Dimensional Torus Research Articles

On adaptive confidence sets for the Wasserstein distances

In the density estimation model, we investigate the problem of constructing adaptive honest confidence sets with radius measured in Wasserstein distance $W_p$, $p\geq1$, and for densities with unknown regularity measured on a Besov scale. As sampling domains, we focus on the $d-$dimensional torus $\mathbb{T}^d$, in which case $1\leq p\leq 2$, and $\mathbb{R}^d$, for which $p=1$. We identify necessary and sufficient conditions for the existence of adaptive confidence sets with diameters of the order of the regularity-dependent $W_p$-minimax estimation rate. Interestingly, it appears that the possibility of such adaptation of the diameter depends on the dimension of the underlying space. In low dimensions, $d\leq 4$, adaptation to any regularity is possible. In higher dimensions, adaptation is possible if and only if the underlying regularities belong to some interval of width at least $d/(d-4)$. This contrasts with the usual $L_p-$theory where, independently of the dimension, adaptation requires regularities to lie in a small fixed-width window. For configurations allowing these adaptive sets to exist, we explicitly construct confidence regions via the method of risk estimation, centred at adaptive estimators. Those are the first results in a statistical approach to adaptive uncertainty quantification with Wasserstein distances. Our analysis and methods extend more globally to weak losses such as Sobolev norm distances with negative smoothness indices.

On Bernstein type quantitative estimates for Ornstein non-inequalities

For the sequence of multi-indices \{\alpha_j\}_{j=1}^m and \beta , we study the inequality \lVert D^{\beta} f \rVert_{L_1(\mathbb{T}^d)} \leq K_N \sum_{j= 1}^{m} \lVert D^{\alpha_j}f \rVert_{L_1(\mathbb{T}^d)}, where f is a trigonometric polynomial of degree at most N on the d -dimensional torus. Assuming some natural geometric property of the set \{\alpha_j\}\cup\{\beta\} , we show that K_{N}\geq C (\ln N)^{\phi}, where \phi<1 depends only on the set \{ \alpha_j \}\cup\{\beta\} .

A pathwise stochastic Landau-Lifshitz-Gilbert equation with application to large deviations

Using a rough path formulation, we investigate existence, uniqueness and regularity for the stochastic Landau-Lifshitz-Gilbert equation with Stratonovich noise on the one dimensional torus. As a main result we show the continuity of the so-called Itô-Lyons map in the energy spaces L∞(0,T;Hk)∩L2(0,T;Hk+1) for any k≥1. The proof proceeds in two steps. First, based on an energy estimate in the aforementioned space together with a compactness argument we prove existence of a unique solution, implying the continuous dependence in a weaker norm. This is then strengthened in the second step where the continuity in the optimal norm is established through an application of the rough Gronwall lemma. Our approach is direct and does not rely on any transformation formula, which permits to treat multidimensional noise. As an easy consequence we then deduce a Wong-Zakai type result, a large deviation principle for the solution and a support theorem.

Quasi-periodic Hamiltonian pitchfork bifurcation in a phenomenological model with 3 degrees of freedom

Litvak-Hinenzon et al. developed the phenomenological model to simulate the horizontal motion of particles in the atmosphere, and a series of their papers (see e.g., Litvak-Hinenzon et al. (Phys. D 164:213-250, 2002; Nonlinearity 15:1149-1177, 2002; SIAM J. Appl. Dyn. Syst. 3:525-573, 2004)) focused on studying the fate of parabolic resonance lower dimensional tori (as part of a quasi-periodic Hamiltonian pitchfork bifurcation (HPB) scenario in the unperturbed phenomenological model) under perturbations. However, the structural stability of quasi-periodic HPB involved therein has not been fully exposed theoretically (Litvak-Hinenzon et al. have only given some numerical explanations). Based on BCKV singularity theory established by Broer et al. (Z. Angew. Math. Phys. 44:389-432, 1993), we consider a more general quasi-periodic HPB triggered by the Z2-invariant universal unfolding Nbckv=ay22−(λ+bI1)x22+cx44with respect to Z2-equivariant BCKV-restricted morphisms of the planar singularity a2y2+c4x4 (the coefficients a,b,c≠0, the I1 is regarded as distinguished parameter with respect to the external parameter λ). We prove a KAM (Kolmogorov–Arnold–Moser) theorem concerning parabolic tori in such quasi-periodic HPB, by which Diophantine parabolic tori (and the whole corresponding Diophantine HPB scenario) survive non-integrable and Z2-invariant Hamiltonian perturbations, parametrized by pertinent large Cantor sets. In the context of Z2-symmetry, our results can be seen as rigorous proof of the structural stability problem of bifurcations of Floquet-tori triggered by the universal unfolding Nbckv with distinguished parameters which is proposed by Broer et al. (Z. Angew. Math. Phys. 44:389-432, 1993). Ultimately, we similarly obtain the structural stability result of quasi-periodic HPB in the phenomenological model mentioned above, which can be utilized as a starting point for a deeper understanding of the various resonances and chaotic dynamics (in the gaps of the Cantor sets), just as Litvak-Hinenzon et al. did for normally parabolic tori undergoing a HPB.

Bootstrap percolation in random geometric graphs

AbstractFollowing Bradonjić and Saniee, we study a model of bootstrap percolation on the Gilbert random geometric graph on the 2-dimensional torus. In this model, the expected number of vertices of the graph is n, and the expected degree of a vertex is $a\log n$ for some fixed $a>1$ . Each vertex is added with probability p to a set $A_0$ of initially infected vertices. Vertices subsequently become infected if they have at least $ \theta a \log n $ infected neighbours. Here $p, \theta \in [0,1]$ are taken to be fixed constants.We show that if $\theta < (1+p)/2$ , then a sufficiently large local outbreak leads with high probability to the infection spreading globally, with all but o(n) vertices eventually becoming infected. On the other hand, for $ \theta > (1+p)/2$ , even if one adversarially infects every vertex inside a ball of radius $O(\sqrt{\log n} )$ , with high probability the infection will spread to only o(n) vertices beyond those that were initially infected.In addition we give some bounds on the $(a, p, \theta)$ regions ensuring the emergence of large local outbreaks or the existence of islands of vertices that never become infected. We also give a complete picture of the (surprisingly complex) behaviour of the analogous 1-dimensional bootstrap percolation model on the circle. Finally we raise a number of problems, and in particular make a conjecture on an ‘almost no percolation or almost full percolation’ dichotomy which may be of independent interest.

Spectral Multiplicity and Nodal Domains of Torus-Invariant Metrics

Abstract Let a $d$-dimensional torus $\mathbb{T}$ act freely and smoothly on a closed manifold $M$ of dimension $n>d$. We show that, for a generic $\mathbb{T}$-invariant Riemannian metric $g$ on $M$, each real $\Delta _{g}$-eigenspace is an irreducible real representation of $\mathbb{T}$ and, therefore, has dimension at most two. We also show that, for the generic $\mathbb{T}$-invariant metric $g$ on $M$, if $u$ is a non-invariant real-valued $\Delta _{g}$-eigenfunction that vanishes on some $\mathbb{T}$-orbit, then the nodal set of $u$ is a connected smooth hypersurface. If $n>d+1$, we show that the complement of the nodal set has exactly two connected components. As a consequence, we obtain new examples of manifolds for which—up to a sequence of Weyl density zero—each eigenfunction has exactly two nodal domains.

Tracing projective modules over noncommutative orbifolds

For an action of a finite cyclic group F on an n -dimensional noncommutative torus A_\theta , we give sufficient conditions when the fundamental projective modules over A_\theta , which determine the range of the canonical trace on A_\theta , extend to projective modules over the crossed product C^* -algebra A_\theta \rtimes F . Our results allow us to understand the range of the canonical trace on A_\theta \rtimes F , and determine it completely for several examples including the crossed products of 2-dimensional noncommutative tori with finite cyclic groups and the flip action of \mathbb Z_2 on any n -dimensional noncommutative torus. As an application, for the flip action of \mathbb Z_2 on a simple n -dimensional torus A_\theta , we determine the Morita equivalence class of A_\theta \rtimes \mathbb Z_2, in terms of the Morita equivalence class of $A_\theta.

An Extensive Power and Performance Analysis for High Dimensional Mesh and Torus Interconnection Networks

The next generation parallel computers are the keys to achieve exascale performance, whereas sequential computers have already been saturated. In order to achieve this mighty target of exascale computing, one of the main challenges is the reduction of power consumption along with achieving suitable performance. Energy efficiency is a key feature to ensure the trade-off between the performances over the required power usages. Hence, to focus on those issues, the target of this article is to analyze the performance versus power usage trade-off for the conventional networks like- Mesh and Torus. High degree networks show much better performance than the low degree of networks. However, high degree networks require higher power usage for their high degree of interconnected links. This article showed that with zero load latency, the 3D Torus could show about 57.07% better performance than a 2D Torus. On the other hand, a 2D Mesh network requires about 24.22% less router power usage than the 3D Mesh, & 5D Torus requires about 66.8% higher router power usage than a 3D Torus network.

Concentration of the number of intersections of random eigenfunctions on flat tori

We show that in two dimensional flat torus the number of intersections between random eigenfunctions of general eigenvalues and a given smooth curve is almost exponentially concentrated around its mean, even when the randomness is not gaussian.

Quasi-periodic solutions for a class of wave equation system

In this paper, we establish an abstract infinite dimensional KAM theorem. As an application, we use the theorem to study the 1D wave equation system { u 1 tt − u 1 xx + σ u 1 + u 1 u 2 2 = 0 u 2 tt − u 2 xx + μ u 2 + u 1 2 u 2 = 0 under Dirichlet boundary conditions, where 0 < σ ∈ [ σ 1 , σ 2 ] ⊂ [ 0 , 1 ] , 0 < μ ∈ [ μ 1 , μ 2 ] ⊂ [ 0 , 1 ] are real parameters. By establishing a block-diagonal normal form, we obtain the existence of a Whitney smooth family of small amplitude quasi-periodic solutions corresponding to finite dimensional invariant tori of an associated infinite dimensional dynamic system.

A construction of G2-manifolds from K3 surfaces with a [formula omitted]-action

A product of a K3 surface S and a flat 3-dimensional torus T3 is a manifold with holonomy SU(2). Since SU(2) is a subgroup of G2, S×T3 carries a torsion-free G2-structure. We assume that S admits an action of Z22 with certain properties. There are several possibilities to extend this action to S×T3. A recent result of Joyce and Karigiannis allows us to resolve the singularities of (S×T3)/Z22 such that we obtain smooth G2-manifolds. We classify the quotients (S×T3)/Z22 under certain restrictions and compute the Betti numbers of the corresponding G2-manifolds. Moreover, we study a class of quotients by a non-abelian group. Several of our examples have new values of (b2,b3).

The Near-Critical Two-Point Function and the Torus Plateau for Weakly Self-avoiding Walk in High Dimensions

We use the lace expansion to study the long-distance decay of the two-point function of weakly self-avoiding walk on the integer lattice $$\mathbb {Z}^d$$ in dimensions $$d>4$$ , in the vicinity of the critical point, and prove an upper bound $$|x|^{-(d-2)}\exp [-c|x|/\xi ]$$ , where the correlation length $$\xi $$ has a square root divergence at the critical point. As an application, we prove that the two-point function for weakly self-avoiding walk on a discrete torus in dimensions $$d{>}4$$ has a “plateau.” We also discuss the significance and consequences of the plateau for the analysis of critical behaviour on the torus.

T-duality building blocks for α′ string corrections

T-duality has been shown to constrain the higher-derivative corrections of string theory. We revisit the problem of understanding the T-duality constraints imposed on the ${\ensuremath{\alpha}}^{\ensuremath{'}}$ corrections using the language of a torsionful connection. We find a convenient way to express the $O(d,d)$ invariants in terms of linear combinations of the metric and B-field for general $d$-dimensional torus compactifications, which we then use to revisit the heterotic and bosonic ten-dimensional string action at order ${\ensuremath{\alpha}}^{\ensuremath{'}}$. We also comment on the four-point functions for corrections of order ${\ensuremath{\alpha}}^{\ensuremath{'}3}$ in the type II string.

Response solutions for quasi-periodically forced harmonic oscillators in Gevrey class

We prove that for quasi-periodically forced harmonic oscillator, if the forcing frequency ω∈R2﹨Q2 is non-resonant and the forcing is partial Gevrey smooth, then there exist response solutions. This generalized the result by [22] where the forcing term is real analytic. The proof is based on modified KAM theory for lower dimensional invariant tori.

Hardy and Rellich inequality on lattices

In this paper, we study the asymptotic behaviour of the sharp constant in discrete Hardy and Rellich inequality on the lattice mathbb {Z}^d as d rightarrow infty . In the process, we proved some Hardy-type inequalities for the operators Delta ^m and nabla (Delta ^m) for non-negative integers m on a d dimensional torus. It turns out that the sharp constant in discrete Hardy and Rellich inequality grows as d and d^2 respectively as d rightarrow infty .

QUASI-PERIODIC SOLUTIONS FOR 1D NONLINEAR WAVE EQUATION

In this paper, one-dimensional (1D) nonlinear wave equation $$u_{tt}-u_{xx}+mu+u^{7}=0$$on the finite $x$-interval $[0,\pi]$ with Dirichlet boundary conditions is considered. It is proved that there are many $3$-dimensional elliptic invariant tori, and thus quasi-periodic solutions for the above equation. This is an extension of the previous work [11] by the same author, where many $2$-dimensional elliptic invariant tori for the above equation are obtained. The proof is based on infinite-dimensional KAM theory and partial Birkhoff normal form.

Reducibility for a linear wave equation with Sobolev smooth fast-driven potential

We prove a reducibility result for a linear wave equation with a time quasi-periodic driving on the one dimensional torus. The driving is assumed to be fast oscillating, but not necessarily of small size. Provided that the external frequency vector is sufficiently large and chosen from a Cantor set of large measure, the original equation is conjugated to a time-independent, block-diagonal one. With the present paper, we extend the previous work [26] to more general assumptions: we replace the analytic regularity in time with Sobolev one; the potential in the Schrödinger operator is a non-trivial smooth function instead of the constant one. The key tool to achieve the result is a localization property of each eigenfunction of the Schrödinger operator close to a subspace of exponentials, with a polynomial decay away from the latter.

How is a graph not like a manifold?

For an equivariantly formal action of a compact torus $T$ on a smooth manifold $X$ with isolated fixed points we investigate the global homological properties of the graded poset $S(X)$ of face submanifolds. We prove that the condition of the $j$-independency of tangent weights at each fixed point implies the $(j+1)$-acyclicity of the skeleta $S(X)_r$ for $r&gt;j+1$. This result provides a necessary topological condition for a GKM graph to be a GKM graph of some GKM manifold. We use particular acyclicity arguments to describe the equivariant cohomology algebra of an equivariantly formal manifold of dimension $2n$ with an $(n-1)$-independent action of the $(n-1)$-dimensional torus, under certain colourability assumptions on its GKM graph. This description relates the equivariant cohomology algebra to the face algebra of a simplicial poset. This observation underlines a certain similarity between actions of complexity $1$ and torus manifolds. Bibliography: 27 titles.

Multilevel Markov Chain Monte Carlo for Bayesian inverse problem for Navier-Stokes equation

&lt;p style='text-indent:20px;'&gt;Bayesian inverse problems for inferring the unknown forcing and initial condition of Navier-Stokes equation play important roles in many practical areas. The computation cost of sampling the posterior probability measure can be exceedingly high. We develop the Finite Element Multilevel Markov Chain Monte Carlo (FE-MLMCMC) sampling method for approximating expectation with respect to the posterior probability measure of quantities of interest for a model problem of Navier-Stokes equation in the two dimensional periodic torus. We first consider the case where the forcing and the initial condition are bounded for all the realizations and depend linearly on a countable set of random variables which are uniformly distributed in a compact interval. We establish the essentially optimal convergence rate of the method and verify it numerically. The method follows from that developed in V. H. Hoang, Ch. Schwab and A. M. Stuart, Inverse problems, vol. 29, 2013 for inferring the coefficients of linear elliptic forward equations under the uniform prior probability measure. In the case of the Gaussian prior probability measure, numerical results, using the MLMCMC method developed for the Gaussian prior in V. H. Hoang, J. H. Quek and Ch. Schwab, Inverse problems, vol. 36, 2020, indicate the essentially optimal convergence rates. However, a rigorous theory for the MLMCMC sampling procedure is not available, due to the non-integrability with respect to the Gaussian prior of the theoretical finite element errors of the forward solvers that are available in the literature.&lt;/p&gt;

Invariant tori for the Hamiltonian derivative wave equation with higher order nonlinearity

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 $ m&gt;0 $ is a potential and \begin{document}$ D: = \sqrt{-\partial_{xx}+m}. $\end{document} We will prove that, for any integer $ b\geq2 $, the above equation admits many small amplitude quasi-periodic solutions corresponding to $ b $-dimensional invariant tori of an associated infinite dimensional Hamiltonian system. The proof is based on infinite dimensional KAM theory and partial Birkhoff normal form.