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We prove sharp spectral projection estimates for general tori in all dimensions at the exponent p c = 2 (n + 1) n - 1 for shrinking windows of width 1 down to windows of length λ - 1 + κ for fixed κ > 0 . This improves and generalizes the work of Blair–Huang–Sogge which proved sharp results for windows of width λ - 1 n + 3 in Blair et al. (J Eur Math Soc. ), and the work of Hickman (Math Res Rep 1:31–45, 2020), Germain–Myerson (Forum Math Sigma 10:Paper No. e24, 20, 2022), and Demeter–Germain ( L 2 to L p bounds for spectral projectors on the Euclidean two-dimensional torus. ) which proved results for windows of all widths but incurred a sub-polynomial loss. Our work uses the approaches of these two groups of authors, combining the bilinear decomposition and microlocal techniques of Blair–Huang–Sogge with the decoupling theory and explicit lattice point lemmas used by Hickman, Germain–Myerson, and Demeter–Germain to remove these losses. [ABSTRACT FROM AUTHOR]

We prove some new Hasse principles related with the property of being split or (strongly) quasi-split for connected smooth affine algebraic groups which are defined over an algebraic extension of a global field. We give also some refinements of the Hasse principles established before for infinite algebraic extensions of global fields. [ABSTRACT FROM AUTHOR]

Benign bony growths called tori can develop in various places throughout the mandible and maxilla, although their exact origin is still unknown. There have been numerous reports of unilateral or bilateral Mandibular Tori (MT); however, no reports of a midline anterior mandibular tori have been made yet. A 27-year-old female patient reported with a complaint of a bony growth on her lower jaw below the tongue. The tori are unusually situated at the midline on the lingual side of the mandibular central incisor. The elements shown to be causal were the gender of the male, the ethnicity of Asians, and developmental in origin. The cornerstone of treatment for Tori is surgical resection. After a year of follow-up, there was no recurrence. The present case highlighted the significance of etiological factors for bony exostosis management and recurrence, which, in conjunction with origin, gender, ethnic and racial factors, may account for the patient’s lack of recurrence.

Benign bony growths called tori can develop in various places throughout the mandible and maxilla, although their exact origin is still unknown. There have been numerous reports of unilateral or bilateral Mandibular Tori (MT); however, no reports of a midline anterior mandibular tori have been made yet. A 27-year-old female patient reported with a complaint of a bony growth on her lower jaw below the tongue. The tori are unusually situated at the midline on the lingual side of the mandibular central incisor. The elements shown to be causal were the gender of the male, the ethnicity of Asians, and developmental in origin. The cornerstone of treatment for Tori is surgical resection. After a year of follow-up, there was no recurrence. The present case highlighted the significance of etiological factors for bony exostosis management and recurrence, which, in conjunction with origin, gender, ethnic and racial factors, may account for the patient's lack of recurrence. [ABSTRACT FROM AUTHOR]

We study the solvability of a class of fully nonlinear equations on the flat torus. The equations arise in the study of some Calabi–Yau type problems in torus bundles. [ABSTRACT FROM AUTHOR]

If one thinks of a Riemannian metric, g 1 , analogously as the gradient of the corresponding distance function, d 1 , with respect to a background Riemannian metric, g 0 , then a natural question arises as to whether a corresponding theory of Sobolev inequalities exists between the Riemannian metric and its distance function. In this paper, we study the sub-critical case p < m 2 where we show a Sobolev inequality exists between a Riemannian metric and its distance function. In particular, we show that an L p 2 bound on a Riemannian metric implies an L q bound on its corresponding distance function. We then use this result to state a convergence theorem and show how this theorem can be useful to prove geometric stability results by proving a version of Gromov’s conjecture for tori with almost non-negative scalar curvature in the conformal case. Examples are given to show that the hypotheses of the main theorems are necessary. [ABSTRACT FROM AUTHOR]

Background: Each combat sport carries different risks of injury due to the specifics of the sport (including the weight categories and sex) and the fighting techniques used according to different sports regulations. The purpose of this study is to examine injury symmetry in judo. Methods: Injuries recorded in 195 people (93 women, 102 men) suffered during top-level judo tournaments were verified. Using the European Judo Union medical questionnaire, information on injuries was obtained from each injured athlete. Results: Based on the analysis of the results, it can be concluded that injuries are almost evenly distributed on the left and right sides of the body in both men and women. In women, there were 129 injuries to the left side (41.2%), 134 injuries to the right (42.8%), and 50 (16.0%) to the middle part of the body. Conclusions: injuries in judo are evenly distributed and slightly more common on the right side. Injuries occur at a greater rate during defending maneuvers than attacking maneuvers in judo. Identification and monitoring of who (tori or uke) and which side of the body sustains an injury are crucial and important in injury prevention. This knowledge makes it possible to modify existing sports regulations by eliminating behaviors (e.g., certain types of defense) to improve the safety of athletes participating in top-level competitions and the training process in terms of ensuring safety in both attack and defense. [ABSTRACT FROM AUTHOR]

Introduction: Oral involuntary changes in total edentulous age often lead to therapeutic difficulties and complications that may interfere with or influence prosthetic iatrogenesis. Purpose: The aim of this clinical study was to identify bone complications of denture bearing areas and to establish their frequency in the group of complete edentulous patients. Material and methods: The study group consisted of 130 patients, 60 males and 70 females,complete edentulous aged between 30 and> 80 years. All data obtained from study group patients were centralized and statistically analyzed. Results and discussions: Of the 130 patients, after clinical examination it was revealed the presence of 86 hyperostotic tuberosities. A total of 49 patients (37.69%) were diagnosed with a maxillary tori and the existence of mandibular torus was found in the case of 26 patients (20%). Conclusions: Careful investigation of the general and local condition, is a particularly valuable tool in detecting bone complications

In this note, we study the normal compact Kähler (possibly singular) threefold X admitting the action of a free abelian group G of maximal rank, all the non-trivial elements of which are of positive entropy. If such X is further assumed to have only terminal singularities, then we prove that it is either a rationally connected projective threefold or bimeromorphic to a quasi-étale quotient of a complex 3-torus. [ABSTRACT FROM AUTHOR]

In this article we reduce the geometric stability conjecture for the scalar torus rigidity theorem to the conformal case via the Yamabe problem. Then we are able to prove the case where a sequence of Riemannian manifolds is conformal to a uniformly controlled sequence of flat tori and satisfies the geometric stability conjecture. We are also able to handle the case where a sequence of Riemannian manifolds is conformal to a sequence of constant negative scalar curvature Riemannian manifolds which converge to a flat torus in C 1 . The full conjecture from the conformal perspective is also discussed as a possible approach to resolving the conjecture. [ABSTRACT FROM AUTHOR]

Background: Each combat sport carries different risks of injury due to the specifics of the sport (including the weight categories and sex) and the fighting techniques used according to different sports regulations. The purpose of this study is to examine injury symmetry in judo. Methods: Injuries recorded in 195 people (93 women, 102 men) suffered during top-level judo tournaments were verified. Using the European Judo Union medical questionnaire, information on injuries was obtained from each injured athlete. Results: Based on the analysis of the results, it can be concluded that injuries are almost evenly distributed on the left and right sides of the body in both men and women. In women, there were 129 injuries to the left side (41.2%), 134 injuries to the right (42.8%), and 50 (16.0%) to the middle part of the body. Conclusions: injuries in judo are evenly distributed and slightly more common on the right side. Injuries occur at a greater rate during defending maneuvers than attacking maneuvers in judo. Identification and monitoring of who (tori or uke) and which side of the body sustains an injury are crucial and important in injury prevention. This knowledge makes it possible to modify existing sports regulations by eliminating behaviors (e.g., certain types of defense) to improve the safety of athletes participating in top-level competitions and the training process in terms of ensuring safety in both attack and defense.

The studies of the classical cops and robber problem are generally aimed at determining the minimum number of cops needed to capture the robber, and proposing algorithms for the capture. This paper is a contribution to this problem, directed to two-dimensional grids, cylinders (i. e., grids with toroidal closure in one dimension), and tori, with a new extension to using teams of any number of cops. We discuss some new features of the problem and propose a new solution to the capture problem on grids that was already solved under a different approach, and then give efficient capture algorithms on cylinders and tori making use of these features. We examine the effect of using teams of any number k of cops and give efficient algorithms for this case, evaluating lower and upper bounds on the capture time t k , and compute the minimum value of k needed for any given capture time. To this aim we extend the concept of work w k = k · t k of an algorithm, inherited from parallel processing, and study a possible speed-up phenomenon using larger teams of cops. [ABSTRACT FROM AUTHOR]

We give a concrete characterization of the rational conjugacy classes of maximal tori in groups of type D n , with specific emphasis on the case of number fields and p -adic fields. This includes the forms associated to quadratic spaces, all of their inner and outer forms as well as the Spin groups, their simply connected covers. In particular, in this work, we handle all (simply connected) outer forms of D 4 . [ABSTRACT FROM AUTHOR]

A system of ordinary differential equations (ODEs) is derived from a discrete system of Vicsek, Czir\'ok et al. 1995, describing the motion of a school of fish. Classes of linear and stationary solutions of the ODEs are found and their stability explored using equivariant bifurcation theory. The existence of periodic and toroidal solutions is also proven under deterministic perturbations and structurally stable heteroclinic connections are found. Applications of the model to the migration of the capelin, a pelagic fish that undertakes an extensive migration in the North Atlantic, are dissussed and simulation of the ODEs presented.

Teaching point: Multiple buccal exostosis are rare but asymptomatic, unequivocal, and always benign conditions of the jaws

We address the question whether there is a three-dimensional bounded domain such that the Neumann–Poincaré operator defined on its boundary has infinitely many negative eigenvalues. It is proved in this paper that tori have such a property. It is done by decomposing the Neumann–Poincaré operator on tori into infinitely many self-adjoint compact operators on a Hilbert space defined on the circle using the toroidal coordinate system and the Fourier basis, and then by proving that the numerical range of infinitely many operators in the decomposition has both positive and negative values. [ABSTRACT FROM AUTHOR]

Buccal exostoses are broad-based, non-malignant surface growth occurring on the outer or facial surface of the maxilla and/or mandible, found usually in the premolar and molar region. Etiology is still not established, but it has been suggested that the bony overgrowth can be because of abnormally increased masticatory forces to the teeth. Compensatory response to periodontal disease has been proposed to explain some cases of exostoses; they tend to appear in early adolescence and may very slowly increase in size with time. They are painless, self-limiting and may increase patient concern about poor esthetics, inability to perform oral hygiene procedures, and compromised periodontal health by causing food lodgment. The following article presents a very rare case of bilateral buccal-sided maxillary and mandibular exostoses in the anterior region which was an accidental finding in a chronic generalized periodontitis patient.

We study a problem of stochastically forced quasi-periodic self-oscillations of nonlinear dynamic systems, which are modelled by an invariant torus in the phase space. For weak noise, an asymptotic of the stationary distribution of random trajectories is studied using the quasipotential. For the constructive analysis of a probabilistic distribution near a torus, we use a quadratic approximation of the quasipotential. A parametric description of this approximation is based on the stochastic sensitivity functions (SSF) technique. Using this technique, we create a new mathematical method for the probabilistic analysis of stochastic flows near the torus. The construction of SSF is reduced to a boundary value problem for a linear differential matrix equation. For the case of the two-torus in the three-dimensional space, a constructive solution of this problem is given. Our theoretical results are illustrated with an example.

In the current study, distance-based topological invariants, namely the Wiener number and the topological roundness index, were computed for graphenic tori and Klein bottles (named toroidal and Klein bottle fullerenes or polyhexes in the pre-graphene literature) described as closed graphs with N vertices and 3N/2 edges, with N depending on the variable length of the cylindrical edge LC of these nano-structures, which have a constant length LM of the Möbius zigzag edge. The presented results show that Klein bottle cubic graphs are topologically indistinguishable from toroidal lattices with the same size (N, LC, LM) over a certain threshold size LC. Both nano-structures share the same values of the topological indices that measure graph compactness and roundness, two key topological properties that largely influence lattice stability. Moreover, this newly conjectured topological similarity between the two kinds of graphs transfers the translation invariance typical of the graphenic tori to the Klein bottle polyhexes with size LC ≥ LC, making these graphs vertex transitive. This means that a traveler jumping on the nodes of these Klein bottle fullerenes is no longer able to distinguish among them by only measuring the chemical distances. This size-induced symmetry transition for Klein bottle cubic graphs represents a relevant topological effect influencing the electronic properties and the theoretical chemical stability of these two families of graphenic nano-systems. The present finding, nonetheless, provides an original argument, with potential future applications, that physical unification theory is possible, starting surprisingly from the nano-chemical topological graphenic space; thus, speculative hypotheses may be drawn, particularly relating to the computational topological unification (that is, complexification) of the quantum many-worlds picture (according to Everett’s theory) with the space-curvature sphericity/roundness of general relativity, as is also currently advocated by Wolfram’s language unification of matter-physical phenomenology.

A 56 year old African-American female with history of well-controlled hypertension and hyperlipidemia presented to the office for an annual physical examination. She did not have any complaints. She reported being compliant with her medications, exercised daily at her local gym, and maintained a low salt diet. She visits her dentist every 6 months and has had a few fillings in her premolars in the past. On physical examination, her vital signs were normal and the entirety of her physical examination was normal with the exception of her oropharynx. Throat examination revealed a 2 × 1 cm midline hard palate swelling that was bony hard in consistency and covered by normally appearing oral mucosa. It was diagnosed as a torus palatinus. The patient was unaware of its presence and denied being informed about it by her dentist on any visit. She was also educated about the diagnosis and safety-netted by being informed about red-flags that would prompt investigation such as change in size or shape.

The objectives of the study is the assessment of torus palatinus (TP) prevalence as well as torus mandibular is (TM) among different gender and age groups. Seven hundred and fifty (750) out door patients (380 male and 370 female) were tested for the tori presence or absence at dentistry collage in Basrah city between January 2019 and June 2019. Sex, gender and location factors were evaluated. The prevalence rate of TP was 4.73% for male and 5.13% for female, and the prevalence rate of TM was 2.36% for male and 3.24% for female. In females, TP prevalence was more compared to males. The TP incidence more commonly in middle aged persons, While the TM more commonly in older persons. [ABSTRACT FROM AUTHOR]

It is shown that the surface with a constant mean curvature encloses the extremal volume among all toroidal surfaces of given area. The exact solution for the corresponding variational problem is derived, and its parametric analysis is performed in the limits of high and small mean curvatures. An absence of smooth torus with constant mean curvature is proved, and the extremal surface is demonstrated to have at least one edge located on the outer side of the torus. [ABSTRACT FROM AUTHOR]

Objective: This systematic review aims to answer the question: 'Is there an association between any specific signs and symptoms of bruxism and the presence of tori?' Material and methods: Observational studies, which evaluated the association between signs and symptoms of bruxism and tori, were selected. Signs and symptoms of bruxism (such as teeth grinding, jaw clenching, abnormal tooth wear, facial muscle hypertrophy, pain, or fatigue) had to be determined by questionnaire or anamnesis and tori within clinical assessment. Search-strategies were developed for five databases, in addition to three gray literature's databases. The risk of bias was evaluated using the ' Meta-Analysis of Statistics Assessment and Review Instrument'. A summary of overall strength of evidence was estimated using GRADE's Summary of findings table. Results: Among 575 studies, five were included. Two studies were categorized as moderate risk of bias and three as high risk of bias. Self-report of teeth grinding and/or clenching presented contradictory results. Presence of abnormal tooth wear increased the odds of having tori, mainly for torus mandibularis. The overall quality of evidence ranged from low to very low. Conclusion: Based on available evidence, the presence of abnormal tooth wear might be associated with tori, mainly torus mandibularis. There is no sufficient evidence to credit or discredit the association of tori and other signs and/or symptoms of bruxism. Clinical relevance: Bruxism diagnosis is a challenge. The association between signs and symptoms of bruxism and tori could help clinicians on the recognition of patients susceptible to bruxism. This knowledge might also aid to the understanding of tori's development and stimulate new relevant research. [ABSTRACT FROM AUTHOR]

A new algebraic method for extracting tori from a minimal point set, made of two oriented points and a simple point, is proposed. We prove a degree bound on the number of such tori; this bound is reached on examples, even when we restrict to smooth tori. Our method is based on pre-computed closed formulae well suited for numerical computations with approximate input data. [ABSTRACT FROM AUTHOR]

Given a connected surface $${\mathbb {F}}^2$$ with Euler characteristic $$\chi $$ and three integers $$b>a\ge 1

ABSTRACT: Objectives: The objectives of this study were to determine the prevalence and clinical characteristics of torus palatinus (TP) and torus mandibularis (TM) in Malaysian dental patients. Methods: Thousand five hundred and thirty-two dental patients were examined for the presence of oral tori at the Faculty of Dentistry outpatient clinic, AIMST University. Factors such as race, age, sex, size, and shape of tori were studied. Results: The prevalence rates were 12% for TP and 2.8% for TM. A variation in the presence of tori among the three races in Malaysia–Chinese, Malays, and Indians–was noted, where the Chinese significantly had a higher prevalence of TP (17.9%) and TM (4.6%). Predominantly, tori were observed >40 years and older age group, and further both TP and TM were seen more commonly in women. Most TP were of smooth type (52.2%) and >2 cm (67.4%), while all TM were bilateral and nodular, plus most were