A unified approach for minimizing composite norms

We propose a first-order augmented Lagrangian algorithm (FALC) to solve the composite norm minimization problem $$\begin{aligned} \begin{array}{ll} \min \limits _{X\in \mathbb{R }^{m\times n}}&\mu _1\Vert \sigma (\mathcal{F }(X)-G)\Vert _\alpha +\mu _2\Vert \mathcal{C }(X)-d\Vert _\beta ,\\ \text{ subject} \text{ to}&\mathcal{A }(X)-b\in \mathcal{Q }, \end{array} \end{aligned}$$ where $\sigma (X)$ denotes the vector of singular values of $X \in \mathbb{R }^{m\times n}$ , the matrix norm $\Vert \sigma (X)\Vert _{\alpha }$ denotes either the Frobenius, the nuclear, or the $\ell _2$ -operator norm of $X$ , the vector norm $\Vert .\Vert _{\beta }$ denotes either the $\ell _1$ -norm, $\ell _2$ -norm or the $\ell _{\infty }$ -norm; $\mathcal{Q }$ is a closed convex set and $\mathcal{A }(.)$ , $\mathcal{C }(.)$ , $\mathcal{F }(.)$ are linear operators from $\mathbb{R }^{m\times n}$ to vector spaces of appropriate dimensions. Basis pursuit, matrix completion, robust principal component pursuit (PCP), and stable PCP problems are all special cases of the composite norm minimization problem. Thus, FALC is able to solve all these problems in a unified manner. We show that any limit point of FALC iterate sequence is an optimal solution of the composite norm minimization problem. We also show that for all $\epsilon >0$ , the FALC iterates are $\epsilon $ -feasible and $\epsilon $ -optimal after $\mathcal{O }(\log (\epsilon ^{-1}))$ iterations, which require $\mathcal{O }(\epsilon ^{-1})$ constrained shrinkage operations and Euclidean projection onto the set $\mathcal{Q }$ . Surprisingly, on the problem sets we tested, FALC required only $\mathcal{O }(\log (\epsilon ^{-1}))$ constrained shrinkage, instead of the $\mathcal{O }(\epsilon ^{-1})$ worst case bound, to compute an $\epsilon $ -feasible and $\epsilon $ -optimal solution. To best of our knowledge, FALC is the first algorithm with a known complexity bound that solves the stable PCP problem.     

Geodesic Ricci solitons on unit tangent sphere bundles

In the paper, (Abbassi and Kowalski, Ann Glob Anal Geom, 38: 11–20, 2010) the authors study Einstein Riemannian $g$ natural metrics on unit tangent sphere bundles. In this study, we equip the unit tangent sphere bundle $T_1 M$ of a Riemannian manifold $(M,g)$ with an arbitrary Riemannian $g$ natural metric $\tilde{G}$ and we show that if the geodesic flow $\tilde{\xi }$ is the potential vector field of a Ricci soliton $(\tilde{G},\tilde{\xi },\lambda )$ on $T_1M,$ then $(T_1M,\tilde{G})$ is Einstein. Moreover, we show that the Reeb vector field of a contact metric manifold is an infinitesimal harmonic transformation if and only if it is Killing. Thus, we consider a natural contact metric structure $(\tilde{G}, \tilde{\eta }, \tilde{\varphi }, \tilde{\xi })$ over $T_1 M$ and we show that the geodesic flow $\tilde{\xi }$ is an infinitesimal harmonic transformation if and only if the structure $(\tilde{G}, \tilde{\eta }, \tilde{\varphi },\tilde{\xi })$ is Sasaki $\eta $ -Einstein. Consequently, we get that $(\tilde{G},\tilde{\xi }, \lambda )$ is a Ricci soliton if and only if the structure $(\tilde{G}, \tilde{\eta }, \tilde{\varphi }, \tilde{\xi })$ is Sasaki-Einstein with $\lambda = 2(n-1) >0.$ This last result gives new examples of Sasaki–Einstein structures.     

A subelliptic analogue of Aronson–Serrin’s Harnack inequality

We study the Harnack inequality for weak solutions of a class of degenerate parabolic quasilinear PDE $$\begin{aligned} \partial _t u={-}X_i^* A_i(x,t,u,Xu)+ B(x,t,u,Xu), \end{aligned}$$ in cylinders $\Omega \times (0,T)$ where $\Omega \subset M$ is an open subset of a manifold $M$ endowed with control metric $d$ corresponding to a system of Lipschitz continuous vector fields $X=(X_1,\ldots ,X_m)$ and a measure $d\sigma $ . We show that the Harnack inequality follows from the basic hypothesis of doubling condition and a weak Poincaré inequality in the metric measure space $(M,d,d\sigma )$ . We also show that such hypothesis hold for a class of Riemannian metrics $g_\epsilon $ collapsing to a sub-Riemannian metric $\lim _{\epsilon \rightarrow 0} g_\epsilon =g_0$ uniformly in the parameter $\epsilon \ge 0$ .     

λ-Central BMO estimates for commutators of multilinear maximal operators

The authors establish λ-central BMO estimates for commutators of maximal multilinear Calderón-Zygmund operators $T_{\Pi \vec b}^*$ and multilinear fractional operators $I_{\alpha \vec b}$ on central Morrey spaces respectively. Similar results still hold for $T_{\vec b} ,T_{\vec b}^*$ and $I_{\alpha ,\vec b}^*$ .     

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 $ .     

On the sup-norm of Maass cusp forms of large level. III

Let $f$ be a Hecke–Maass cuspidal newform of square-free level $N$ and Laplacian eigenvalue $\lambda $ . It is shown that $\left||f \right||_\infty \ll _{\lambda ,\epsilon } N^{-\frac{1}{6}+\epsilon } \left||f \right||_2$ for any $\epsilon >0$ .     

On Estimating the Asymptotic Variance of Stationary Point Processes

We investigate a class of kernel estimators $\widehat{\sigma}^2_n$ of the asymptotic variance σ ² of a d-dimensional stationary point process $\Psi = \sum_{i\ge 1}\delta_{X_i}$ which can be observed in a cubic sampling window $W_n = [-n,n]^d\,$ . σ ² is defined by the asymptotic relation $Var(\Psi(W_n)) \sim \sigma^2 \,(2n)^d$ (as n →? ∞) and its existence is guaranteed whenever the corresponding reduced covariance measure $\gamma^{(2)}_{red}(\cdot)$ has finite total variation. Depending on the rate of decay (polynomially or exponentially) of the total variation of $\gamma^{(2)}_{red}(\cdot)$ outside of an expanding ball centered at the origin, we determine optimal bandwidths b _n (up to a constant) minimizing the mean squared error of $\widehat{\sigma}^2_n$ . The case when $\gamma^{(2)}_{red}(\cdot)$ has bounded support is of particular interest. Further we suggest an isotropised estimator $\widetilde{\sigma}^2_n$ suitable for motion-invariant point processes and compare its properties with $\widehat{\sigma}^2_n$ . Our theoretical results are illustrated and supported by a simulation study which compares the (relative) mean squared errors of $\widehat{\sigma}^2_n$ for planar Poisson, Poisson cluster, and hard-core point processes and for various values of n b _n .     

Pareto–Fenchel {\epsilon} -subdifferential sum rule and {\epsilon} -efficiency

The paper is centered around a sum rule for the efficient (Pareto) ${\epsilon}$ -subdifferential of two convex vector mappings, having the property to be exact under a qualification condition. Such a formula has not been explored previously. Our formula which holds under the Attouch?CBrézis as well as Moreau?CRockafellar conditions, reveals strangely a primordial presence of the convex (Fenchel) ${\epsilon}$ -subdifferential. This appearance turns out to be rather favorable. This effectively permits to derive approximate efficiency conditions in terms of Pareto subgradient and vectorial normal cone, which completely characterizes an ${\epsilon}$ -efficient solution in constrained convex vector optimization in (partially) ordered spaces. Our sum rule also allows a fundamental deduction of relation between Pareto and Fenchel ${\epsilon}$ -subdifferentials, which, in reality, brings out a certain gap linking ${\epsilon}$ -efficiency with ${\epsilon}$ -optimality. Scalarization approaches in connection with ${\epsilon}$ -subdifferentials are first established by simple proofs. This principle has contributed for a large part, not only for discovering the sum formula, but also for establishing some punctual necessary and/or sufficient conditions for Pareto ${\epsilon}$ -subdifferentiability.     

Averaged Pointwise Bounds for Deformations of Schrödinger Eigenfunctions

Let (M,g) be an n-dimensional, compact Riemannian manifold and ${P_0(\hbar) = -\hbar{^2} \Delta_g + V(x)}$ be a semiclassical Schrödinger operator with ${\hbar \in (0,\hbar_0]}$ . Let ${E(\hbar) \in [E-o(1),E+o(1)]}$ and ${(\phi_{\hbar})_{\hbar \in (0,\hbar_0]}}$ be a family of L ²-normalized eigenfunctions of ${P_0(\hbar)}$ with ${P_0(\hbar) \phi_{\hbar} = E(\hbar) \phi_{\hbar}}$ . We consider magnetic deformations of ${P_0(\hbar)}$ of the form ${P_u(\hbar) = - \Delta_{\omega_u}(\hbar) + V(x)}$ , where ${\Delta_{\omega_u}(\hbar) = (\hbar d + i \omega_u(x))^*({\hbar}d + i \omega_u(x))}$ . Here, u is a k-dimensional parameter running over ${B^k(\epsilon)}$ (the ball of radius ${\epsilon}$ ), and the family of the magnetic potentials ${(w_u)_{u\in B^k(\epsilon)}}$ satisfies the admissibility condition given in Definition 1.1. This condition implies that k ≥ n and is generic under this assumption. Consider the corresponding family of deformations of ${(\phi_{\hbar})_{\hbar \in (0, \hbar_0]}}$ , given by ${(\phi^u_{\hbar})_{\hbar \in(0, \hbar_0]}}$ , where $$\phi_{\hbar}^{(u)}:= {\rm e}^{-it_0 P_u(\hbar)/\hbar}\phi_{\hbar}$$ for ${|t_0|\in (0,\epsilon)}$ ; the latter functions are themselves eigenfunctions of the ${\hbar}$ -elliptic operators ${Q_u(\hbar): ={\rm e}^{-it_0P_u(\hbar)/\hbar} P_0(\hbar) {\rm e}^{it_0 P_u(\hbar)/\hbar}}$ with eigenvalue ${E(\hbar)}$ and ${Q_0(\hbar) = P_{0}(\hbar)}$ . Our main result, Theorem1.2, states that for ${\epsilon >0 }$ small, there are constants ${C_j=C_j(M,V,\omega,\epsilon) > 0}$ with j = 1,2 such that $$C_{1}\leq \int\limits_{\mathcal{B}^k(\epsilon)} |\phi_{\hbar}^{(u)}(x)|^2 \, {\rm d}u \leq C_{2}$$ , uniformly for ${x \in M}$ and ${\hbar \in (0,h_0]}$ . We also give an application to eigenfunction restriction bounds in Theorem 1.3.     

Matrix Representations of the Low Order Real Clifford Algebras

In this paper we construct the matrix subalgebras ${L_{r,s}(\mathbb{R})}$ of the real matrix algebra ${M_{2^{r+s}} (\mathbb{R})}$ when 2 ≤ r + s ≤ 3 and we show that each ${L_{r,s}(\mathbb{R})}$ is isomorphic to the real Clifford algebra ${\mathcal{C} \ell_{r,s}}$ . In particular, we prove that the algebras ${L_{r,s}(\mathbb{R})}$ can be induced from ${L_{0,n}(\mathbb{R})}$ when 2 ≤ r + s = n ≤ 3 by deforming vector generators of ${L_{0,n}(\mathbb{R})}$ to multiply the specific diagonal matrices. Also, we construct two subalgebras ${T_4(\mathbb{C})}$ and ${T_2(\mathbb{H})}$ of matrix algebras ${M_4(\mathbb{C})}$ and ${M_2(\mathbb{H})}$ , respectively, which are both isomorphic to the Clifford algebra ${\mathcal{C} \ell_{0,3}}$ , and apply them to obtain the properties related to the Clifford group Γ_0,3.     

Asymptotic behaviour of first passage time distributions for Lévy processes

Let $X$ be a real valued Lévy process that is in the domain of attraction of a stable law without centering with norming function $c.$ As an analogue of the random walk results in Vatutin and Wachtel (Probab Theory Relat Fields 143(1–2):177–217, 2009) and Doney (Probab Theory Relat Fields 152(3–4):559–588, 2012), we study the local behaviour of the distribution of the lifetime $\zeta $ under the characteristic measure $\underline{n}$ of excursions away from $0$ of the process $X$ reflected in its past infimum, and of the first passage time of $X$ below $0,$ $T_{0}=\inf \{t>0:X_{t}<0\},$ under $\mathbb{P }_{x}(\cdot ),$ for $x>0,$ in two different regimes for $x,$ viz. $x=o(c(\cdot ))$ and $x>D c(\cdot ),$ for some $D>0.$ We sharpen our estimates by distinguishing between two types of path behaviour, viz. continuous passage at $T_{0}$ and discontinuous passage. In order to prove our main results we establish some sharp local estimates for the entrance law of the excursion process associated to $X$ reflected in its past infimum.     

Subgame perfect equilibria in stopping games

Stopping games (without simultaneous stopping) are multi-player sequential games in which at every stage one of the players is chosen according to a stochastic process, and that player decides whether to continue the interaction or to stop it, whereby the terminal payoff vector is obtained by another stochastic process. We prove that if the payoff process is integrable, a $\delta $ -approximate subgame perfect ${\epsilon }$ -equilibrium exists for every $\delta ,\epsilon >0$ ; that is, there exists a strategy profile that is an ${\epsilon }$ -equilibrium in all subgames, except possibly in a set of subgames that occurs with probability at most $\delta $ (even after deviation by some of the players).     

Diophantine approximation of the orbit of 1 in the dynamical system of beta expansions

We consider the distribution of the orbits of the number 1 under the $\beta $ -transformations $T_\beta $ as $\beta $ varies. Mainly, the size of the set of $\beta >1$ for which a given point can be well approximated by the orbit of 1 is measured by its Hausdorff dimension. The dimension of the following set $E\big (\{\ell _n\}_{n\ge 1}, x_0\big )=\Big \{\,\beta >1: |T^n_{\beta }1-x_0|<\beta ^{-\ell _n}, \hbox { for infinitely many}, \, n\in \mathbb{N }\,\Big \}$ is determined, where $x_0$ is a given point in $[0,1]$ and $\{\ell _n\}_{n\ge 1}$ is a sequence of integers tending to infinity as $n\rightarrow \infty $ . For the proof of this result, the notion of the recurrence time of a word in symbolic space is introduced to characterise the lengths and the distribution of cylinders (the set of $\beta $ with a common prefix in the expansion of 1) in the parameter space $\{\,\beta \in \mathbb{R }: \beta >1\,\}$ .     

Creature forcing and large continuum: the joy of halving

For ${f,g\in\omega^\omega}$ let ${c^\forall_{f,g}}$ be the minimal number of uniform g-splitting trees needed to cover the uniform f-splitting tree, i.e., for every branch ν of the f-tree, one of the g-trees contains ν. Let ${c^\exists_{f,g}}$ be the dual notion: For every branch ν, one of the g-trees guesses ν(m) infinitely often. We show that it is consistent that ${c^\exists_{f_\epsilon,g_\epsilon}{=}c^\forall_{f_\epsilon,g_\epsilon}{=}\kappa_\epsilon}$ for continuum many pairwise different cardinals ${\kappa_\epsilon}$ and suitable pairs ${(f_\epsilon,g_\epsilon)}$ . For the proof we introduce a new mixed-limit creature forcing construction.     

Non-isomorphic semipartial geometries

In De Winter and Thas (Des Codes Cryptogr, 32, 153–166, 2004) a semipartial geometry ${\mathcal{S}(\overline{\mathcal{U})}}$ was constructed from any Buekenhout–Metz unital ${\mathcal{U}}$ in PG(2,q²), and it was shown that, although having the same parameters, ${\mathcal{S}(\overline{\mathcal{U})}\not\cong T_2^*(\mathcal{U})}$ , where ${T_2^*\mathcal{U}}$ is the semipartial geometry arising from the linear representation of ${\mathcal{U}}$ . In this note, we will first briefly overview what is known on the geometry ${\mathcal{S}(\overline{\mathcal{U})}}$ (providing shortened unpublished proofs for most results). Then we answer the following question of G. Ebert affirmatively: “Do non-isomorphic Buekenhout–Metz unitals ${\mathcal{U}_1}$ and ${\mathcal{U}_2}$ yield non-isomorphic semipartial geometries ${\mathcal{S}(\overline{\mathcal{U}}_1)}$ and ${\mathcal{S}(\overline{\mathcal{U}}_2)}$ ?”.     

f-algebras with a \sigma -bounded approximate unit

Let $X$ be a lattice ordered algebra ( $\ell $ -algebra). A positive element $x\in $ $X$ is said to be totally bounded if $x^{2}\le x$ . The $\ell $ -algebra $X$ is said to have a $\sigma $ -bounded approximate unit if for each positive linear functional $f$ on $X$ the set $\left\{ f(x)\text{: } x \text{ totally } \text{ bounded }\right\} $ is bounded in $\mathbb R $ . In this paper we study the class of $f$ -algebras with a $\sigma $ -bounded approximate unit which contains the class of all unital $f$ -algebras. In particular It is shown that an $f$ -algebra $X$ has a $\sigma $ -bounded approximate unit if and only if the order bidual $X^{\sim \sim }$ is a unital $f$ -algebra.     

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.     

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.     

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 ${\mathbb{P\ast P}}$ .     

\epsilon -Optimality conditions of vector optimization problems with set-valued maps based on the algebraic interior in real linear spaces

In this paper, firstly, the necessary and sufficient optimality conditions for $\epsilon $ -global properly efficient elements of set-valued optimization problems, respectively, are established in linear spaces. Secondly, an equivalent characterization of $\epsilon $ -global proper saddle point is presented. Finally, the necessary and sufficient conditions for $\epsilon $ -global properly saddle point of a Lagrangian set-valued map are obtained. The results in this paper generalize some known results in the literature.