A note on the perturbations of compact quantum metric spaces

In this short note,we consider the perturbation of compact quantum metric spaces.We first show that for two compact quantum metric spaces(A,P) and(B,Q) for which A and B are subspaces of an order-unit space C and P and Q are Lip-norms on A and B respectively,the quantum Gromov–Hausdorff distance between(A,P) and(B,Q) is small under certain conditions.Then some other perturbation results on compact quantum metric spaces derived from spectral triples are also given.     

Tangential polynomials and matrix KdV elliptic solitons

Let (X, q) be an elliptic curve marked at the origin. Starting from any cover π: Γ → X of an elliptic curve X marked at d points {π _i } of the fiber π ^?1(q) and satisfying a particular criterion, Krichever constructed a family of d × d matrix KP solitons, that is, matrix solutions, doubly periodic in x, of the KP equation. Moreover, if Γ has a meromorphic function f: Γ → P¹ with a double pole at each p _i , then these solutions are doubly periodic solutions of the matrix KdV equation U _t = 1/4(3UU _x + 3U _x U + U _xxx ). In this article, we restrict ourselves to the case in which there exists a meromorphic function with a unique double pole at each of the d points {p _i }; i.e. Γ is hyperelliptic and each pi is a Weierstrass point of Γ. More precisely, our purpose is threefold: (1) present simple polynomial equations defining spectral curves of matrix KP elliptic solitons; (2) construct the corresponding polynomials via the vector Baker–Akhiezer function of X; (3) find arbitrarily high genus spectral curves of matrix KdV elliptic solitons.     

Non-trivial m-quasi-Einstein metrics on quadratic Lie groups

We call a metric m-quasi-Einstein if \({Ric_X^m}\) (a modification of the m-Bakry–Emery Ricci tensor in terms of a suitable vector field X) is a constant multiple of the metric tensor. It is a generalization of Einstein metrics which contain Ricci solitons. In this paper, we focus on left-invariant vector fields and left-invariant Riemannian metrics on quadratic Lie groups. First we prove that any left-invariant vector field X such that the left-invariant Riemannian metric on a quadratic Lie group is m-quasi-Einstein is a Killing vector field. Then we construct infinitely many non-trivial m-quasi-Einstein metrics on solvable quadratic Lie groups G(n) for m finite.     

On the Probabilistic Nature of Quantum Mechanics and the Notion of Closed Systems

The notion of “closed systems” in Quantum Mechanics is discussed. For this purpose, we study two models of a quantum mechanical system P spatially far separated from the “rest of the universe” Q. Under reasonable assumptions on the interaction between P and Q, we show that the system P behaves as a closed system if the initial state of P ∨ Q belongs to a large class of states, including ones exhibiting entanglement between P and Q. We use our results to illustrate the non-deterministic nature of quantum mechanics. Studying a specific example, we show that assigning an initial state and a unitary time evolution to a quantum system is generally not sufficient to predict the results of a measurement with certainty.     

Combinatorial Yang–Baxter maps arising from the tetrahedron equation

We survey the matrix product solutions of the Yang–Baxter equation recently obtained from the tetrahedron equation. They form a family of quantum R-matrices of generalized quantum groups interpolating the symmetric tensor representations of U_q(A_n?1⁽¹⁾) and the antisymmetric tensor representations of \({U_{ - {q^{ - 1}}}}\left( {A_{n - 1}^{\left( 1 \right)}} \right)\). We show that at q = 0, they all reduce to the Yang–Baxter maps called combinatorial R-matrices and describe the latter by an explicit algorithm.     

Inelastic interaction of nearly equal solitons for the quartic gKdV equation

This paper describes the interaction of two solitons with nearly equal speeds for the quartic (gKdV) equation

$\partial_tu+\partial_x(\partial_x^2u+u^4)=0,\quad t,x\in \mathbb{R}.$

(0.1)

We call soliton a solution of (0.1) of the form u(t,x)=Q _c (x?ct?y ₀), where c>0, y ₀∈? and \(Q_{c}''+Q_{c}^{4}=cQ_{c}\). Since (0.1) is not an integrable model, the general question of the collision of two given solitons \(Q_{c_{1}}(x-c_{1}t)\), \(Q_{c_{2}}(x-c_{2}t)\) with c ₁≠c ₂ is an open problem.

We focus on the special case where the two solitons have nearly equal speeds: let U(t) be the solution of (0.1) satisfying

$\lim_{t\to-\infty}\|{U}(t)-Q_{c_1^-}(.-c_1^-t)-Q_{c_2^-}(.-c_2^-t)\|_{H^1}=0,$

for \(\mu_{0}=(c_{2}^{-}-c_{1}^{-})/(c_{1}^{-}+c_{2}^{-})>0\) small. By constructing an approximate solution of (0.1), we prove that, for all time t∈?,

$\begin{array}{l}\displaystyle{U}(t)={Q}_{c_1(t)}(x-y_1(t))+{Q}_{c_2(t)}(x-y_2(t))+{w}(t)\\[6pt]\displaystyle\quad\mbox{where }\|w(t)\|_{H^1}\leq|\ln\mu_0|\mu_0^2,\end{array}$

with y ₁(t)?y ₂(t)>2|ln?μ ₀|+C, for some C∈?. These estimates mean that the two solitons are preserved by the interaction and that for all time they are separated by a large distance, as in the case of the integrable KdV equation in this regime.

However, unlike in the integrable case, we prove that the collision is not perfectly elastic, in the following sense, for some C>0,

$\lim_{t\to+\infty}c_1(t)>c_2^-\biggl(1+\frac{\mu_0^5}{C}\biggr),\quad \lim_{t\to+\infty}c_2(t)

and \({w}(t)\not\to0\) in H ¹ as t→+∞.

     

Lie bialgebra structures on derivation Lie algebra over quantum tori

We investigate Lie bialgebra structures on the derivation Lie algebra over the quantum torus. It is proved that, for the derivation Lie algebra W over a rank 2 quantum torus, all Lie bialgebra structures on W are the coboundary triangular Lie bialgebras. As a by-product, it is also proved that the first cohomology group H ¹(W, W ? W) is trivial.     

Exact soliton solutions of a D-dimensional nonlinear Schrödinger equation with damping and diffusive terms

In the present study, we apply function transformation methods to the D-dimensional nonlinear Schrödinger (NLS) equation with damping and diffusive terms. As special cases, this method applies to the sine-Gordon, sinh-Gordon, and other equations. Also, the results show that these equations depend on only one function that can be obtained analytically by solving an ordinary differential equation. Furthermore, certain exact solutions of these three equations are shown to lead to the exact soliton solutions of a D-dimensional NLS equation with damping and diffusive terms. Finally, our results imply that the planar solitons, N multiple solitons, propagational breathers, and quadric solitons are solutions to the sine-Gordon, sinh-Gordon, and D-dimensional NLS equations.     

Schrödinger soliton from Lorentzian manifolds

In this paper, we introduce a new notion named as Schrödinger soliton. The so-called Schrödinger solitons are a class of solitary wave solutions to the Schrödinger flow equation from a Riemannian manifold or a Lorentzian manifold M into a Kähler manifold N. If the target manifold N admits a Killing potential, then the Schrödinger soliton reduces to a harmonic map with potential from M into N. Especially, when the domain manifold M is a Lorentzian manifold, the Schrödinger soliton is a wave map with potential into N. Then we apply the geometric energy method to this wave map system, and obtain the local well-posedness of the corresponding Cauchy problem as well as global existence in 1+1 dimension. As an application, we obtain the existence of Schrödinger soliton solution to the hyperbolic Ishimori system.     

Equiconvergence of the trigonometric Fourier series and the expansion in eigenfunctions of the Sturm-Liouville operator with a distribution potential

We consider the Sturm-Liouville operator L = ?d ²/dx ² + q(x) with the Dirichlet boundary conditions in the space L ₂[0, π] under the assumption that the potential q(x) belongs to W ₂ ^?1 [0, π]. We study the problem of uniform equiconvergence on the interval [0, π] of the expansion of a function f(x) in the system of eigenfunctions and associated functions of the operator L and its Fourier sine series expansion. We obtain sufficient conditions on the potential under which this equiconvergence holds for any function f(x) of class L ₁. We also consider the case of potentials belonging to the scale of Sobolev spaces W ₂ ^?θ [0, π] with ½ < θ ≤ 1. We show that if the antiderivative u(x) of the potential belongs to some space W ₂ ^θ [0, π] with 0 < θ < 1/2, then, for any function in the space L ₂[0, π], the rate of equiconvergence can be estimated uniformly in a ball lying in the corresponding space and containing u(x). We also give an explicit estimate for the rate of equiconvergence.     

Mixed problem for the wave equation with integrable potential in the case of two-point boundary conditions of distinct orders

We study a mixed problem for the wave equation with integrable potential and with two-point boundary conditions of distinct orders for the case in which the corresponding spectral problem may have multiple spectrum. Based on the resolvent approach in the Fourier method and the Krylov convergence acceleration trick for Fourier series, we obtain a classical solution u(x, t) of this problem under minimal constraints on the initial condition u(x, 0) = ?(x). We use the Carleson–Hunt theorem to prove the convergence almost everywhere of the formal solution series in the limit case of ?(x) ∈ L _p[0, 1], p > 1, and show that the formal solution is a generalized solution of the problem.     

The number of energy levels of a particle in an MQW structure

A method is proposed for calculating the number of energy levels of a quantum particle moving in a one-dimensional piecewise constant potential field of a certain kind, which is called a multiple quantum well (MQW) structure. It consists of several layers, namely, of potential wells with a zero potential separated by walls with a potential U > 0. The external walls have an infinite width. The method is based on a recently obtained multilayer equation that makes it possible to calculate the eigenvalues of the energy E of a quantum particle in an arbitrary MQW structure. The equation has the form F _j ^* (E) = 0, where F _j ^* (E) is a rather complex function that is constructed from the given MQW structure and depends on the index j of an arbitrarily chosen bounded layer. The key property is that the functions F _j ^* (E) corresponding to the external bounded layers are strictly monotone on the intervals of their continuity. For internal bounded layers, these functions may not be monotone. The formula for the number of energy levels holds in the case of general position. This means that it is not valid for specially chosen well and wall widths (these cases are rather rare and are called resonant). An example is presented in which the number of energy levels does not increase when the potential well is doubled.     

Isomorphisms of Nonnoetherian Down-Up Algebras

We solve the isomorphism problem for nonnoetherian down-up algebras A(α, 0, γ) by lifting isomorphisms between some of their noncommutative quotients. The quotients we consider are either quantum polynomial algebras in two variables for γ =?0 or quantum versions of the Weyl algebra A ₁ for nonzero γ. In particular we obtain that no other down-up algebra is isomorphic to the monomial algebra A(0, 0, 0). We prove in the second part of the article that this is the only monomial algebra within the family of down-up algebras. Our method uses homological invariants that determine the shape of the possible quivers and we apply the abelianization functor to complete the proof.     

The Darboux transformation for the Wadati–Konno–Ichikawa system

Based on a conservation law, we construct a hodograph transformation for the Wadati–Konno–Ichikawa (WKI) equation, which implies that the WKI equation is equivalent to a modified WKI (mWKI) equation. Applying the Darboux transformation to the mWKI equation, we show that in both the focusing and defocusing cases, the mWKI equation admits an analytic bright soliton solution from the vacuum and the collisions of n solitons are elastic based on the asymptotic analysis. In addition, we find that the mWKI equation still admits the breather and rogue wave solutions, although a modulation instability does not exist for it.     

Quantum Ergodic Restriction Theorems. I: Interior Hypersurfaces in Domains with Ergodic Billiards

Quantum ergodic restriction (QER) is the problem of finding conditions on a hypersurface H so that restrictions \({\phi_j |_H}\) to H of Δ-eigenfunctions of Riemannian manifolds (M, g) with ergodic geodesic flow are quantum ergodic on H. We prove two kinds of results: First (i) for any smooth hypersurface H in a piecewise-analytic Euclidean domain, the Cauchy data \({(\phi_j|H,\partial_{\nu}^H \phi_j|H)}\) is quantum ergodic if the Dirichlet and Neumann data are weighted appropriately. Secondly, (ii) we give conditions on H so that the Dirichlet (or Neumann) data is individually quantum ergodic. The condition involves the almost nowhere equality of left and right Poincaré maps for H. The proof involves two further novel results: (iii) a local Weyl law for boundary traces of eigenfunctions, and (iv) an ‘almost-orthogonality’ result for Fourier integral operators whose canonical relations almost nowhere commute with the geodesic flow.     

Last Multipliers for Riemannian Geometries,Dirichlet Forms and Markov Diffusion Semigroups

We start this study with last multipliers and the Liouville equation for a symmetric and non-degenerate tensor field Z of (0, 2)-type on a given Riemannian geometry (M, g) as a measure of how far away is Z from being divergence-free (and hence \(g^C\)-harmonic) with respect to g. The some topics are studied also for the Riemannian curvature tensor of (M, g) and finally for a general tensor field of (1, k)-type. Several examples are provided, some of them in relationship with Ricci solitons. Inspired by the Riemannian setting, we introduce last multipliers in the abstract framework of Dirichlet forms and symmetric Markov diffusion semigroups. For the last framework, we use the Bakry-Emery carré du champ associated to the infinitesimal generator of the semigroup.     

Markov decision processes with noise-corrupted and delayed state observations

We consider the partially observed Markov decision process with observations delayed by k time periods. We show that at stage t, a sufficient statistic is the probability distribution of the underlying system state at stage t - k and all actions taken from stage t - k through stage t - 1. We show that improved observation quality and/or reduced data delay will not decrease the optimal expected total discounted reward, and we explore the optimality conditions for three important special cases. We present a measure of the marginal value of receiving state observations delayed by (k - 1) stages rather than delayed by k stages. We show that in the limit as k →∞ the problem is equivalent to the completely unobserved case. We present numerical examples which illustrate the value of receiving state information delayed by k stages.     

Riesz transform characterization of Hardy spaces associated with Schrödinger operators with compactly supported potentials

Let L=?Δ+V be a Schrödinger operator on ? ^d , d≥3. We assume that V is a nonnegative, compactly supported potential that belongs to L ^p (? ^d ), for some p>d /2. Let K _t be the semigroup generated by ?L. We say that an L ¹(? ^d )-function f belongs to the Hardy space \(H^{1}_{L}\) associated with L if sup?_t>0|K _t f| belongs to L ¹(? ^d ). We prove that \(f\in H^{1}_{L}\) if and only if R _j f∈L ¹(? ^d ) for j=1,…,d, where R _j =(?/? x _j )L ^?1/2 are the Riesz transforms associated with L.     

Variations on average character degrees and p-nilpotence

We prove that if p is an odd prime, G is a solvable group, and the average value of the irreducible characters of G whose degrees are not divisible by p is strictly less than 2(p + 1)/(p + 3), then G is p-nilpotent. We show that there are examples that are not p-nilpotent where this bound is met for every prime p. We then prove a number of variations of this result.