Tori are bony growths in the mouth caused in general by genetic and environmental factors. Oral tori may grow over time and interfere with oral hygiene, speech, mastication and the application of dentures. The aim of this study is to evaluate the prevalence and patterns of torus mandibularis and torus palatinus according age and gender among patients attending the Diamond Dental Hospital in Tirana. Methodology. A case-series study conducted at DDH from February 1-30 April 2024. Written consent was taken from each participant and the questionnaire comprises some variables. The patients were examined intraoral only by one examiner and were taken photos from participants having positive findings. Results. The prevalence of oral tori in Albania was very high 48.4% and the peak incidence was 18-29 age group (54.2%). The most common type was TM 39.3%. The most prevalent form for TP was flat (71.4%) and for TM was solitary bilateral (47.9%). Conclusions. The prevalence of oral tori was found to be very high and not related to gender or bruxism. Dental professionals should be aware of the high occurrence of oral tori and their importance in dental practice. Nevertheless, further assessment of the prevalence of tori in a larger sample is required.

The Gol'denveizer problem of a torus can be described as follows: a toroidal shell is loaded under axial forces and the outer and inner equators are loaded with opposite balanced forces. Gol'denveizer pointed out that the membrane theory of shells is unable to predict deformation in this problem, as it yields diverging stress near the crowns. Although the problem has been studied by Audoly and Pomeau (2002) with the membrane theory of shells, the problem is still far from resolved within the framework of bending theory of shells. In this paper, the bending theory of shells is applied to formulate the Gol'denveizer problem of a torus. To overcome the computational difficulties of the governing complex-form ordinary differential equation (ODE), the complex-form ODE is converted into a real-form ODE system. Several numerical studies are carried out and verified by finite-element analysis. Investigations reveal that the deformation and stress of an elastic torus are sensitive to the radius ratio, and the Gol'denveizer problem of a torus can only be fully understood based on the bending theory of shells.

Gol'denveizer's problem of a torus has been analyzed by Audoly and Pomeau (2002) and Sun (2021). However, all of the investigations of Gol'denveizer's problem of an elastic torus have been linear. In this paper, the finite element method is used to more accurately address this problem. Furthermore, Sun (2021) cannot be solved by nonlinear analysis. We research the nonlinear mechanical properties of Gol'denveizer's problem of circular and elliptic tori, and relevant nephograms are given. We study the buckling of Gol'denveizer's problem of an elastic torus, and propose failure patterns and force-displacement curves of tori in the nonlinear range. Investigations reveal that circular tori have more rich buckling phenomena as the parameter a increases. Gol'denveizer's problem of the buckling of an elliptic torus is analyzed, and we find a new buckling phenomenon called a "skirt." As a/b increases, the collapse load of an elliptic torus of the Gol'denveizer problem is enhanced gradually.

Snap-fit is a commonly used jointing/connention mechanism, and its easy installation and difficult removal are the result of the coordinated interaction between friction, geometric shape, and elasticity. This paper presents a detailed study on the assembly/disassembly forces of thin-walled torus snap-fit through finite element simulation, experimental testing, and data fitting of approximate analytical solutions. The research reveals that the non-zero Gaussian curvature of the torus has a significant impact on the mechanical performance of the torus snap fit. The findings in this study are of great significance for future design of high-performance structural connections.

The nonlinear deformation and stress analysis of a circular torus is a difficult undertaking due to its complicated topology and the variation of the Gauss curvature. A nonlinear deformation (only one term in strain is omitted) of Mindlin torus was formulated in terms of the generalized displacement, and a general Maple code was written for numerical simulations. Numerical investigations show that the results obtained by nonlinear Mindlin, linear Mindlin, nonlinear Kirchhoff-Love, and linear Kirchhoff-Love models are close to each other. The study further reveals that the linear Kirchhoff-Love modeling of the circular torus gives good accuracy and provides assurance that the nonlinear deformation and stress analysis (not dynamics) of a Mindlin torus can be replaced by a simpler formulation, such as a linear Kirchhoff-Love theory of the torus, which has not been reported in the literature.

The formulation used by the most of studies on an elastic torus are either Reissner mixed formulation or Novozhilov's complex-form one, however, for vibration and some displacement boundary related problem of the torus, application of those formulations has encountered great difficulty. It is highly demanded to have a displacement-type formulation for the torus. In this paper, I will simulate some typical problems and free vibration of the torus. The numerical results are verified by both finite element analysis and H. Reissner's formulation. My investigations show that both deformation and stress response of an elastic torus are sensitive to the radius ratio, and suggest that the analysis of a torus should be done by using the bending theory of a shell, and also reveal that the inner torus is stronger than outer torus due to the property of their Gaussian curvature. Regarding the free vibration of the torus, our analysis indicates that both initial in u and w direction must be included otherwise will cause big errors in eigenfrequency.

The formulation used by most of the studies on an elastic torus are either Reissner mixed formulation or Novozhilov's complex-form one, however, for vibration and some displacement boundary related problem of a torus, those formulations face a great challenge. It is highly demanded to have a displacement-type formulation for the torus. In this paper, I will carry on my previous work [ B.H. Sun, Closed-form solution of axisymmetric slender elastic toroidal shells. J. of Engineering Mechanics, 136 (2010) 1281-1288.], and with the help of my own maple code, I am able to simulate some typical problems and free vibration of the torus. The numerical results are verified by both finite element analysis and H. Reissner's formulation. My investigations show that both deformation and stress response of an elastic torus are sensitive to the radius ratio, and suggest that the analysis of a torus should be done by using the bending theory of a shell, and also reveal that the inner torus is stronger than outer torus due to the property of their Gaussian curvature. Regarding the free vibration of a torus, our analysis indicates that both initial in u and w direction must be included otherwise will cause big errors in eigenfrequency. One of the most intestine discovery is that the crowns of a torus are the turning point of the Gaussian curvature at the crown where the mechanics' response of inner and outer torus is almost separated.

The Gol'denveizer problem of a torus was studied analytically by Audoly and Pomeau (2002), and the accuracy of the Audoly and Pomeau linear law was confirmed numerically by Sun (2021). However, the law does not include the major radius R of the torus. To find the influence of the major radius, we used finite element numerical simulation to simulate different cases, and we propose a modified Audoly and Pomeau linear law for vertical deformation, which includes R. A linear law of horizontal deformation is presented as well. Our studies show that the Audoly and Pomeau linear law has high accuracy. With modified vertical and horizontal deformation, a displacement-compatible relation between them is formulated.

For a given material, different shapes correspond to different rigidities. In this paper, the radii of the oblique elliptic torus are formulated, a nonlinear displacement formulation is presented and numerical simulations are carried out for circular, normal elliptic, and oblique tori, respectively. Our investigation shows that both the deformation and the stress response of an elastic torus are sensitive to the radius ratio, and indicate that the analysis of a torus should be done by using the bending theory of shells rather than membrance theory. A numerical study demonstrates that the inner region of the torus is stiffer than the outer region due to the Gauss curvature. The study also shows that an elastic torus deforms in a very specific manner, as the strain and stress concentration in two very narrow regions around the top and bottom crowns. The desired rigidity can be achieved by adjusting the ratio of minor and major radii and the oblique angle.

Using the orbit decomposition, a new enumerative polynomial P(x) is introduced for abstract (simplicial) complexes of a given type, e.g., trees with a fixed number of vertices or triangulations of the torus with a fixed graph. The polynomial has the following three useful properties. (I) The value P(1) is equal to the total number of unlabeled complexes (of a given type). (II) The value of the derivative P'(1) is equal to the total number of nontrivial automorphisms when counted across all unlabeled complexes. (III) The integral of P(x) from 0 to 1 is equal to the total number of vertex-labeled complexes, divided by the order of the acting group. The enumerative polynomial P(x) is demonstrated for trees and then is applied to the triangulations of the torus with the vertex-labeled complete four-partite graph G = K_{2,2,2,2}, in which specific case P(x) = x^{31}. The graph G embeds in the torus as a triangulation, T(G). The automorphism group of G naturally acts on the set of triangulations of the torus with the vertex-labeled graph G. For the first time, by a combination of algebraic and symmetry techniques, all vertex-labeled triangulations of the torus (twelve in number) with the graph G are classified intelligently without using computing technology, in a uniform and systematic way. It is helpful to notice that the graph G can be converted to the Cayley graph of the quaternion group Q_8 with the three imaginary quaternions i, j, k as generators.

The Hyper-Torus Universe Model (HTUM) proposes a novel framework to unify quantum mechanics, cosmology, and consciousness. This paper introduces HTUM's core concept: a higher-dimensional hyper-torus encompassing all possible states of existence. HTUM suggests that the universe functions as a quantum system where all outcomes are inherently connected, with consciousness playing a significant role in actualizing reality. The model introduces several key innovations, including the universe's 4-dimensional toroidal structure (4DTS), providing a new geometric interpretation of spacetime. HTUM presents dark matter and dark energy as nonlinear probabilistic phenomena within this toroidal framework, offering a unique perspective on their nature and interactions. The model proposes a mechanism for wave function collapse through universal self-observation, bridging quantum mechanics and gravity. It introduces a Topological Vacuum Energy Modulator (TVEM) function, offering a potential resolution to the cosmological constant problem. Furthermore, HTUM integrates consciousness as a fundamental aspect of the universe, influencing quantum state actualization and potentially resolving the hard problem of consciousness. The model explores the implications of a timeless singularity and the emergence of time as a product of causal relationships within the toroidal structure. This framework addresses critical challenges in modern physics, including the nature of quantum entanglement, the origin and fate of the universe, and the relationship between mind and matter. The paper discusses the mathematical formulation of HTUM, its implications for quantum mechanics and cosmology, and its potential to bridge the gap between science and philosophy. We present detailed predictions for the cosmic microwave background (CMB) power spectrum, gravitational wave signatures, and large-scale structure formation, demonstrating how HTUM differs from the standard ΛCDM model. Our analysis reveals distinctive oscillatory patterns in the relative difference between HTUM and ΛCDM predictions, with deviations reaching up to ±20% at certain scales. The paper also explores the unification of fundamental forces within the HTUM framework, proposing novel approaches to particle physics and quantum field theory. It introduces the concept of unified mathematical operations, suggesting a more holistic approach to mathematical thinking that aligns with the model's interconnected view of the universe. Additional testable predictions and proposed experimental approaches are discussed, including advanced space-based tests and laboratory experiments probing quantum gravity effects. These specific, quantifiable predictions offer multiple avenues for empirical validation, highlighting HTUM's potential to be distinguished from ΛCDM and other alternative cosmological models. HTUM represents a significant new perspective in our understanding of the universe, inviting further research and exploration into the nature of reality, consciousness, and the fundamental structure of the cosmos. By providing a comprehensive framework that unifies various aspects of physics, cosmology, and consciousness studies, HTUM offers a promising path toward a more complete understanding of our universe.

\abstract{The existence of vorticies is well known in fluid mechanics and particularly in turbulence theory. In the context of the millennium problem of the Clay Institute on the existence (or not) of Global solutions in time of the Periodic Navier Stokes equations (PNS) results are now proposed to confirm that Global solutions do not exist in the general case of PNS defined on $\mathbb{T}^3$. This corresponds to part ($D$) proposition in the Navier Stokes problem as posted by C.Fefferman. Recent work has shown that for the Euler equations defined on the 3-Torus there exists a H\"older continuous solution for the third component of velocity $v_3$ at the center of each cell or cube of the 3-Torus Lattice. It has also been determined that as the kinematic viscosity changes from the corresponding value at $\nu=0$ to the fully viscous case for the Periodic Navier Stokes equations (PNS) at the value $\nu=1$ the solution reaches a peak ($\frac{\partial v_3}{\partial y_3} = 0$ ) in $y_3$. (third component of $\vec{r}=(y_1,y_2,y_3)$ in Cartesian co-ordinates) Here in the viscous case there exists a dipole which is not centered at the center of the cell of the Lattice. This immediately implies that since the dipole by definition has an equal in magnitude positive and negative peak in the third component of velocity, then the dipole Riemann cut-off surface is covered by a closed surface which is the sphere $\|\vec{r}\|=1$ and where a given cell of dimensions $[-1,1]^3$ is circumscribed on a sphere of radius $1$. For such a closed surface containing a dipole it necessarily follows that the flux at the surface of the sphere of $v_3$ wrt to surface normal $\vec{n}$ is zero including at $y_i, i=1\dots 3$ is zero at the points where the surface of sphere touches the cube walls. At the finite time singularity on the sphere a rotation boundary condition is deduced. Here $\frac{\partial v_3}{\partial \vec{n}}= \vec{n}\cdot \nabla v_3$. Also a boundary condition on the sphere shows rotation of sphere. Finally as a result, on the sphere a solution for $v_3$ is obtained which is proven to be H\"older continuous and it is proven that it is possible to extend H\"older continuity on the sphere uniquely to all of the interior of the ball.}