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The Laguerre calculus is widely used for the inversion of differential operators on the Heisenberg group. Applying the Laguerre calculus established on nilpotent Lie groups of step two in Chang et al. [The Laguerre calculus on the nilpotent Lie group of step two. Preprint; 2019. Available from: ], we find the explicit formulas for the heat kernel of sub-Laplace operator and the fundamental solution of power of sub-Laplace operator on nilpotent Lie groups of step two. Calin, Chang and Markina [Generalized Hamilton–Jacobi equation and heat kernel on step two nilpotent Lie groups. In: Gustafsson B, Vasil'ev A, editors. Analysis and mathematical physics. Basel: Birkhüser; 2009 (Trends in mathematics)] also get the formulas for the heat kernel of sub-Laplace operator on nilpotent Lie groups of step two by using the Hamiltonian and Lagrangian formalisms that are related to geometric mechanics. In this paper, we use a totally different method to prove our main results by using the Laguerre calculus, which is more direct from the point of view of Fourier analysis. [ABSTRACT FROM AUTHOR]

We present new criteria for asymptotic stability of two classes of nonlinear neutral delay differential equations. By using two auxiliary functions on a contraction condition, we extend the results in [12]. Also we give two examples that illustrate our results. [ABSTRACT FROM AUTHOR]

We survey scaling arguments, both asymptotic (involving regular variation) and exact (involving self-similarity), in various areas of mathematical analysis and mathematical physics. [ABSTRACT FROM AUTHOR]

The spatiotemporal nonlinear Schrödinger equation with power-law nonlinearity in -symmetric potentials is investigated, and two families of analytical three-dimensional spatiotemporal structure solutions are obtained. The stability of these solutions is tested by the linear stability analysis and the direct numerical simulation. Results indicate that solutions are stable below some thresholds for the imaginary part of -symmetric potentials in the self-focusing medium, while they are always unstable for all parameters in the self-defocusing medium. Moreover, some dynamical properties of these solutions are discussed, such as the phase switch, power and transverse power-flow density. The span of phase switch gradually enlarges with the decrease of the competing parameter k in -symmetric potentials. The power and power-flow density are all positive, which implies that the power flow and exchange from the gain toward the loss domains in the cell. [ABSTRACT FROM AUTHOR]

We obtain conditions for the absolute instability of trivial solutions of the nonlinear differential equations. [ABSTRACT FROM AUTHOR]

Bilevel programs (BL) form a special class of optimization problems. They appear in many models in economics, game theory and mathematical physics. BL programs show a more complicated structure than standard finite problems. We study the so-called KKT-approach for solving bilevel problems, where the lower level minimality condition is replaced by the KKT- or the FJ-condition. This leads to a special structured mathematical program with complementarity constraints. We analyze the KKT-approach from a generic viewpoint and reveal the advantages and possible drawbacks of this approach for solving BL problems numerically. [ABSTRACT FROM AUTHOR]

Abstract: In this work, by using the fixed point index method, some existence results for positive solutions of certain three-point boundary value problems are obtained under a non-well-ordered upper and lower solution condition. [Copyright &y& Elsevier]

In this paper we introduce a new kind of Lax-Oleinik type operator with parameters associated with positive definite Lagrangian systems for both the time-periodic case and the time-independent case. On one hand, the family of new Lax-Oleinik type operators with an arbitrary $${u \in C(M, \mathbb{R}^1)}$$ as initial condition converges to a backward weak KAM solution in the time-periodic case, while it was shown by Fathi and Mather that there is no such convergence of the Lax-Oleinik semigroup. On the other hand, the family of new Lax-Oleinik type operators with an arbitrary $${u \in C(M, \mathbb{R}^1)}$$ as initial condition converges to a backward weak KAM solution faster than the Lax-Oleinik semigroup in the time-independent case. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we use the exp-function method to construct some new soliton solutions of the Benjamin–Bona–Mahony and modified Benjamin–Bona–Mahony equations. These equations have important and fundamental applications in mathematical physics and engineering sciences. The exp-function method is used to find the soliton solution of a wide class of nonlinear evolution equations with symbolic computation. This method provides the concise and straightforward solution in a very easier way. The results obtained in this paper can be viewed as a refinement and improvement of the previously known results. [Copyright &y& Elsevier]

We consider the question of uniqueness of weak solutions of the extrinsic polyharmonic map heat flows of arbitrary order in the corresponding critical dimension and show that weak solutions with non-increasing energy are unique. In addition, under a natural regularity assumption on the time-dependence of the energy functional, we prove that the only way for non-uniqueness to occur is by reverse bubbling. [ABSTRACT FROM AUTHOR]

The article discusses some analogues of the Borsuk problem. It notes the occurrence of an infinite subsequence of points of a single part because the number of parts is finite. It points out that the points' rationality does not offer an advantage in partition into smaller parts. It reveals the similarity of the proofs of the lemmas of the problem using the Dirichlet principle to obtain inequalities. It also notes the availability of extending counterexamples to high dimensions with diameters that ceased to be rational.

We consider the quenched and the averaged (or annealed) large deviation rate functions I and I for space-time and (the usual) space-only RWRE on $${\mathbb{Z}^d}$$ . By Jensen's inequality, I ≤ I. In the space-time case, when d ≥ 3 + 1, I and I are known to be equal on an open set containing the typical velocity ξ. When d = 1 + 1, we prove that I and I are equal only at ξ. Similarly, when d = 2 + 1, we show that I < I on a punctured neighborhood of ξ. In the space-only case, we provide a class of non-nestling walks on $${\mathbb{Z}^d}$$ with d = 2 or 3, and prove that I and I are not identically equal on any open set containing ξ whenever the walk is in that class. This is very different from the known results for non-nestling walks on $${\mathbb{Z}^d}$$ with d ≥ 4. [ABSTRACT FROM AUTHOR]

Linearizing nonlinear differential equations and finding exact solutions has been an interesting subject in the theory of differential equations. For many nonlinear differential equations, so far no general treatment on linearizing transformations and obtaining exact solutions has been formulated. In this article we use the generalized linearizing transformation to derive the first integral of a generalized reaction-diffusion equation with higher order nonlinear rate of growth, which enables us to linearize the associated second-order nonlinear ordinary differential equation to a linear case. Under certain parametric conditions we can also find travelling wave solutions explicitly for the generalized reaction-diffusion equation through using the derived first integral. [ABSTRACT FROM AUTHOR]

Galilean conformal algebra (GCA) is an Inönü–Wigner (IW) contraction of a conformal algebra, while Newton–Hooke string algebra is an IW contraction of an Anti-de Sitter (AdS) algebra, which is the isometry of an AdS space. It is shown that the GCA is a boundary realization of the Newton–Hooke string algebra in the bulk AdS. The string lies along the direction transverse to the boundary, and the worldsheet is AdS2. The one-dimensional conformal symmetry so(2,1) and rotational symmetry so(d) contained in the GCA are realized as the symmetry on the AdS2 string worldsheet and rotational symmetry in the space transverse to the AdS2 in AdSd+2, respectively. It follows from this correspondence that 32 supersymmetric GCAs can be derived as IW contractions of superconformal algebras, psu(2,2|4), osp(8|4), and osp(8*|4). We also derive less supersymmetric GCAs from su(2,2|2), osp(4|4), osp(2|4), and osp(8*|2). [ABSTRACT FROM AUTHOR]

We consider integrable open spin chains related to the quantum affine algebras Uq(o(3)) and Uq(A2(2)). We discuss the symmetry algebras of these chains with the local C3 space related to the Birman–Wenzl–Murakami algebra. The symmetry algebra and the Birman–Wenzl–Murakami algebra centralize each other in the representation space H=x1NC3 of the system, and this determines the structure of the spin system spectra. Consequently, the corresponding multiplet structure of the energy spectra is obtained. [ABSTRACT FROM AUTHOR]

This paper deals with the following equation mt=(1/2)(1/mk)xxx-(1/2)(1/mk)x, which is proposed by Z. J. Qiao [J. Math. Phys. 48, 082701 (2007)] and Qiao and Liu [Chaos, Solitons Fractals 41, 587 (2009)]. By adopting the phase analysis method of planar dynamical systems and the theory of the singular traveling wave systems to the traveling wave solutions of the equation, it is shown that for different k, the equation may have infinitely many solitary wave solutions, periodic wave solutions, kink/antikink wave solutions, cusped solitary wave solutions, and breaking loop solutions. We discuss in a detail the cases of k=-2,-12,12,2, and parametric representations of all possible bounded traveling wave solutions are given in the different (c,g)-parameter regions. [ABSTRACT FROM AUTHOR]

The need to consider n-ary algebraic structures, generalizing Lie and Poisson algebras, has become increasingly important in physics, and it should therefore be of interest to study the mathematical concepts related to n-ary algebras. The purpose of this paper is to investigate ternary multiplications (as deformations of n-Lie structures) constructed from the binary multiplication of a Hom–Lie algebra, a linear twisting map, and a trace function satisfying certain compatibility conditions. We show that the relation between the kernels of the twisting maps and the trace function plays an important role in this context and provide examples of Hom–Nambu–Lie algebras obtained using this construction. [ABSTRACT FROM AUTHOR]

By using of the invariant theory, we have studied the generalized time-dependent giant spin model. The dynamical and geometric phases are given, respectively. The Aharonov–Anandan phase is also obtained under the cyclical evolution. [ABSTRACT FROM AUTHOR]

We consider a superintegrable quantum potential in two-dimensional Euclidean space with a second and a third order integral of motion. The potential is written in terms of the fourth Painlevé transcendent. We construct for this system a cubic algebra of integrals of motion. The algebra is realized in terms of parafermionic operators and we present Fock-type representations which yield the corresponding energy spectra. We also discuss this potential from the point of view of higher order supersymmetric quantum mechanics and obtain ground state wave functions. [ABSTRACT FROM AUTHOR]

Abstract: The semifields of order which are two-dimensional over their left nucleus and six-dimensional over their center have been geometrically partitioned into six classes by using the associated linear sets in . One of these classes has been partitioned further into three subclasses. In this paper the generic multiplication is determined for each of these three subclasses, and several examples of new semifields are constructed that belong to these subclasses. For two of the subclasses, no examples were previously known. [Copyright &y& Elsevier]

Let (N; ℝ, θ) be a centrally ergodic W* dynamical system. When N is not a factor, we show that for each nonzero real number t, the crossed product induced by the time t automorphism θt is not a factor if and only if there exist a rational number r and an eigenvalue s of the restriction of θ to the center of N, such that rst = 2π. In the C* setting, minimality seems to be the notion corresponding to central ergodicity. We show that if (A, ℝ, α) is a minimal unital C* dynamical system and A is not simple, then, for each nonzero real number t, the crossed product induced by the time t automorphism αt is not simple if there exist a rational number r and an eigenvalue s of the restriction of to the center of A, such that rst = 2π. The converse is true if, in addition, A is commutative or prime. [ABSTRACT FROM AUTHOR]

We analyze Shannon information of scale-free networks in terms of their assortativeness, and identify classes of networks according to the dependency of the joint remaining degree distribution on the assortativeness. We conjecture that these classes comprise minimalistic and maximalistic networks in terms of Shannon information. For the studied classes, the information is shown to depend non-linearly on the absolute value of the assortativeness, with the dominant term of the relationship being a power-law. We exemplify this dependency using a range of real-world networks. Optimization of scale-free networks according to information they contain depends on the landscape of parameters’ search-space, and we identify two regions of interest: a slope region and a stability region. In the slope region, there is more freedom to generate and evaluate candidate networks since the information content can be changed easily by modifying only the assortativeness, while even a small change in the power-law’s scaling exponent brings a reward in a higher rate of information change. This feature may explain why the exponents of real-world scale-free networks are within a certain range, defined by the slope and stability regions. [ABSTRACT FROM AUTHOR]

Abstract: The Exp-function method is used to seek exact solutions of the generalized-Zakharov equations. The validity and reliability of the method is tested by its applications on a class of nonlinear evolution equations of special interest in mathematical physics. As a result, many exact traveling wave solutions are obtained which include single and combined generalized solitonary solutions. The method is straightforward and concise, and it can also be applied to other nonlinear evolution equations in mathematical physics. [Copyright &y& Elsevier]

The article presents an elementary proof of the random walk hitting time theorem. This theorem states that a left-continuous random walk on (Z) that start at a nonnegative integer (k) wherein the probability that hits the origin (n) is equal to k/n. It discusses the hitting time theorem which is equivalent to the ballot theorem that dates back to Whitford and Bertrand. It presents equations and solutions to prove these theorems. It also presents minor modification or assumptions in the hitting time theorem.

Abstract: A general model of a heterogeneous control system is introduced in the form of a first order distributed system with nonlocal dynamics and exogenous side-conditions. The heterogeneity is represented by a parameter taking values in an abstract measurable space, so that both continuous and discrete heterogeneity, as well as probabilistic heterogeneity without density, are included. A distributed and a lumped controls are involved, the latter appearing also in the side conditions. An existence theorem is proved for the uncontrolled system, and the sensitivity of the solution with respect to the control variables is estimated. The main result is an optimality condition in the form of the Pontryagin local maximum principle. A global maximum principle holds for the distributed control under an additional condition that rules out discrete measurable heterogeneity spaces. A number of possible applications are outlined: age-structured systems, size-structured systems, (nonlocal) advection–reaction equations, static parametric heterogeneity in epidemiology, and two-stage control systems with uncertain switching time. [Copyright &y& Elsevier]

We use viability techniques for solving Dirichlet problems with inequality constraints (obstacles) for a class of Hamilton-Jacobi equations. The hypograph of the "solution" is defined as the "capture basin" under an auxiliary control system of a target associated with the initial and boundary conditions, viable in an environment associated with the inequality constraint. From the tangential condition characterizing capture basins, we prove that this solution is the unique "upper semicontinuous" solution to the Hamilton-Jacobi-Bellman partial differential equation in the Barron-Jensen/Frankowska sense. We show how this framework allows us to translate properties of capture basins into corresponding properties of the solutions to this problem. For instance, this approach provides a representation formula of the solution which boils down to the Lax-Hopf formula in the absence of constraints. [ABSTRACT FROM AUTHOR]

Keisler in [7] proved that for a strong limit cardinal κ and a singular cardinal λ, the transfer relation κ → λ holds. We analyze the λ -like models produced in the proof of Keisler's transfer theorem when κ is further assumed to be regular. Our main result shows that with this extra assumption, Keisler's proof can be modified to produce a λ -like model M with built-in Skolem functions that satisfies the following two properties: (1) M is generated by a subset C of order-type λ. (2) M can be written as union of an elementary end extension chain 〈Ni: i < δ 〉 such that for each i < δ, there is an initial segment Ci of C with Ci ⊆ Ni, and Ni ∩ (C \Ci) = ∅. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [ABSTRACT FROM AUTHOR]

Abstract: Mathematical arguments are presented that give a unique answer to the question in the title. Subsequently, the mathematical analysis is extended using results of detailed model calculations that, in addition, throw further light on the consequences of the analysis. Finally, through a comparison with various recent studies, many of the latter are given a new interpretation. [Copyright &y& Elsevier]

The presented paper develops a new method for solving Einstein equations explicitly. Not only the solutions are new but also the mathematical framework of their construction which is given by a nonstandard function theory built over nonstandard algebras (in the following denoted as GAPs) and will be called here as “GAP Theory.” It is shown that the GAP theory succeeds in regions, where group theory fails: the group structure is too small to solve Einstein equations explicitly. [ABSTRACT FROM AUTHOR]

For the critical XXZ model, we consider the space $$\mathcal{W}_{[\alpha]}$$ of operators which are products of local operators with a disorder operator. We introduce two anti-commutative families of operators $${\bf {b}}(\zeta), {\bf {c}}(\zeta)$$ which act on $$\mathcal{W}_{[\alpha]}$$ . These operators are constructed as traces over representations of the q-oscillator algebra, in close analogy with Baxter’s Q-operators. We show that the vacuum expectation values of operators in $$\mathcal{W}_{[\alpha]}$$ can be expressed in terms of an exponential of a quadratic form of $${\bf {b}}(\zeta), {\bf {c}}(\zeta)$$ . [ABSTRACT FROM AUTHOR]

Answering a question left open in Métivier and Zumbrun (2005), we show for general symmetric hyperbolic boundary problems with constant coefficients, including in particular systems with characteristics of variable multiplicity, that the uniform Lopatinski condition implies strong L2 well-posedness, with no further structural assumptions. The result applies, more generally, to any system that is strongly L2 well-posed for at least one boundary condition. The proof is completely elementary, avoiding reference to Kreiss symmetrizers or other specific techniques. On the other hand, it is specific to the constant-coefficient case; at least, it does not translate in an obvious way to the variable-coefficient case. The result in the hyperbolic case is derived from a more general principle that can be applied, for example, to parabolic or partially parabolic problems like the Navier-Stokes or viscous MHD equations linearized about a constant state or even a viscous shock. [ABSTRACT FROM AUTHOR]

In this paper we investigate the critical exponents of two families of Pucci's extremal operators. The notion of critical exponent that we have chosen for these fully nonlinear operators which are not variational is that of threshold between existence and nonexistence of the solutions for semilinear equations with pure power nonlinearities. Interesting new exponents appear in this context. [ABSTRACT FROM AUTHOR]

We consider viscosity solutions of fully nonlinear elliptic equations [image omitted] for which u ≡ 0 is a solution. We assume F is smooth and uniformly elliptic only in a neighborhood of the points (0, 0, 0, x), and show that u is smooth in the interior if ‖u‖L∞ is sufficiently small. [ABSTRACT FROM AUTHOR]

The background for this article is the question of modification of the geometric configuration of an elastic structure by means of "volume"-type actuation. In this actuation mode stresses are applied to the elastic body by injection/extraction of a fluid into, or from, a large number of vacuoles in the elastic "matrix" material. Previous articles by the author, and others, have examined this process and studied its effectiveness in the context of a "naive" continuous model. The present paper develops modified models in the two-dimensional case by applying the "multiple scales" homogenization method to elastic bodies with increasingly small and numerous vacuoles arranged in a rectangularly periodic structure. The process leads to a modification of the naive model potentially useful in future elastic system control research. [ABSTRACT FROM AUTHOR]

We study the existence of nodal solutions to the boundary value problem − Δ u = up − 1u in a bounded, smooth domain Ω in ℝ2, with homogeneous Dirichlet boundary condition, when p is a large exponent. We prove that, for p large enough, there exist at least two pairs of solutions which change sign exactly once and whose nodal lines intersect the boundary of Ω. [ABSTRACT FROM AUTHOR]

Although complex scalar functions, varying over the infinite plane r= {x, y}, commonly possess vortices (points of phase singularity), it is possible to devise exceptional functions with no vortices. Similarly, although real scalar functions representing surfaces as small departures from the plane commonly possess umbilic (locally spherical) points, it is possible to devise exceptional functions that do not possess them. Examples of both are given. [ABSTRACT FROM AUTHOR]

Given a formal power series g(x) = Σ∞n=0 bn xn and a nonunit ƒ(x) = Σ∞n=1 an xn,it is well-known that the derivative of the composition of formal power series satisfies the Chain Rule, that is, (g o ƒ)′ = g′ (ƒ) · ƒ′. It is also proved that the right distributive law for formal power series exists if the composed series, such as ƒ above, is a nonunit. This paper provides a generalized Chain Rule without the requirement of nonunitness for the composed formal power series. A generalized right distributive law for formal power series is proved in this paper too. [ABSTRACT FROM AUTHOR]

We investigate the large N behavior of the time the simple random walk on the discrete cylinder [InlineMediaObject not available: see fulltext.] needs to disconnect the discrete cylinder. We show that when d≥2, this time is roughly of order N 2 d and comparable to the cover time of the slice [InlineMediaObject not available: see fulltext.], but substantially larger than the cover timer of the base by the projection of the walk. Further we show that by the time disconnection occurs, a massive ``clogging'' typically takes place in the truncated cylinders of height [InlineMediaObject not available: see fulltext.]. These mechanisms are in contrast with what happens when d=1. [ABSTRACT FROM AUTHOR]

Abstract: We examine the current status of the physical version of the Church-Turing Thesis (PhCT for short) in view of latest developments in spacetime theory. This also amounts to investigating the status of hypercomputation in view of latest results on spacetime. We agree with [D. Deutsch, A. Ekert, R. Lupacchini, Machines, logic and quantum physics, Bulletin of Symbolic Logic 6 (3) (2000) 265–283] that PhCT is not only a conjecture of mathematics but rather a conjecture of a combination of theoretical physics, mathematics and, in some sense, cosmology. Since the idea of computability is intimately connected with the nature of time, relevance of spacetime theory seems to be unquestionable. We will see that recent developments in spacetime theory show that temporal developments may exhibit features that traditionally seemed impossible or absurd. We will see that recent results point in the direction that the possibility of artificial systems computing non-Turing computable functions may be consistent with spacetime theory. All these trigger new open questions and new research directions for spacetime theory, cosmology, and computability. [Copyright &y& Elsevier]

We consider the orthofermion algebra and construct an explicit realization of orthofermion operators. This explicit realization is shown to be invariant with respect to transformations which form a quantum group. [ABSTRACT FROM AUTHOR]

Abstract: An algorithm is proposed to solve the constrained energy problem from potential theory. Numerical examples are presented, showing the accuracy of the algorithm. The algorithm is also compared with another numerical method for the same problem. [Copyright &y& Elsevier]

We consider a quantum integrable inhomogeneous model based on the Brauer algebra B(1) and discuss the properties of its ground state eigenvector. In particular we derive various sum rules, and show how some of its entries are related to multidegrees of algebraic varieties. [ABSTRACT FROM AUTHOR]

We extend the theory of the chemical potential associated to a compact separable gauge group, to the case of disordered quantum systems. This is done in the framework of operator algebras. Among the other results, we show that the chemical potential does not depend on the disorder. The situation of the n–torus is treated in some detail. Indeed, the chemical potential is intrinsically defined in terms of the direct integral decomposition of the Connes–Radon–Nikodym cocycle associated to the (normal extension of the) KMS state ω and its trasforms ω○ ρ by the (localized) automorphisms ρ of the observable algebra, carrying the Abelian charges of the model under consideration. This description parallels the analogous one relative to the usual (i.e. non disordered) quantum models. [ABSTRACT FROM AUTHOR]

A new so-called truncation error reduction method (TERM) is developed in this work. This is an iterative process which initially uses a coarse grid (2h) to estimate the truncation error and then reduces the error on the original grid (h). The purpose of this method is to use multigrid technique to achieve high-order accuracy on simple stencils. [ABSTRACT FROM AUTHOR]

In this paper we find a description, up to equivalence, of all irreducible representations for a class of algebras generated by four linearly related idempotents. [ABSTRACT FROM AUTHOR]

We give sufficient conditions for ergodicity of the Markovian semigroups associated to Dirichlet forms on standard forms of von Neumann algebras constructed by the method proposed by Park. We apply our result to show that the diffusion type Markovian semigroups for quantum spin systems are ergodic in the region of high temperatures where the uniqueness of the KMS state holds. [ABSTRACT FROM AUTHOR]

We construct a new geometric framework based on the concepts of left and right jet-bundles of a classical space-time V in order to analyze the impulsive behavior of a unilateral constraint S. The setup allows deep insights into how one can choose an ideality criterion for the constraint S when the hypothesis of conservation of kinetic energy is assumed. We show that the conservation of kinetic energy alone univocally determines the impulsive reaction when the codimension of S is 1, and that it leaves the impulsive reaction partially undetermined when the codimension of S is greater than 1. If the codimension of S is greater than 1, we prove that an additional minimality requirement determines a physically meaningful constitutive characterization of S. We show that both the Newton-like and the Poisson-like approaches to the description of the reactive impulse are equivalent, in the sense that both give the same results about the ideality criterion. Moreover, we prove that the same results hold using the classical approach based on reflection operators, possible only in case of codimension 1. We present also several physically meaningful examples. [ABSTRACT FROM AUTHOR]

We extend to the K-SAT and p-XOR-SAT optimization problems the results recently achieved, by introducing the concept of random multi-overlap structure, for the Viana-Bray model of diluted mean field spin glass. More precisely we can prove a generalized bound and an extended variational principle for the free energy per site in the thermodynamic limit. Moreover a trial function implementing ultrametric breaking of replica symmetry is exhibited. The ultrametric structure exhibits the same factorization property as the optimal structures for the Viana-Bray model and the Sherrington-Kirkpatrick nondiluted model. [ABSTRACT FROM AUTHOR]

We consider the one-dimensional Stark–Wannier type operators [\[H=-\dfrac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}-Fx-q(x)+v(x),\quad F>0,\]]] where q is a smooth function slowly growing at infinity and v is periodic, [\[$v\in L_{1}(\mathbb{T})$]], with the Fourier coefficients of the form (ln |n|)-β, 0<β<1/2, as n→∞. We show that for suitable q and F the spectrum of the corresponding operator is purely singular continuous. This proves the sharpness of the a.c. spectrum stability result obtained in Comm. Math. Phys. 234 (2003), 359–381. [ABSTRACT FROM AUTHOR]