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)

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)

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)

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)

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)

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)

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.

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)

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.

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 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.

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)

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)

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)

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)

We present a version of system Fω, called image, in which the layer of type constructors is essentially the traditional one of Fω, whereas provability of types is classical. The proof-term calculus accounting for the classical reasoning is a variant of Barbanera and Berardi’s symmetric λ-calculus.We prove that the whole calculus is strongly normalising. For the layer of type constructors, we use Tait and Girard’s reducibility method combined with orthogonality techniques. For the layer of terms, we use Barbanera and (...) Berardi’s method based on a symmetric notion of reducibility candidate. We prove that orthogonality does not capture the fixpoint construction of symmetric candidates.We establish the consistency of image, and relate the calculus to the traditional system Fω, also when the latter is extended with axioms for classical logic. (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)

Extreme learning machine is a recent scheme for single hidden layer feed forward networks. It has attracted much interest in the machine intelligence and pattern recognition fields with numerous real-world applications. The ELM structure has several advantages, such as its adaptability to various problems with a rapid learning rate and low computational cost. However, it has shortcomings in the following aspects. First, it suffers from the irrelevant variables in the input data set. Second, choosing the optimal number of neurons in (...) the hidden layer is not well defined. In case the hidden nodes are greater than the training data, the ELM may encounter the singularity problem, and its solution may become unstable. To overcome these limitations, several methods have been proposed within the regularization framework. In this article, we considered a greedy method for sparse approximation of the output weight vector of the ELM network. More specifically, the orthogonal matching pursuit algorithm is embedded to the ELM. This new technique is named OMP-ELM. OMP-ELM has several advantages over regularized ELM methods, such as lower complexity and immunity to the singularity problem. Experimental works on nine commonly used regression problems indicate that the investigated OMP-ELM method confirms these advantages. Moreover, OMP-ELM is compared with the ELM method, the regularized ELM scheme, and artificial neural networks. (shrink)

Both Mehr et al.'s credible signaling hypothesis and Savage et al.'s music and social bonding hypothesis emphasize the role of multilevel social structures in the evolution of music. Although empirical evidence preferentially supports the social bonding hypothesis, rhythmic music may enable bonding in a way uniquely fitted to the normative and language-based character of multilevel human societies.

From the Hilbert space formalism we note that five simple conditions are satisfied by the orthogonality relation between the (pure) states of a quantum system. We argue, by proving a mathematical theorem, that they capture the essentials of this relation. Based on this, we investigate the rationale behind these conditions in the form of six physical hypotheses. Along the way, we reveal an implicit theoretical assumption in theories of physics and prove a theorem which formalizes the idea that the (...) Superposition Principle makes quantum physics different from classical physics. The work follows the paradigm of mathematical foundations of quantum theory, which I will argue by methodological reflection that it exemplifies a formal approach to analysing concepts in theories. (shrink)

In the literature there has been evidence that a kind of relational structure called a quantum Kripke frame captures the essential characteristics of the orthogonality relation between pure states of quantum systems, and thus is a good qualitative mathematical model of quantum systems. This paper adds another piece of evidence by providing a tensor-product construction of two finite-dimensional quantum Kripke frames. We prove that this construction is exactly the qualitative counterpart of the tensor-product construction of two finite-dimensional Hilbert spaces (...) over the complex numbers, and thus show that composition of quantum systems, especially the phenomenon of quantum entanglement, can be modelled in the framework of quantum Kripke frames. The assumptions used in our construction hint that we need complex numbers in quantum theory. Moreover, for this construction, we give a new and interesting characterization of linear maps of trace 0 in terms of the orthogonality relation. (shrink)

In this contribution, we use the connection between stable polynomials and orthogonal polynomials on the real line to construct sequences of Hurwitz polynomials that are robustly stable in terms of several uncertain parameters. These sequences are constructed by using properties of orthogonal polynomials, such as the well-known three-term recurrence relation, as well as by considering linear combinations of two orthogonal polynomials with consecutive degree. Some examples are presented.

It is well known that the non-orthogonality relation between the states of a quantum system is reflexive and symmetric, and the modal logic \ is sound and complete with respect to the class of sets each equipped with a reflexive and symmetric binary relation. In this paper, we consider two properties of the non-orthogonality relation: Separation and Superposition. We find sound and complete modal axiomatizations for the classes of sets each equipped with a reflexive and symmetric relation that (...) satisfies each one of these two properties and both, respectively. We also show that the modal logics involved are decidable. (shrink)

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)

We prove that if V = L then there is a $\Pi _{1}^{1}$ maximal orthogonal (i.e., mutually singular) set of measures on Cantor space. This provides a natural counterpoint to the well-known theorem of Preiss and Rataj [16] that no analytic set of measures can be maximal orthogonal.

Rational social theorists have failed to confirm that observations of social reality equal social reality. Yet they argue that teams, organizations and social systems should minimize interdependence and competition, echoed by social psychologists to make data iid. But the evidence indicates that competitive teams maximize interdependence; self-reports of social reality correlate poorly with social behavior; and only competition measures interdependent social states. Rational expectations aside, we report progress towards a science of interdependence for human–machine teams. Our model of interdependence works (...) like an uncertainty principle in the sense that tradeoffs arise from the uncertainty caused by measuring interdependent actors in orthogonal roles; e.g., in the tradeoff between teams and individuals, teams are more productive but more opaque. Previously, we described interdependence as bistable stories of social reality; the motivation to reject alternative interpretations, increasing uncertainty and errors; and the inability to factor social states. Now we explore education as a surrogate for intelligence in teams. We hypothesized that teams rely on the education of its members to produce more patents. We found that the average schooling in a nation is significantly related to its total patents produced. (shrink)

Rational social theorists have failed to confirm that observations of social reality equal social reality. Yet they argue that teams, organizations and social systems should minimize interdependence and competition, echoed by social psychologists to make data iid. But the evidence indicates that competitive teams maximize interdependence; self-reports of social reality correlate poorly with social behavior; and only competition measures interdependent social states. Rational expectations aside, we report progress towards a science of interdependence for human–machine teams. Our model of interdependence works (...) like an uncertainty principle in the sense that tradeoffs arise from the uncertainty caused by measuring interdependent actors in orthogonal roles; e.g., in the tradeoff between teams and individuals, teams are more productive but more opaque. Previously, we described interdependence as bistable stories of social reality; the motivation to reject alternative interpretations, increasing uncertainty and errors; and the inability to factor social states. Now we explore education as a surrogate for intelligence in teams. We hypothesized that teams rely on the education of its members to produce more patents. We found that the average schooling in a nation is significantly related to its total patents produced. (shrink)

Rational social theorists have failed to confirm that observations of social reality equal social reality. Yet they argue that teams, organizations and social systems should minimize interdependence and competition, echoed by social psychologists to make data iid. But the evidence indicates that competitive teams maximize interdependence; self-reports of social reality correlate poorly with social behavior; and only competition measures interdependent social states. Rational expectations aside, we report progress towards a science of interdependence for human–machine teams. Our model of interdependence works (...) like an uncertainty principle in the sense that tradeoffs arise from the uncertainty caused by measuring interdependent actors in orthogonal roles; e.g., in the tradeoff between teams and individuals, teams are more productive but more opaque. Previously, we described interdependence as bistable stories of social reality; the motivation to reject alternative interpretations, increasing uncertainty and errors; and the inability to factor social states. Now we explore education as a surrogate for intelligence in teams. We hypothesized that teams rely on the education of its members to produce more patents. We found that the average schooling in a nation is significantly related to its total patents produced. (shrink)

PurposeThe purpose of this study was to introduce an orthogonal experimental design to improve the efficiency of building and optimizing models for freezing of gait prediction.MethodsA random forest model was developed to predict FOG by using acceleration signals and angular velocity signals to recognize possible precursor signs of FOG. An OED was introduced to optimize the feature extraction parameters.ResultsThe main effects and interaction among the feature extraction hyperparameters were analyzed. The false-positive rate, hit rate, and mean prediction time were 27%, (...) 68%, and 2.99 s, respectively.ConclusionThe OED was an effective method for analyzing the main effects and interactions among the feature extraction parameters. It was also beneficial for optimizing the feature extraction parameters of the FOG prediction model. (shrink)

In the 1920s, Heyting attempted at axiomatizing constructive geometry. Recently, von Plato used different concepts to axiomatize the geometry: he used 14 axioms to describe the axiomatization for apartness geometry. Then he added axioms A1 and A2 to his apartness geometry to get his affine geometry, then he added axioms O1, O2, O3 and O4 to the affine geometry to get orthogonality. In total, this gives 22 axioms. von Plato used four relations to describe the concept of orthogonality (...) in O1, O2 and O4. That is, all the three relations of two lines, which are convergence, unorthogonality and difference, and the relation of a point and a line. ANDP is an automated natural deduction prover developed over the years at our institute. After doing a lot of experiments using ANDP, much shorter and more intuitive axioms were found for axioms O1, O2 and O4, respectively. For example, O2 can be replaced by one of its four conjuncts. This paper shows that it is enough to use two relations on lines, which are convergence and unorthogonality, to describe the concept of orthogonality. (shrink)