Symmetric orthonormal complex wavelets with masks of arbitrarily high linear-phase moments and sum rules

In this paper, we investigate compactly supported symmetric orthonormal dyadic complex wavelets such that the symmetric orthonormal refinable functions have high linear-phase moments and the antisymmetric wavelets have high vanishing moments. Such wavelets naturally lead to real-valued symmetric tight wavelet frames with some desirable moment properties, and are related to coiflets which are real-valued and are of interest in numerical algorithms. For any positive integer m, employing only the Riesz lemma without solving any nonlinear equations, we obtain a 2π-periodic trigonometric polynomial \(\hat a\) with complex coefficients such that

1. (i)
   \(\hat a\) is an orthogonal mask: \(|\hat a(\xi)|^2+|\hat a(\xi+\pi)|^2=1\).
    
2. (ii)
   \(\hat a\) has m?+?1???odd _m sum rules: \(\hat a(\xi+\pi)=O(|\xi|^{m+1-odd_m})\) as ξ→0, where \(odd_m:=\frac{1-(-1)^m}{2}\).
    
3. (iii)
   \(\hat a\) has m?+?odd _m linear-phase moments: \(\hat a(\xi)=e^{{{\mathrm{i}}} c\xi}+O(|\xi|^{m+odd_m})\) as ξ→0 with phase c?=???1/2.
    
4. (iv)
   \(\hat a\) has symmetry and coefficient support [2???2m,2m???1]: \(\hat a(\xi)=\sum_{k=2-2m}^{2m-1} h_k e^{-{{\mathrm{i}}} k\xi}\) with h_1???k ?=?h _k .
    
5. (v)
   \(\hat a(\xi)\ne 0\) for all ξ?∈?(???π,π).
    

Define \(\hat \phi(\xi):=\prod_{j=1}^\infty \hat a(2^{-j}\xi)\) and \(\hat \psi(2\xi)=e^{-{{\mathrm{i}}} \xi} {\overline{\hat a(\xi+\pi)}}\hat \phi(\xi)\). Then ψ is a compactly supported antisymmetric orthonormal wavelet with m?+?1???odd _m vanishing moments, and ? is a compactly supported symmetric orthonormal refinable function with the special linear-phase moments: \(\int_{{{\mathbb R}}} \phi(x)dx=1\) and \(\int_{{{\mathbb R}}} (x-1/2)^j \phi(x) dx=0\) for all j?=?1,...,m?+?odd _m ???1. Both functions ? and ψ are supported on [2???2m,2m???1].The mask of a coiflet has real coefficients and satisfies (i), (ii), and (iii), often with a general phase c and the additional condition that the order of the linear-phase moments is equal (or close) to the order of the sum rules. On the one hand, as Daubechies showed in [3, 5] that except the Haar wavelet, any compactly supported dyadic orthonormal real-valued wavelets including coiflets cannot have symmetry. On the other hand, solving nonlinear equations, [4, 12] constructed many interesting real-valued dyadic coiflets without symmetry. But it remains open whether there is a family of real-valued orthonormal wavelets such as coiflets whose masks can have arbitrarily high linear-phase moments. This partially motivates this paper to study the complex wavelet case with symmetry property. Though symmetry can be achieved by considering complex wavelets, the symmetric Daubechies complex orthogonal masks in [11] generally have no more than 2 linear-phase moments. In this paper, we shall study and construct orthonormal dyadic complex wavelets and masks with symmetry, linear-phase moments, and sum rules. Examples and two general construction procedures for symmetric orthogonal masks with high linear-phase moments and sum rules are given to illustrate the results in this paper. We also answer an open question on construction of symmetric Daubechies complex orthogonal masks in the literature.     

Remarks on the nonexistence of biharmonic maps

In this short note we study a nonexistence result of biharmonic maps from a complete Riemannian manifold into a Riemannian manifold with nonpositive sectional curvature. Assume that \({\phi : (M, g) \to (N, h)}\) is a biharmonic map, where (M, g) is a complete Riemannian manifold and (N, h) a Riemannian manifold with nonpositive sectional curvature, we will prove that \({\phi}\) is a harmonic map if one of the following conditions holds: (i) \({|d\phi|}\) is bounded in L^q(M) and \({\int_{M}|\tau(\phi)|^{p}dv_{g} < \infty}\), for some \({1 \leq q \leq \infty}\), \({1 < p < \infty}\); or (ii) \({Vol(M) = \infty}\) and \({\int_{M}|\tau(\phi)|^{p}dv_{g} < \infty}\), for some \({1 < p < \infty}\). In addition, if N has strictly negative sectional curvature, we assume that \({rank\phi(q) \geq 2}\) for some \({q \in M}\) and \({\int_{M}|\tau(\phi)|^{p}dv_{g} < \infty}\), for some \({1 < p < \infty}\). These results improve the related theorems due to Baird et al. (cf. Ann Golb Anal Geom 34:403–414, 2008), Nakauchi et al. (cf. Geom. Dedicata 164:263–272, 2014), Maeta (cf. Ann Glob Anal Geom 46:75–85, 2014), and Luo (cf. J Geom Anal 25:2436–2449, 2015).     

Remarks on the Trotter–Kato Product Formula for Unitary Groups

Let A and B be non-negative self-adjoint operators in a separable Hilbert space such that their form sum C is densely defined. It is shown that the Trotter product formula holds for imaginary parameter values in the L ²-norm, that is, one has

$ \lim_{n\to+\infty} \int\limits^T_{-T} \left\|\left(e^{-itA/n}e^{-itB/n} \right)^nh - e^{-itC}h\right\|^2dt = 0 $

for each element h of the Hilbert space and any T > 0. This result is extended to the class of holomorphic Kato functions, to which the exponential function belongs. Moreover, for a class of admissible functions: \({\phi(\cdot),\psi(\cdot):{\mathbb R}_+ \longrightarrow {\mathbb C}}\), where \({{\mathbb R}_+ := [0,\infty)}\), satisfying in addition \({{\Re{\rm e}}\,(\phi(y))\ge 0, {\Im{\rm m}}\,(\phi(y) \le 0}\) and \({{\Im{\rm m}}\,(\psi(y)) \le 0}\) for \({y \in {\mathbb R}_+}\), we prove that

$ \,\mbox{\rm s-}\hspace{-2pt} \lim_{n\to\infty}(\phi(tA/n)\psi(tB/n))^n = e^{-itC} $

holds true uniformly on \({[0,T]\ni t}\) for any T > 0.

     

Properties of traveling waves for integrodifference equations with nonmonotone growth functions

In this paper, we will establish some new properties of traveling waves for integrodifference equations with the nonmonotone growth functions. More precisely, for c ≥ c _*, we show that either lim_x?+￥ f(x)=u*{\lim\limits_{\xi\rightarrow+\infty} \phi(\xi)=u*} or 0 < liminf_{x? + ￥} f(x) < u* < limsup_x?+￥f(x) ￡ b,{0 < \liminf\limits_{\xi \rightarrow + \infty} \phi(\xi) < u* < \limsup \limits_{\xi\rightarrow+\infty}\phi(\xi)\leq b,} that is, the wave converges to the positive equilibrium or oscillates about it at +∞. Sufficient conditions can assure that both results will arise. We can also obtain that any traveling wave with wave speed c > c* possesses exponential decay at −∞. These results can be well applied to three types of growth functions arising from population biology. By choosing suitable parameter numbers, we can obtain the existence of oscillating waves. Our analytic results are consistent with some numerical simulations in Kot (J Math Biol 30:413–436, 1992), Li et al. (J Math Biol 58:323–338, 2009) and complement some known ones.     

On the maximum TSP with <Emphasis Type="Italic">γ</Emphasis>-parameterized triangle inequality

The maximum TSP with γ-parameterized triangle inequality is defined as follows. Given a complete graph G = (V, E, w) in which the edge weights satisfy w(uv) ≤ γ · (w(ux) + w(xv)) for all distinct nodes \({u,x,v \in V}\), find a tour with maximum weight that visits each node exactly once. Recently, Zhang et al. (Theor Comput Sci 411(26–28):2537–2541, 2010) proposed a \({\frac{\gamma+1}{3\gamma}}\)-approximation algorithm for \({\gamma\in\left[\frac{1}{2},1\right)}\). In this paper, we show that the approximation ratio of Kostochka and Serdyukov’s algorithm (Upravlyaemye Sistemy 26:55–59, 1985) is \({\frac{4\gamma+1}{6\gamma}}\), and the expected approximation ratio of Hassin and Rubinstein’s randomized algorithm (Inf Process Lett 81(5):247–251, 2002) is \({\frac{3\gamma+\frac{1}{2}}{4\gamma}-O\left(\frac{1}{\sqrt{n}}\right)}\), for \({\gamma\in\left[\frac{1}{2},+\infty\right)}\). These improve the result in Zhang et al. (Theor Comput Sci 411(26–28):2537–2541, 2010) and generalize the results in Hassin and Rubinstein and Kostochka and Serdyukov (Inf Process Lett 81(5):247–251, 2002; Upravlyaemye Sistemy 26:55–59, 1985).     

Asymptotic behavior of the p-torsion functions as p goes to 1

Let \({\Omega}\) be a Lipschitz bounded domain of \({\mathbb{R}^N}\), \({N\geq2}\), and let \({u_p\in W_0^{1,p}(\Omega)}\) denote the p-torsion function of \({\Omega}\), p > 1. It is observed that the value 1 for the Cheeger constant \({h(\Omega)}\) is threshold with respect to the asymptotic behavior of u_p, as \({p\rightarrow 1^+}\), in the following sense: when \({h(\Omega) > 1}\), one has \({\lim_{p\rightarrow 1^+}\left\|u_{p}\right\| _{L^\infty(\Omega)}=0}\), and when \({h(\Omega) < 1}\), one has \({\lim_{p\rightarrow 1^+}\left\|u_p\right\| _{L^\infty(\Omega)}=\infty}\). In the case \({h(\Omega)=1}\), it is proved that \({\limsup_{p\rightarrow1^+}\left\|u_p\right\|_{L^\infty(\Omega)}<\infty}\). For a radial annulus \({\Omega_{a,b}}\), with inner radius a and outer radius b, it is proved that \({\lim_{p\rightarrow 1^+}\left\|u_p\right\| _{L^\infty(\Omega_{a,b})}=0}\) when \({h(\Omega_{a,b})=1}\).     

On exceptional sets in Erdős–Rényi limit theorem revisited

For \(x\in [0,1],\) the run-length function \(r_n(x)\) is defined as the length of the longest run of 1’s amongst the first n dyadic digits in the dyadic expansion of x. Let H denote the set of monotonically increasing functions \(\varphi :\mathbb {N}\rightarrow (0,+\infty )\) with \(\lim _{n\rightarrow \infty }\varphi (n)=+\infty \). For any \(\varphi \in H\), we prove that the set

$$\begin{aligned} E_{\max }^\varphi =\left\{ x\in [0,1]:\liminf \limits _{n\rightarrow \infty }\frac{r_n(x)}{\varphi (n)}=0, \limsup \limits _{n\rightarrow \infty }\frac{r_n(x)}{\varphi (n)}=+\infty \right\} \end{aligned}$$

either has Hausdorff dimension one and is residual in [0, 1] or is empty. The result solves a conjecture posed in Li and Wu (J Math Anal Appl 436:355–365, 2016) affirmatively.

     

Concentration-compactness principle for Trudinger–Moser inequalities on Heisenberg groups and existence of ground state solutions

Let \(\mathbb {H}^{n}=\mathbb {C}^{n}\times \mathbb {R}\) be the n-dimensional Heisenberg group, \(Q=2n+2\) be the homogeneous dimension of \(\mathbb {H}^{n}\). We extend the well-known concentration-compactness principle on finite domains in the Euclidean spaces of Lions (Rev Mat Iberoam 1:145–201, 1985) to the setting of the Heisenberg group \(\mathbb {H}^{n}\). Furthermore, we also obtain the corresponding concentration-compactness principle for the Sobolev space \({ HW}^{1,Q}(\mathbb {H}^{n}) \) on the entire Heisenberg group \(\mathbb {H}^{n}\). Our results improve the sharp Trudinger–Moser inequality on domains of finite measure in \(\mathbb {H}^{n}\) by Cohn and Lu (Indiana Univ Math J 50(4):1567–1591, 2001) and the corresponding one on the whole space \(\mathbb {H}^n\) by Lam and Lu (Adv Math 231:3259–3287, 2012). All the proofs of the concentration-compactness principles for the Trudinger–Moser inequalities in the literature even in the Euclidean spaces use the rearrangement argument and the Polyá–Szegö inequality. Due to the absence of the Polyá–Szegö inequality on the Heisenberg group, we will develop a different argument. Our approach is surprisingly simple and general and can be easily applied to other settings where symmetrization argument does not work. As an application of the concentration-compactness principle, we establish the existence of ground state solutions for a class of Q- Laplacian subelliptic equations on \(\mathbb {H}^{n}\):

$$\begin{aligned} -\mathrm {div}\left( \left| \nabla _{\mathbb {H}}u\right| ^{Q-2} \nabla _{\mathbb {H}}u\right) +V(\xi ) \left| u\right| ^{Q-2}u=\frac{f(u) }{\rho (\xi )^{\beta }} \end{aligned}$$

with nonlinear terms f of maximal exponential growth \(\exp (\alpha t^{\frac{Q}{Q-1}})\) as \(t\rightarrow +\infty \). All the results proved in this paper hold on stratified groups with the same proofs. Our method in this paper also provide a new proof of the classical concentration-compactness principle for Trudinger-Moser inequalities in the Euclidean spaces without using the symmetrization argument.

     

Approximation orders of the unit in the <Emphasis Type="Italic">β</Emphasis>-dynamical systems

For any real number β > 1, let S _n (β) be the partial sum of the first n items of the β-expansion of 1. It was known that the approximation order of 1 by S _n (β) is β ^?n for Lebesgue almost all β > 1. We consider the size of the set of β > 1 for which 1 can be approximated with the other orders \({\beta^{-\varphi(n)}}\) , where \({\varphi}\) is a positive function defined on \({\mathbb N}\) . More precisely, the size of the sets

$$\left\{\beta\in \mathfrak{B}:\limsup_{n\rightarrow\infty}\frac{\log_{\beta}(1-S_n(\beta))}{\varphi(n)}=-1\right\}$$

and

$$\left\{\beta\in \mathfrak{B}:\liminf_{n\rightarrow\infty}\frac{\log_{\beta}(1-S_n(\beta))}{\varphi(n)}=-1\right\}$$

are determined, where \({\mathfrak{B}=\{ \beta>1:\beta \text{ is not a simple Parry number}\}}\) .

     

A Nonconventional Local Limit Theorem

Local limit theorems have their origin in the classical De Moivre–Laplace theorem, and they study the asymptotic behavior as \(N\rightarrow \infty \) of probabilities having the form \(P\{ S_N=k\}\) where \(S_N=\sum ^N_{n=1}F(\xi _n)\) is a sum of an integer-valued function F taken on i.i.d. or Markov-dependent sequence of random variables \(\{\xi _j\}\). Corresponding results for lattice-valued and general functions F were obtained, as well. We extend here this type of results to nonconventional sums of the form \(S_N=\sum _{n=1}^NF(\xi _n,\xi _{2n}, \ldots ,\xi _{\ell n})\) which continues the recent line of research studying various limit theorems for such expressions.     

Ground State Solutions for Asymptotically Periodic Kirchhoff-Type Equations with Asymptotically Cubic or Super-cubic Nonlinearities

This paper is concerned with the following Kirchhoff-type equation

$$\begin{aligned} -\left( a+b\int _{\mathbb {R}^3}|\nabla {u}|^2\mathrm {d}x\right) \triangle u+V(x)u=f(x, u), \quad x\in \mathbb {R}^{3}, \end{aligned}$$

where \(V\in \mathcal {C}(\mathbb {R}^{3}, (0,\infty ))\), \(f\in \mathcal {C}({\mathbb {R}}^{3}\times \mathbb {R}, \mathbb {R})\), V(x) and f(x, t) are periodic or asymptotically periodic in x. Using weaker assumptions \(\lim _{|t|\rightarrow \infty }\frac{\int _0^tf(x, s)\mathrm {d}s}{|t|^3}=\infty \) uniformly in \(x\in \mathbb {R}^3\) and

$$\begin{aligned}&\left[ \frac{f(x,\tau )}{\tau ^3}-\frac{f(x,t\tau )}{(t\tau )^3} \right] \mathrm {sign}(1-t) +\theta _0V(x)\frac{|1-t^2|}{(t\tau )^2}\ge 0, \quad \\&\quad \forall x\in \mathbb {R}^3,\ t>0, \ \tau \ne 0 \end{aligned}$$

with a constant \(\theta _0\in (0,1)\), instead of the common assumption \(\lim _{|t|\rightarrow \infty }\frac{\int _0^tf(x, s)\mathrm {d}s}{|t|^4}=\infty \) uniformly in \(x\in \mathbb {R}^3\) and the usual Nehari-type monotonic condition on \(f(x,t)/|t|^3\), we establish the existence of Nehari-type ground state solutions of the above problem, which generalizes and improves the recent results of Qin et al. (Comput Math Appl 71:1524–1536, 2016) and Zhang and Zhang (J Math Anal Appl 423:1671–1692, 2015). In particular, our results unify asymptotically cubic and super-cubic nonlinearities.

     

Multilicity results for some nonlinear ellitic roblems with asymtotically $${{\varvec{}}}$$-linear terms

Taking any \(p > 1\), we consider the asymptotically p-linear problem

$$\begin{aligned} \left\{ \begin{array}{ll} - {{\mathrm{div}}}(a(x,u,\nabla u)) + A_t(x,u,\nabla u)\ = \ \lambda ^\infty |u|^{p-2}u + g^\infty (x,u) &{}\quad \hbox {in}\;\Omega ,\\ u\ = \ 0 &{}\quad \hbox {on}\;\partial \Omega , \end{array} \right. \end{aligned}$$

where \(\Omega \) is a bounded domain in \(\mathbb R^N\), \(N\ge 2\), \(A(x,t,\xi )\) is a real function on \(\Omega \times \mathbb R\times \mathbb R^N\) which grows with power p with respect to \(\xi \) and has partial derivatives \(A_t(x,t,\xi ) = \frac{\partial A}{\partial t}(x,t,\xi )\), \(a(x,t,\xi ) = \nabla _\xi A(x,t,\xi )\). If \(A(x,t,\xi ) \rightarrow A^\infty (x,t)\) and \(\frac{g^\infty (x,t)}{|t|^{p-1}} \rightarrow 0\) as \(|t| \rightarrow +\infty \), suitable assumptions, variational methods and either the cohomological index theory or its related pseudo-index one, allow us to prove the existence of multiple nontrivial bounded solutions in the non-resonant case, i.e. if \(\lambda ^\infty \) is not an eigenvalue of the operator associated to \(\nabla _\xi A^\infty (x,\xi )\). In particular, while in [14] the model problem \(A(x,t,\xi ) = \mathcal{A}(x,t) |\xi |^p\) with \(p > N\) is studied, here our goal is twofold: extending such results not only to a more general family of functions \(A(x,t,\xi )\), but also to the more difficult case \(1 < p \le N\).

     

Optimal stationary linear control of the Wiener process

In the present paper, we consider the following stochastic control problem: to minimize the average expected total cost $$J(x,u) = \mathop {\lim \inf }\limits_{T \to \infty } (1/T)E_x^u \int_0^T {\left[ {\phi (\xi _t ) + |u_t (\xi )|} \right]} dt,$$ 〈subject to $$d\xi _t = u_1 (\xi )dt + dw_t , \xi _0 = x, |u| \leqslant 1,$$ (w _t) a Wiener process, with all measurable functions on the past of the state process {ξ _s ;s≤t} and bounded by unity, admissible as controls. It is proved that, under very mild conditions on the running cost function φ(·), the optimal law is of the form $$\begin{gathered} u_t^* (\xi ) = - sign\xi _t , |\xi _t | > b, \hfill \\ u_t^* (\xi ) = 0, |\xi _t | > b. \hfill \\ \end{gathered} $$ The cutoff pointb and the performance rate of the optimal lawu* are simultaneously determined in terms of the function φ(·) through a simple system of integrotranscendental equations.     

The range of approximate unitary equivalence classes of homomorphisms from AH-algebras

Let C be a unital AH-algebra and A be a unital simple C*-algebras with tracial rank zero. It has been shown that two unital monomorphisms \({\phi, \psi: C\to A}\) are approximately unitarily equivalent if and only if

$ [\phi]=[\psi]\quad {\rm in}\quad KL(C,A)\quad {\rm and}\quad \tau\circ \phi=\tau\circ \psi \quad{\rm for\, all}\tau\in T(A),$

where T(A) is the tracial state space of A. In this paper we prove the following: Given \({\kappa\in KL(C,A)}\) with \({\kappa(K_0(C)_+\setminus\{0\})\subset K_0(A)_+\setminus\{0\}}\) and with κ([1 _C ]) = [1 _A ] and a continuous affine map \({\lambda: T(A)\to T_{\mathfrak f}(C)}\) which is compatible with κ, where \({T_{\mathfrak f}(C)}\) is the convex set of all faithful tracial states, there exists a unital monomorphism \({\phi: C\to A}\) such that

$[\phi]=\kappa\quad{\rm and}\quad \tau\circ \phi(c)=\lambda(\tau)(c)$

for all \({c\in C_{s.a.}}\) and \({\tau\in T(A).}\) Denote by \({{\rm Mon}_{au}^e(C,A)}\) the set of approximate unitary equivalence classes of unital monomorphisms. We provide a bijective map

$\Lambda: {\rm Mon}_{au}^e (C,A)\to KLT(C,A)^{++},$

where KLT(C, A)⁺⁺ is the set of compatible pairs of elements in KL(C, A)⁺⁺ and continuous affine maps from T(A) to \({T_{\mathfrak f}(C).}\) Moreover, we found that there are compact metric spaces X, unital simple AF-algebras A and \({\kappa\in KL(C(X), A)}\) with \({\kappa(K_0(C(X))_+\setminus\{0\})\subset K_0(A)_+\setminus\{0\}}\) for which there is no homomorphism h: C(X) → A so that [h] = κ.

     

Almost Everywhere Convergence of Fejér Means of Two-dimensional Triangular Walsh–Fourier Series

In 1987 Harris proved (Proc Am Math Soc 101(4):637–643, 1987)—among others—that for each \(1\le p<2\) there exists a two-dimensional function \(f\in L^p\) such that its triangular Walsh–Fourier series diverges almost everywhere. In this paper we investigate the Fejér (or (C, 1)) means of the triangle two variable Walsh–Fourier series of \(L^1\) functions. Namely, we prove the a.e. convergence \(\sigma _n^{\bigtriangleup }f = \frac{1}{n}\sum _{k=0}^{n-1}S_{k, n-k}f\rightarrow f\) (\(n\rightarrow \infty \)) for each integrable two-variable function f.     

On the approximation of dynamical indicators in systems with nonuniformly hyperbolic behavior

Let f be a \(C^{1+\alpha }\) diffeomorphism of a compact Riemannian manifold and \(\mu \) an ergodic hyperbolic measure with positive entropy. We prove that for every continuous potential \(\phi \) there exists a sequence of basic sets \(\Omega _n\) such that the topological pressure \(P(f|\Omega _n,\phi )\) converges to the free energy \(P_{\mu }(\phi ) = h(\mu ) + \int \phi {d\mu }\). We also prove that for a suitable class of potentials \(\phi \) there exists a sequence of basic sets \(\Omega _n\) such that \(P(f|\Omega _n,\phi ) \rightarrow P(\phi )\).     

Min-max and robust polynomial optimization

We consider the robust (or min-max) optimization problem

$J^*:=\max_{\mathbf{y}\in{\Omega}}\min_{\mathbf{x}}\{f(\mathbf{x},\mathbf{y}): (\mathbf{x},\mathbf{y})\in\mathbf{\Delta}\}$

where f is a polynomial and \({\mathbf{\Delta}\subset\mathbb{R}^n\times\mathbb{R}^p}\) as well as \({{\Omega}\subset\mathbb{R}^p}\) are compact basic semi-algebraic sets. We first provide a sequence of polynomial lower approximations \({(J_i)\subset\mathbb{R}[\mathbf{y}]}\) of the optimal value function \({J(\mathbf{y}):=\min_\mathbf{x}\{f(\mathbf{x},\mathbf{y}): (\mathbf{x},\mathbf{y})\in \mathbf{\Delta}\}}\). The polynomial \({J_i\in\mathbb{R}[\mathbf{y}]}\) is obtained from an optimal (or nearly optimal) solution of a semidefinite program, the ith in the “joint + marginal” hierarchy of semidefinite relaxations associated with the parametric optimization problem \({\mathbf{y}\mapsto J(\mathbf{y})}\), recently proposed in Lasserre (SIAM J Optim 20, 1995-2022, 2010). Then for fixed i, we consider the polynomial optimization problem \({J^*_i:=\max\nolimits_{\mathbf{y}}\{J_i(\mathbf{y}):\mathbf{y}\in{\Omega}\}}\) and prove that \({\hat{J}^*_i(:=\displaystyle\max\nolimits_{\ell=1,\ldots,i}J^*_\ell)}\) converges to J* as i → ∞. Finally, for fixed ? ≤ i, each \({J^*_\ell}\) (and hence \({\hat{J}^*_i}\)) can be approximated by solving a hierarchy of semidefinite relaxations as already described in Lasserre (SIAM J Optim 11, 796–817, 2001; Moments, Positive Polynomials and Their Applications. Imperial College Press, London 2009).

     

Martingale Approximation and Optimality of Some Conditions for the Central Limit Theorem

Let (X _i ) be a stationary and ergodic Markov chain with kernel Q and f an L ² function on its state space. If Q is a normal operator and f=(I?Q)^1/2 g (which is equivalent to the convergence of \(\sum_{n=1}^{\infty}\frac{\sum_{k=0}^{n-1}Q^{k}f}{n^{3/2}}\) in L ²), we have the central limit theorem [cf. (Derriennic and Lin in C.R. Acad. Sci. Paris, Sér. I 323:1053–1057, 1996; Gordin and Lif?ic in Third Vilnius conference on probability and statistics, vol. 1, pp. 147–148, 1981)]. Without assuming normality of Q, the CLT is implied by the convergence of \(\sum_{n=1}^{\infty}\frac{\|\sum_{k=0}^{n-1}Q^{k}f\|_{2}}{n^{3/2}}\), in particular by \(\|\sum_{k=0}^{n-1}Q^{k}f\|_{2}=o(\sqrt{n}/\log^{q}n)\), q>1 by Maxwell and Woodroofe (Ann. Probab. 28:713–724, 2000) and Wu and Woodroofe (Ann. Probab. 32:1674–1690, 2004), respectively. We show that if Q is not normal and f∈(I?Q)^1/2 L ², or if the conditions of Maxwell and Woodroofe or of Wu and Woodroofe are weakened to \(\sum_{n=1}^{\infty}c_{n}\frac{\|\sum_{k=0}^{n-1}Q^{k}f\|_{2}}{n^{3/2}}<\infty\) for some sequence c _n ↘0, or by \(\|\sum_{k=0}^{n-1}Q^{k}f\|_{2}=O(\sqrt{n}/\log n)\), the CLT need not hold.     

Finite time blowup of the <Emphasis Type="Italic">n</Emphasis>-harmonic flow on <Emphasis Type="Italic">n</Emphasis>-manifolds

In this paper, we generalize the no-neck result of Qing and Tian (in Commun Pure Appl Math 50:295–310, 1997) to show that there is no neck during blowing up for the n-harmonic flow as \(t\rightarrow \infty \). As an application of the no-neck result, we settle a conjecture of Hungerbühler (in Ann Scuola Norm Sup Pisa Cl Sci 4:593–631, 1997) by constructing an example to show that the n-harmonic map flow on an n-dimensional Riemannian manifold blows up in finite time for \(n\ge 3\).     

A New Result on Multiplicity of Nontrivial Solutions for the Nonhomogenous Schrödinger–Kirchhoff Type Problem in <Emphasis Type="Italic">R</Emphasis><Superscript><Emphasis Type="Italic">N</Emphasis></Superscript>

In this paper, we consider the following nonhomogenous Schrödinger–Kirchhoff type problem

$$\left\{ \begin{array}{ll} - (a+b\int_{R^{N}}|\nabla u|^{2}dx)\triangle u + V(x)u =f(x,u)+g(x), & \,\,\,{\rm for} \, x \in R^N, \\ u(x)\rightarrow0, & \,\, {\rm as}\, |x|\rightarrow\infty,\end{array}\right.$$

(0.1)

where constants a > 0, b ≥ 0, N = 1, 2 or 3, \({V\in C(R^{N},R)}\), \({f\in C(R^{N} \times R, R)}\) and \({g\in L^{2}(R^{N})}\). Under more relaxed assumptions on the nonlinear term f that are much weaker than those in Chen and Li (Nonlinear Anal RWA 14:1477–1486, 2013), using some new proof techniques especially the verification of the boundedness of Palais–Smale sequence, a new result on multiplicity of nontrivial solutions for the problem (1.1) is obtained, which sharply improves the known result of Theorem 1.1 in Chen and Li (Nonlinear Anal RWA 14:1477–1486, 2013).