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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]

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]

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]

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]

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]

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]

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

There is a two-parametric family of rotational symmetric CMC surfaces;more precisely, for every real number H and every C ≥ 2(H +√1 + H2) there is arotational symmetry surface ΣH,C with mean curvature H. Perdomo (Asian J Math14:73-108, 2010) showed that for every H between cot ΣЎm and m2-22√m2-1there existsan embedded rotational symmetric example with non-constant principal curvaturesthat is invariant under the cyclic groupZm. Recently Andrews and Li (J Differ Geom99:169-189, 2015) showed that these embedded CMC tori are the only embeddedgenus 1 surfaces with CMC on the sphere. In this paper we complete the study of thisfamily of CMC surfaces and we show that for every integer m > 2, there is a properlyimmersed example in this family that contains a great circle and is invariant underthe cyclic group Zm. We will say that these examples contain the axis of symmetry.We also show that every non-isoparametric surface ΣH,C is either properly immersedand invariant under the cyclic group Zm for some integer m > 1 or it is dense in theregion bounded by two isoparametric tori if the surface ΣH,C does not contain theaxis of symmetry or it is dense in the region bounded by a totally umbilical surface ifthe surface ΣH,C contains the axis of symmetry.√. [ABSTRACT FROM AUTHOR]

To study the dynamics of quasi-periodic bifurcations, we consider a system of two nonlinearly coupled oscillators using averaging, continuation and numerical bifurcation techniques. This relatively simple system displays considerable complexity. Assuming the internal resonance to be 1:2, we find a 2 π-periodic solution which undergoes a supercritical Neimark-Sacker bifurcation, yielding a stable torus. Choosing a route in parameter space, we show by numerical bifurcation techniques how the torus gets destroyed by dynamical and topological changes in the involved manifolds (Krauskopf and Osinga in J Comput Phys 146:404-419, ). The 1:6 resonance turns out to be prominent in parameter space, and we detected a cascade of period doublings within the corresponding resonance tongue yielding a strange attractor. The phenomena agree with the Ruelle-Takens (Commun. Math. Phys. 20:167-192, , Commun. Math. Phys. 23:343-344, ) scenario leading to strange attractors. Other periodic regimes are present in this system, and there is interesting evidence that two different regimes interact with each other, yielding yet another type of strange attractor. In this context, certain π-periodic solutions emerge that are studied by continuation following the Poincaré-Lindstedt method using Mathieu functions; when the implicit function theorem breaks down, the analysis is supplemented by numerical bifurcation techniques. [ABSTRACT FROM AUTHOR]

A system of ordinary differential equations (ODEs) is derived from a discrete system of Vicsek, Czirók et al. [ Phys. Rev. Lett. 75(6):1226–1229, 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 discussed and simulation of the ODEs presented. [ABSTRACT FROM AUTHOR]

We examine the solutions to a damped, quasiperiodic (QP) Mathieu equation with cubic nonlinearities. The system is suspended in a four-dimensional phase space ...² × T² in which there exist attracting, knotted 2-tori called torus braids. We develop a topological classification scheme in which a torus braid is characterized by closed braids that exist in two Poincaré sections, ...² × S¹ × {...} and ...² × {...} × S¹. Based on the classification scheme, we develop numerical invariants that describe the linkedness of attractors and provide information about the global dynamics. Numerical simulations show that changes of a single parameter lead to a global bifurcation through which the attracting torus loses stability and locally "doubles," shedding a torus of different equivalence class. We call this a topological torus bifurcation of the doubling variety (TTBD). We provide a topological analysis of the doubling produced by TTBDs and we examine the qualitative dynamical changes that result. We also examine the effect of TTBDs on the spectrum of Lyapunov exponents and the time series power spectrum. [ABSTRACT FROM AUTHOR]

Invariant manifolds like tori, spheres and cylinders play an important part in dynamical systems. In engineering, tori correspond with the important phenomenon of multi-frequency oscillations. Normal hyperbolicity guarantees the robustness of these manifolds but in many applications weaker forms of hyperbolicity present more realistic cases and interesting phenomena. We will review the theory and present a number of examples using normalization-averaging techniques. [ABSTRACT FROM AUTHOR]

The incremental and dynamic construction of interconnection networks from smaller components often leaves the fundamental problem of assigning addresses to processors to be contended with at power-up time. The problem is fundamental, for virtually all parallel algorithms known to the authors assume that the processors know their global coordinates within the newly created entity. We refer to this problem as the initialization problem. Rather surprisingly, the initialization problem has not received much attention in the literature. Our main contribution is to present parallel algorithms for the initialization problem on a number of network topologies, including complete binary trees, meshes of trees, pyramids, linear arrays, rings, meshes, tori, higher dimensional meshes and tori, hypercubes, butterflies, linear arrays with a global bus, rings with a global bus and meshes with multiple broadcasting, under various assumptions about edge labels, leader existence, and a priori knowledge of the number of nodes in the network. With two exceptions, the proposed algorithms are optimal. [ABSTRACT FROM AUTHOR]