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The electric polarizability and the spread of the total position tensors are used to characterize the metallic vs insulator nature of large (finite) systems. Finite clusters are usually treated within the open boundary condition formalism. This introduces border effects, which prevent a fast convergence to the thermodynamic limit and can be eliminated within the formalism of periodic boundary conditions. Recently, we introduced an original approach to periodic boundary conditions, named Clifford boundary conditions. It considers a finite fragment extracted from a periodic system and the modification of its topology into that of a Clifford torus. The quantity representing the position is modified in order to fulfill the system periodicity. In this work, we apply the formalism of Clifford boundary conditions to the case of carbon nanotubes, whose treatment results in a particularly simple zigzag geometry. Indeed, we demonstrate that at the Hückel level, these nanotubes, either finite or periodic, are formally equivalent to a collection of non-interacting dimerized linear chains, thus simplifying their treatment. This equivalence is used to describe some nanotube properties as the sum of the contributions of the independent chains and to identify the origin of peculiar behaviors (such as conductivity). Indeed, if the number of hexagons along the circumference is a multiple of three, a metallic behavior is found, namely a divergence of both the (per electron) polarizability and total position spread of at least one linear chain. These results are in agreement with those in the literature from tight-binding calculations. [ABSTRACT FROM AUTHOR]

We consider on the torus the scaling limit of stochastic 2D (inviscid) fluid dynamics equations with transport noise to deterministic viscous equations. Quantitative estimates on the convergence rates are provided by combining analytic and probabilistic arguments, especially heat kernel properties and maximal estimates for stochastic convolutions. Similar ideas are applied to the stochastic 2D Keller-Segel model, yielding explicit choice of noise to ensure that the blow-up probability is less than any given threshold. Our approach also gives rise to some mixing property for stochastic linear transport equations and dissipation enhancement in the viscous case. [ABSTRACT FROM AUTHOR]

We determine which bipartite graphs embedded in a torus are move-reduced. In addition, we classify equivalence classes of such move-reduced graphs under square/spider moves. This extends the class of minimal graphs on a torus studied by Goncharov–Kenyon, and gives a toric analog of Postnikov's and Thurston's results on a disk. [ABSTRACT FROM AUTHOR]

Let F be a Morse–Bott foliation on the solid torus T = S 1 × D 2 into 2-tori parallel to the boundary and one singular central circle. Gluing two copies of T by some diffeomorphism between their boundaries, one gets a lens space L p , q with a Morse–Bott foliation F p , q obtained from F on each copy of T and thus consisting of two singular circles and parallel 2-tori. In the previous paper Khokliuk and Maksymenko (J Homotopy Relat Struct 18:313–356. https://doi.org/10.1007/s40062-023-00328-z, 2024) there were computed weak homotopy types of the groups D lp (F p , q) of leaf preserving (i.e. leaving invariant each leaf) diffeomorphisms of such foliations. In the present paper it is shown that the inclusion of these groups into the corresponding group D + fol (F p , q) of foliated (i.e. sending leaves to leaves) diffeomorphisms which do not interchange singular circles are homotopy equivalences. [ABSTRACT FROM AUTHOR]

We rigorously derive the reduced dynamical law for quantized vortex dynamics of the nonlinear wave equation on the torus when the core size of vortex $ \varepsilon\to 0 $. It is proved that the reduced dynamical law is a system consisting of second-order nonlinear ordinary differential equations driven by the renormalized energy on the torus, and the initial data of the reduced dynamical law is determined by the positions of vortices and the limit momentum of the solution of the nonlinear wave equation. We will also investigate the effect of the limit momentum on the vortex dynamics via numerical simulation. [ABSTRACT FROM AUTHOR]

Asaoka & Irie [3] recently proved a $ C^{\infty} $ closing lemma of Hamiltonian diffeomorphisms of closed surfaces. We reformulated their techniques into a more general perturbation lemma for area-preserving diffeomorphism and proved a $ C^{\infty} $ closing lemma for area-preserving diffeomorphisms on a torus $ \mathbb{T}^2 $ that is isotopic to identity. i.e., we show that the set of periodic orbits is dense for a generic diffeomorphism isotopic to identity area-preserving diffeomorphism on $ \mathbb{T}^2 $. The main tool is the flux vector of area-preserving diffeomorphisms, which may not be zero as in the Hamiltonian cases. [ABSTRACT FROM AUTHOR]

We study the mixing time of a random walk on the torus, alternated with a Lebesgue measure preserving Bernoulli map. Without the Bernoulli map, the mixing time of the random walk alone is O(1/\varepsilon ^2), where \varepsilon is the step size. Our main results show that for a class of Bernoulli maps, when the random walk is alternated with the Bernoulli map \varphi the mixing time becomes O(\lvert \ln \varepsilon \rvert). We also study the dissipation time of this process, and obtain O(\lvert \ln \varepsilon \rvert) upper and lower bounds with explicit constants. [ABSTRACT FROM AUTHOR]

We report high angular resolution observations, made with the Atacama Large Millimeter Array in band 6, of high excitation molecular lines of CH 3 CN and SO2 and of the H29 α radio recombination line toward the G345.0061+01.794 B HC H ii region in order to investigate the physical and kinematical characteristics of its surroundings. Emission was detected in all observed components of the J = 14 →13 rotational ladder of CH 3 CN and in the 304,26–303,27, and 324,28–323,29 lines of SO2. The peak of the velocity-integrated molecular emission is located ∼0.″4 northwest of the peak of the continuum emission. The first-order moment images and channel maps show a velocity gradient of 1.1 km s−1 arcsec−1 across the source and a distinctive spot of blueshifted emission toward the peak of the zero-order moment. The rotational temperature is found to decrease from 252±24 K at the peak position to 166±16 K at its edge, indicating that our molecular observations are probing a hot molecular core that is internally excited. The emission in the H29 α line arises from a region of 0.″65 in size, where its peak coincides with that of the dust continuum. We model the kinematical characteristics of the "central blue spot" feature as due to infalling motions, suggesting a central mass of 172.8±8.8 M ⊙. Our observations indicate that this HC H ii region is surrounded by a compact structure of hot molecular gas, which is rotating and infalling toward a central mass, that is most likely confining the ionized region. The observed scenario is reminiscent of a "butterfly pattern" with an approximately edge-on torus and ionized gas roughly parallel to its rotation axis. [ABSTRACT FROM AUTHOR]

In an attempt to generalise knot matrix models for non-torus knots, which currently remains an open problem, we derived expressions for the Harer–Zagier transform—a discrete Laplace transform—of the HOMFLY–PT polynomial for some infinite families of twisted hyperbolic knots. Among them, we found a family of pretzel knots for which, like for torus knots, the transform has a fully factorised form, while for the remaining families considered it consists of sums of factorised terms. Their zero loci show a remarkable structure, which mostly lies on the unit circle and deviates from it only in pairs. [ABSTRACT FROM AUTHOR]

This paper focuses on establishing the existence of smooth solutions for the incompressible Landau-Lifshitz-Gilbert equation with Dzyaloshinskii-Moriya (DM) interaction and V flow in two-dimensional domains, including the torus and Euclidean space. The primary objective is to establish global existence results by investigating the conditions under which a smooth solution exists for all time, provided that the L² norm of the initial magnetization gradient is sufficiently small. Rigorous mathematical proofs for these global existence results are provided by combining analytical techniques and energy estimates. These findings enhance our understanding of solution behavior and regularity in the Landau-Lifshitz-Gilbert equation with DM interaction, shedding light on its dynamics in different spatial domains. [ABSTRACT FROM AUTHOR]

We prove exact controllability for quasi-linear Hamiltonian Schrödinger equations on tori of dimension greater or equal then two. The result holds true for sufficiently small initial conditions satisfying natural minimal regularity assumptions, provided that the region of control satisfies the geometric control condition. [ABSTRACT FROM AUTHOR]

Using high-precision technologies for studying Hamiltonian systems, it is shown that the symmetrical and planar approach of four C164 nanotori with zero spin energy ends in the formation of two kinematic pairs. These pairs are in an intermediate state of dynamic equilibrium until the inclinations of the tori own axes exceed their limiting values. Further, as a result of the connection of the two bases of the tori, a new state of dynamic equilibrium is formed, characterized by the absolute maximum of the van der Waals interaction. [ABSTRACT FROM AUTHOR]

Riemannian cubics in tension are critical points of the linear combination of two objective functionals, namely the squared L 2 norms of the velocity and acceleration of a curve on a Riemannian manifold. We view this variational problem of finding a curve as a multi-objective optimization problem and construct the Pareto fronts for some given instances where the manifold is a sphere and where the manifold is a torus. The Pareto front for the curves on the torus turns out to be particularly interesting: the front is disconnected and it reveals two distinct Riemannian cubics with the same boundary data, which is the first known nontrivial instance of this kind. We also discuss some convexity conditions involving the Pareto fronts for curves on general Riemannian manifolds. [ABSTRACT FROM AUTHOR]

We describe a relationship between the monopole Floer homology of three‐manifolds and the geometry of Riemann surfaces. For an automorphism φ$\varphi$ of a compact Riemann surface Σ$\Sigma$ with quotient P1$\mathbb {P}^1$, there is a natural correspondence between theta characteristics L$L$ on Σ$\Sigma$ which are invariant under φ$\varphi$ and self‐conjugate spinc${\text{spin}}^c$ structures sL$\mathfrak {s}_L$ on the mapping torus Mφ$M_{\varphi }$ of φ$\varphi$. We show that the monopole Floer homology groups of (Mφ,sL)$(M_{\varphi },\mathfrak {s}_L)$ are explicitly determined by the eigenvalues of the (lift of the) action of φ$\varphi$ on H0(L)$H^0(L)$, the space of holomorphic sections of L$L$, and discuss several consequences of this description. Our result is based on a detailed analysis of the transversality properties of the Seiberg–Witten equations for suitable small perturbations. [ABSTRACT FROM AUTHOR]

The loop group |$G((z^{-1}))$| of a simple complex Lie group |$G$| has a natural Poisson structure. We introduce a natural family of Poisson commutative subalgebras |$\overline{{\textbf{B}}}(C) \subset{{\mathcal{O}}}(G((z^{-1}))$| depending on the parameter |$C\in G$| called classical universal Bethe subalgebras. To every antidominant cocharacter |$\mu $| of the maximal torus |$T \subset G$|⁠ , one can associate the closed Poisson subspace |${{\mathcal{W}}}_{\mu }$| of |$G((z^{-1}))$| (the Poisson algebra |${{\mathcal{O}}}({{\mathcal{W}}}_{\mu })$| is the classical limit of so-called shifted Yangian |$Y_{\mu }(\mathfrak{g})$| defined in [ 1 , Appendix B]). We consider the images of |$\overline{{\textbf{B}}}(C)$| in |${{\mathcal{O}}}({{\mathcal{W}}}_{\mu })$|⁠ , denoted by |$\overline{B}_{\mu }(C)$|⁠ , that should be considered as classical versions of (not yet defined in general) Bethe subalgebras in shifted Yangians. For regular |$C$| centralizing |$\mu $|⁠ , we compute the Poincaré series of these subalgebras. For |$\mathfrak{g}=\mathfrak{g}\mathfrak{l}_{n}$|⁠ , we define the natural quantization |${\textbf{Y}}^{\textrm{rtt}}(\mathfrak{g}\mathfrak{l}_{n})$| of |${{\mathcal{O}}}(\operatorname{Mat}_{n}((z^{-1})))$| and universal Bethe subalgebras |${\textbf{B}}(C) \subset{\textbf{Y}}^{\textrm{rtt}}(\mathfrak{g}\mathfrak{l}_{n})$|⁠. Using the RTT realization of |$Y_{\mu }(\mathfrak{g}\mathfrak{l}_{n})$| (invented by Frassek, Pestun, and Tsymbaliuk), we obtain the natural surjections |${\textbf{Y}}^{\textrm{rtt}}(\mathfrak{g}\mathfrak{l}_{n}) \twoheadrightarrow Y_{\mu }(\mathfrak{g}\mathfrak{l}_{n})$|⁠ , which quantize the embedding |${{\mathcal{W}}}_{\mu } \subset \operatorname{Mat}_{n}((z^{-1}))$|⁠. Taking the images of |${\textbf{B}}(C)$| in |$Y_{\mu }(\mathfrak{g}\mathfrak{l}_{n})$|⁠ , we recover Bethe subalgebras |$B_{\mu }(C) \subset Y_{\mu }(\mathfrak{g}\mathfrak{l}_{n})$| proposed by Frassek, Pestun, and Tsymbaliuk. [ABSTRACT FROM AUTHOR]

Let $X$ be a smooth geometrically connected projective curve over the field of fractions of a discrete valuation ring $R$ , and $\mathfrak {m}$ a modulus on $X$ , given by a closed subscheme of $X$ which is geometrically reduced. The generalized Jacobian $J_\mathfrak {m}$ of $X$ with respect to $\mathfrak {m}$ is then an extension of the Jacobian of $X$ by a torus. We describe its Néron model, together with the character and component groups of the special fibre, in terms of a regular model of $X$ over $R$. This generalizes Raynaud's well-known description for the usual Jacobian. We also give some computations for generalized Jacobians of modular curves $X_0(N)$ with moduli supported on the cusps. [ABSTRACT FROM AUTHOR]

We provide a short proof that the dimensions of the mod p homology groups of the unordered configuration space B_k(T) of k points in a closed torus are the same as its Betti numbers for p>2 and k\leq p. Hence the integral homology has no p-power torsion in this range. The same argument works for the once-punctured genus g surface with g\geq 0, thereby recovering a result of Brantner-Hahn-Knudsen via Lubin-Tate theory. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence of response solutions for a class of quasi-periodically forced harmonic oscillators$ \ddot{x}+\lambda^2 x = \epsilon f(t, x, \dot{x}) $where $ \lambda\in\mathcal{O} $ with $ \mathcal{O} $ being a closed interval not containing zero, the forcing term $ f $ is of class $ C^l (l>2n) $ and quasi-periodic in $ t $ with the frequency $ \omega = (\omega_1, \cdots, \omega_n)\in\mathbb{R}^n $. In addition, the forced term $ f $ is reversible with respect to the involution $ G:(x, y)\rightarrow (-x, y) $. In order to study this equation, we investigate the existence of invariant tori for the following elliptic lower dimensional reversible system$ \dot{\theta} = \omega, \quad\dot{x} = \Omega y+f(\theta, x, y; \Omega), \quad\dot{y} = -\Omega x+g(\theta, x, y;\Omega), $where $ (\theta, x, y)\in\mathbb{T}^n\times\mathbb{R}^m\times \mathbb{R}^m $, $ \omega\in\mathbb{R}^n $ is fixed satisfying the diophantine condition and $ \Omega = (\Omega_1, \cdots, \Omega_m)\in\mathbb{R}^m $ are real parameters. The perturbations $ f $ and $ g $ are of class $ C^l (l>2n) $. Moreover, this system is reversible with respect to the involution $ M:(\theta, x, y)\rightarrow (-\theta, x, -y) $. By KAM iterative method, we prove the existence of invariant tori for the above reversible system. By applying this result, we show that there exists a positive Lebesgue measure set of $ \lambda $ contained in $ \mathcal{O} $ such that the harmonic oscillators has response solutions. [ABSTRACT FROM AUTHOR]

On flat torus with standard metric $ g_b $, two metrics $ g_1 $ and $ g_2 $ with unit volume in the conformal class of $ g_b $ are said to be equivalent if they have the same Gaussian curvature, denoted by $ {g_1\;} \sim g_2 $. The main purpose of this article was to study the boundary of the moduli space $ [g_b]/\sim $. Our main conclusion was that as the energy minimizing metrics of the equivalent classes along a certain direction approach the boundary point, the corresponding metrics must be blown up at exactly one point, which sharpens Struwe's estimate (JEMS 2020). [ABSTRACT FROM AUTHOR]

A fundamental result in toric topology identifies the cohomology ring of the moment-angle complex \mathcal {Z}_K associated to a simplicial complex K with the Koszul homology of the Stanley–Reisner ring of K. By studying cohomology operations induced by the standard torus action on the moment-angle complex, we extend this to a topological interpretation of the minimal free resolution of the Stanley–Reisner ring. The exterior algebra module structure in cohomology induced by the torus action recovers the linear part of the minimal free resolution, and we show that higher cohomology operations induced by the action (in the sense of Goresky–Kottwitz–MacPherson [Invent. Math. 131 (1998), pp. 25–83]) can be assembled into an explicit differential on the resolution. Describing these operations in terms of Hochster's formula, we recover and extend a result due to Katthän [Mathematics 7 (2019), no. 7, p. 605]. We then apply all of this to study the equivariant formality of torus actions on moment-angle complexes. For these spaces, we obtain complete algebraic and combinatorial characterisations of which subtori of the naturally acting torus act equivariantly formally. [ABSTRACT FROM AUTHOR]

This paper is concerned with the incompressible limit of the non-isentropic ideal magnetohydrodynamic equations with periodic boundary conditions and general initial data. Weak convergence results of the smooth solutions are established by filtering method in the torus. [ABSTRACT FROM AUTHOR]

We compute the equivariant partition function of the six-dimensional M-string SCFTs on a background with the topology of a product of a two-dimensional torus and an ALE singularity. We determine the result by exploiting BPS strings probing the singularity, whose worldvolume theories we determine via a chain of string dualities. A distinguished feature we observe is that for this class of background the BPS strings' worldsheet theories become relative field theories that are sensitive to finer discrete data generalizing to 6d the familiar choices of flat connections at infinity for instantons on ALE spaces. We test our proposal against a conjectural 6d N = (1, 0) generalization of the Nekrasov master formula, as well as against known results on ALE partition functions in four dimensions. [ABSTRACT FROM AUTHOR]

We investigate whether it is possible to distinguish chaotic time series from random time series using network theory. In this perspective, we selected four methods to generate graphs from time series: the natural, the horizontal, the limited penetrable horizontal visibility graph, and the phase space reconstruction method. These methods claim that the distinction of chaos from randomness is possible by studying the degree distribution of the generated graphs. We evaluated these methods by computing the results for chaotic time series from the 2D Torus Automorphisms, the chaotic Lorenz system, and a random sequence derived from the normal distribution. Although the results confirm previous studies, we found that the distinction of chaos from randomness is not generally possible in the context of the above methodologies. [ABSTRACT FROM AUTHOR]

We develop a new deconvolution method to recover the precise Crab Nebula image taken by the Hitomi HXT, suppressing the artifact due to the bright Crab pulsar. We extend the Richardson–Lucy method, introducing two components corresponding to the nebula and pulsar with regularization for smoothness and flux, respectively, and performing simultaneous deconvolution of multi-pulse-phase images. The structures, including the torus and jets, seen in the deconvolved nebula image at the lowest energy band of 3.6–15 keV appear consistent with those identified in the high-resolution Chandra X-ray image. Above 15 keV, we confirm NuSTAR's findings that the nebula size decreases in higher energy bands. We find that the north-east side of the nebula is fainter in higher energy bands. Our deconvolution method is applicable for any telescope images of faint diffuse objects containing a bright point source. [ABSTRACT FROM AUTHOR]

The object of this paper is to provide a new and systematic tauberian approach to quantitative long time behaviour of C_{0}-semigroups \left (\mathcal {V}(t)\right)_{t \geqslant 0} in L^{1}(\mathbb {T}^{d}\times \mathbb {R}^{d}) governing conservative linear kinetic equations on the torus with general scattering kernel \boldsymbol {k}(v,v') and degenerate (i.e. not bounded away from zero) collision frequency \sigma (v)=\int _{\mathbb {R}^{d}}\boldsymbol {k}(v',v)\boldsymbol {m}(\mathrm {d}v'), (with \boldsymbol {m}(\mathrm {d}v) being absolutely continuous with respect to the Lebesgue measure). We show in particular that if N_{0} is the maximal integer s \geqslant 0 such that \begin{equation*} \frac {1}{\sigma (\cdot)}\int _{\mathbb {R}^{d}}\boldsymbol {k}(\cdot,v)\sigma ^{-s}(v)\boldsymbol {m}(\mathrm {d}v) \in L^{\infty }(\mathbb {R}^{d}), \end{equation*} then, for initial datum f such that \displaystyle \int _{\mathbb {T}^{d}\times \mathbb {R}^{d}}|f(x,v)|\sigma ^{-N_{0}}(v)\mathrm {d}x\boldsymbol {m}(\mathrm {d}v) <\infty it holds \begin{equation*} \left \|\mathcal {V}(t)f-\varrho _{f}\Psi \right \|_{L^{1}(\mathbb {T}^{d}\times \mathbb {R}^{d})}=\dfrac {{\varepsilon }_{f}(t)}{(1+t)^{N_{0}-1}}, \qquad \varrho _{f}≔\int _{\mathbb {R}^{d}}f(x,v)\mathrm {d}x\boldsymbol {m}(\mathrm {d}v), \end{equation*} where \Psi is the unique invariant density of \left (\mathcal {V}(t)\right)_{t \geqslant 0} and \lim _{t\to \infty }{\varepsilon }_{f}(t)=0. We in particular provide a new criteria of the existence of invariant density. The proof relies on the explicit computation of the time decay of each term of the Dyson-Phillips expansion of \left (\mathcal {V}(t)\right)_{t \geqslant 0} and on suitable smoothness and integrability properties of the trace on the imaginary axis of Laplace transform of remainders of large order of this Dyson-Phillips expansion. Our construction resorts also on collective compactness arguments and provides various technical results of independent interest. Finally, as a by-product of our analysis, we derive essentially sharp "subgeometric" convergence rate for Markov semigroups associated to general transition kernels. [ABSTRACT FROM AUTHOR]

In this paper, we construct a compactification of the space of Bridgeland stability conditions on a smooth projective curve, as an analogue of Thurston compactifications in Teichmüller theory. In the case of elliptic curves, we compare our results with the classical one of the torus via homological mirror symmetry and give the Nielsen–Thurston classification of autoequivalences using the compactification. Furthermore, we observe an interesting phenomenon in the case of the projective line. [ABSTRACT FROM AUTHOR]

We examine the logarithmic Gromov–Witten cycles of a toric variety relative to its full toric boundary. The cycles are expressed as products of double ramification cycles and natural tautological classes in the logarithmic Chow ring of the moduli space of curves. We introduce a simple new technique that relates the Gromov–Witten cycles of rigid and rubber geometries; the technique is based on a study of maps to the logarithmic algebraic torus. By combining this with recent work on logarithmic double ramification cycles, we deduce that all logarithmic Gromov–Witten pushforwards, for maps to a toric variety relative to its full toric boundary, lie in the tautological ring of the moduli space of curves. A feature of the approach is that it avoids the as yet undeveloped logarithmic virtual localization formula, instead relying directly on piecewise polynomial functions to capture the structure that would be provided by such a formula. The results give a common generalization of work of Faber and Pandharipande, and more recent work of Holmes and Schwarz as well as Molcho and Ranganathan. The proof passes through general structure results on the space of stable maps to the logarithmic algebraic torus, which may be of independent interest. [ABSTRACT FROM AUTHOR]

We are concerned with the isentropic compressible Navier–Stokes system in the two-dimensional torus, with rough data and vacuum; the initial velocity belongs to the Sobolev space H 1 and the initial density is only bounded and nonnegative. Arbitrary regions of vacuum are admissible, and no compatibility condition is required. Under these assumptions and for large enough bulk viscosity, global solutions have been constructed in Danchin and Mucha (Commun Pure Appl Math 76:3437–3492, 2023). The main goal of the paper is to establish that these solutions converge exponentially fast to a constant state, and to specify the convergence rate in terms of the viscosity coefficients. We prove similar exponential decay results for the solutions to the inhomogeneous incompressible Navier–Stokes equations, thereby extending to the torus the recent paper (Danchin et al. in Ann Anal Non Lin IHP, 2023) where bounded domains are considered. [ABSTRACT FROM AUTHOR]

We obtain new asymptotic results about systems of N particles governed by Riesz interactions involving k-nearest neighbors of each particle as N → ∞ . These results include a generalization to weighted Riesz potentials with external field. Such interactions offer an appealing alternative to other approaches for reducing the computational complexity of an N-body interaction. We find the first-order term of the large N asymptotics and characterize the limiting distribution of the minimizers. We also obtain results about the Γ -convergence of such interactions, and describe minimizers on the 1-dimensional flat torus in the absence of external field, for all N. [ABSTRACT FROM AUTHOR]

We prove that (under appropriate orientation conditions, depending on R) a Hamiltonian isotopy ψ 1 of a symplectic manifold (M , ω) fixing a relatively exact Lagrangian L setwise must act trivially on R ∗ (L) , where R ∗ is some generalised homology theory. We use a strategy inspired by that of Hu et al. (Geom Topol 15:1617–1650, 2011), who proved an analogous result over Z / 2 and over Z under stronger orientation assumptions. However the differences in our approaches let us deduce that if L is a homotopy sphere, ψ 1 | L is homotopic to the identity. Our technical set-up differs from both theirs and that of Cohen et al. (in: Algebraic topology, Springer, Berlin, 2019) and Cohen (in: The Floer memorial volume, Birkhäuser, Basel). We also prove (under similar conditions) that ψ 1 | L acts trivially on R ∗ (L L) , where L L is the free loop space of L. From this we deduce that when L is a surface or a K (π , 1) , ψ 1 | L is homotopic to the identity. Using methods of Lalonde and McDuff (Topology 42:309–347, 2003), we also show that given a family of Lagrangians all of which are Hamiltonian isotopic to L over a sphere or a torus, the associated fibre bundle cohomologically splits over Z / 2 . [ABSTRACT FROM AUTHOR]

This paper gives a geometric interpretation of bordered Heegaard Floer homology for manifolds with torus boundary. If M is such a manifold, we show that the type D structure \widehat {CFD}(M) may be viewed as a set of immersed curves decorated with local systems in \partial M. These curves-with-decoration are invariants of the underlying three-manifold up to regular homotopy of the curves and isomorphism of the local systems. Given two such manifolds and a homeomorphism h between the boundary tori, the Heegaard Floer homology of the closed manifold obtained by gluing with h is obtained from the Lagrangian intersection Floer homology of the curve-sets. This machinery has several applications: We establish that the dimension of \widehat {HF} decreases under a certain class of degree one maps (pinches) and we establish that the existence of an essential separating torus gives rise to a lower bound on the dimension of \widehat {HF}. In particular, it follows that a prime rational homology sphere Y with \widehat {HF}(Y)<5 must be geometric. Other results include a new proof of Eftekhary's theorem that L-space homology spheres are atoroidal; a complete characterization of toroidal L-spaces in terms of gluing data; and a proof of a conjecture of Hom, Lidman, and Vafaee on satellite L-space knots. [ABSTRACT FROM AUTHOR]

For an algebraic torus S , Blinstein and Merkurjev have given an estimate of 3-rd unramified cohomology H ¯ n r 3 (F (S) , Q / Z (2)) obtained from a flasque resolution of S. Based on their work, for the norm one torus W = R K / F (1) G m with K / F abelian, we compute the 3-rd unramified cohomology H ¯ n r 3 (F (W) , Q / Z (2)). [ABSTRACT FROM AUTHOR]

The Joint European Torus (JET) is playing an important role in preparing for the operation of the future world's largest tokomak, ITER. In this respect, the tritium campaign (C40) and second deuterium–tritium experiment (DTE2, C41) took place in the JET during the years 2021 and 2022. In this work, a corresponding irradiation scenario was utilized for the activation calculations of eight material foils located at the JET outer long-term irradiation station (OLTIS). Neutron-induced activities and dose rates at a 30 cm distance after shutdown at specified cooling intervals were calculated with the FISPACT-II code, employing the EAF-2010 nuclear and TENDL-2021 data libraries. The Monte Carlo MCNP6.2 particle transport code equipped with the FENDL-3.1d nuclear data library was used for the calculation of the neutron flux densities. [ABSTRACT FROM AUTHOR]

Let K be a rationally null-homologous knot in a 3-manifold Y, equipped with a non-zero framing λ, and let Yλ​(K) denote the result of λ-framed surgery on Y. Ozsváth and Szabó gave a formula for the Heegaard Floer homology groups of Yλ​(K) in terms of the knot Floer complex of (Y,K). We strengthen this formula by adding a second filtration that computes the knot Floer complex of the dual knot Kλ​ in Yλ​, i.e., the core circle of the surgery solid torus. In the course of proving our refinement we derive a combinatorial formula for the Alexander grading which may be of independent interest. [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]

We consider the problem of conjugating a flow on R d × R to an "integrable" flow q ˙ = ω and p ˙ = c p , if possible, with c ≠ 0. Our emphasis is on a constructive approach, using a KAM type iteration and analyticity. An application to the periodically forced Van der Pol oscillator is given. [ABSTRACT FROM AUTHOR]

We consider a class of finite-dimensional and infinite-dimensional locally coupled systems of periodic Hill-type equations with impacts. First, for finite dimensional systems, with assumptions of superlinearity on the restoring forces and boundedness on the coupling terms, we show the twist properties of the Poincaré maps on the suitably large torus in each sub-phase plane. Then, by a high-dimensional zero point theorem, it is proved the existence of infinitely many symmetric harmonic and subharmonic bouncing solutions. Second, we consider an infinite-dimensional system as a limit case of a sequence of finite-dimensional systems. By a detailed discussions for taking limits, it is proved that a sequence of even and periodic bouncing solutions of some finite-dimensional systems can converge to an even and periodic bouncing solution of the infinite-dimensional system. [ABSTRACT FROM AUTHOR]

Inspired by string theory compactifications and torus 2 topology, we consider a general interacting D(3 + n)-brane model, with n being the number of extra dimensions, built from a flat torus ℝ 2 n / ℤ 2 n compactification. In particular, we present the squared n = 2 torus topological features and investigate the associated low-energy D-brane physics. [ABSTRACT FROM AUTHOR]

The Boltzmann equation of kinetic theory gives a statistical description of a gas of interacting particles. It is well known that the Boltzmann equation is related to the Euler and Navier–Stokes equations in the field of gas dynamics. In this paper we are concerned with the incompressible Navier–Stokes–Fourier limit of the Boltzmann equation. We prove the incompressible Navier–Stokes–Fourier limit globally in time and the time decay rate of the solution to the rescaled Boltzmann equation in a torus. For ɛ small, by using the truncated expansion and L x , v 2 – L x , v ∞ method, we prove such a limit for the general potentials γ ∈ (− 3 , 1 ] . [ABSTRACT FROM AUTHOR]

We consider the rich dynamics and bifurcations exhibited by a three-dimensional quadratic map. Torus doubling bifurcations are central to bifurcation theory. Such bifurcations can only occur in a higher dimensional map of dimensions greater than or equal to three. After subsequent doublings, formation of Shilnikov attractors and hyperchaotic attractors are usually observed. It is shown that the map under consideration shows both resonant and ergodic torus doubling bifurcation. We show that the system exhibits resonant torus doubling bifurcation in which the doubled mode-locked periodic orbits lies on a Möbius strip. Additionally, we also illustrate the doubling of ergodic tori and analyze the bifurcation via the use of the second Poincaré section and the method of Lyapunov bundles. The analysis involves the techniques of construction of one-dimensional manifolds, a one-parameter continuation of saddle periodic orbits, and multi-dimensional Newton–Raphson method to locate the saddle periodic orbits. [ABSTRACT FROM AUTHOR]

An implicit Euler finite-volume scheme for a nonlocal cross-diffusion system on the one-dimensional torus, arising in population dynamics, is proposed and analyzed. The kernels are assumed to be in detailed balance and satisfy a weak cross-diffusion condition. The latter condition allows for negative off-diagonal coefficients and for kernels defined by an indicator function. The scheme preserves the nonnegativity of the densities, conservation of mass, and production of the Boltzmann and Rao entropies. The key idea is to "translate" the entropy calculations for the continuous equations to the finite-volume scheme, in particular to design discretizations of the mobilities, which guarantee a discrete chain rule even in the presence of nonlocal terms. Based on this idea, the existence of finite-volume solutions and the convergence of the scheme are proven. As a by-product, we deduce the existence of weak solutions to the continuous cross-diffusion system. Finally, we present some numerical experiments illustrating the behavior of the solutions to the nonlocal and associated local models. [ABSTRACT FROM AUTHOR]

An optimal control problem in the space of probability measures and the viscosity solutions of the corresponding dynamic programming equations defined using the intrinsic linear derivative are studied. The value function is shown to be Lipschitz continuous with respect to a smooth Fourier--Wasserstein metric. A comparison result between the Lipschitz viscosity sub- and supersolutions of the dynamic programming equation is proved using this metric, characterizing the value function as the unique Lipschitz viscosity solution. [ABSTRACT FROM AUTHOR]

Supertubes are supersymmetric configurations in string theory in which branes are extending along a closed curve. For a supertube of codimension two, its dipole charge is characterized by the duality monodromy around the closed curve. When multiple codimension-2 supertubes are present, the monodromies around different supertubes can be non-commuting, namely non-Abelian. Non-Abelian configurations of supertubes are expected to play an important role in non-perturbative physics of string theory, especially black holes. In this paper, in the framework of five-dimensional supergravity, we construct exact solutions describing codimension-2 supertubes in three-dimensional space. We use an extension formula to construct a three-dimensional solution from a two-dimensional seed solution. The two-dimensional seed is an F-theory like configuration in which a torus is nontrivially fibered over a complex plane. In the first example, there is a stack of circular supertubes around which there is a non-trivial monodromy. In some cases this can be thought of as a microstate of a black hole in AdS2 × S2. The second example is an axi-symmetric solution with two stacks of circular supertubes with non-Abelian monodromies. In addition, there is a continuous distribution of charges on the symmetry axis. [ABSTRACT FROM AUTHOR]

We use the 3D magnetohydrodynamic (MHD) code M3D-C1 [Jardin et al., Comput. Sci. Discovery 5, 014002 (2012)] to examine the MHD stability and subsequent evolution of NSTX shot 129169. This discharge had a period with a non-monotonic safety factor profile, q (reversed shear), which was terminated by a MHD event that abruptly lowered the central safety factor, q0, and greatly reduced the peakedness of the pressure profile. We show that the equilibrium just before the MHD event occurred was linearly unstable to many pressure-driven infernal modes. Modes with toroidal mode number n ≥ 3 all had rational surfaces very close to the minimum value of q. However, a non-resonant pressure-driven (1, 1) mode was also present, and this dominated the nonlinear evolution. The final state in the simulation, after the MHD activity subsided, had a reduced and flattened pressure profile and a nearly monotonic q-profile, in qualitative agreement with experimental results. The initial state was also unstable to the resistive interchange criteria in the reversed-shear region, but the final state was stable everywhere. The "double tearing mode" (DTM) does not appear to play a role in the MHD activity of this discharge. In Appendix A, we show that in a torus, the DTM is strongly stabilized by pressure, but it is destabilized in cylindrical geometry (which has been the most extensively analyzed in the literature). [ABSTRACT FROM AUTHOR]

The gravity-driven settlement process of bi-disperse particle clouds is investigated by using a computational fluid dynamic–discrete element method coupled model. The purpose of this work is to discuss the effect of particle size ratio and cloud size on cloud settling, with an emphasis on cloud morphological changes and dynamic mechanisms, as well as the spatial distribution and movement rules of each dispersed phase. The simulation results demonstrate that the settling of bi-dispersed clouds is accompanied by cloud breakup and the separation of large and small particles. The rise in particle size ratio will make cloud breakup more difficult, and the increase in cloud size facilitates the separation of large and small particles in the cloud. Essentially, cloud breakup and separation are closely related to weak inter-particle forces, and the thresholds for these two events are, respectively, proposed. Decomposable clouds will continuously flatten into toruses before breaking up, with an increasing aspect ratio γ that suggests a weakening of the interactions. When the aspect ratio exceeds a critical value of 2, the cloud breaks up. The non-dimensional distance l d * between large and small particles, describing the interaction between two dispersed phases, correlates with their separation. The critical distance for separation is 29. Furthermore, the sequence in which the two critical conditions are met determines the order in which cloud breakup and particle separation take place. [ABSTRACT FROM AUTHOR]