D-dimensional Torus Research Articles

Specific Properties of the ODE’s Flow in Dimension Two Versus Dimension Three

This paper deals with the asymptotics of the ODE’s flow induced by a regular vector field b on the d-dimensional torus \({\mathbb {R}}^d/{\mathbb {Z}}^d\). First, we start by revisiting the Franks-Misiurewicz theorem which claims that the Herman rotation set of any two-dimensional continuous flow is a closed line segment of \({\mathbb {R}}^2\). Various general examples illustrate this result, among which a complete study of the Stepanoff flow associated with a vector field \(b=a\,\zeta \), where \(\zeta \) is a constant vector in \({\mathbb {R}}^2\). Furthermore, several extensions of the Franks-Misiurewicz theorem are obtained in the two-dimensional ODE’s context. On the one hand, we provide some interesting stability properties in the case where the Herman rotation set has a commensurable direction. On the other hand, we present new results highlighting the exceptional character of the opposite case, i.e. when the Herman rotation set is a closed line segment with \(0_{{\mathbb {R}}^2}\) at one end and with an irrational slope, if it is not reduced to a single point. Besides this, given a pair \((\mu ,\nu )\) of invariant probability measures for the flow, we establish new Fourier relations between the determinant \(\det \,(\widehat{\mu b}(j),\widehat{\nu b}(k))\) and the determinant \(\det \,(j,k)\) for any pair (j, k) of non null integer vectors, which can be regarded as an extension of the Franks-Misiurewicz theorem. Next, in contrast with dimension two, any three-dimensional closed convex polyhedron with rational vertices is shown to be the rotation set associated with a suitable vector field b. Finally, in the case of an invariant measure \(\mu \) with a regular density and a non null mass \(\mu (b)\) with respect to b, we show that the homogenization of the two-dimensional transport equation with the oscillating velocity \(b(x/\varepsilon )\) as \(\varepsilon \) tends to 0, leads us to a nonlocal limit transport equation, but with the effective constant velocity \(\mu (b)\).

A partial uniqueness result and an asymptotically sharp nonuniqueness result for the Zhikov problem on the torus

We consider the stationary diffusion equation \(-\mathrm {div} (\nabla u + bu )=f\) in d-dimensional torus \(\mathbb {T}^d\), where \(f\in H^{-1}\) is a given forcing and \(b\in L^p\) is a divergence-free drift. Zhikov (Funkts Anal Prilozhen, 38(3):15–28, 2004) considered this equation in the case of a bounded, Lipschitz domain \(\Omega \subset \mathbb {R}^d\), and proved existence of solutions for \(b\in L^{2d/(d+2)}\), uniqueness for \(b\in L^2\), and has provided a point-singularity counterexample that shows nonuniqueness for \(b\in L^{3/2-}\) and \(d=3,4,5\). We apply a duality method and a DiPerna–Lions-type estimate to show uniqueness of the solutions constructed by Zhikov for \(b\in W^{1,1}\). We use a Nash iteration to demonstrate sharpness of this result, and also show that solutions in \(H^1\cap L^{p/(p-1)}\) are flexible for \(b\in L^p\), \(p\in [1,2(d-1)/(d+1))\); namely we show that the set of \(b\in L^p\) for which nonuniqueness in the class \(H^1\cap L^{p/(p-1)}\) occurs is dense in the divergence-free subspace of \(L^p\).

Formula omitted]-Numbers of embeddings of weighted Wiener algebras

In this paper we study the asymptotic behavior of Kolmogorov, approximation, Bernstein and Weyl numbers of embeddings Amixs,r(Td)→L2(Td) and Amixs,r(Td)→A(Td), where Amixs,r(Td) is a weighted Wiener algebra of mixed smoothness s and A(Td) is the Wiener algebra itself, both defined on the d-dimensional torus Td. Our main interest consists in the calculation of the associated asymptotic constants.

Concentration inequalities for log-concave distributions with applications to random surface fluctuations

We derive two concentration inequalities for linear functions of log-concave distributions: an enhanced version of the classical Brascamp–Lieb concentration inequality and an inequality quantifying log-concavity of marginals in a manner suitable for obtaining variance and tail probability bounds. These inequalities are applied to the statistical mechanics problem of estimating the fluctuations of random surfaces of the ∇φ type. The classical Brascamp–Lieb inequality bounds the fluctuations whenever the interaction potential is uniformly convex. We extend these bounds to the case of convex potentials whose second derivative vanishes only on a zero measure set, when the underlying graph is a d-dimensional discrete torus. The result applies, in particular, to potentials of the form U(x)=|x|p with p>1 and answers a question discussed by Brascamp–Lieb–Lebowitz (In Statistical Mechanics (1975) 379–390, Springer). Additionally, new tail probability bounds are obtained for the family of potentials U(x)=|x|p+x2, p>2. This result answers a question mentioned by Deuschel and Giacomin (Stochastic Process. Appl. 89 (2000) 333–354).

Convergence of a particle approximation for the quasi-stationary distribution of a diffusion process: Uniform estimates in a compact soft case

We establish the convergences (with respect to the simulation time t; the number of particles N; the timestep γ) of a Moran/Fleming-Viot type particle scheme toward the quasi-stationary distribution of a diffusion on the d-dimensional torus, killed at a smooth rate. In these conditions, quantitative bounds are obtained that, for each parameter (t →∞, N →∞ or γ → 0) are independent from the two others.

Sparse Fourier transforms on rank-1 lattices for the rapid and low-memory approximation of functions of many variables

This paper considers fast and provably accurate algorithms for approximating smooth functions on the d-dimensional torus, \(f: \mathbb {T}^d \rightarrow \mathbb {C}\), that are sparse (or compressible) in the multidimensional Fourier basis. In particular, suppose that the Fourier series coefficients of f, \(\{c_\mathbf{k} (f) \}_{\mathbf{k} \in \mathbb {Z}^d}\), are concentrated in a given arbitrary finite set \(\mathscr {I} \subset \mathbb {Z}^d\) so that $$\begin{aligned} \min _{\text{\O}mega \subset \mathscr {I} ~s.t.~ \left| \text{\O}mega \right| =s }\left\| f - \sum _{\mathbf{k} \in \text{\O}mega } c_\mathbf{k} (f) \, \mathbb {e}^{ -2 \pi \mathbb {i} {{\mathbf {k}}}\cdot \circ } \right\| _2 < \epsilon \Vert f \Vert _2 \end{aligned}$$holds for \(s \ll \left| \mathscr {I} \right| \) and \(\epsilon \in (0,1)\) small. In such cases we aim to both identify a near-minimizing subset \(\text{\O}mega \subset \mathscr {I}\) and accurately approximate its associated Fourier coefficients \(\{ c_\mathbf{k} (f) \}_{\mathbf{k} \in \text{\O}mega }\) as rapidly as possible. In this paper we present both deterministic and explicit as well as randomized algorithms for solving this problem using \(\mathscr {O}(s^2 d \log ^c (|\mathscr {I}|))\)-time/memory and \(\mathscr {O}(s d \log ^c (|\mathscr {I}|))\)-time/memory, respectively. Most crucially, all of the methods proposed herein achieve these runtimes while simultaneously satisfying theoretical best s-term approximation guarantees which guarantee their numerical accuracy and robustness to noise for general functions. These results are achieved by modifying several different one-dimensional Sparse Fourier Transform (SFT) methods to subsample a function along a reconstructing rank-1 lattice for the given frequency set \(\mathscr {I} \subset \mathbb {Z}^d\) in order to rapidly identify a near-minimizing subset \(\text{\O}mega \subset \mathscr {I}\) as above without having to use anything about the lattice beyond its generating vector. This requires the development of new fast and low-memory frequency identification techniques capable of rapidly recovering vector-valued frequencies in \(\mathbb {Z}^d\) as opposed to recovering simple integer frequencies as required in the univariate setting. Two different multivariate frequency identification strategies are proposed, analyzed, and shown to lead to their own best s-term approximation methods herein, each with different accuracy versus computational speed and memory tradeoffs.

On the variance of the nodal volume of arithmetic random waves

Abstract Rudnick and Wigman (2008) conjectured that the variance of the volume of the nodal set of arithmetic random waves on the d-dimensional torus is O ⁢ ( E / 𝒩 ) {O(E/\mathcal{N})} , as E → ∞ {E\to\infty} , where E is the energy and 𝒩 {\mathcal{N}} is the dimension of the eigenspace corresponding to E. Previous results have established this with stronger asymptotics when d = 2 {d=2} and d = 3 {d=3} . In this brief note we prove an upper bound of the form O ⁢ ( E / 𝒩 1 + α ⁢ ( d ) - ϵ ) {O(E/\mathcal{N}^{1+\alpha(d)-\epsilon})} , for any ϵ &gt; 0 {\epsilon&gt;0} and d ≥ 4 {d\geq 4} , where α ⁢ ( d ) {\alpha(d)} is positive and tends to zero with d. The power saving is the best possible with the current method (up to ϵ) when d ≥ 5 {d\geq 5} due to the proof of the ℓ 2 {\ell^{2}} -decoupling conjecture by Bourgain and Demeter.

Idempotent Fourier multipliers acting contractively on H^p spaces

We describe the idempotent Fourier multipliers that act contractively on H^p spaces of the d-dimensional torus mathbb {T}^d for dge 1 and 1le p le infty . When p is not an even integer, such multipliers are just restrictions of contractive idempotent multipliers on L^p spaces, which in turn can be described by suitably combining results of Rudin and Andô. When p=2(n+1), with n a positive integer, contractivity depends in an interesting geometric way on n, d, and the dimension of the set of frequencies associated with the multiplier. Our results allow us to construct a linear operator that is densely defined on H^p(mathbb {T}^infty ) for every 1 le p le infty and that extends to a bounded operator if and only if p=2,4,ldots ,2(n+1).

The VC-dimension of axis-parallel boxes on the Torus

We show in this paper that the VC-dimension of the family of d-dimensional axis-parallel boxes and cubes on the d-dimensional torus are both asymptotically dlog2⁡d. This is especially surprising as in most other examples the VC-dimension usually grows linearly with d in similar settings.

Crystallographic interacting topological phases and equivariant cohomology: to assume or not to assume

For symmorphic crystalline interacting gapped systems we derive a classification under adiabatic evolution. This classification is complete for non-degenerate ground states. For the degenerate case we discuss some invariants given by equivariant characteristic classes. We do not assume an emergent relativistic field theory nor that phases form a topological spectrum. We also do not restrict to systems with short-range entanglement, stability against stacking with trivial systems nor assume the existence of quasi-particles as is done in SPT and SET classifications respectively. Using a slightly generalized Bloch decomposition and Grassmanians made out of ground state spaces, we show that the P-equivariant cohomology of a d-dimensional torus gives rise to different interacting phases, where P denotes the point group of the crystalline structure. We compare our results to bosonic symmorphic crystallographic SPT phases and to non-interacting fermionic crystallographic phases in class A. Finally we discuss the relation of our assumptions to those made for crystallographic SPT and SET phases.

Onset of the wave turbulence description of the longtime behavior of the nonlinear Schr\xf6dinger equation

Consider the cubic nonlinear Schrödinger equation set on a d-dimensional torus, with data whose Fourier coefficients have phases which are uniformly distributed and independent. We show that, on average, the evolution of the moduli of the Fourier coefficients is governed by the so-called wave kinetic equation, predicted in wave turbulence theory, on a nontrivial timescale.

The Navier\u2013Stokes Equation with Time Quasi-Periodic External Force: Existence and Stability of Quasi-Periodic Solutions

We prove the existence of small amplitude, time-quasi-periodic solutions (invariant tori) for the incompressible Navier–Stokes equation on the d-dimensional torus mathbb T^d, with a small, quasi-periodic in time external force. We also show that they are orbitally and asymptotically stable in H^s (for s large enough). More precisely, for any initial datum which is close to the invariant torus, there exists a unique global in time solution which stays close to the invariant torus for all times. Moreover, the solution converges asymptotically to the invariant torus for t rightarrow + infty , with an exponential rate of convergence O( e^{- alpha t }) for any arbitrary alpha in (0, 1).

Closed-form solutions to the dynamics of confined biased lattice random walks in arbitrary dimensions.

Biased lattice random walks (BLRW) are used to model random motion with drift in a variety of empirical situations in engineering and natural systems such as phototaxis, chemotaxis, or gravitaxis. When motion is also affected by the presence of external borders resulting from natural barriers or experimental apparatuses, modelling biased random movement in confinement becomes necessary. To study these scenarios, confined BLRW models have been employed but so far only through computational techniques due to the lack of an analytic framework. Here, we lay the groundwork for such an analytical approach by deriving the Green's functions, or propagators, for the confined BLRW in arbitrary dimensions and arbitrary boundary conditions. By using these propagators we construct explicitly the time-dependent first-passage probability in one dimension for reflecting and periodic domains, while in higher dimensions we are able to find its generating function. The latter is used to find the mean first-passage passage time for a d-dimensional box, d-dimensional torus or a combination of both. We show the appearance of surprising characteristics such as the presence of saddles in the spatiotemporal dynamics of the propagator with reflecting boundaries, bimodal features in the first-passage probability in periodic domains and the minimization of the mean first-return time for a bias of intermediate strength in rectangular domains. Furthermore, we quantify how in a multitarget environment with the presence of a bias shorter mean first-passage times can be achieved by placing fewer targets close to boundaries in contrast to many targets away from them.

Threshold behavior of bootstrap percolation

Consider a graph G and an initial random configuration, where each node is black with probability p and white otherwise, independently. In discrete-time rounds, each node becomes black if it has at least r black neighbors and white otherwise. We prove that this basic process exhibits a threshold behavior with two phase transitions when the underlying graph is a d-dimensional torus and identify the threshold values.

Unconditional local well-posedness for periodic NLS

Nonlinear Schrödinger equations with nonlinearities |u|2ku on the d-dimensional torus are considered for arbitrary positive integers k and d. The solution of the Cauchy problem is shown to be unique in the class CtHxs for a certain range of scale-subcritical regularities s, which is almost optimal in the case d≥4 or k≥2. The proof is based on various multilinear estimates and the infinite normal form reduction argument.

Uncertainty Principles for Fourier Multipliers

The admittable Sobolev regularity is quantified for a function, w, which has a zero in the d-dimensional torus and whose reciprocal $$u=1/w$$ is a (p, q)-multiplier. Several aspects of this problem are addressed, including zero-sets of positive Hausdorff dimension, matrix valued Fourier multipliers, and non-symmetric versions of Sobolev regularity. Additionally, we make a connection between Fourier multipliers and approximation properties of Gabor systems and shift-invariant systems. We exploit this connection and the results on Fourier multipliers to refine and extend versions of the Balian–Low uncertainty principle in these settings.

Approximation of differentiable and analytic functions by splines on the torus

The objective of this article is to construct optimal distribution of data points in the sense of n-widths. We present a concrete method of interpolation using sk-splines which is optimal. For a fixed continuous kernel K on the d-dimensional torus, we consider a generalization of the univariate sk-spline to the torus, associated with the kernel K. An estimate that provides the rate of convergence of a given function by its interpolating sk-splines is proved, in the norm of Lq for functions of the type f=K⁎φ, where φ∈Lp and 1≤p≤2≤q≤∞,1/p−1/q≥1/2. The rate of convergence is obtained on analogues of Sobolev classes and on classes of analytic functions. These rates have the same order as the respective trigonometric approximations, in the special case p=1, q=2.

Mutation timing in a spatial model of evolution

Motivated by models of cancer formation in which cells need to acquire k mutations to become cancerous, we consider a spatial population model in which the population is represented by the d-dimensional torus of side length L. Initially, no sites have mutations, but sites with i−1 mutations acquire an ith mutation at rate μi per unit area. Mutations spread to neighboring sites at rate α, so that t time units after a mutation, the region of individuals that have acquired the mutation will be a ball of radius αt. We calculate, for some ranges of the parameter values, the asymptotic distribution of the time required for some individual to acquire k mutations. Our results, which build on previous work of Durrett, Foo, and Leder, are essentially complete when k=2 and when μi=μ for all i.

Gravitational waves in warped compactifications

We study gravitational waves propagating on a warped Minkowski space-time with D − 4 compact extra dimensions. While Kaluza-Klein scales are typically too high for any current detection, we analyse how the warp factor changes the Kaluza-Klein spectrum of gravitational waves. To that end we provide a complete and explicit expression for the warp factor, as well as the Green’s function, on a d-dimensional torus. This expression differs from that of braneworld models and should find further uses in string compactifications. We then evaluate the Kaluza-Klein spectrum of gravitational waves. Our preliminary numerical results indicate not only a deviation from the standard toroidal spectrum, but also that the first masses get lowered due to the warp factor.

Unique Ergodicity for a Class of Stochastic Hyperbolic Equations with Additive Space-Time White Noise

In this paper, we consider a certain class of second order nonlinear PDEs with damping and space-time white noise forcing, posed on the d-dimensional torus. This class includes the wave equation for d=1 and the beam equation for dle 3. We show that the Gibbs measure is the unique invariant measure for this system. Since the flow does not satisfy the strong Feller property, we introduce a new technique for showing unique ergodicity. This approach may be also useful in situations in which finite-time blowup is possible.