We have studied the dynamics and symmetries of a particle constrained to move in a torus knot. The Hamiltonian system turns out to be Second Class in Dirac’s formulation and the Dirac brackets yield novel noncommutative structures. The equations of motion are obtained for a path in general where the knot is present in the particle orbit but it is not restricted to a particular torus. We also study the motion when it is restricted to a specific (...) class='Hi'>torus. The rotational symmetries are studied as well. We have also considered the behavior of small fluctuations of the particle motion about a fixed torus knot. (shrink)

Domination is a structural complexity of chemical molecular graphs. A dominating set in a graphG=V,Eis a subsetS⊆Vsuch that each vertex inV\Sis adjacent to at least one vertex inS. The domination numberγGof a graphGis the minimum size of a dominating set inG. In this paper, computer-aided approaches for obtaining bounds for domination number on torus graphs are here considered, and many new exact values and bounds are obtained.

In this study, we first introduce polygonal cylinder and torus using Cartesian products and topologically identifications and then find their Wiener and hyper-Wiener indices using a quick, interesting technique of counting. Our suggested mathematical structures could be of potential interests in representation of computer networks and enhancing lattice hardware security.

We consider a universe of finite Morley rank and the following definable objects: a field math formula, a non-trivial action of a group math formula on a connected abelian group V, and a torus T of G such that math formula. We prove that every T-minimal subgroup of V has Morley rank math formula. Moreover V is a direct sum of math formula-minimal subgroups of the form math formula, where W is T-minimal and ζ is an element of G (...) of order 4 inverting T. (shrink)

In the new physics and in the new field of cosmometry, 1 it is the fundamental pattern that results in the motion from which all is created. Everything starts with the point of infinite potential. The tetrahedron at the point gives birth to the cuboctahedron ; its motion and structure result in the creation of the torus structure. The torus structure is self-referencing on a moment by moment basis since all must pass through the center. (...) But isn't self-referencing the basis for consciousness? It is said that all of creation has awareness, but at different levels. We know plants are aware of threats and of death to other living creatures by the work of Cleve Backster. We know we influence random number generators from the PEAR studies at Princeton. We know baby chicks will influence the movement of robots programed to do a random walk from the work of Rene Peoc'h. Quantum physics has embraced geometry with the work in the equations which explain the Feynman diagrams where particles come in and out of existence; those equations form a structure called the Amplituhedron - which is a quarter of a star tetrahedron. We also know that the tetrahedron can form the star tetrahedron and that each point of the star tetrahedron can form its own star tetrahedron, which can go on infinitely. That is, there is infinity in the finite. As in the fractal and holographic universe, each point or tetrahedron is connected to every other point. If each point has awareness due to formation of the torus, then Hermes teaching of "As above, as below" has meaning in the torus structure. If the torus is the fundamental unit of self-reference, is that the fundamental unit from which consciousness arises? The torus or double torus appears to be fundamental to all of creation - from galaxies to planets to atoms to photons. (shrink)

An argument of A. Borel [Bor—61, Proposition 3.1] shows that every compact connected Lie group is homeomorphic to the Cartesian product of its derived subgroup and a torus. We prove a parallel result for definably compact definably connected groups definable in an o-minimal expansion of a real closed field. As opposed to the Lie case, however, we provide an example showing that the derived subgroup may not have a definable semidirect complement.

Let M be a smooth, compact manifold of dimension n ≥ 5 and sectional curvature | K | ≤ 1. Let Met (M) = Riem(M)/Diff(M) be the space of Riemannian metrics on M modulo isometries. Nabutovsky and Weinberger studied the connected components of sublevel sets (and local minima) for certain functions on Met (M) such as the diameter. They showed that for every Turing machine T e , e ∈ ω, there is a sequence (uniformly effective in e) of homology (...) n-spheres {P k e } k ∈ ω which are also hypersurfaces, such that P k e is diffeomorphic to the standard n-sphere S n (denoted P k e ≈ diff S n ) iff T e halts on input k, and in this case the connected sum N k e =M ♯ P k e ≈ diff M , so N k e ∈ Met(M), and N k e is associated with a local minimum of the diameter function on Met(M) whose depth is roughly equal to the settling time σ e (k) of T e on inputs y i } ∈ ω of c.e. sets so that for all i the settling time of the associated Turing machine for A i dominates that for A i + 1 , even when the latter is composed with an arbitrary computable function. From this, Nabutovsky and Weinberger showed that the basins exhibit a "fractal" like behavior with extremely big basins, and very much smaller basins coming off them, and so on. This reveals what Nabutovsky and Weinberger describe in their paper on fractals as "the astonishing richness of the space of Riemannian metrics on a smooth manifold, up to reparametrization." From the point of view of logic and computability, the Nabutovsky-Weinberger results are especially interesting because: (1) they use c.e. sets to prove structural complexity of the geometry and topology, not merely undecidability results as in the word problem for groups, Hilbert's Tenth Problem, or most other applications; (2) they use nontrivial information about c.e. sets, the Soare sequence {A i } i ∈ ω above, not merely G öodel's c.e. noncomputable set K of the 1930's; and (3) without using computability theory there is no known proof that local minima exist even for simple manifolds like the torus T 5 (see §). (shrink)

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

There is sometimes a sense in which one theory posits ‘less structure’ than another. Philosophers of science have recently appealed to this idea both in the debate about equivalence of theories and in discussions about structural parsimony. But there are a number of different proposals currently on the table for how to compare the ‘amount of structure’ that different theories posit. The aim of this paper is to compare these proposals against one another and evaluate them on their (...) own merits. (shrink)

ABSTRACT In this wide-ranging interview Professor Douglas V. Porpora discusses a number of issues. First, how he became a Critical Realist through his early work on the concept of structure. Second, drawing on his Reconstructing Sociology, his take on the current state of American sociology. This leads to discussion of the broader range of his work as part of Margaret Archer’s various Centre for Social Ontology projects, and on moral-macro reasoning and the concept of truth in political discourse.

ABSTRACTThis article argues that received accounts of the concept of human dignity face more difficulties than has been appreciated, when explaining the connection between human dignity and the duty of respect that dignity is supposed to generate. It also argues that a novel, relational, account has the adequate structure to explain such connection.

This volume considers how current transitions in postsecondary education are impacting Higher Education institutions and subjects in a number of Northern nations, as well as how these transitions are indicative of the wider shift from the welfare to the market state. The university is now considered a key site for training and wealth generation in the so-called 'knowledge economy' that operates in a globalising, high tech world. Further, these transitions are underpinned by neo-liberal economic ideas that assume that the public (...) sector is a drag on the economy unless it is subject to the rules, regulations and assumptions that govern the private sector. This excellent volume - an important contribution to Education as well as Economics and Politics - furthers our understandings of universities as marketable entities as part of the globalized economy. (shrink)

Aimed at graduate students, research logicians and mathematicians, this much-awaited text covers over 40 years of work on relative classification theory for nonstandard models of arithmetic. The book covers basic isomorphism invariants: families of type realized in a model, lattices of elementary substructures and automorphism groups.

The Everett interpretation of quantum mechanics divides naturally into two parts: first, the interpretation of the structure of the quantum state, in terms of branching, and second, the interpretation of this branching structure in terms of probability. This is the first of two reviews of the Everett interpretation, and focuses on structure, with particular attention to the role of decoherence theory. Written in terms of the quantum histories formalism, decoherence theory just is the theory of branching (...) class='Hi'>structure, in Everett's sense. (shrink)

La structure du definiens fait l’objet de deux traitements apparemment contradictoires dans les textes aristotéliciens : certains textes avancent que l’énoncé par lequel est définie une chose ne peut pas prendre la forme d’une prédication, parce que cet énoncé doit exprimer uniquement ce qu’est la chose qui est objet de la définition et non pas lui attribuer des termes, tandis que d’autres textes, notamment dans la Métaphysique, semblent suggérer précisément l’inverse, à savoir que le definiens doit prendre la forme (...) d’une prédication. Le présent article se propose de lever cette tension apparente en prolongeant la thèse de Jacques Brunschwig selon laquelle la « prédication hylémorphique », c’est-à-dire la prédication de la forme à la matière, doit être distinguée de la prédication classique. Nous montrerons que la prédication, lorsqu’elle est hylémorphique, se distingue en ce qu’elle permet d’assurer l’unité des termes qu’elle articule – matière et forme sont en effet « une seule et même chose ». Or c’est précisément là ce qui fait défaut à la prédication classique, qui attribue toujours « une chose à une autre », et c’est ce qui lui interdit donc d’être la structure du definiens. La tension entre les textes portant sur la structure prédicative ou non du definiens peut alors être levée en affirmant que le definiens est bien une prédication, qui ne peut pas être une prédication au sens standard du terme, mais qui doit être une prédication d’une forme à une matière. (shrink)

Thomas Kuhn’s The Structure of Scientific Revolutions offers an insightful and engaging theory of science that speaks to scholars across many disciplines. Though initially widely misunderstood, it had a profound impact on the way intellectuals and educated laypeople thought about science. K. Brad Wray traces the influences on Kuhn as he wrote Structure, including his ‘Aristotle epiphany’, his interactions, and his studies of the history of chemistry. Wray then considers the impact of Structure on the social sciences, (...) on the history of science, and on the philosophy of science, where the problem of theory change has set the terms of contemporary realism/anti-realism debates. He examines Kuhn’s frustrations with the Strong Programme sociologists’ appropriations of his views, and debunks several popular claims about what influenced Kuhn as he wrote Structure. His book is a rich and comprehensive assessment of one of the most influential works in the modern sciences. (shrink)

The question of social structure and its relationship to human agency remains one of the central problems of social theory. One of the most promising attempts to provide a solution has been Margaret Archer's morphogenetic approach, which invokes emergence to justify treating social structure as causally effective. Archer's argument, however, has been criticised by a number of authors who suggest that the examples she cites can be explained in reductionist terms and thus that they fail to sustain her (...) claim for the independent causal effectiveness of social structure. This paper offers an alternative argument to support the emergentist claim for the causal effectiveness of social structure, and shows how this argument refutes a representative critique of social emergence. (shrink)

We have provided a model-theoretic proof for the decidability of the additive structure of integers together with the function f mapping x to $$\lfloor \varphi x\rfloor $$ where $$\varphi $$ is the golden ratio.

This article deals with a tripartite structure applied to a Tannaitic source. This structure is designed in the form of a difficulty and a short resolution, bringing one example for each of the three parts of the Tannaitic source. The uniform formative-stylistic phrasing gives the discussion of the Tannaitic source's three parts a tripartite form. The Tannaitic source utilises a chaining of examples from the first to the second part and from the second to the third part.