Torus

An algorithm based on a recurrent neural Wang’s network and the WTA (“Winner takes all”) principle is applied to the construction of Hamiltonian cycles in graphs of distributed computer systems (CSs). The algorithm is used for: 1) regular graphs (2D- and 3D-tori, and hypercubes) of distributed CSs and 2) 2D-tori disturbed by removing an arbitrary edge. The neural network parameters for the construction of Hamiltonian cycles and suboptimal cycles with a length close to that of Hamiltonian ones are determined. Our experiments show that the iterative method (Jacobi, Gauss-Seidel, or SOR) used for solving the system of differential equations describing a neural network strongly affects the process of cycle construction and depends on the number of torus nodes.

Potential merit of a toroidal helical configuration
Configuration of a toroidal helix
Deriving coordinates for a toroidal helix (helical toroid)
Dynamic possibilities of 3D visualizations of a helical toroid with multiple windings
Movement of spheres along the pathways of a toroidal helix
Potential cognitive implications of toroidal helical movement
Technomimicry, metaphors and symbolism
Category juggling reframed through visualization dynamics
Requisite controversy engendered and encoded by a counter-coil pattern?
Towards a 3D visualization of toroidal counter-coiling dynamics
Transforming vehicles of identity between global and toroidal forms
Fearful attraction of a hole
Representational challenge of the periodic table of chemical elements
Cognitive heart dynamics framed by two tori in 3D

"Moreover, brethren, I would not that ye should be ignorant, how that all our fathers were under the cloud, and all passed through the sea; and were all immersed unto Moses in the cloud and in the sea; and did all eat the same spiritual food; and did all drink the same spiritual drink; for they drank of that spiritual Rock that went with them; and that Rock was the Messiah." Traina-Holy Name Bible I Corinthians 10:1-5

Torus interconnect -- as used in supercomputers
Cubic organization of I Ching trigrams -- an eightfold way
Pattern language and polyhedral mapping
Higher dimensionality, polyhedral packing and transformation
Brain organization, cognition, comprehension -- and music
Oppositional logic and its requisite polyhedral geometry
Reflexivity in multi-loop thinking and higher order learning
Toroidal constraint -- nuclear fusion as metaphor of cognitive fusion
Supercomputers, hypercomputing and superquestions?
Framing the space for conscious creativity?
Dancing cognitively inside the box -- and beyond

As demonstrated by Columbia University's "Proto-CIRCUS" plasma containment device, a flexible "Villarceau torus" – an array of interlocking rigid Villarceau-circle coils with adjustable or flexible mountings – can be used to create toroidal fields with variable geometry. We review some basic results relevant to Villarceau circles and fixed and variable Villarceau coils.

In the current research, atempts are made to formulate new hypotheses on the processes that could have determined the shape of the sail of the umbrella vault of Pazzi Chapel in Santa Croce church in Florence. The descriptons given so far, verifed through three-dimensional models, despite whether drew near to geometry, did not give any explanaton of building process, which is a core issue for explaining the peculiar shape of vaults. The analyses are done by correlatng detailed surveys and geometric analyses of the vault, comprising the curves, surfaces and the possible masonry texture with the hypothesized form. The vault is double-sailed and the form of the inner sail (conchoid surface) is obtained by a three-dimensional transformaton of an outher sail (toric surface), with respect to the fxed point (dome’s oculus) and constant lenght (torus radius), like the Nicomedes conchoid in 2d.

Background and Aim: Oral tori are bony growth present in the oral cavity and are not considered as pathological lesions. Smaller
tori do not cause any problems, but larger sized tori can result in significant problems. The aim of this study was to determine
the prevalence of torus palatinus (TP) and torus mandibularis (TM) in the Indian population.
Study Design: A total of 3087 patients were examined between August, 2010 and October, 2012 for the presence of TP and
TM by inspection and palpation. The data were collected and analyzed using SPSS 12.0 (SPSS Inc., Chicago, USA).
Results: A total of 36 patients (1.3%) presented with TP, while 214 patients (6.9%) presented with TM. Tori were more frequent
in males and this difference was statistically significant (P < 0.05). TP was more common in 31-40 years age group (13 patients,
36.1%), whereas TM was more in the 41-50 years age group (81 patients, 37.9%). The most commonly observed type of TP
was flat shaped, whereas bilateral solitary types were the most common TM.
Conclusion: The results of this study showed a significantly higher prevalence of tori in males. No difference in the prevalence
of TP and TM was noted regarding the age groups. Although, not pathologically significant, these bony overgrowths very often
need surgical removal, especially when prosthesis are indicated.

Let $\bf M$ be a smooth compact oriented Riemannian manifold, and let $\Delta_{\bf M}$ be the Laplace-Beltrami operator on ${\bf M}$. Say $0 \neq f \in \mathcal{S}(\RR^+)$, and that $f(0) = 0$. For $t > 0$, let $K_t(x,y)$ denote the kernel of $f(t^2 \Delta_{\bf M})$. We show that $K_t$ is well-localized near the diagonal, in the sense that it satisfies estimates akin to those satisfied by the kernel of the convolution operator $f(t^2\Delta)$ on $\RR^n$. We define continuous ${\cal S}$-wavelets on ${\bf M}$, in such a manner that $K_t(x,y)$ satisfies this definition, because of its localization near the diagonal. Continuous ${\cal S}$-wavelets on ${\bf M}$ are analogous to continuous wavelets on $\RR^n$ in $\mathcal{S}(\RR^n)$. In particular, we are able to characterize the H$\ddot{o}$lder continuous functions on ${\bf M}$ by the size of their continuous ${\mathcal{S}}-$wavelet transforms, for H$\ddot{o}$lder exponents strictly between 0 and 1. If $\bf M$ is the torus $\TT^2$ or the sphere $S^2$, and $f(s)=se^{-s}$ (the ``Mexican hat'' situation), we obtain two explicit approximate formulas for $K_t$, one to be used when $t$ is large, and one to be used when $t$ is small.

The longitudinal motions and vertical accelerations of a floating torus as well as wave motion inside the torus are studied by model tests in regular deep-water waves. Comparisons are made with linear and partly with second-order potential-flow theory for the smallest examined experimental wave height-to-wave length ratio 1/120. Reasonable agreement is obtained, in particular for the linear problem. The importance of 3D flow, hydroelasticity and strong hydrodynamic frequency dependency is documented. Experimental precision errors and bias errors, for instance, due to tank-wall interference are discussed. Numerical errors due to viscous effects are found to be secondary. Experiments show that the third and fourth harmonic accelerations of the torus matter and cannot be explained by a perturbation method with the wave steepness as a small parameter. Wave-induced loads on a fish farm with a circular plastic collar motivate our studies. The interaction with other parts of the fish farm, such as the netting structure, bottom rings, chains, ropes and a realistic mooring system, are important but are not investigated. Both wave and current loads matter but current is neglected. Dynamic behavior of a fish farm in real conditions is a complex scenario (see Fig. 1), which for the floater can involve large relative vertical motion compared to the cross-sectional dimensions, large local accelerations, hydroelastic effects and violent wave structure interaction with local wave overtopping and out of water phenomena. Dedicated model tests and numerical simulations by Kristiansen and Faltinsen (2015) investigated the mooring loads through a detailed and broad numerical study and assessed the physical effects which are relevant for the prediction of the mooring loads. Amongst them, a rigid floater significantly alters the mooring loads if compared with a realistic elastic floater. He et al. (2015) demonstrated by model tests with live fish occupying a representative volume 2.5% of the net cage that fish could have a non-negligible influence on the mooring loads in waves and current. The considered floater is a torus even though it is more common in practice to operate with two adjacent tori. Wavelengths of practical interest are of the order of the torus diameter but long relative to the cross-sectional diameter. Engineering tools for net cages often estimate the wave-induced floater loads using strip theory with linear potential flow and drag-force corrections from Morison equation, which disregard important 3D flow, frequency dependency and nonlinear effects.

Patients with exostosis (bony nodules on the vestibular face of the mandible bone) and mandibular torus (bony nodules on the lingual face of the mandible bone) frequently present in bruxism, but not all patients with bruxism present torus. With the strong relation between bruxism and anxiety already consecrated in the literature, the idea arose that the anxiety could have some participation in this process. Patients with torus present more frequently dental tightness, anxiety, TMD (temporomandibular disorder) or bruxism. The presence of torus and its size may be related to parafunction. In the literature there are torus
in both sexes. It is evident in adulthood between 20 and 30 years and is rare in children younger than 10 years.

Previous 3D experiments with representations of symbols in 2D Alternative hexagram representation-as a resonance hybrid? Configuring the set of hexagrams as a toroidal Ouroboros Experimental interactive animations of the Ouroboros-Yi Jing "integration" Cognitive engagement with fundamental symbol dynamics Indications of how to experiment with the 3D animations Technical constraints of 3D applications and their cognitive implications Further 3D design possibilities and metaphors

Reality distortion, psychosocial torsion and psychological torque
Intuitive recognition of toroidal cycles?
Imagining life on a torus, or within it -- as lived in reality?
Being "flat"-- or "global" and "well-rounded"?
Transforming the world into a doughnut: a vital clarification
Toroidal conceptualization
Toroidal embodiment, knottedness and being a torus?
Mistaken recognition of holes repressed by a global focus?
Torc-bearing, Playing-ball, Scoring and Nesting
Aesthetic reconciliation of contrasting toroidal metaphors?
Transformative interplay between flat, global and toroidal?

Pseudo-differential and Fourier series operators on the n-torus are analyzed by using global representations by Fourier series instead of local representations in coordinate charts. Toroidal symbols are investigated and the correspondence between toroidal and Euclidean symbols of pseudo-differential operators is established. Periodization of operators and hyperbolic partial differential equations is discussed. Fourier series operators, which are analogues of Fourier integral operators on the torus, are introduced, and formulae for their compositions with pseudo-differential operators are derived. It is shown that pseudo-differential and Fourier series operators are bounded on $L^2$ under certain conditions on their phases and amplitudes.

Abstract Vibration analysis of a gossamer or inflated structure poses special problems, usually not encountered in a conventional metallic or composite structure. In an inflated structure, internal pressure is a major source of strength and rigidity. In the past, most of the studies conducted on the vibration analysis of gossamer structures used inaccurate or approximate theories in modelling the internal pressure. The inexactness in these theories arises due to (1) exclusion of the follower pressure loads, and (2) approximations in the geometric non-linearity. Taking cues from the earlier work done in this area and using line-of-curvature co-ordinates, we re-derive the governing equations for vibration analysis of a shell under pressure, and point out the shortcomings of the previous approximate theories. The same governing equations were derived earlier by Budiansky using tensors. Thereafter, a free-vibration analysis of an inflated torus with free boundary condition is performed using the accurate and the approximate shell theories. It can be seen that the natural frequencies and the mode shapes obtained from the approximate theories are significantly different from those obtained from the accurate shell theory. Since the boundary condition of the torus is free, the vibration analysis should yield six zero frequencies corresponding to the six rigid-body modes. It is shown here that while the accurate theory does give six zero frequencies, the approximate theories do not.

The grid and the mesh of trees (or MOT) are among the best-known parallel architectures in the literature. Both of them enjoy efficient VLSI layouts, simplicity of topology, and a large number of parallel algorithms that can efficiently execute on them. One drawback of these architectures is that algorithms that perform best on one of them do not perform very well on the other. Thus there is a gap between the algorithmic capabilities of these two architectures. We propose a new class of parallel architectures, called the mesh ...

Abstract. Let I be a monomial ideal in a polynomial ring A= k [x1,..., xn] over a field k of characteristic 0, Tt/k (I) be the module of I-preserving k-derivations on A and G be the n-dimensional algebraic torus on k. We compute the weight spaces of Tt/k (I) considered as ...

We propose an effective and low cost method to increase the yield and the lifetime of torus NoCs. The method consists in detecting and diagnosing NoC interconnect faults using BIST structures and activating alternative paths for the faulty links. Alternative paths use the inherent redundancy of the torus topology, thus leading to minimal performance, area, and power overhead. We assume