Relative average widths of Sobolev spaces in L 2(ℝ d )

In this paper, in order to consider the problems of relative width on ? ^d , we proposed definitions of relative average width which combine the ideas of the relative width and the average width. We established the smallest number M which make the following equality $$ \overline K _\sigma (U(W_2^\alpha ),M(W_2^\alpha ),L_2 ({\mathbb{R}}^d )) = \overline d _\sigma (U(W_2^\alpha ),L_2 ({\mathbb{R}}^d )) $$ hold, where U(W ₂ ^α ) is the Riesz potential or Bessel potential of the unit ball in L ²(? ^k ) and the notations $ \overline K _\sigma $ (·, ·,L ₂(? ^d )) and $ \overline d _\sigma $ (·, L ₂(? ^d )) denote respectively the relative average width in the sense of Kolmogorov and the average width in the sense of Kolmogorov in their given order. In 2001, Subbotin and Telyakovskii got similar results on the relative width of Kolmogorov type. We also proved that $$ \overline K _\sigma (U(W_2^\alpha ) \cap B(L_2 (\mathbb{R}^d )),U(W_2^\beta ) \cap B(L_2 (\mathbb{R}^d ))L_2 (\mathbb{R}^d )) = \overline d _\sigma (U(W_2^\alpha ),L_2 (\mathbb{R}^d )), $$ where 0 × β × α.     

Non-Uniformity and Generalised Sacks Splitting

We show that there do not exist computable functions f ₁(e, i), f ₂(e, i), g ₁(e, i), g ₂(e, i) such that for all e, i ∈ ω, (1) $ {\left( {W_{{f_{1} {\left( {e,i} \right)}}} - W_{{f_{2} {\left( {e,i} \right)}}} } \right)} \leqslant _{{\rm T}} {\left( {W_{e} - W_{i} } \right)}; $ (2) $ {\left( {W_{{g_{1} {\left( {e,i} \right)}}} - W_{{g_{2} {\left( {e,i} \right)}}} } \right)} \leqslant _{{\rm T}} {\left( {W_{e} - W_{i} } \right)}; $ (3) $ {\left( {W_{e} - W_{i} } \right)} \not\leqslant _{{\rm T}} {\left( {W_{{f_{1} {\left( {e,i} \right)}}} - W_{{f_{2} {\left( {e,i} \right)}}} } \right)} \oplus {\left( {W_{{g_{1} {\left( {e,i} \right)}}} - W_{{g_{2} {\left( {e,i} \right)}}} } \right)}; $ (4) $ {\left( {W_{e} - W_{i} } \right)} \not\leqslant _{{\rm T}} {\left( {W_{{f_{1} {\left( {e,i} \right)}}} - W_{{f_{2} {\left( {e,i} \right)}}} } \right)}{\text{unless}}{\left( {W_{e} - W_{i} } \right)} \leqslant _{{\rm T}} {\emptyset};{\text{and}} $ (5) $ {\left( {W_{e} - W_{i} } \right)} \leqslant _{{\rm T}} {\left( {W_{{g_{1} {\left( {e,i} \right)}}} - W_{{g_{2} {\left( {e,i} \right)}}} } \right)}{\text{unless}}{\left( {W_{e} - W_{i} } \right)} \leqslant _{{\rm T}} {\emptyset}. $ It follows that the splitting theorems of Sacks and Cooper cannot be combined uniformly.     

On the Distance to the Closest Matrix with Triple Zero Eigenvalue

The 2-norm distance from a matrix A to the set ${\mathcal{M}}$ of n × n matrices with a zero eigenvalue of multiplicity ≥3 is estimated. If $$Q(\gamma _1 ,\gamma _2 ,\gamma _3 ) = \left( {\begin{array}{*{20}c} A & {\gamma _1 I_n } & {\gamma _3 I_n } \\ 0 & A & {\gamma _2 I_n } \\ 0 & 0 & A \\ \end{array} } \right), n \geqslant 3,$$ then $$\rho _2 (A,{\mathcal{M}}) \geqslant {\mathop {max}\limits_{\gamma _1 ,\gamma _2 \geqslant 0,\gamma _3 \in {\mathbb{C}}}} \sigma _{3n - 2} (Q(\gamma _1 ,\gamma _2 ,\gamma _3 )),$$ where σⁱ(·)is the ith singular value of the corresponding matrix in the decreasing order of singular values. Moreover, if the maximum on the right-hand side is attained at the point $\gamma ^ * = (\gamma _1^ * ,\gamma _2^ * ,\gamma _3^ * )$ , where $\gamma _1^ * \gamma _2^ * \ne 0$ , then, in fact, one has the exact equality $$\rho _2 (A,{\mathcal{M}}) = \sigma _{3n - 2} (Q(\gamma _1^ * ,\gamma _2^ * ,\gamma _3^ * )).$$ This result can be regarded as an extension of Malyshev's formula, which gives the 2-norm distance from A to the set of matrices with a multiple zero eigenvalue.     

On a result of Hardy and Littlewood concerning Fourier series

Let f ∈ L ¹( $ \mathbb{T} $ ) and assume that $$ f\left( t \right) \sim \frac{{a_0 }} {2} + \sum\limits_{k = 1}^\infty {\left( {a_k \cos kt + b_k \sin kt} \right)} $$ Hardy and Littlewood [1] proved that the series $ \sum\limits_{k = 1}^\infty {\frac{{a_k }} {k}} $ converges if and only if the improper Riemann integral $$ \mathop {\lim }\limits_{\delta \to 0^ + } \int_\delta ^\pi {\frac{1} {x}} \left\{ {\int_{ - x}^x {f(t)dt} } \right\}dx $$ exists. In this paper we prove a refinement of this result.     

Regularity for solutions to anisotropic obstacle problems

For Ω a bounded subset of R n,n 2,ψ any function in Ω with values in R∪{±∞}andθ∈W1,(q i)(Ω),let K(q i)ψ,θ(Ω)={v∈W1,(q i)(Ω):vψ,a.e.and v-θ∈W1,(q i)0(Ω}.This paper deals with solutions to K(q i)ψ,θ-obstacle problems for the A-harmonic equation-divA(x,u(x),u(x))=-divf(x)as well as the integral functional I(u;Ω)=Ωf(x,u(x),u(x))dx.Local regularity and local boundedness results are obtained under some coercive and controllable growth conditions on the operator A and some growth conditions on the integrand f.     

Existence of homoclinic solutions for second order Hamiltonian systems with general potentials

In this paper we are concerned with the existence of homoclinic solutions for the following second order non-autonomous Hamiltonian systems HS $$ \ddot{q}-L(t)q+W_{q}(t,q)=0, $$ where $W\in C^{1}(\mathbb{R}\times\mathbb{R}^{n},\mathbb{R})$ and $L\in C(\mathbb{R},\mathbb{R}^{n^{2}})$ is a symmetric and positive definite matrix for all $t\in\mathbb{R}$ . Assuming that the potential W satisfies some weaken global Ambrosetti-Rabinowitz conditions and L meets the coercive condition, we show that (HS) has at least one nontrivial homoclinic solution via using the Mountain Pass Theorem. Some recent results in the literature are generalized and significantly improved.     

Sharp weak-type inequalities for Hilbert transform and Riesz projection

The paper is devoted to the study of the weak norms of the classical operators in the vector-valued setting.

1. Let S, H denote the singular integral involution operator and the Hilbert transform on $L^p \left( {\mathbb{T}, \ell _\mathbb{C}^2 } \right)$ , respectively. Then for 1 ≤ p ≤ 2 and any f, $$\left\| {\mathcal{S}f} \right\|_{p,\infty } \leqslant \left( {\frac{1} {\pi }\int_{ - \infty }^\infty {\frac{{\left| {\tfrac{2} {\pi }\log \left| t \right|} \right|^p }} {{t^2 + 1}}dt} } \right)^{ - 1/p} \left\| f \right\|p,$$ $$\left\| {\mathcal{H}f} \right\|_{p,\infty } \leqslant \left( {\frac{1} {\pi }\int_{ - \infty }^\infty {\frac{{\left| {\tfrac{2} {\pi }\log \left| t \right|} \right|^p }} {{t^2 + 1}}dt} } \right)^{ - 1/p} \left\| f \right\|p.$$ Both inequalities are sharp.
2. Let P ₊ and P _? stand for the Riesz projection and the co-analytic projection on $L^p \left( {\mathbb{T}, \ell _\mathbb{C}^2 } \right)$ , respectively. Then for 1 ≤ p ≤ 2 and any f, $$\left\| {P + f} \right\|_{p,\infty } \leqslant \left\| f \right\|_p ,$$ $$\left\| {P - f} \right\|_{p,\infty } \leqslant \left\| f \right\|_p .$$ Both inequalities are sharp.
3. We establish the sharp versions of the estimates above in the nonperiodic case.

The results are new even if the operators act on complex-valued functions. The proof rests on the construction of an appropriate plurisubharmonic function and probabilistic techniques.     

Some new estimates of the Fourier transform in \mathbb{L}_2 (ℝ)

Given a function $\mathbb{L}_2 $ (?), its Fourier transform $g(x) = \hat f(x) = F[f](x) = \frac{1} {{\sqrt {2\pi } }}\int\limits_{ - \infty }^{ + \infty } {f(x)e^{ - ixt} dt} ,f(t) = F^{ - 1} [g](t) = \frac{1} {{\sqrt {2\pi } }}\int\limits_{ - \infty }^{ + \infty } {g(x)e^{ - ixt} dx} $ and the inverse Fourier transform are considered in the space f ε $\mathbb{L}_2 $ (?). New estimates are presented for the integral $\int\limits_{|t| \geqslant N} {|g(t)|^2 dt} = \int\limits_{|t| \geqslant N} {|\hat f(t)|^2 dt} ,N \geqslant 1,$ in the vase of f ε $\mathbb{L}_2 $ (?) characterized by the generalized modulus of continuity of the kth order constructed with the help of the Steklov function. Some other estimates associated with this integral are proved.     

On a certain sum-product estimate in fields of prime order

Let Fp be the finite field of p elements with p prime.If A is a subset of Fp and g is an element of F*p with order ν,then max{|A + g·A|,|A·A|} (ν/(ν + |A|2) )1/12|A|13/12.     

On the perturbation of weighted group inverse of rectangular matrices

Cen (Math. Numer. Sin. 29:39–48, 2007) defined a weighted group inverse of rectangular matrices. For given matrices A∈C ^m×n and W∈C ^n×m , if X∈C ^m×n satisfies $$( W_{1} )\ AWXWA=A, \qquad ( W_{2} ) \ XWAWX=X,\qquad ( W_{3} )\ AWX=XWA $$ then X is called the W-weighted group inverse, which is denoted by $A_{W}^{\#}$ . In this paper, for given rectangular matrices A and E and B=A+E, we investigate the perturbation of the weighted group inverse $A_{W}^{\#}$ and present the upper bounds for $\|B_{W}^{\#} \|$ .     

On the Dimension of Bivariate Weak Spline Space Over Regular Rectilinear Partition

In this paper, we mainly study the dimensions of bivariate weak spline spaces ${W_k^\mu(I_{1}\Delta)}$ (k ≥ 2μ+1) and ${W_{2}^{1} (I_{1}^{*}\Delta)}$ by using the smoothing cofactor-conformality method, where I ₁Δ and ${I_{1}^{*} \Delta}$ are regular rectilinear partitions with appointed point sets. Some future works relative to bivariate weak splines are also listed at the end of this paper.     

A decomposition of the universal embedding space for the near polygon {{\mathbb H}_n}

Let ${{\mathbb H}_n, n \geq 1}$ , be the near 2n-gon defined on the 1-factors of the complete graph on 2n?+?2 vertices, and let e denote the absolutely universal embedding of ${{\mathbb H}_n}$ into PG(W), where W is a ${\frac{1}{n+2} \left(\begin{array}{c}2n+2 \\ n+1\end{array}\right)}$ -dimensional vector space over the field ${{\mathbb F}_2}$ with two elements. For every point z of ${{\mathbb H}_n}$ and every ${i \in {\mathbb N}}$ , let Δ _i (z) denote the set of points of ${{\mathbb H}_n}$ at distance i from z. We show that for every pair {x, y} of mutually opposite points of ${{\mathbb H}_n, W}$ can be written as a direct sum ${W_0 \oplus W_1 \oplus \cdots \oplus W_n}$ such that the following four properties hold for every ${i \in \{0,\ldots,n \}}$ : (1) ${\langle e(\Delta_i(x) \cap \Delta_{n-i}(y)) \rangle = {\rm PG}(W_i)}$ ; (2) ${\left\langle e \left( \bigcup_{j \leq i} \Delta_j(x) \right) \right\rangle = {\rm PG}(W_0 \oplus W_1 \oplus \cdots \oplus W_i)}$ ; (3) ${\left\langle e \left( \bigcup_{j \leq i} \Delta_j(y) \right) \right\rangle = {\rm PG}(W_{n-i}\oplus W_{n-i+1} \oplus \cdots \oplus W_n)}$ ; (4) ${\dim(W_i) = |\Delta_i(x) \cap \Delta_{n-i}(y)| = \left(\begin{array}{c}n \\ i\end{array}\right)^2 - \left(\begin{array}{c}n \\ i-1\end{array}\right) \cdot \left(\begin{array}{c}n \\ i+1\end{array}\right)}$ .     

Stability of the Derivative of a Canonical Product

With each sequence \(\alpha =(\alpha _n)_{n\in \mathbb{N }}\) of pairwise distinct and non-zero points which are such that the canonical product $$\begin{aligned} P_\alpha (z) := \lim _{r\rightarrow \infty }\prod _{|\alpha _n|\le r}\left( 1-\frac{z}{\alpha _n}\right) \end{aligned}$$ converges, the sequence $$\begin{aligned} \alpha ^{\prime } := \bigl (P_\alpha ^{\prime }(\alpha _n)\bigr )_{n\in \mathbb{N }} \end{aligned}$$ is associated. We give conditions on the difference \(\beta -\alpha \) of two sequences which ensure that \(\beta ^{\prime }\) and \(\alpha ^{\prime }\) are comparable in the sense that $$\begin{aligned} \exists \,c,C>0:\quad c|\alpha ^{\prime }_n| \le |\beta ^{\prime }_n| \le C|\alpha ^{\prime }_n|, \quad n\in \mathbb{N }. \end{aligned}$$ The values \(\alpha ^{\prime }_n\) play an important role in various contexts. As a selection of applications we present: an inverse spectral problem, a class of entire functions and a continuation problem.     

Necessary and sufficient conditions for the asymptotic distribution of the largest entry of a sample correlation matrix

Let {X _k,i ; i ≥ 1, k ≥ 1} be a double array of nondegenerate i.i.d. random variables and let {p _n ; n ≥ 1} be a sequence of positive integers such that n/p _n is bounded away from 0 and ∞. In this paper we give the necessary and sufficient conditions for the asymptotic distribution of the largest entry ${L_{n}={\rm max}_{1\leq i < j\leq p_{n}}|\hat{\rho}^{(n)}_{i,j}|}$ of the sample correlation matrix ${{\bf {\Gamma}}_{n}=(\hat{\rho}_{i,j}^{(n)})_{1\leq i,j\leq p_{n}}}$ where ${\hat{\rho}^{(n)}_{i,j}}$ denotes the Pearson correlation coefficient between (X _1,i , ..., X _n,i )′ and (X _1,j ,...,X _n,j )′. Write ${F(x)= \mathbb{P}(|X_{1,1}|\leq x), x\geq0}$ , ${W_{c,n}={\rm max}_{1\leq i < j\leq p_{n}}|\sum_{k=1}^{n}(X_{k,i}-c)(X_{k,j}-c)|}$ , and ${W_{n}=W_{0,n},n\geq1,c\in(-\infty,\infty)}$ . Under the assumption that ${\mathbb{E}|X_{1,1}|^{2+\delta} < \infty}$ for some δ > 0, we show that the following six statements are equivalent: $$ {\bf (i)} \quad \lim_{n \to \infty} n^{2}\int\limits_{(n \log n)^{1/4}}^{\infty}\left( F^{n-1}(x) - F^{n-1}\left(\frac{\sqrt{n \log n}}{x}\right) \right) dF(x) = 0,$$ $$ {\bf (ii)}\quad n \mathbb{P}\left ( \max_{1 \leq i < j \leq n}|X_{1,i}X_{1,j} | \geq \sqrt{n \log n}\right ) \to 0 \quad{\rm as}\,n \to \infty,$$ $$ {\bf (iii)}\quad \frac{W_{\mu, n}}{\sqrt {n \log n}}\stackrel{\mathbb{P}}{\rightarrow} 2\sigma^{2},$$ $$ {\bf (iv)}\quad \left ( \frac{n}{\log n}\right )^{1/2} L_{n} \stackrel{\mathbb{P}}{\rightarrow} 2,$$ $$ {\bf (v)}\quad \lim_{n \rightarrow \infty}\mathbb{P}\left (\frac{W_{\mu, n}^{2}}{n \sigma^{4}} - a_{n}\leq t \right ) = \exp \left \{ - \frac{1}{\sqrt{8\pi}} e^{-t/2}\right \}, - \infty < t < \infty,$$ $$ {\bf (vi)}\quad \lim_{n \rightarrow \infty}\mathbb{P}\left (n L_{n}^{2} - a_{n}\leq t \right ) = \exp \left \{ - \frac{1}{\sqrt{8 \pi}} e^{-t/2}\right \}, - \infty < t < \infty$$ where ${\mu=\mathbb{E}X_{1,1}, \sigma^{2}=\mathbb{E}(X_{1,1} - \mu)^{2}}$ , and a _n  = 4 log p _n ? log log p _n . The equivalences between (i), (ii), (iii), and (v) assume that only ${\mathbb{E}X_{1,1}^{2} < \infty}$ . Weak laws of large numbers for W _n and L _n , n ≥  1, are also established and these are of the form ${W_{n}/n^{\alpha}\stackrel{\mathbb{P}}{\rightarrow} 0}\,(\alpha > 1/2)$ and ${n^{1-\alpha}L_{n}\stackrel{\mathbb{P}}{\rightarrow} 0}\,(1/2 < \alpha \leq 1)$ , respectively. The current work thus provides weak limit analogues of the strong limit theorems of Li and Rosalsky as well as a necessary and sufficient condition for the asymptotic distribution of L _n obtained by Jiang. Some open problems are also posed.     

Extinction profile of the logarithmic diffusion equation

Let ${N \geq 3}$ and u be the solution of u _t = Δ log u in ${\mathbb{R}^N \times (0, T)}$ with initial value u ₀ satisfying ${B_{k_1}(x, 0) \leq u_{0} \leq B_{k_2}(x, 0)}$ for some constants k ₁ > k ₂ > 0 where ${B_k(x, t) = 2(N - 2)(T - t)_{+}^{N/(N - 2)}/(k + (T - t)_{+}^{2/(N - 2)}|x|^{2})}$ is the Barenblatt solution for the equation and ${u_0 - B_{k_0} \in L^{1}(\mathbb{R}^{N})}$ for some constant k ₀ > 0 if ${N \geq 4}$ . We give a new different proof on the uniform convergence and ${L^1(\mathbb{R}^N)}$ convergence of the rescaled function ${\tilde{u}(x, s) = (T - t)^{-N/(N - 2)}u(x/(T - t)^{-1/(N - 2)}, t), s = -{\rm log}(T - t)}$ , on ${\mathbb{R}^N}$ to the rescaled Barenblatt solution ${\tilde{B}_{k_0}(x) = 2(N - 2)/(k_0 + |x|^{2})}$ for some k ₀ > 0 as ${s \rightarrow \infty}$ . When ${N \geq 4, 0 \leq u_0(x) \leq B_{k_0}(x, 0)}$ in ${\mathbb{R}^N}$ , and ${|u_0(x) - B_{k_0}(x, 0)| \leq f \in L^{1}(\mathbb{R}^{N})}$ for some constant k ₀ > 0 and some radially symmetric function f, we also prove uniform convergence and convergence in some weighted L ¹ space in ${\mathbb{R}^N}$ of the rescaled solution ${\tilde{u}(x, s)}$ to ${\tilde{B}_{k_0}(x)}$ as ${s \rightarrow \infty}$ .     

Differentiation and the Balian-Low Theorem

The Balian-Low theorem (BLT) is a key result in time-frequency analysis, originally stated by Balian and, independently, by Low, as: If a Gabor system $\{e^{2\pi imbt} \, g(t-na)\}_{m,n \in \mbox{\bf Z}}$ with $ab=1$ forms an orthonormal basis for $L^2({\bf R}),$ then $\left(\int_{-\infty}^\infty |t \, g(t)|^2 \, dt\right) \, \left(\int_{-\infty}^\infty |\gamma \, \hat g(\gamma)|^2 \, d\gamma\right) = +\infty.$ The BLT was later extended from orthonormal bases to exact frames. This paper presents a tutorial on Gabor systems, the BLT, and related topics, such as the Zak transform and Wilson bases. Because of the fact that $(g')^{\wedge}(\gamma) = 2 \pi i \gamma \, \hat g(\gamma)$ , the role of differentiation in the proof of the BLT is examined carefully. The major new contributions of this paper are the construction of a complete Gabor system of the form $\{e^{2\pi ib_mt\} \, g(t-a_n)}$ such that $\{(a_n,b_m)\}$ has density strictly less than 1, an Amalgam BLT that provides distinct restrictions on Gabor systems $\{e^{2\pi imbt} \, g(t-na)\}$ that form exact frames, and a new proof of the BLT for exact frames that does not require differentiation and relies only on classical real variable methods from harmonic analysis.     

On oscillatory and asymptotic behavior of fourth order nonlinear neutral delay dynamic equations with positive and negative coefficients

In this paper, oscillatory and asymptotic properties of solutions of nonlinear fourth order neutral dynamic equations of the form $(r(t)(y(t) + p(t)y(\alpha _1 (t)))^{\Delta ^2 } )^{\Delta ^2 } + q(t)G(y(\alpha _2 (t))) - h(t)H(y(\alpha _3 (t))) = 0(H)$ and $(r(t)(y(t) + p(t)y(\alpha _1 (t)))^{\Delta ^2 } )^{\Delta ^2 } + q(t)G(y(\alpha _2 (t))) - h(t)H(y(\alpha _3 (t))) = f(t),(NH)$ are studied on a time scale $\mathbb{T}$ under the assumption that $\int\limits_{t_0 }^\infty {\tfrac{t} {{r(t)}}\Delta t = \infty } $ and for various ranges of p(t). In addition, sufficient conditions are obtained for the existence of bounded positive solutions of the equation (NH) by using Krasnosel’skii’s fixed point theorem.     

One-radius results for supermedian functions on {\mathbb R^d} , d ≤ 2

A classical result states that every lower bounded superharmonic function on ${\mathbb{R}^{2}}$ is constant. In this paper the following (stronger) one-circle version is proven. If ${f : \mathbb{R}^{2} \to (-\infty,\infty]}$ is lower semicontinuous, lim inf_|x|→∞ f (x)/ ln |x| ≥ 0, and, for every ${x \in \mathbb{R}^{2}}$ , ${1/(2\pi) \int_0^{2\pi} f(x + r(x)e^{it}) \, dt \le f(x)}$ , where ${r : \mathbb{R}^{2} \to (0,\infty)}$ is continuous, ${{\rm sup}_{x \in \mathbb{R}^{2}} (r(x) - |x|) < \infty},$ , and ${{\rm inf}_{x \in \mathbb{R}^{2}} (r(x)-|x|)=-\infty}$ , then f is constant. Moreover, it is shown that, assuming r ≤ c| · | + M on ${\mathbb{R}^d}$ , d ≤ 2, and taking averages on ${\{y \in \mathbb{R}^{d} : |y-x| \le r(x)\}}$ , such a result of Liouville type holds for supermedian functions if and only if c ≤ c ₀, where c ₀ = 1, if d = 2, whereas 2.50 < c ₀ < 2.51, if d = 1.     

Properties of finite unrefinable chains of ring topologies

Let R(+, ·) be a nilpotent ring and $ \left( {\mathfrak{M}, < } \right) $ be the lattice of all ring topologies on R(+, ·) or the lattice of all such ring topologies on R(+, ·) in each of which the ring R possesses a basis of neighborhoods of zero consisting of subgroups. Let ?? and ??? be ring topologies from $ \mathfrak{M} $ such that $ \tau = {\tau_0}{ \prec_\mathfrak{M}}{\tau_1}{ \prec_\mathfrak{M}} \cdots { \prec_\mathfrak{M}}{\tau_n} = \tau ^{\prime} $ . Then k????n for every chain $ \tau = {\tau ^{\prime}_0} < {\tau ^{\prime}_1} < \cdots < {\tau ^{\prime}_k} = \tau ^{\prime} $ of topologies from $ \mathfrak{M} $ , and also n?=?k if and only if $ {\tau ^{\prime}_i}{ \prec_\mathfrak{M}}{\tau ^{\prime}_{i + 1}} $ for all 0????i?<?k.