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In this paper, we consider a one-dimensional completely degenerate oscillator subjected to an analytically ϵ -dependent quasi-periodic perturbation, whose frequencies satisfy a Diophantine condition. By the KAM method, we show that one of the following results holds true: 1. For all sufficiently small ϵ and all initial values φ ∈ T 1 , there exists a family of analytically (ϵ , φ) -parameterized response solutions, which corresponds to the persistence of the resonant Lagrangian torus of the equivalent Hamiltonian system. 2. For all sufficiently small ϵ , there exists a response solution, moreover, for an uncountable number of sufficiently small ϵ , there exists another response solution. In this case, the resonant Lagrangian torus of the equivalent Hamiltonian system is destroyed and it splits into a hyperbolic or hyperbolic-type degenerate lower dimensional torus for all sufficiently small ϵ , and another (possibly elliptic) lower dimensional torus for an uncountable number of sufficiently small ϵ . [ABSTRACT FROM AUTHOR]

This paper studies a fully parabolic chemotaxis system with competitive kinetics and fractional diffusion of order α 1 , α 2 ∈ (0 , 2) { u t = − d 1 Λ α 1 u − χ 1 ∇ ⋅ (u ∇ w) + μ 1 u (1 − u − a 1 v) , x ∈ T 2 , t > 0 , v t = − d 2 Λ α 2 v − χ 2 ∇ ⋅ (v ∇ w) + μ 2 v (1 − v − a 2 u) , x ∈ T 2 , t > 0 , w t = d 3 Δ w − γ 1 w + γ 2 u + γ 3 v , x ∈ T 2 , on two dimensional periodic torus T 2 . It is proved that (1) has a unique global bounded solution for all appropriately regular nonnegative initial data u 0 , v 0 and w 0 . Moreover, if μ 1 and μ 2 ( μ 1 or μ 2 ) are large enough, we can reach the following conclusions by constructing appropriate energy functional: (i) 0 < a 1 , a 2 < 1 (u , v , w) (⋅ , t) → (1 − a 1 1 − a 1 a 2 , 1 − a 2 1 − a 1 a 2 , γ 2 (1 − a 1) + γ 2 (1 − a 2) γ 1 (1 − a 1 a 2)) uniformly in T 2 as t → ∞. (ii) a 1 ≥ 1 , 0 < a 2 < 1 (u , v , w) (⋅ , t) → (0 , 1 , γ 3 γ 1 ) uniformly in T 2 as t → ∞. (iii) 0 < a 1 < 1 , a 2 ≥ 1 (u , v , w) (⋅ , t) → (1 , 0 , γ 2 γ 1 ) uniformly in T 2 as t → ∞. [ABSTRACT FROM AUTHOR]

An organ transplant involves complex psychodynamic processing that has been explored particularly in terms of object relationship theory. In the present study, we develop a model of transplantation based primarily on Lacan's explanations of the torus. In a prospective study, we examined 40 patients, 2 weeks, 3 months and 6 months post‐transplant. Based on the analysis of a dream, we identified the so‐called 'transplantation complex' in the form of earlier, for example, oral‐sadistic fantasies. In a further step, we examined the extent to which direct and indirect indications of this complex were found in the interviews of the total sample. In the present paper, these results are related to Lacan's graph of the torus. Our results display the dialectic of a repetitive demand, especially on the lungs, and the desire for an object that is basically lost or Oedipally forbidden. This dialectic may explain both the different quality of life and the frequent occurrence of feelings of guilt reported by the patients. We also show the function of identification, for example, with a single trait of the donor. Overall, the model of the torus may be a way of understanding the processing of the transplant in a deeper and more coherent psychodynamic manner. [ABSTRACT FROM AUTHOR]

In this paper, the implementation and validation of the torus mapping, a method to dealias radial wind observations by meteorological radars is presented. We apply the procedure to data from the Operational program for exchange of weather radar information (OPERA) over central part of Europe for the whole year 2021. The quality of the resulting radar winds is assessed by a comparison to colocated radiosonde and aircraft observations and also through analysis of first guess departures using a regional numerical weather prediction model. Performance of quality control on dealiased radial wind data is also evaluated. We show that the torus mapping method is a robust procedure on data with a sufficiently low amount of measurement noise. It produces datasets of comparable quality for a wide range of Nyquist velocity values and has potential for operational applications. [ABSTRACT FROM AUTHOR]

This paper proposes a novel approach to quantizing Nambu brackets in classical mechanics using operator formalism. The approach employs the "Planck derivative" to represent Nambu brackets, from which we derive a commutation relation for their quantization. Notably, this commutation relation aligns with that emerging from the T-duality of closed strings in a twisted torus with a B-field, thereby hinting at a potential connection with double field theory. [ABSTRACT FROM AUTHOR]

Recently, Lie bialgebra structures of the derivation Lie algebra over quantum torus were determined. In this paper, we use the general method of quantization by a Drinfel'd twist to quantize these algebras with their Lie bialgebra structures and present the explicit formulae. [ABSTRACT FROM AUTHOR]

Let X be a Q$\mathbb {Q}$‐factorial compact Kähler klt threefold admitting an action of a free abelian group G, which is of positive entropy and of maximal rank. After running the G‐equivariant log minimal model program, we show that such X is either rationally connected or bimeromorphic to a Q‐complex torus. In particular, we fix an issue in the proof of our previous paper [23, Theorem 1.3]. [ABSTRACT FROM AUTHOR]

For each contact diffeomorphism ϕ : (Q, ξ) → (Q, ξ) of (Q, ξ), we equip its mapping torus Mϕ with a locally conformal symplectic form of Banyaga's type, which we call the lcs mapping torus of the contact diffeomorphism ϕ. In the present paper, we consider the product Q × S1 = Mid (corresponding to ϕ = id) and develop basic analysis of the associated J-holomorphic curve equation, which has the form ∂ ˉ π w = 0 , w ∗ λ ∘ j = f ∗ d θ for the map u = (w, f) : Σ ˙ → Q × S 1 for a λ-compatible almost complex structure J and a punctured Riemann surface (Σ ˙ , j). In particular, w is a contact instanton in the sense of [31], [32].We develop a scheme of treating the non-vanishing charge by introducing the notion of charge class in H 1 (Σ ˙ , Z) and develop the geometric framework for the study of pseudoholomorphic curves, a correct choice of energy and the definition of moduli spaces towards the construction of a compactification of the moduli space on the lcs-fication of (Q, λ) (more generally on arbitrary locally conformal symplectic manifolds). [ABSTRACT FROM AUTHOR]

By applying Gröbner basis theory to spectral sequences algebras, we develop a new computational methodology applicable to any Leray-Serre spectral sequence for which the cohomology of the base space is the quotient of a finitely generated polynomial algebra. We demonstrate the procedure by deducing the cohomology of the free loop space of flag manifolds, presenting a significant extension over previous knowledge of the topology of free loop spaces. A complete flag manifold is the quotient of a Lie group by its maximal torus. The rank of a flag manifold is the dimension of the maximal torus of the Lie group. The rank 2 complete flag manifolds are SU(3)/T 2, Sp(2)/T 2, Spin(4)/T², Spin(5)/T² and G2/T². In this paper we calculate the cohomology of the free loop space of the rank 2 complete flag manifolds. [ABSTRACT FROM AUTHOR]

In this paper we prove Lieb-Thirring inequalities for magnetic Schrödinger operators on the torus, where the constants in the inequalities depend on the magnetic flux. [ABSTRACT FROM AUTHOR]

In formulating a non-orientable analogue of the Milnor Conjecture on the 4-genus of torus knots, Batson [Math. Res. Lett. 21 (2014), pp. 423–436] developed an elegant construction that produces a smooth non-orientable spanning surface in B^4 for a given torus knot in S^3. While Lobb [Math. Res. Lett. 26 (2019), pp. 1789] showed that Batson's surfaces do not always minimize the non-orientable 4-genus, we prove that they do minimize among surfaces that share their normal Euler number. We also determine the possible pairs of normal Euler number and first Betti number for non-orientable surfaces whose boundary lies in a class of torus knots for which Batson's surfaces are non-orientable 4-genus minimizers. [ABSTRACT FROM AUTHOR]

In this paper, we consider the forced incompressible Navier–Stokes equations with vanishing viscosity on the three-dimensional torus. We show that there are (classical) solutions for which the dissipation rate of the kinetic energy is bounded away from zero, uniformly in the viscosity parameter, while the body forces are uniformly bounded in some reasonable regularity class. [ABSTRACT FROM AUTHOR]

All knots are known to be hyperbolic, satellite, or torus knots, and one important family is Lorenz links, or T-links, which arise from dynamics. However, it remains difficult to determine the geometric type of a Lorenz link from a description via dynamics or as a T-link. In this paper, we consider those T-links that are torus links. We show that T-links obtained by full twists along torus links can never be torus links, aside from a family of cases. This addresses a question of Birman and Kofman [J. Topol. 2 (2009), pp. 227–248]. [ABSTRACT FROM AUTHOR]

This paper proves the persistence of degenerate hyperbolic lower-dimensional invariant tori in Hamiltonian systems with Bruno non-degeneracy conditions, whose frequency vector is a small dilation of the prescribed one. The proof is based on the stability of real roots of approximating real odd-order polynomials. [ABSTRACT FROM AUTHOR]

The T T ¯ deformation is a solvable irrelevant deformation whose properties depend on the sign of the deformation parameter μ. In particular, T T ¯ -deformed CFTs with μ < 0 have been proposed to be holographically dual to Einstein gravity where the metric satisfies Dirichlet boundary conditions at a finite cutoff surface. In this paper, we put forward a holographic proposal for T T ¯ -deformed CFTs with μ > 0, in which case the bulk geometry is constructed by gluing a patch of AdS3 to the original spacetime. As evidence, we show that the T T ¯ trace flow equation, the spectrum on the cylinder, and the partition function on the torus and the sphere, among other results, can all be reproduced from bulk calculations in glue-on AdS3. [ABSTRACT FROM AUTHOR]

Recently introduced connections between quantum codes and Narain CFTs provide a simple ansatz to express a modular-invariant function Z τ τ ¯ in terms of a multivariate polynomial satisfying certain additional properties. These properties include algebraic identities, which ensure modular invariance of Z τ τ ¯ , and positivity and integrality of coefficients, which imply positivity and integrality of the 픲(1)n × 픲(1)n character expansion of Z τ τ ¯ . Such polynomials come naturally from codes, in the sense that each code of a certain type gives rise to the so-called enumerator polynomial, which automatically satisfies all necessary properties, while the resulting Z τ τ ¯ is the partition function of the code CFT — the Narain theory unambiguously constructed from the code. Yet there are also "fake" polynomials satisfying all necessary properties, that are not associated with any code. They lead to Z τ τ ¯ satisfying all modular bootstrap constraints (modular invariance and positivity and integrality of character expansion), but whether they are partition functions of any actual CFT is unclear. We consider the group of the six simplest fake polynomials and denounce the corresponding Z's as fake: we show that none of them is the torus partition function of any Narain theory. Moreover, four of them are not partition functions of any unitary 2d CFT; our analysis for other two is inconclusive. Our findings point to an obvious limitation of the modular bootstrap approach: not every solution of the full set of torus modular bootstrap constraints is due to an actual CFT. In the paper we consider six simple examples, keeping in mind that thousands more can be constructed. [ABSTRACT FROM AUTHOR]

This paper presents an investigation on the global hypoellipticity problem for a class of systems of pseudo‐differential operators on the torus. The approach consists in establishing conditions on the matrix symbol of the system such that it can be transformed into a suitable triangular form involving a nilpotent upper triangular matrix. Hence, the global hypoellipticity is studied by analyzing the behavior of the eigenvalues and their averages. [ABSTRACT FROM AUTHOR]

In high dimension, the bifurcation theory of periodic orbits of nonlinear dynamics systems are difficult to establish in general. In this paper, by performing the curvilinear coordinate frame and constructing a Poincaré map, we obtain some sufficient conditions of the bifurcation of periodic solutions of some 2n-dimensional systems for the unperturbed system in two cases: one is a decoupled n-degree-of-freedom nonlinear Hamiltonian system and the other has an isolated invariant torus. We use a new method and study new types of systems compared with the existing results. As an application we study the bifurcation and number of periodic solutions of an ice covered suspension system. Under a certain parametrical condition, the number of periodic solutions of this system can be 2 or 1 with the variation of parameter p 2 . [ABSTRACT FROM AUTHOR]

In this paper, we prove an abstract KAM (Kolmogorov–Arnold–Moser) theorem for infinite dimensional reversible Schrödinger systems. Using this KAM theorem together with partial Birkhoff normal form method, we obtain the existence of quasi-periodic solutions for a class of completely resonant reversible coupled nonlinear Schrödinger systems on two dimensional torus. [ABSTRACT FROM AUTHOR]

This paper studies local rigidity for some isometric toral extensions of partially hyperbolic \mathbb {Z}^k (k\geqslant 2) actions on the torus. We prove a C^\infty local rigidity result for such actions, provided that the smooth perturbations of the actions satisfy the intersection property. We also give a local rigidity result within a class of volume preserving actions. Our method mainly uses a generalization of the Kolmogorov-Arnold-Moser iterative scheme. [ABSTRACT FROM AUTHOR]

Nanoscale materials have gained considerable interest because of their special properties and wide range of applications. Many types of boron nitride at the nanoscale have been realized, including nanotubes, nanocones, fullerenes, tori, and graphene sheets. The connection of these structures at the nanoscale leads to merged structures that have enhanced features and applications. Modeling the joining between nanostructures has been adopted by different methods. Namely, carbon nanostructures have been joined by minimizing the elastic energy in symmetric configurations. In other words, the only considerable curvature in the elastic energy is the axial curvature. Accordingly, because it has nanoscale structures similar to those in carbon, BN can also be joined and connected by using this method. On the other hand, different methods have been proposed to consider the rotational curvature because it has a similar size. Based on that argument, the Willmore energy, which depends on both curvatures, has been minimized to join carbon nanostructures. This energy is used to identify the joining region, especially for a three-dimensional structure. In this paper, we expand the use of Willmore energy to cover the joining of boron nitride nanostructures. Therefore, because catenoids are absolute minimizers of this energy, pieces of catenoids can be used to connect nanostructures. In particular, we joined boron nitride fullerene to three other BN nanostructures: nanotube, fullerene, and torus. For now, there are no experimental or simulation data for comparison with the theoretical connecting structures predicted by this study, which is some justification for the suggested simple model shown in this research. Ultimately, various nanoscale BN structures might be connected by considering the same method, which may be considered in future work. [ABSTRACT FROM AUTHOR]

Adaptive diagnosis is an approach in which tests can be scheduled dynamically during the diagnosis process based on the previous test outcomes. Naturally, reducing the number of test rounds as well as the total number of tests is a major goal of an efficient adaptive diagnosis algorithm. The adaptive diagnosis of multiprocessor systems under the PMC model has been widely investigated, while adaptive diagnosis using comparison model has been independently discussed only for three networks, including hypercube, torus, and completely connected networks. In addition, adaptive diagnosis of general Hamiltonian networks is more meaningful than that of special graph. In this paper, the problem of adaptive fault diagnosis in Hamiltonian networks under the comparison model is explored. First, we propose an adaptive diagnostic scheme which takes five to six test rounds. Second, we derive a dynamic upper bound of the number of fault nodes instead of setting a value like normal. Finally, we present an algorithm such that at least one sequence obtained from cycle partition can be picked out and all nodes in this sequence can be identified based on the previous upper bound. [ABSTRACT FROM AUTHOR]

In this paper we study large-time behavior evolution problems on the n-dimensional torus T n , n ≥ 1 . Here we analyze the solutions to these problems, studying their regularity and obtaining estimates of them. The main tools we use is the toroidal Fourier transform, together with Fourier series and a version of the Hardy-Littlewood inequality, applied to our case of the n-dimensional torus T n . We use this inequality to find an estimate of solutions to evolution problems. [ABSTRACT FROM AUTHOR]

Consider the Kirchhoff equation ∂ tt u - Δ u (1 + ∫ T d | ∇ u | 2) = 0 on the d-dimensional torus T d . In a previous paper we proved that, after a first step of quasilinear normal form, the resonant cubic terms show an integrable behavior, namely they give no contribution to the energy estimates. This leads to the question whether the same structure also emerges at the next steps of normal form. In this paper, we perform the second step and give a negative answer to the previous question: the quintic resonant terms give a nonzero contribution to the energy estimates. This is not only a formal calculation, as we prove that the normal form transformation is bounded between Sobolev spaces. [ABSTRACT FROM AUTHOR]

In this paper, we recover a class of displacement interpolations of probability measures, in the sense of the Optimal Transport theory, by means of semiclassical measures associated with solutions of Schrödinger equation defined on the flat torus. Moreover, we prove the completing viewpoint by proving that a family of displacement interpolations can always be viewed as a path of time-dependent semiclassical measures. [ABSTRACT FROM AUTHOR]

In the paper we give some basic properties of the superderivations of Lie superalgebras. Under certain condition, for solvable Lie superalgebras with given nilradicals, we give estimates for upper bounds to the dimensions of complementary subspaces to the nilradicals. Moreover, under the condition that an analogue of Lie's theorem is true, the description of solvable Lie superalgebras of maximal rank is obtained. Namely, we prove that an arbitrary solvable Lie superalgebra of maximal rank under the mentioned condition is isomorphic to the maximal solvable extension of nilradical of maximal rank. Finally, along with effective method of construction of solvable Lie superalgebras of maximal rank we present the description of special type of maximal solvable extension of nilpotent Lie superalgebras. [ABSTRACT FROM AUTHOR]

Beyond Hölder’s type, this paper mainly concerns the persistence and the remaining regularity of an individual frequency-preserving KAM torus in a finitely differentiable Hamiltonian system, even allowing the non-integrable part to be critically finitely smooth. To achieve this goal, besides investigating the Jackson approximation theorem towards only the modulus of continuity, we demonstrate an abstract regularity theorem adapting to the new iterative scheme. Via these tools, we obtain a KAM theorem with sharp differentiability hypotheses, asserting that the persistent torus keeps the prescribed universal Diophantine frequency unchanged. Further, the non-Hölder regularity for the invariant KAM torus as well as the conjugation is explicitly shown by introducing asymptotic analysis. To our knowledge, this is the first approach to KAM on these aspects in a continuous sense, and we also provide two systems, which cannot be studied by previous KAM but by ours. [ABSTRACT FROM AUTHOR]

The k -matching preclusion number of a graph is the minimum number of edges whose deletion results in the remaining graph that has neither perfect k -matchings nor almost perfect k -matchings. For many networks, their optimal k -matching preclusion sets are precisely those edges incident with a single vertex. In this paper, we introduce the concept of conditional k -matching preclusion, in which isolated vertices are not permitted in fault networks. We establish the conditional k -matching preclusion numbers and all possible minimum conditional k -matching preclusion sets for n -dimensional torus networks with n ≥ 3. In addition, we investigate the relationship between all optimal sets for three kinds of (conditional) matching preclusion problems. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new ‘Thom class’ formulation of topological T-duality for principal torus bundles. This definition is equivalent to the established one of Bunke, Rumpf, and Schick but has the virtue of removing the global assumptions on the H-flux required in the old definition. With the new definition, we provide easier and more transparent proofs of the classification of T-duals and generalize the local formulation of T-duality for circle bundles by Bunke, Schick, and Spitzweck to the torus case. [ABSTRACT FROM AUTHOR]

In this paper, we present a way of combining the computation of invariant tori and their stable and unstable manifolds with the multiple shooting technique. We start by showing some of the results of Jorba (Nonlinearity 14(5):943–976, 2001) that should be modified in order to introduce the multiple shooting technique in these computations. After that, by a direct application in the planar elliptic restricted three-body problem (PERTBP), how to modify the equations and methods to compute the above-mentioned objects is introduced. In particular, the structure of the (systems of) equations and matrices involved in these computations is shown. An application of these computations can be found in Duarte and Jorba (Invariant manifolds of tori near L 1 and L 2 in the planar elliptic restricted three-body problem II. The Dynamics of Comet Oterma, Preprint 2023), where the dynamics of comet 39P/Oterma is modelled as a PERTBP. [ABSTRACT FROM AUTHOR]

In this paper we establish the well-posedness of the Cauchy problem for a class of pseudo-differential hyperbolic equations on the torus. The class considered here includes a space-like fractional order Laplacians. By applying the toroidal pseudo-differential calculus we establish regularity estimates, existence and uniqueness in the scale of the standard Sobolev spaces on the torus. [ABSTRACT FROM AUTHOR]

In this paper, we consider the polynomial and exponential convergence rates of the weighted Birkhoff averages of irrational rotations on tori. It is shown that these can be achieved for finite and infinite dimensional tori which correspond to the quasiperiodic and almost periodic dynamical systems respectively, under certain balance between the nonresonant condition and the decay rate of the Fourier coefficients. Diophantine rotations with finite and infinite dimensions are provided as examples. For the first time, we prove the universality of exponential convergence and arbitrary polynomial convergence in the quasiperiodic case and almost periodic case under analyticity respectively. [ABSTRACT FROM AUTHOR]

In this paper, we report the discovery of some novel dynamical scenarios for quasi-periodic shrimp-shaped structures embedded within chaotic phases in bi-parameter space of a discrete predator–prey system. By constructing high-resolution, two-dimensional stability diagrams based on Lyapunov exponents, we observe the abundance of both periodic and quasi-periodic shrimp-shaped organized domains in a certain parameter space of the system. A comprehensive comparative analysis is conducted to elucidate the similarities and differences between these two types of shrimps. Our analysis reveals that, unlike periodic shrimp, quasi-periodic shrimp induces (i) torus bubbling transition to chaos and (ii) multistability with multi-tori, torus-chaotic, and multi-chaotic coexisting attractors, resulting from the crossing of its two inner antennae. The basin sets of the coexisting attractors are analyzed, and we observe the presence of intriguing basin boundaries. We also verify that, akin to periodic shrimp structures, quasi-periodic shrimps also maintain the three-times self-similarity scaling. Furthermore, we encounter the occurrence of spiral organization for the self-distribution of quasi-periodic shrimps within a large chaotic domain. We believe that these novel findings will significantly enhance our understanding of shrimp-shaped structures and the intricate dynamics exhibited by their distribution in chaotic regimes. [ABSTRACT FROM AUTHOR]

Let p be an odd prime and let ρ : ℤ / p → GL n ⁡ (ℤ) be an action of ℤ / p on a lattice and let Γ := ℤ n ⋊ ρ ℤ / p be the corresponding semidirect product. The torus bundle M := T ρ n × ℤ / p S ℓ over the lens space S ℓ / ℤ / p has fundamental group Γ. When ℤ / p fixes only the origin of ℤ n , Davis and Lück (2021) compute the L-groups L m 〈 j 〉 ⁢ (ℤ ⁢ [ Γ ]) and the structure set 풮 geo , s ⁢ (M) . In this paper, we extend these computations to all actions of ℤ / p on ℤ n . In particular, we compute L m 〈 j 〉 ⁢ (ℤ ⁢ [ Γ ]) and 풮 geo , s ⁢ (M) in a case where E ¯ ⁢ Γ has a non-discrete singular set. [ABSTRACT FROM AUTHOR]

The Parrondo game is a game where a combination of losing strategies becomes a winning strategy. This phenomenon is called the Parrondo paradox. The Parrondo game based on quantum walk and the search algorithm via quantum walk has been widely studied, respectively. This paper presents a Parrondo game for quantum search based on quantum walk for the first time. Moreover, we confirm that Parrondo's paradox exists for our model on the one- and two-dimensional tori by numerical simulations. Afterward, we show the range in which the paradox occurs is symmetric about the origin on the d-dimensional torus (d ≥ 1) with even vertices and one marked vertex. [ABSTRACT FROM AUTHOR]

Let A∈Matm×n(R)$A\in \operatorname{Mat}_{m\times n}(\mathbb {R})$ be a matrix. In this paper, we investigate the set BadA⊂Tm$\operatorname{Bad}_A\subset \mathbb {T}^m$ of badly approximable targets for A$A$, where Tm$\mathbb {T}^m$ is the m$m$‐torus. It is well known that BadA$\operatorname{Bad}_A$ is a winning set for Schmidt's game and hence is a dense subset of full Hausdorff dimension. We investigate the relationship between the measure of BadA$\operatorname{Bad}_A$ and Diophantine properties of A$A$. On the one hand, we give the first examples of a nonsingular A$A$ such that BadA$\operatorname{Bad}_A$ has full measure with respect to some nontrivial algebraic measure on the torus. For this, we use transference theorems due to Jarnik and Khintchine, and the parametric geometry of numbers in the sense of Roy. On the other hand, we give a novel Diophantine condition on A$A$ that slightly strengthens nonsingularity, and show that under the assumption that A$A$ satisfies this condition, BadA$\operatorname{Bad}_A$ is a null‐set with respect to any nontrivial algebraic measure on the torus. For this, we use naive homogeneous dynamics, harmonic analysis, and a novel concept that we refer to as mixing convergence of measures. [ABSTRACT FROM AUTHOR]

In this paper, we classify the gradings up to equivalence on the filiform Lie superalgebras Lm,n over an algebraically closed field of characteristic 0. Moreover, by the relation between a grading and a standard grading, we describe all pairwise inequivalent gradings and their universal groups of Lm,n. [ABSTRACT FROM AUTHOR]

In this paper, we prove Strichartz estimates for many body Schrödinger equations in the periodic setting, specifically on tori 핋 d {\mathbb{T}^{d}} , where d ≥ 3 {d\geq 3} . The results hold for both rational and irrational tori, and for small interacting potentials in a certain sense. Our work is based on the standard Strichartz estimate for Schrödinger operators on periodic domains, as developed in [J. Bourgain and C. Demeter, The proof of the l 2 l^{2} decoupling conjecture, Ann. of Math. (2) 182 2015, 1, 351–389]. As a comparison, this result can be regarded as a periodic analogue of [Y. Hong, Strichartz estimates for N-body Schrödinger operators with small potential interactions, Discrete Contin. Dyn. Syst. 37 2017, 10, 5355–5365] though we do not use the same perturbation method. We also note that the perturbation method fails due to the derivative loss property of the periodic Strichartz estimate. [ABSTRACT FROM AUTHOR]

We study the behaviour of global minimizers of a continuum Landau–de Gennes energy functional for nematic liquid crystals, in three-dimensional axially symmetric domains diffeomorphic to a ball (a nematic droplet) and in a restricted class of S 1 -equivariant configurations. It is known from our previous paper (Dipasquale et al. in J Funct Anal 286:110314, 2024) that, assuming smooth and uniaxial (e.g. homeotropic) boundary conditions and a physically relevant norm constraint in the interior (Lyuksyutov constraint), minimizing configurations are either of torus or of split type. Here, starting from a nematic droplet with the homeotropic boundary condition, we show how singular (split) solutions or smooth (torus) solutions (or even both) for the Euler–Lagrange equations do appear as energy minimizers by suitably deforming either the domain or the boundary data. As a consequence, we derive symmetry breaking results for the minimization among all competitors. [ABSTRACT FROM AUTHOR]

In this paper we further develop the theory of equivariant Seiberg–Witten–Floer cohomology of the two authors, with an emphasis on Brieskorn homology spheres. We obtain a number of applications. First, we show that the knot concordance invariants θ(c)$\theta ^{(c)}$ defined by the first author satisfy θ(c)(Ta,b)=(a−1)(b−1)/2$\theta ^{(c)}(T_{a,b}) = (a-1)(b-1)/2$ for torus knots, whenever c$c$ is a prime not dividing ab$ab$. Since θ(c)$\theta ^{(c)}$ is a lower bound for the slice genus, this gives a new proof of the Milnor conjecture. Second, we prove that a free cyclic group action on a Brieskorn homology 3‐sphere Y=Σ(a1,⋯,ar)$Y = \Sigma (a_1, \dots, a_r)$ does not extend smoothly to any homology 4‐ball bounding Y$Y$. In the case of a non‐free cyclic group action of prime order, we prove that if the rank of HFred+(Y)$HF_{red}^+(Y)$ is greater than p$p$ times the rank of HFred+(Y/Zp)$HF_{red}^+(Y/\mathbb {Z}_p)$, then the Zp$\mathbb {Z}_p$‐action on Y$Y$ does not extend smoothly to any homology 4‐ball bounding Y$Y$. Third, we prove that for all but finitely many primes a similar non‐extension result holds in the case that the bounding 4‐manifold has positive‐definite intersection form. Finally, we also prove non‐extension results for equivariant connected sums of Brieskorn homology spheres. [ABSTRACT FROM AUTHOR]

Knot filtered embedded contact homology was first introduced by Hutchings in 2015; it has been computed for the standard transverse unknot in irrational ellipsoids by Hutchings and for the Hopf link in lens spaces L(n,n−1)$L(n,n-1)$ via a quotient by Weiler. While toric constructions can be used to understand the ECH chain complexes of many contact forms adapted to open books with binding the unknot and Hopf link, they do not readily adapt to general torus knots and links. In this paper, we generalize the definition and invariance of knot filtered embedded contact homology to allow for degenerate knots with rational rotation numbers. We then develop new methods for understanding the embedded contact homology chain complex of positive torus knotted fibrations of the standard tight contact 3‐sphere in terms of their presentation as open books and as Seifert fiber spaces. We provide Morse–Bott methods, using a doubly filtered complex and the energy filtered perturbed Seiberg–Witten Floer theory developed by Hutchings and Taubes, and use them to compute the T(2,q)$T(2,q)$ knot filtered embedded contact homology, for q$q$ odd and positive. [ABSTRACT FROM AUTHOR]

In this paper, we calculate the Hausdorff dimension of the fractal set x ∈ d : ∏ 1 ≤ i ≤ d | T β i n (x i) − x i | < ψ (n)   for infinitely many  n ∈ ℕ , where T β i is the standard β i -transformation with β i > 1 , ψ is a positive function on ℕ and | ⋅ | is the usual metric on the torus . Moreover, we investigate a modified version of the shrinking target problem, which unifies the shrinking target problems and quantitative recurrence properties for matrix transformations of tori. Let T be a d × d non-singular matrix with real coefficients. Then, T determines a self-map of the d -dimensional torus d : = ℝ d / ℤ d . For any 1 ≤ i ≤ d , let ψ i be a positive function on ℕ and Ψ (n) : = (ψ 1 (n) , ... , ψ d (n)) with n ∈ ℕ. We obtain the Hausdorff dimension of the fractal set { x ∈ d : T n (x) ∈ L (f n (x) , Ψ (n))   for infinitely many  n ∈ ℕ } , where L (f n (x , Ψ (n))) is a hyperrectangle and { f n } n ≥ 1 is a sequence of Lipschitz vector-valued functions on d with a uniform Lipschitz constant. [ABSTRACT FROM AUTHOR]

In this paper, we introduce an abstract KAM (Kolmogorov–Arnold–Moser) theorem. As an application, we study the two-dimensional completely resonant beam system under periodic boundary conditions. Using the KAM theorem together with partial Birkhoff normal form method, we obtain a family of Whitney smooth small–amplitude quasi–periodic solutions for the equation system. [ABSTRACT FROM AUTHOR]

In this paper, we study the online nearest neighbor random tree in dimension d ∈ N (called d-NN tree for short) defined as follows. We fix the torus T n d of dimension d and area n and equip it with the metric inherited from the Euclidean metric in R d . Then, embed consecutively n vertices in T n d uniformly at random and independently, and let each vertex but the first one connect to its (already embedded) nearest neighbor. Call the resulting graph G n . We show multiple results concerning the degree sequence of G n . First, we prove that typically the number of vertices of degree at least k ∈ N in the d-NN tree decreases exponentially with k and is tightly concentrated by a new Lipschitz-type concentration inequality that may be of independent interest. Second, we obtain that the maximum degree of G n is of logarithmic order. Third, we give explicit bounds for the number of leaves that are independent of the dimension and also give estimates for the number of paths of length two. Moreover, we show that typically the height of a uniformly chosen vertex in G n is (1 + o (1)) log n and the diameter of T n d is (2 e + o (1)) log n , independently of the dimension. Finally, we define a natural infinite analog G ∞ of G n and show that it corresponds to the local limit of the sequence of finite graphs (G n) n ≥ 1 . Furthermore, we prove almost surely that G ∞ is locally finite, that the simple random walk on G ∞ is recurrent, and that G ∞ is connected. [ABSTRACT FROM AUTHOR]

In this paper, we discuss index theory for Toeplitz operators on a discrete quarter-plane of two-variable rational matrix function symbols. By using Gohberg–Kreĭn theory for matrix factorizations, we extend the symbols defined originally on a two-dimensional torus to some three-dimensional sphere and derive a formula to express their Fredholm indices through extended symbols. Variants for families of (self-adjoint) Fredholm quarter-plane Toeplitz operators and those preserving real structures are also included. For some bulk-edge gapped single-particle Hamiltonians of finite hopping range on a discrete lattice with a codimension-two right angle corner, topological invariants related to corner states are provided through extensions of bulk Hamiltonians. [ABSTRACT FROM AUTHOR]

A domatic (resp. total domatic) k-coloring of a graph G is an assignment of k colors to the vertices of G such that each vertex contains vertices of all k colors in its closed neighborhood (resp. open neighborhood). The domatic (resp. total domatic) number of G, denoted d(G) (resp. d t (G) ), is the maximum k for which G has a domatic (resp. total domatic) k-coloring. In this paper, we show that for two non-trivial graphs G and H, the domatic and total domatic numbers of their Cartesian product G □ H is bounded above by max { | V (G) | , | V (H) | } and below by max { d (G) , d (H) } . Both these bounds are tight for an infinite family of graphs. Further, we show that if H is bipartite, then d t (G □ H) is bounded below by 2 min { d t (G) , d t (H) } and d (G □ H) is bounded below by 2 min { d (G) , d t (H) } . As a consequences, we get new bounds for the domatic and total domatic number of hypercubes and Hamming graphs of certain dimensions, and exact values for some n-dimensional tori which turns out to be a generalization of a result due to Gravier from 2002. [ABSTRACT FROM AUTHOR]

In this paper, we develop and analyze a model that studies the interaction between a specialist predator (one that relies exclusively on a single prey species), a generalist predator (one that takes advantage of alternative food sources in addition to consuming the focal prey species), and their common prey in a two-trophic ecosystem featuring three timescales. We assume that the prey operates on a faster timescale, while the specialist and generalist predators operate on slow and superslow timescales respectively. Treating the predation efficiency of the generalist predator as the primary varying parameter and the proportion of its diet formed by the prey species under study as the secondary parameter, we obtain a host of rich and interesting dynamics, including relaxation oscillations, mixed-mode oscillations (MMOs), subcritical elliptic bursting patterns, torus canards, and mixed-type torus canards. By grouping the timescales into two classes and using the timescale separation between classes, we apply one-fast/two-slow and two-fast/one-slow analysis techniques to gain insights about the dynamics. Using the geometric properties and flows of the singular subsystems, in combination with bifurcation analysis and numerical continuation of the full system, we classify the oscillatory dynamics and discuss the transitions from one type of dynamics to the other. The types of oscillatory patterns observed in this model are novel in population models featuring three-timescales; some of which qualitatively resemble natural cycles in small mammals and insects. Furthermore, oscillatory dynamics displaying torus canards, mixed-type torus canards, and MMOs experiencing a delayed loss of stability near one of the invariant sheets of the self-intersecting critical manifold before getting attracted to the adjacent attracting sheet of the critical manifold have not been previously reported in three-timescale models. [ABSTRACT FROM AUTHOR]

In this paper, we will study the Hamiltonian derivative wave equation with higher order nonlinearity \begin{document}$ y_{tt}-y_{xx}+my+(Dy)^5 = 0, \quad x\in\mathbb{T}: = \mathbb{R}/2\pi\mathbb{Z}, $\end{document} where \begin{document}$ m>0 $\end{document} is a potential and$ D: = \sqrt{-\partial_{xx}+m}. $We will prove that, for any integer \begin{document}$ b\geq2 $\end{document}, the above equation admits many small amplitude quasi-periodic solutions corresponding to \begin{document}$ b $\end{document}-dimensional invariant tori of an associated infinite dimensional Hamiltonian system. The proof is based on infinite dimensional KAM theory and partial Birkhoff normal form. [ABSTRACT FROM AUTHOR]

In the paper, we are concerned with the nonlinear Cauchy problem on the Vlasov–Poisson–Landau/Boltzmann system around global Maxwellians in a torus or a finite channel. The main goal is to establish the global existence and large time behaviour of small amplitude solutions for a class of low regularity initial data. The molecular interaction type is restricted to the case of hard potentials for two classical collision operators because of the effect of the self-consistent forces. The result extends the one by Duan–Liu–Sakamoto-Strain (Duan et al 2021 Commun. Pure Appl. Math. 74 932–1020) for the pure Landau/Boltzmann equation to the case of the VPL and VPB systems. [ABSTRACT FROM AUTHOR]

With the development of cloud computing and big data, secure multi-party computation, which can collaborate with multiple parties to deal with a large number of transactions, plays an important role in protecting privacy. Private set intersection (PSI), a form of multi-party secure computation, is a formidable cryptographic technique that allows the sender and the receiver to calculate their intersection and not reveal any more information. As the data volume increases and more application scenarios emerge, PSI with multiple participants is increasingly needed. Homomorphic encryption is an encryption algorithm designed to perform a mathematical-style operation on encrypted data, where the decryption result of the operation is the same as the result calculated using unencrypted data. In this paper, we present a cloud-assisted multi-key PSI (CMPSI) system that uses fully homomorphic encryption over the torus (TFHE) encryption scheme to encrypt the data of the participants and that uses a cloud server to assist the computation. Specifically, we design some TFHE-based secure computation protocols and build a single cloud server-based private set intersection system that can support multiple users. Moreover, security analysis and performance evaluation show that our system is feasible. The scheme has a smaller communication overhead compared to existing schemes. [ABSTRACT FROM AUTHOR]