Optimal L^2, H^1 and L^\infty analysis of finite volume methods for the stationary Navier–Stokes equations with large data

Previous work on the stability and convergence analysis of numerical methods for the stationary Navier–Stokes equations was carried out under the uniqueness condition of the solution, which required that the data be small enough in certain norms. In this paper an optimal analysis for the finite volume methods is performed for the stationary Navier–Stokes equations, which relaxes the solution uniqueness condition and thus the data requirement. In particular, optimal order error estimates in the $H^1$ -norm for velocity and the $L^2$ -norm for pressure are obtained with large data, and a new residual technique for the stationary Navier–Stokes equations is introduced for the first time to obtain a convergence rate of optimal order in the $L^2$ -norm for the velocity. In addition, after proving a number of additional technical lemmas including weighted $L^2$ -norm estimates for regularized Green’s functions associated with the Stokes problem, optimal error estimates in the $L^\infty $ -norm are derived for the first time for the velocity gradient and pressure without a logarithmic factor $O(|\log h|)$ for the stationary Naiver–Stokes equations.     

Hausdorff measures of subgroups of \mathbb{R }/\mathbb{Z } and \mathbb{R }

For a sequence $\underline{u}=(u_n)_{n\in \mathbb{N }}$ of integers, let $t_{\underline{u}}(\mathbb{T })$ be the group of all topologically $\underline{u}$ -torsion elements of the circle group $\mathbb{T }:=\mathbb{R }/\mathbb{Z }$ . We show that for any $s\in ]0,1[$ and $m\in \{0,+\infty \}$ there exists $\underline{u}$ such that $t_{\underline{u}}(\mathbb{T })$ has Hausdorff dimension $s$ and $s$ -dimensional Hausdorff measure equal to $m$ (no other values for $m$ are possible). More generally, for dimension functions $f,g$ with $f(t)\prec g(t), f(t)\prec \!\!\!\prec t$ and $g(t)\prec \!\!\!\prec t$ we find $\underline{u}$ such that $t_{\underline{u}}(\mathbb{T })$ has at the same time infinite $f$ -measure and null $g$ -measure.     

Complexity of unconstrained L_2-L_p minimization

We consider the unconstrained $L_q$ - $L_p$ minimization: find a minimizer of $\Vert Ax-b\Vert ^q_q+\lambda \Vert x\Vert ^p_p$ for given $A \in R^{m\times n}$ , $b\in R^m$ and parameters $\lambda >0$ , $p\in [0, 1)$ and $q\ge 1$ . This problem has been studied extensively in many areas. Especially, for the case when $q=2$ , this problem is known as the $L_2-L_p$ minimization problem and has found its applications in variable selection problems and sparse least squares fitting for high dimensional data. Theoretical results show that the minimizers of the $L_q$ - $L_p$ problem have various attractive features due to the concavity and non-Lipschitzian property of the regularization function $\Vert \cdot \Vert ^p_p$ . In this paper, we show that the $L_q$ - $L_p$ minimization problem is strongly NP-hard for any $p\in [0,1)$ and $q\ge 1$ , including its smoothed version. On the other hand, we show that, by choosing parameters $(p,\lambda )$ carefully, a minimizer, global or local, will have certain desired sparsity. We believe that these results provide new theoretical insights to the studies and applications of the concave regularized optimization problems.     

A weak second term identity of the regularized Siegel-Weil formula for unitary groups

Following W. T. Gan and S. Takeda, we obtain a weak second term identity of the regularized Siegel-Weil formula for the unitary dual pair $(U(n,n),U(V))$ , where $V$ is a split hermitian space of dimension $2r$ with $r+1\le n \le 2r-1$ . As an application, we obtain a Rallis inner product formula for theta lifts from $U(W)$ to $U(V)$ for a skew-hermitian space $W$ of dimension $n$ .     

Convergence of the eigenvalues of the p-Laplace operator as p goes to 1

Let $(\lambda ^k_p)_k$ be the usual sequence of min-max eigenvalues for the $p$ -Laplace operator with $p\in (1,\infty )$ and let $(\lambda ^k_1)_k$ be the corresponding sequence of eigenvalues of the 1-Laplace operator. For bounded $\Omega \subseteq \mathbb{R }^n$ with Lipschitz boundary the convergence $\lambda ^k_p\rightarrow \lambda ^k_1$ as $p\rightarrow 1$ is shown for all $k\in \mathbb{N }$ . The proof uses an approximation of $BV(\Omega )$ -functions by $C_0^\infty (\Omega )$ -functions in the sense of strict convergence on $\mathbb{R }^n$ .     

Compression-based distance between string data and its application to literary work classification based on authorship

There are many well-known document classification/clustering algorithms. In this paper, compression-based distances between documents are focused on, in particular, the normalized compression distance (NCD). The NCD is a popular and powerful metric between strings. A new distance $D_\alpha $ with one parameter $\alpha $ between strings is designed on the basis of the NCD, and several properties of $D_\alpha $ are studied. It is also proved that every pair of strings $(x,y)$ can be plotted on the contour graphs of NCD and $D_\alpha $ (and some other compression-based distances) in a 2-dimensional plane. The distance $D_\alpha (x,y)$ is defined to take a relatively small value if a string $x$ is a portion of a string $y.$ Literary works $x$ and $y$ are usually assumed to be written by the same author(s) if $x$ is a portion of $y.$ Therefore, it may be appropriate to consider the performance of $D_\alpha $ for literary work classification based on authorship, as a benchmark. An algorithm to determine an appropriate value of $\alpha $ is presented using the contour graphs, and this algorithm does not require the knowledge of the names of the authors of each work. According to experimental results of the area under receiver operating characteristics curves and clustering, $D_\alpha $ with such an appropriate value of $\alpha $ performs somewhat better in literary work classification based on authorship.     

Donsker-type theorems for nonparametric maximum likelihood estimators

Let ${\mathcal{P}}$ be a nonparametric probability model consisting of smooth probability densities and let ${\hat{p}_{n}}$ be the corresponding maximum likelihood estimator based on n independent observations each distributed according to the law ${\mathbb{P}}$ . With $\hat{\mathbb{P}}_{n}$ denoting the measure induced by the density ${\hat{p}_{n}}$ , define the stochastic process ${\hat{\nu}}_{n}: f\longmapsto \sqrt{n} \int fd({\hat{\mathbb{P}}}_{n} -\mathbb{P})$ where f ranges over some function class ${\mathcal{F}}$ . We give a general condition for Donsker classes ${\mathcal{F}}$ implying that the stochastic process $\hat{\nu}_{n}$ is asymptotically equivalent to the empirical process in the space ${\ell ^{\infty }(\mathcal{F})}$ of bounded functions on ${ \mathcal{F}}$ . This implies in particular that $\hat{\nu}_{n}$ converges in law in ${\ell ^{\infty }(\mathcal{F})}$ to a mean zero Gaussian process. We verify the general condition for a large family of Donsker classes ${\mathcal{ F}}$ . We give a number of applications: convergence of the probability measure ${\hat{\mathbb{P}}_{n}}$ to ${\mathbb{P}}$ at rate ${\sqrt{n}}$ in certain metrics metrizing the topology of weak(-star) convergence; a unified treatment of convergence rates of the MLE in a continuous scale of Sobolev-norms; ${\sqrt{n}}$ -efficient estimation of nonlinear functionals defined on ${\mathcal{P}}$ ; limit theorems at rate ${\sqrt{n}}$ for the maximum likelihood estimator of the convolution product ${\mathbb{P\ast P}}$ .     

C^*-algebras of Toeplitz type associated with algebraic number fields

We associate with the ring $R$ of algebraic integers in a number field a C*-algebra ${\mathfrak T }[R]$ . It is an extension of the ring C*-algebra ${\mathfrak A }[R]$ studied previously by the first named author in collaboration with X. Li. In contrast to ${\mathfrak A }[R]$ , it is functorial under homomorphisms of rings. It can also be defined using the left regular representation of the $ax+b$ -semigroup $R\rtimes R^\times $ on $\ell ^2 (R\rtimes R^\times )$ . The algebra ${\mathfrak T }[R]$ carries a natural one-parameter automorphism group $(\sigma _t)_{t\in {\mathbb R }}$ . We determine its KMS-structure. The technical difficulties that we encounter are due to the presence of the class group in the case where $R$ is not a principal ideal domain. In that case, for a fixed large inverse temperature, the simplex of KMS-states splits over the class group. The “partition functions” are partial Dedekind $\zeta $ -functions. We prove a result characterizing the asymptotic behavior of quotients of such partial $\zeta $ -functions, which we then use to show uniqueness of the $\beta $ -KMS state for each inverse temperature $\beta \in (1,2]$ .     

Stability results for the N-dimensional Schiffer conjecture via a perturbation method

Given a eigenvalue $\mu _{0m}^2$ of $-\Delta $ in the unit ball $B_1$ , with Neumann boundary conditions, we prove that there exists a class $\mathcal{D}$ of $C^{0,1}$ -domains, depending on $\mu _{0m} $ , such that if $u$ is a no trivial solution to the following problem $ \Delta u+\mu u=0$ in $\Omega , u=0$ on $\partial \Omega $ , and $ \int \nolimits _{\partial \Omega }\partial _{\mathbf{n}}u=0$ , with $\Omega \in \mathcal{D}$ , and $\mu =\mu _{0m}^2+o(1)$ , then $\Omega $ is a ball. Here $\mu $ is a eigenvalue of $-\Delta $ in $\Omega $ , with Neumann boundary conditions.     

Convergence of inexact Newton methods for generalized equations

For solving the generalized equation $f(x)+F(x) \ni 0$ , where $f$ is a smooth function and $F$ is a set-valued mapping acting between Banach spaces, we study the inexact Newton method described by $$\begin{aligned} \left( f(x_k)+ D f(x_k)(x_{k+1}-x_k) + F(x_{k+1})\right) \cap R_k(x_k, x_{k+1}) \ne \emptyset , \end{aligned}$$ where $Df$ is the derivative of $f$ and the sequence of mappings $R_k$ represents the inexactness. We show how regularity properties of the mappings $f+F$ and $R_k$ are able to guarantee that every sequence generated by the method is convergent either q-linearly, q-superlinearly, or q-quadratically, according to the particular assumptions. We also show there are circumstances in which at least one convergence sequence is sure to be generated. As a byproduct, we obtain convergence results about inexact Newton methods for solving equations, variational inequalities and nonlinear programming problems.     

A regularized Newton method without line search for unconstrained optimization

In this paper, we propose a regularized Newton method without line search. The proposed method controls a regularization parameter instead of a step size in order to guarantee the global convergence. We show that the proposed algorithm has the following convergence properties. (a) The proposed algorithm has global convergence under appropriate conditions. (b) It has superlinear rate of convergence under the local error bound condition. (c) An upper bound of the number of iterations required to obtain an approximate solution \(x\) satisfying \(\Vert \nabla f(x) \Vert \le \varepsilon \) is \(O(\varepsilon ^{-2})\) , where \(f\) is the objective function and \(\varepsilon \) is a given positive constant.     

How large dimension guarantees a given angle?

We study the following two problems: (1) Given $n\ge 2$ and $0\le \alpha \le 180^\circ $ , how large Hausdorff dimension can a compact set $A\subset \mathbb{R }^n$ have if $A$ does not contain three points that form an angle $\alpha $ ? (2) Given $\alpha $ and $\delta $ , how large Hausdorff dimension can a subset $A$ of a Euclidean space have if $A$ does not contain three points that form an angle in the $\delta $ -neighborhood of $\alpha $ ? An interesting phenomenon is that different angles show different behaviour in the above problems. Apart from the clearly special extreme angles $0$ and $180^\circ $ , the angles $60^\circ , 90^\circ $ and $120^\circ $ also play special role in problem (2): the maximal dimension is smaller for these angles than for the other angles. In problem (1) the angle $90^\circ $ seems to behave differently from other angles.     

A quadrilateral Morley element for biharmonic equations

In this paper, we propose a Morley-type finite element for quadrilateral meshes to solve biharmonic problems. For each quadrilateral $Q$ , the finite element space is defined by the span of $P_2(Q)$ plus two functions in $P_3(Q)$ . Each of the cubic polynomials is the product of a pair of equations of opposite edges and the equation of the bimedian between them. The degrees of freedom consist of the values at vertices and integrals of normal derivatives over edges. Optimal orders of convergence are proved both in discrete $H^2$ and $H^1$ seminorms. Several numerical tests confirm the convergence analysis.     

A unified algorithm for mixed l_{2,p}-minimizations and its application in feature selection

Recently, matrix norm $l_{2,1}$ has been widely applied to feature selection in many areas such as computer vision, pattern recognition, biological study and etc. As an extension of $l_1$ norm, $l_{2,1}$ matrix norm is often used to find jointly sparse solution. Actually, computational studies have showed that the solution of $l_p$ -minimization ( $0<p<1$ ) is sparser than that of $l_1$ -minimization. The generalized $l_{2,p}$ -minimization ( $p\in (0,1]$ ) is naturally expected to have better sparsity than $l_{2,1}$ -minimization. This paper presents a type of models based on $l_{2,p}\ (p\in (0, 1])$ matrix norm which is non-convex and non-Lipschitz continuous optimization problem when $p$ is fractional ( $0<p<1$ ). For all $p$ in $(0, 1]$ , a unified algorithm is proposed to solve the $l_{2,p}$ -minimization and the convergence is also uniformly demonstrated. In the practical implementation of algorithm, a gradient projection technique is utilized to reduce the computational cost. Typically different $l_{2,p}\ (p\in (0,1])$ are applied to select features in computational biology.     

Characterization of some simple groups by the multiplicity pattern

Let $G$ be a finite group and let ${\mathrm{Irr}}(G)$ denote the set of all complex irreducible characters of $G.$ Let ${\mathrm{cd}}(G)$ be the set of all character degrees of $G.$ For each positive integer $d,$ the multiplicity of $d$ in $G$ is defined to be the number of irreducible characters of $G$ having the same degree $d.$ The multiplicity pattern ${\mathrm{mp}}(G)$ is the vector whose first coordinate is $|G:G^{\prime }|$ and for $i\ge 1,$ the $(i+1)$ th-coordinate of ${\mathrm{mp}}(G)$ is the multiplicity of the $i$ th-smallest nontrivial character degree of $G.$ In this paper, we show that every nonabelian simple group with at most $7$ distinct character degrees is uniquely determined by the multiplicity pattern.     

Hypercentralizing generalized skew derivations on left ideals in prime rings

Let $\mathcal{R }$ be a prime ring of characteristic different from $2, \mathcal{Q }_r$ the right Martindale quotient ring of $\mathcal{R }, \mathcal{C }$ the extended centroid of $\mathcal{R }, \mathcal{I }$ a nonzero left ideal of $\mathcal{R }, F$ a nonzero generalized skew derivation of $\mathcal{R }$ with associated automorphism $\alpha $ , and $n,k \ge 1$ be fixed integers. If $[F(r^n),r^n]_k=0$ for all $r \in \mathcal{I }$ , then there exists $\lambda \in \mathcal{C }$ such that $F(x)=\lambda x$ , for all $x\in \mathcal{I }$ . More precisely one of the following holds: (1) $\alpha $ is an $X$ -inner automorphism of $\mathcal{R }$ and there exist $b,c \in \mathcal{Q }_r$ and $q$ invertible element of $\mathcal{Q }_r$ , such that $F(x)=bx-qxq^{-1}c$ , for all $x\in \mathcal{Q }_r$ . Moreover there exists $\gamma \in \mathcal{C }$ such that $\mathcal{I }(q^{-1}c-\gamma )=(0)$ and $b-\gamma q \in \mathcal{C }$ ; (2) $\alpha $ is an $X$ -outer automorphism of $\mathcal{R }$ and there exist $c \in \mathcal{Q }_r, \lambda \in \mathcal{C }$ , such that $F(x)=\lambda x-\alpha (x)c$ , for all $x\in \mathcal{Q }_r$ , with $\alpha (\mathcal{I })c=0$ .     

Application of the multipole method to direct and inverse problems for the Grad-Shafranov equation with a nonlocal condition

Two homogeneous Dirichlet problems for the Grad-Shafranov equation with an affine right-hand side are considered in plane simply connected domains with a piecewise smooth boundary. The problems are denoted by and $(\mathfrak{D})$ , and $(\mathfrak{A})$ the second involves a nonlocal condition. The corresponding inverse problems $(\mathfrak{D}^{ - 1} )$ and $(\mathfrak{A}^{ - 1} )$ of finding unknown parameters on the right-hand side of the equation from a given normal derivative of the solution to the respective direct problem are also considered. These problems arise in the computation of plasma flow characteristics in a tokamak. It is shown that the sought parameters can be found from two given quantities: (i) the normal derivative in the corresponding direct problem, which is physically interpreted as the magnetic field at an arbitrary single point $\tilde x$ of a special subset $\tilde \Gamma $ of the boundary Γ, and (ii) the integral of the normal derivative over Γ, which is physically interpreted as the total current through the tokamak cross section. Both problems are shown to be uniquely solvable, and necessary and sufficient conditions for unique solvability are given. A method for finding the desired parameters is proposed, which includes a technique for finding the subset $\tilde \Gamma $ . The results are based, first, on the multipole method, which ensures the high-order accurate computation of the normal derivatives of the solution to direct problems $(\mathfrak{D})$ and $(\mathfrak{A})$ , and, second, on the asymptotics of these derivatives as one of the parameters on the right-hand side of the equation tends to infinity.     

An Index Formula in Connection with Meromorphic Approximation

Let $\Phi $ be a continuous $n\times n$ matrix-valued function on the unit circle $\mathbb T $ such that the $(k-1)$ st singular value of the Hankel operator with symbol $\Phi $ is greater than the $k$ th singular value. In this case, it is well-known that $\Phi $ has a unique superoptimal meromorphic approximant $Q$ in $H^{\infty }_{(k)}$ ; that is, $Q$ has at most $k$ poles in the unit disc $\mathbb D $ (in the sense that the McMillan degree of $Q$ in $\mathbb D $ is at most $k$ ) and $Q$ minimizes the essential suprema of singular values $s_{j}\left((\Phi -Q)(\zeta )\right)\!, j\ge 0$ , with respect to the lexicographic ordering. For each $j\ge 0$ , the essential supremum of $s_{j}\left((\Phi -Q)(\zeta )\right)$ is called the $j$ th superoptimal singular value of degree $k$ of $\Phi $ . We prove that if $\Phi $ has $n$ non-zero superoptimal singular values of degree $k$ , then the Toeplitz operator $T_{\Phi -Q}$ with symbol $\Phi -Q$ is Fredholm and has index $$ \mathrm{ind}T_{\Phi -Q}=\dim \ker T_{\Phi -Q}=2k+\dim \mathcal E , $$ where $\mathcal E =\{ \xi \in \ker H_{Q}: \Vert H_{\Phi }\xi \Vert _{2}=\Vert (\Phi -Q)\xi \Vert _{2}\}$ and $H_{\Phi }$ denotes the Hankel operator with symbol $\Phi $ . This result can in fact be extended from continuous matrix-valued functions to the wider class of $k$ -admissible matrix-valued functions, i.e. essentially bounded $n\times n$ matrix-valued functions $\Phi $ on $\mathbb T $ for which the essential norm of the Hankel operator $H_{\Phi }$ is strictly less than the smallest non-zero superoptimal singular value of degree $k$ of $\Phi $ .     

p-Groups with few conjugacy classes of normalizers

For a group $G$ , denote by $\omega (G)$ the number of conjugacy classes of normalizers of subgroups of $G$ . Clearly, $\omega (G)=1$ if and only if $G$ is a Dedekind group. Hence if $G$ is a 2-group, then $G$ is nilpotent of class $\le 2$ and if $G$ is a $p$ -group, $p>2$ , then $G$ is abelian. We prove a generalization of this. Let $G$ be a finite $p$ -group with $\omega (G)\le p+1$ . If $p=2$ , then $G$ is of class $\le 3$ ; if $p>2$ , then $G$ is of class $\le 2$ .     

On the unique representation of very strong algebraic geometry codes

This paper addresses the question of retrieving the triple ${(\mathcal X,\mathcal P, E)}$ from the algebraic geometry code ${\mathcal C = \mathcal C_L(\mathcal X, \mathcal P, E)}$ , where ${\mathcal X}$ is an algebraic curve over the finite field ${\mathbb F_q, \,\mathcal P}$ is an n-tuple of ${\mathbb F_q}$ -rational points on ${\mathcal X}$ and E is a divisor on ${\mathcal X}$ . If ${\deg(E)\geq 2g+1}$ where g is the genus of ${\mathcal X}$ , then there is an embedding of ${\mathcal X}$ onto ${\mathcal Y}$ in the projective space of the linear series of the divisor E. Moreover, if ${\deg(E)\geq 2g+2}$ , then ${I(\mathcal Y)}$ , the vanishing ideal of ${\mathcal Y}$ , is generated by ${I_2(\mathcal Y)}$ , the homogeneous elements of degree two in ${I(\mathcal Y)}$ . If ${n >2 \deg(E)}$ , then ${I_2(\mathcal Y)=I_2(\mathcal Q)}$ , where ${\mathcal Q}$ is the image of ${\mathcal P}$ under the map from ${\mathcal X}$ to ${\mathcal Y}$ . These three results imply that, if ${2g+2\leq m < \frac{1}{2}n}$ , an AG representation ${(\mathcal Y, \mathcal Q, F)}$ of the code ${\mathcal C}$ can be obtained just using a generator matrix of ${\mathcal C}$ where ${\mathcal Y}$ is a normal curve in ${\mathbb{P}^{m-g}}$ which is the intersection of quadrics. This fact gives us some clues for breaking McEliece cryptosystem based on AG codes provided that we have an efficient procedure for computing and decoding the representation obtained.