameters of the proposed Orthogonal Learning Firefly Algorithm are also initially thoroughly explored and tuned. The performance of the developed algorithm (...) is examined and compared with canonical FA and above-mentioned FFPSO. Comparisons have been conducted on well-known CEC 2017 benchmark functions, and the results have been evaluated for statistical significance using the Friedman rank test. (shrink)

Hierocles, a Stoic of the second century CE, famously deployed an image of the ‘concentric circles’ that surround each of us. The image should not be read as advocating absolute impartiality (in the style of classical utilitarianism) or as illustrating the Stoic theory of oikeiōsis. Instead, it is designed to illustrate how it is ‘appropriate to act’ in certain cases. Like other Stoics, Hierocles bases his investigation of appropriate acts on what is ‘in accordance with nature’. According to his (...) view, each of us has a duty of mutual aid towards every other human being—but a weightier duty towards those who are socially closer than those who are more distant. The fundamental principle is that each of us has the duty of being an ‘ally’ of every other human being in serving the natural purposes of all the circles to which we and that other human being both belong. (shrink)

The paper is devoted to the presentation and analysis of the philosophical views concerning logic and mathematics of the leading members of Cracow Circle, i.e., of Jan Salamucha, Jan Franciszek Drewnowski and Józef Maria Bocheński. Their views on the problem of possible applicability of logical tools in metaphysical and theological researches is also discussed.

We define and study the notion of an indexed frame. This is a bi-dimensional structure consisting of a Cartesian product equipped with relations which only relate pairs if they coincide in one of their components. We show that these structures are quite ubiquitous in modal logic, showing up in the literature as products of Kripke frames, subset spaces, or temporal frames for STIT logics. We show that indexed frames are completely characterised by their ‘orthogonal’ relations, and we provide their (...) sound and complete logic. Using these ‘orthogonality’ results, we provide necessary and sufficient conditions for an arbitrary Kripke frame to be isomorphic to certain well-known bi-dimensional structures. (shrink)

The standard argument to the conclusion that artificial intelligence (AI) constitutes an existential risk for the human species uses two premises: (1) AI may reach superintelligent levels, at which point we humans lose control (the ‘singularity claim’); (2) Any level of intelligence can go along with any goal (the ‘orthogonality thesis’). We find that the singularity claim requires a notion of ‘general intelligence’, while the orthogonality thesis requires a notion of ‘instrumental intelligence’. If this interpretation is correct, they cannot be (...) joined as premises and the argument for the existential risk of AI turns out invalid. If the interpretation is incorrect and both premises use the same notion of intelligence, then at least one of the premises is false and the orthogonality thesis remains itself orthogonal to the argument to existential risk from AI. In either case, the standard argument for existential risk from AI is not sound.—Having said that, there remains a risk of instrumental AI to cause very significant damage if designed or used badly, though this is not due to superintelligence or a singularity. (shrink)

This paper begins by asking a simple question: can a farmer own and fully utilise precisely five tractors and precisely six tractors at the same time? Of course not. He can own five or he can own six but he cannot own five and six at the same. The answer to this simple question eventually led this author to Alfred Marshall's historically-ignored, linguistically-depicted 'cardboard model' where my goal was to construct a picture based on his written words. More precisely, in (...) this paper the overall goal is to convert Marshall's ('three-dimensional') words into a three-dimensional picture so that the full import of his insight can be appreciated by all readers. After a brief digression necessary to introduce Euclidean three-dimensional space, plus a brief digression to illustrate the pictorial problem with extant theory, the paper turns to Marshall's historically- ignored words. Specifically, it slowly constructs a visual depiction of Marshall's 'cardboard model'. Unfortunately (for all purveyors of extant economic theory), this visual depiction suddenly opens the door to all manner of Copernican heresy. For example, it suddenly becomes obvious that we can join the lowest points on a firm's series of SRAC curves and thereby form its LRAC curve; it suddenly becomes obvious that the firm's series of SRAC curves only appear to intersect because mainstream theory has naively forced our three-dimensional economic reality into a two-dimensional economic sketch; and it suddenly becomes obvious that a two-dimensional sketch is analytically useless because the 'short run' (SR) never turns into the 'long run' (LR) no matter how long we wait. (shrink)

CONJECTURE: Alongside the innate physical sucking reflex for obtaining milk to be digested, decomposed and used all over the body for growth, repair, and energy, there is a genetically determined information-sucking reflex, which seeks out, sucks in, and decomposes information, which is later recombined in many ways, growing the information-processing architecture and many diverse recombinable competences.

We consider the problem of quadrilaterizing an orthogonal polygon P, that is to decompose P into nonoverlapping convex quadrangles without adding new vertices. In this paper we present a CREW-algorithm for this problem which runs in O time using Θ processors if the rectangle decomposition of P is given, and Θ processors if not. Furthermore we will show that the latter result is optimal if the polygon is allowed to contain holes.

In quantum mechanics it is usually assumed that mutually exclusives states of affairs must be represented by orthogonal vectors. Recent attempts to solve the measurement problem, most notably the GRW theory, require the relaxation of this assumption. It is shown that a consequence of relaxing this assumption is that arithmatic does not apply to ordinary macroscopic objects. It is argued that such a radical move is unwarranted given the current state of understanding of the foundations of quantum mechanics.

It is conjectured that humans and some other altricial species instead use innate mechanisms for decomposing situations into components that can be explicitly learnt about, and stored in such a way that the competence can be re-used in combination with other learnt competences, in perceiving novel situations and performing novel actions.

We introduce and study a variety of modal logics of parallelism, orthogonality, and affine geometries, for which we establish several completeness, decidability and complexity results and state a number of related open, and apparently difficult problems. We also demonstrate that lack of the finite model property of modal logics for sufficiently rich affine or projective geometries (incl. the real affine and projective planes) is a rather common phenomenon.

"'There is only one thing that is certain, namely that we can have nothing certain; and therefore it is not certain that we can have nothing certain,' Samuel Butler once said, expressing in that mindbloggler all the elements required to form a classical paradox. Throughout the ages wise men and jesters alike have been intrigued by such mental twists and riddles which defy common sense and yet appear to be true." -- Dust jacket.

An artificial general intelligence (AGI) might have an instrumental drive to modify its utility function to improve its ability to cooperate, bargain, promise, threaten, and resist and engage in blackmail. Such an AGI would necessarily have a utility function that was at least partially observable and that was influenced by how other agents chose to interact with it. This instrumental drive would conflict with the strong orthogonality thesis since the modifications would be influenced by the AGI’s intelligence. AGIs in highly (...) competitive environments might converge to having nearly the same utility function, one optimized to favorably influencing other agents through game theory. Nothing in our analysis weakens arguments concerning the risks of AGI. (shrink)

Hermeneutic studies of science locate a circle at the heart of scientific practice: scientists only gain knowledge of what they, in some sense, already know. This may seem to threaten the rational validity of science, but one can argue that this circle is a virtuous rather than a vicious one. A virtuous circle is one in which research conclusions are already present in the premises, but only in an indeterminate and underdeveloped way. In order to defend the validity of science, (...) the hermeneuticist must describe a method by which a vague and confused initial knowledge of nature gets transformed into a clear and determinate knowledge of nature. I consider three such methods. The first is regressus demonstrativa, favoured by the physicians of Padua during the fifteenth and sixteenth centuries. The second is mathēsis, introduced by Martin Heidegger in his discussion of seventeenth-century science. The third is Denkstil, a key concept in Ludwik Fleck’s history of syphilology. I conclude by listing three desiderata for a hermeneutic science studies: that it be anti-metaphysical, historical, and sociological. --- Reprinted in: Erich Otto Graf, Martin Schmid & Johannes Fehr (eds.), Fleck and the Hermeneutics of Science (Collegium Helveticum Heft 14) (Zürich, 2016), pp. 85-93. (shrink)

Model organisms, the use of green fluorescent proteins, and orthogonal transfer RNA are examples of artificial causes being used in biology. Recent work characterizing the research interests of biologists in terms of a common set of values has ruled out artificial causes as biologically interesting. For instance, Kenneth Waters argues that biologists are primarily interested in causes that actually obtain. Similarly, Marcel Weber argues that biologists are primarily concerned with biologically normal interventions. Both views express a widely received attitude (...) about the interests and goals of biologists as being primarily concerned with the contingent facts of our world. While I agree with this general attitude about the contingent nature of biology, I argue that neither view fully accounts for the diversity that distinguishes the discipline. Along with actual and biologically normal causes, biologists are also interested in artificial causes for technological and observational purposes. I maintain that research interest in artificial causes provides some pragmatic reasons for thinking that research programs in cellular biology aren’t driven by a common core set of values. (shrink)

pt. 1. The need for a philosophy of pain and suffering. The circle of life on the earth. The misuse of fear. Pain. The problems of old age. Rejuvenation. Death and dying. The hereafter. The circle of recurrence.--pt. 2. Doctors and doctoring. Surgery. Psycho-therapy. The two worlds.

One of the major purposes of this article is to show that friendship was one of Nietzsche's central concerns and that he shared Aristotle's belief that it takes higher and lower forms. Yet Nietzsche's interest in friendship is overlooked in much of the secondary literature. An important reason for this is that this interest is most evident in the works of his middle period, and these tend to be neglected in commentaries on Nietzsche. In the works of the middle period, (...) Nietzsche suggests that there is a close connection between friendship and selfhood, contending that an individual's friendships reflect something about his or her identity. Following Aristotle, he believes that friendship can make a significant contribution to self-knowledge and self-improvement, which are both closely associated with his notion of self-overcoming. Nietzsche encourages individuals to adopt an aesthetic approach to the self: they should refashion themselves by consolidating their strengths, minimising their weaknesses and developing themselves in new directions. The works of the middle period suggest, however, that not only can friendship foster self-overcoming, but that the talent for friendship is one of the marks of a higher human being. Recognising this requires some reconsideration of Nietzsche's putative individualism and the belief that he holds great individuals to be utterly independent and indifferent to the judgements of others. It also requires a revision of the common interpretation that he is unremittingly sceptical about pity and other forms of fellow-feeling. Yet while Nietzsche generalises about friendship in the works of the middle period and contrasts its superior and inferior forms, he remains sensitive to its particularity. He never adopts a wholly formulaic approach to this relationship, but recognises that difference and responsiveness to particularity are among its central characteristics. (shrink)

How can a business institution function as an ethical institution within a wider system if the context of the wider system is inherently unethical? If the primary goal of an institution, no matter how ethical it sets out to be, is to function successfully within a market system, how can it reconcile making a profit and keeping its ethical goals intact? While it has been argued that some ethical businesses do exist, e.g., Johnson and Johnson, the argument I would like (...) to put forth is that no matter how ethical a business institution is, or how ethical its goals are, its capacity to act in an ethical manner is restricted by the wider system in which it must operate, the market system. Unless there is a fundamental change in the notion of the market system itself, the capacity for individual businesses to act in an ethical manner will always be restricted. My argument is divided into two parts. The first part is to show the inherent bias towards unethical outcomes that is inherent in the market system. The second part is to suggest how to reorient the general economic framework in order to make ethical institutions more possible. The question then becomes, how to define economic behavior in terms other than competition for profit. (shrink)

Discusses the work of G.I. Gurdjieff and his establishment of the Institute for the Harmonious Development of man, and examines the contributions of Gurdjieff's two major disciples, P.D. Ouspensky and A.R. Orage.

This article is intended as a contribution to the current debates about the relationship between politics and the philosophy of science in the Vienna Circle. I reconsider this issue by shifting the focus from philosophy of science as theory to philosophy of science as practice. From this perspective I take as a starting point the Vienna Circle’s scientific world-conception and emphasize its practical nature: I reinterpret its tenets as a set of recommendations that express the particular epistemological attitude in which (...) both the Vienna Circle’s (doing) philosophy of science and its political engagement were rooted. -/- Regarding politics, and referring to new primary sources, I reconstruct how the scientific world-conception placed the Vienna Circle within a neoliberal-socialist political network that pursued concrete political aims. In light of my reconstruction I shall argue that neither the Vienna Circle’s alleged ethical noncognitivism nor its alleged adhesion to the Weberian ideal of a value-free science rules out the possibility of ascribing to the Vienna Circle a politically engaged philosophy of science: the case of the Vienna Circle shows how philosophy of science, as a public activity, can itself become a form of political engagement, even without necessarily entailing a theory of objective values. (shrink)

Classically, an osculating circle at a point of a planar curve is introduced technically, often with formula giving its radius and the coordinates of its center. In this note, we propose a new and intuitive definition of this concept: among all the circles which have, on the considered point, the same tangent as the studied curve and thus seem equal to the curve through a microscope, the osculating circle is this that seems equal to the curve through a microscope (...) within microscope. (shrink)

Traditional models of skill learning posit that skilled action unfolds in an automatic manner and that control will prove deleterious to movement and performance proficiency. These perspectives assume that automated processes are characterised by low levels of control and vice versa. By contrast, a number of authors have recently put forward hybrid theories of skilled action which have sought to capture the close integration between fine-grained automatic motor routines and intentional states. Drawing heavily on the work of Bebko et al. (...) and Christensen et al., we argue that controlled and automated processes must operate in parallel if skilled performers are to address the wide range of challenges that they are faced with in training and competition. More specifically, we show how skilled performers use controlled processes to update and improve motor execution in training contexts and to stabilise performance under pressurised conditions. (shrink)

Foucault rejects the subject as a center, i.e. as a transparent self-conscious being, who gives meaning to his actions. However, ideas about subjects that think and will autonomously go on functioning within modern culture. Discourses on subjectivity call for an archeological and genealogical explanation. This compels Foucault to resort increasingly to subjectivity: as product and target of power, but also as a source of resistance and as an agent. After all, Foucault defines power as ‘actions about actions’. In the end, (...) Foucault started to circumscribe the teleology of his philosophical ethos as the production of new forms of subjectivity, in terms of freedom and autonomy. Concern is more with a continuous circling transgression, apprehending subjectivity as an aimless self-negation, than it is about a return of the subject. (shrink)

Safe-by-Design is an approach to engineering that aims to integrate the value of safety in the design and development of new technologies. It does so by integrating knowledge of potential dangers in the design process and developing methods to design undesirable effects out of the innovation. Recent discussions have highlighted several challenges in conceptualizing safety and integrating the value into the design process. Therefore, some have argued to design for the _responsibility_ for safety, instead of for safety itself. However, this (...) idea has not been developed further. In this article, we develop an approach to Safe-by-Design, grounded in care ethics, which builds on the idea of designing for responsibility and can deal with the complexity that is inherent to the conceptualization of safety. We describe five ways in which care ethics contributes to the conceptualization of Safe-by-Design: (1) It suggests the development of ‘circles of care’ in which stakeholders share the responsibility for safety; (2) it recognizes the importance of considering safety as something that is situated in the surroundings of a technology, instead of as a property of the technology itself; (3) it acknowledges that achieving safety is labour that requires an ongoing commitment; (4) it emphasizes that the way in which we relate to technology impacts its safety; and (5) it recognizes the role of emotions in assessing safety. All these elements combined lead to a broader understanding of safety and a philosophically more substantial and practically more appealing conceptualization of Safe-by-Design. (shrink)

Person reidentification aims to recognize a specific pedestrian from uncrossed surveillance camera views. Most re-id methods perform the retrieval task by comparing the similarity of pedestrian features extracted from deep learning models. Therefore, learning a discriminative feature is critical for person reidentification. Many works supervise the model learning with one or more loss functions to obtain the discriminability of features. Softmax loss is one of the widely used loss functions in re-id. However, traditional softmax loss inherently focuses on the feature (...) separability and fails to consider the compactness of within-class features. To further improve the accuracy of re-id, many efforts are conducted to shrink within-class discrepancy as well as between-class similarity. In this paper, we propose a circle-based ratio loss for person re-identification. Concretely, we normalize the learned features and classification weights to map these vectors in the hypersphere. Then we take the ratio of the maximal intraclass distance and the minimal interclass distance as an objective loss, so the between-class separability and within-class compactness can be optimized simultaneously during the training stage. Finally, with the joint training of an improved softmax loss and the ratio loss, the deep model could mine discriminative pedestrian information and learn robust features for the re-id task. Comprehensive experiments on three re-id benchmark datasets are carried out to illustrate the effectiveness of the proposed method. Specially, 83.12% mAP on Market-1501, 71.66% mAP on DukeMTMC-reID, and 66.26%/63.24% mAP on CUHK03 labeled/detected are achieved, respectively. (shrink)

1. We show that if p is a real type which is internal in a set $\sigma$ of partial types in a simple theory, then there is a type p' interbounded with p, which is finitely generated over $\sigma$ , and possesses a fundamental system of solutions relative to $\sigma$ . 2. If p is a possibly hyperimaginary Lascar strong type, almost \sigma-internal$ , but almost orthogonal to $\sigma^{\omega}$ , then there is a canonical non-trivial almost hyperdefinable polygroup which (...) multi-acts on p while fixing $\sigma$ generically. In case p is $\sigma-internal$ and T is stable, this is the binding group of p over \sigma$. (shrink)