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Introduction
Current research interests:
Renormalization group methods, spectral and dynamical problems in quantum field theory.
Mathematical foundations of deep learning, interpretability problem.
Scaling limits of quantum dynamics, nonlinear PDEs, Hamiltonian dynamics.
Current institution
Additional affiliations
January 2017 - August 2024
October 2014 - January 2017
Education
June 1999 - May 2001
June 1996 - June 1999
Publications
Publications (89)
In this paper, we study a cold gas of $N \gg 1$ weakly interacting fermions. We describe the time evolution of the momentum distribution of states close to the Fermi ball by simultaneously analyzing the dynamical behavior of excited particles and holes. Our main result states that, for small values of the coupling constant, and for appropriate init...
We prove that overparametrized neural networks are able to generalize with a test error that is independent of the level of overparametrization, and independent of the Vapnik-Chervonenkis (VC) dimension. We prove explicit bounds that only depend on the metric geometry of the test and training sets, on the regularity properties of the activation fun...
We derive explicit equations governing the cumulative biases and weights in Deep Learning with ReLU activation function, based on gradient descent for the Euclidean cost in the input layer, and under the assumption that the weights are, in a precise sense, adapted to the coordinate system distinguished by the activations. We show that gradient desc...
We prove that the usual gradient flow in parameter space that underlies many training algorithms for neural networks in deep learning can be continuously deformed into an adapted gradient flow which yields (constrained) Euclidean gradient flow in output space. Moreover, if the Jacobian of the outputs with respect to the parameters is full rank (for...
Our previous work [37] presented a rigorous derivation of quantum Boltzmann equations near a Bose-Einstein condensate (BEC). Here, we extend it with a complete characterization of the leading order fluctuation dynamics. For this purpose, we correct the latter via an appropriate Bogoliubov rotation, in partial analogy to the approach by Grillakis-Mac...
We determine sufficient conditions for overparametrized deep learning (DL) networks to guarantee the attainability of zero loss in the context of supervised learning, for the L 2 cost and generic training data. We present an explicit construction of the zero loss minimizers without invoking gradient descent. On the other hand, we point out that inc...
We analyze geometric aspects of the gradient descent algorithm in Deep Learning (DL), and give a detailed discussion of the circumstance that, in underparametrized DL networks, zero loss minimization cannot generically be attained. As a consequence, we conclude that the distribution of training inputs must necessarily be non-generic in order to pro...
We explicitly construct zero loss neural network classifiers. We write the weight matrices and bias vectors in terms of cumulative parameters, which determine truncation maps acting recursively on input space. The configurations for the training data considered are (i) sufficiently small, well separated clusters corresponding to each class, and (ii...
We consider the scenario of supervised learning in Deep Learning (DL) networks, and exploit the arbitrariness of choice in the Riemannian metric relative to which the gradient descent flow can be defined (a general fact of differential geometry).
In the standard approach to DL, the gradient flow on the space of parameters (weights and biases) is d...
We analyze geometric aspects of the gradient descent algorithm in Deep Learning (DL), and give a detailed discussion of the circumstance that in underparametrized DL networks, zero loss minimization can generically not be attained. As a consequence, we conclude that the distribution of training inputs must necessarily be non-generic in order to pro...
In this paper, we study cubic and quintic nonlinear Schrödinger systems on three-dimensional tori, with initial data in an adapted Hilbert space \(H^s_{{\underline{\lambda }}},\) and all of our results hold on rational and irrational rectangular, flat tori. In the cubic and quintic case, we prove local well-posedness for both focusing and defocusin...
In this paper, we approach the cost (loss) minimization in underparametrized shallow neural networks through explicit construction of upper bounds, without any use of gradient descent. A key focus is on elucidating the geometric structure of approximate and precise minimizers. We consider shallow neural networks with one hidden layer, a ReLU activa...
In this paper, we explicitly determine local and global minimizers of the $\cL^2$ cost function in underparametrized Deep Learning (DL) networks. Our main goal is to obtain a rigorous mathematical understanding of their geometric structure and properties; we accomplish this by a direct construction, without invoking the gradient descent flow anywhe...
In this work, we study the quantum fluctuation dynamics in a Bose gas on a torus \(\Lambda =(L{\mathbb {T}})^3\) that exhibits Bose–Einstein condensation, beyond the leading order Hartree–Fock–Bogoliubov (HFB) theory. Given a Bose–Einstein condensate (BEC) with density \(N\gg 1\) surrounded by thermal fluctuations with density 1, we assume that the...
We prove the existence of a class of large global scattering solutions of Boltzmann's equation with constant collision kernel in two dimensions. These solutions are found for $L^2$ perturbations of an underlying initial data which is Gaussian jointly in space and velocity. Additionally, the perturbation is required to satisfy natural physical const...
In this article, we use quasifree reduction to derive the time-dependent Hartree-Fock-Bogoliubov (HFB) equations describing the dynamics of quantum fluctuations around a Bose-Einstein condensate in $\mathbb R^d$. We prove global well-posedness for the HFB equations for sufficiently regular pair interaction potentials, and establish key conservation...
In this paper, we study cubic and quintic nonlinear Schr\"odinger systems on 3-dimensional tori, with initial data in an adapted Hilbert space $H^s_{\underline{\lambda}},$ and all of our results hold on rational and irrational rectangular, flat tori. In the cubic and quintic case, we prove local well-posedness for both focusing and defocusing syste...
this work, we study the quantum fluctuation dynamics in a Bose gas on a torus $\Lambda=(L\mathbb{T})^3$ that exhibits Bose-Einstein condensation, beyond the leading order Hartree-Fock-Bogoliubov (HFB) fluctuations. Given a Bose-Einstein condensate (BEC) with density $N$ surrounded by thermal fluctuations with density 1, we assume that the system is...
We provide a new analysis of the Boltzmann equation with a constant collision kernel in two space dimensions. The scaling-critical Lebesgue space is Lx,v2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsid...
We provide a new analysis of the Boltzmann equation with constant collision kernel in two space dimensions. The scaling-critical Lebesgue space is $L^2_{x,v}$; we prove global well-posedness and a version of scattering, assuming that the data $f_0$ is sufficiently smooth and localized, and the $L^2_{x,v}$ norm of $f_0$ is sufficiently small. The pr...
We use the dispersive properties of the linear Schr\"{o}dinger equation to prove local well-posedness results for the Boltzmann equation and the related Boltzmann hierarchy, set in the spatial domain $\mathbb{R}^d$ for $d\geq 2$. The proofs are based on the use of the (inverse) Wigner transform along with the spacetime Fourier transform. The norms...
We review some results of our paper arXiv:1602.05171v2 on the "nonlinear quasifree approximation" to the many-body Schr\"odinger dynamics of Bose gases. In that paper, we derive, with the help of this approximation, the time-dependent Hartree-Fock-Bogoliubov (HFB) equations, providing an approximate description of the dynamics of quantum fluctuatio...
In this paper, we continue our study of the Boltzmann equation by use of tools originating from the analysis of dispersive equations in quantum dynamics. Specifically, we focus on properties of solutions to the Boltzmann equation with collision kernel equal to a constant in the spatial domain $\mathbb{R}^d$, $d\geq 2$, which we use as a model in th...
In this paper, we continue our study of the Boltzmann equation by use of tools originating from the analysis of dispersive equations in quantum dynamics. Specifically, we focus on properties of solutions to the Boltzmann equation with collision kernel equal to a constant in the spatial domain $\mathbb{R}^d$, $d\geq 2$, which we use as a model in th...
We consider the dynamics of a heavy quantum tracer particle coupled to a non-relativistic boson field in ${\mathbb R}^3$. The pair interactions of the bosons are of mean-field type, with coupling strength proportional to $\frac1N$ where $N$ is the expected particle number. Assuming that the mass of the tracer particle is proportional to $N$, we der...
We consider the dynamics of a heavy quantum tracer particle coupled to a non-relativistic boson field in ${\mathbb R}^3$. The pair interactions of the bosons are of mean-field type, with coupling strength proportional to $\frac1N$ where $N$ is the expected particle number. Assuming that the mass of the tracer particle is proportional to $N$, we der...
We consider the Schr\"odinger equation with a time-independent weakly random potential of a strength $\epsilon\ll 1$, with Gaussian statistics. We prove that when the initial condition varies on a scale much larger than the correlation length of the potential, the compensated wave function converges to a deterministic limit on the time scale $t\sim...
In this paper, we study the mean field quantum fluctuation dynamics for a
system of infinitely many fermions with delta pair interactions in the vicinity
of an equilibrium solution (the Fermi sea) at zero temperature, in dimensions
$d=2,3$, and prove global well-posedness of the corresponding Cauchy problem.
Our work extends some of the recent impo...
We use the dispersive properties of the linear Schr\"{o}dinger equation to prove local well-posedness results for the Boltzmann equation and the related Boltzmann hierarchy, set in the spatial domain $\mathbb{R}^d$ for $d\geq 2$. The proofs are based on the use of the (inverse) Wigner transform along with the spacetime Fourier transform. The norms...
We consider the Schr\"odinger equation with a time-independent weakly random potential of a strength $\epsilon\ll 1$, with Gaussian statistics. We prove that when the initial condition varies on a scale much larger than the correlation length of the potential, the compensated wave function converges to a deterministic limit on the time scale $t\sim...
In this paper, we study the dynamics of a system of infinitely many fermions in dimensions $d\geq3$ near thermal equilibrium and prove scattering in the case of small perturbation around equilibrium in a certain generalized Sobolev space of density operators. This work is a continuation of our previous paper, and extends the important recent result...
In this paper, we study the dynamics of a system of infinitely many fermions in dimensions $d\geq3$ near thermal equilibrium and prove scattering in the case of small perturbation around equilibrium in a certain generalized Sobolev space of density operators. This work is a continuation of our previous paper, and extends the important recent result...
In this article, we use quasifree reduction to derive the time-dependent Hartree-Fock-Bogoliubov (HFB) equations describing the dynamics of quantum fluctuations around a Bose-Einstein condensate in $\mathbb R^d$. We prove global well-posedness for the HFB equations for sufficiently regular pair interaction potentials, and establish key conservation...
We present a new, simpler proof of the unconditional uniqueness of solutions
to the cubic Gross-Pitaevskii hierarchy in $\R^3$. One of the main tools in our
analysis is the quantum de Finetti theorem. Our uniqueness result is equivalent
to the one established in the celebrated works of Erd\"os, Schlein and Yau,
\cite{esy1,esy2,esy3,esy4}.
We derive the cubic defocusing GP hierarchy in ℜ3 from a bosonic N-particle Schrödinger equation as N → ∞, in the strong topology corresponding to the space [Inline formula] of sequences of marginal density matrices. In particular, we thereby eliminate the requirement of regularity [Inline formula] for the initial data used in previous work. Moreov...
We show global well-posedness in energy norm of the semi-relativistic Schrödinger–Poisson system of equations with attractive Coulomb interaction in [Formula: see text] in the presence of pseudo-relativistic diffusion. We also discuss sufficient conditions to have well-posedness in [Formula: see text]. In the absence of dissipation, we show that th...
We prove the existence of scattering states for the defocusing cubic
Gross-Pitaevskii (GP) hierarchy in ${\mathbb R}^3$. Moreover, we show that an
energy growth condition commonly used in the well-posedness theory of the GP
hierarchy is, in a specific sense, necessary. In fact, we prove that without
the latter, there exist initial data for the focu...
We study the cubic defocusing Gross-Pitaevskii (GP) hierarchy in $\R^3$, and
prove global well-posedness of solutions. In particular, we prove that positive
semidefiniteness is preserved over time; so far, this property has been known
for special cases, including solutions obtained from the BBGKY hierarchy of an
$N$-body Schr\"odinger system as $N\...
The Cauchy problem for the semi-relativistic Schr\"odinger-Poisson system of
equations is studied in ${\mathbb R}^n$, $n\geq 1$, for a wide class of
nonlocal interactions. Furthermore, the asymptotic behavior of the solution as
the mass tends to infinity is rigorously discussed, which corresponds to a
non-relativistic limit.
We study the stationary states of the semi-relativistic Schr\"odinger-Poisson
system in the repulsive (plasma physics) Coulomb case. In particular, we
establish the existence and the nonlinear stability of a wide class of
stationary states by means of the energy-Casimir method. We generalize the
global well-posedness result of our previous work for...
In the framework of non-relativistic QED, we show that the renormalized mass
of the electron (after having taken into account radiative corrections) appears
as the kinematic mass in its response to an external potential force.
Specifically, we study the dynamics of an electron in a slowly varying external
potential and with slowly varying initial c...
We show global existence and uniqueness of strong solutions for the
Schrodinger-Poisson system in the repulsive Coulomb case with relativistic
kinetic energy.
We derive the defocusing cubic Gross-Pitaevskii (GP) hierarchy in dimension
$d=3$, from an $N$-body Schr\"{o}dinger equation describing a gas of
interacting bosons in the GP scaling, in the limit $N\rightarrow\infty$. The
main result of this paper is the proof of convergence of the corresponding
BBGKY hierarchy to a GP hierarchy in the spaces intro...
This article consists of two parts. In the first part, we review the most
recent proofs establishing quadratic Morawetz inequalities for the nonlinear
Schr\"odinger equation (NLS). We also describe the applications of these
estimates to the problem of quantum scattering. In the second part, we
generalize some of the methods developed for the NLS by...
We prove a Beale-Kato-Majda type criterion for the loss of regularity for
solutions of the incompressible Euler equations in $H^{s}(\R^3)$, for
$s>\frac52$. Instead of double exponential estimates of Beale-Kato-Majda type,
we obtain a single exponential bound on $\|u(t)\|_{H^s}$ involving the length
parameter introduced by P. Constantin in \cite{co...
We study the dynamics of the thermal momentum distribution function for an interacting, homogeneous Fermi gas on ℤ3 in the presence of an external weak static random potential, where the pair interactions between the fermions are modeled
in dynamical Hartree-Fock theory. We determine the Boltzmann limits associated to different scaling regimes defi...
We consider solutions of the focusing cubic and quintic Gross-Pitaevskii (GP)
hierarchies. We identify an observable corresponding to the average energy per
particle, and we prove that it is a conserved quantity. We prove that all
solutions to the focusing GP hierarchy at the $L^2$-critical or
$L^2$-supercritical level blow up in finite time if the...
In this paper, we determine the exact expression for the hydrogen binding energy in the Pauli–Fierz model up to the order
α
5 log α
−1, where α denotes the fine structure constant, and prove rigorous bounds on the remainder term of the order o(α
5 log α
−1). As a consequence, we prove that the binding energy is not a real analytic function of α, an...
We construct infraparticle scattering states for Compton scattering in the standard model of non-relativistic QED. In our construction, an infrared cutoff initially introduced to regularize the model is removed completely. We rigorously establish the properties of infraparticle scattering theory predicted in the classic work of Bloch and Nordsieck...
In this paper, we review some of our recent results in the study of the dynamics of
interacting Bose gases in the Gross-Pitaevskii (GP) limit. Our investigations focus on the
well-posedness of the associated Cauchy problem for the infinite particle system described
by the GP hierarchy.
We prove the limiting absorption principle for a dressed electron at a fixed
total momentum in the standard model of non-relativistic quantum
electrodynamics. Our proof is based on an application of the smooth
Feshbach-Schur map in conjunction with Mourre's theory.
We consider the cubic and quintic Gross-Pitaevskii (GP) hierarchies in
$d\geq1$ dimensions, for focusing and defocusing interactions. We present a new
proof of existence of solutions that does not require the a priori bound on the
spacetime norm, which was introduced in the work of Klainerman and Machedon,
\cite{klma}, and used in our earlier work,...
We consider the cubic and quintic Gross-Pitaevskii (GP) hierarchy in $d$ dimensions, for focusing and defocusing interactions. We introduce new higher order conserved energy functionals that allow us to prove global existence and uniqueness of solutions for defocusing GP hierarchies, with arbitrary initial data in the energy space. Moreover, we pro...
In this paper, we determine the exact expression for the hydrogen binding
energy in the Pauli-Fierz model up to the order $O(\alpha^5\log\alpha^{-1})$,
where $\alpha$ denotes the finestructure constant, and prove rigorous bounds on
the remainder term of the order $o(\alpha^5\log\alpha^{-1})$. As a consequence,
we prove that the binding energy is no...
We consider the dynamical Gross-Pitaevskii (GP) hierarchy on $\R^d$, $d\geq1$, for cubic, quintic, focusing and defocusing interactions. For both the focusing and defocusing case, and any $d\geq1$, we prove local existence and uniqueness of solutions in certain Sobolev type spaces $\cH_\xi^\alpha$ of sequences of marginal density matrices. The regu...
We study the infrared problem in the usual model of quantum electrodynamics with nonrelativistic matter. We prove spectral and regularity properties characterizing the mass shell of an electron and one-electron infraparticle states of this model. Our results are crucial for the construction of infraparticle scattering states, which are treated in a...
We investigate the dynamics of a boson gas with three-body interactions in
dimensions $d=1,2$. We prove that in the limit of infinite particle number, the
BBGKY hierarchy of $k$-particle marginals converges to a limiting
(Gross-Pitaevskii (GP)) hierarchy for which we prove existence and uniqueness
of solutions. Factorized solutions of the GP hierar...
We investigate the dynamics of a boson gas with three-body interactions in dimensions $d=1,2$. We prove that in the limit of infinite particle number, the BBGKY hierarchy of $k$-particle marginals converges to a limiting (Gross-Pitaevskii (GP)) hierarchy for which we prove existence and uniqueness of solutions. Factorized solutions of the GP hierar...
We consider a spin-$\frac12$ electron in a translation-invariant model of non-relativistic Quantum Electrodynamics (QED). Let $H(\vp,\sig)$ denote the fiber Hamiltonian corresponding to the conserved total momentum $\vp\in\R^3$ of the Pauli electron and the photon field, regularized by a fixed ultraviolet cutoff in the interaction term, and an infr...
We discuss some basic aspects of the dynamics of a homogenous Fermi gas in a weak random potential, under negligence of the particle pair interactions. We derive the kinetic scaling limit for the momentum distribution function with a translation invariant initial state and prove that it is determined by a linear Boltzmann equation. Moreover, we pro...
We study the infrared problem in the usual model of QED with non-relativistic matter. We prove spectral and regularity properties characterizing the mass shell of an electron and one-electron infraparticle states of this model. Our results are crucial for the construction of infraparticle scattering states, which are treated in a separate paper.
In this note, we determine the ground state energy of the translation invariant Pauli-Fierz model to subleading order $O(\alpha^3)$ with respect to powers of the finestructure constant $\alpha$, and prove rigorous error bounds of order $O(\alpha^{4})$. A main objective of our argument is its brevity.
This work addresses the problem of infrared mass renormalization for a non-relativistic electron minimally coupled to the quantized electromagnetic field (the standard, translationally invariant system of an electron in non-relativistic QED). We assume that the interaction of the electron with the quantized electromagnetic field is subject to an ul...
We report on the author’s recent work [J. Stat. Phys. 120, No. 1-2, 279–337 (2005; Zbl 1142.82008)] concerning lower bounds on the localization length of eigenfunctions in the three-dimensional Anderson model at weak disorders, that uses an extension of methods developed by L. Erdös and H.-T. Yau. Our results are similar to those obtained by W. Sch...
We consider dressed 1-electron states in a translation-invariant model of non-relativistic QED. To start with a well-defined model, the interaction Hamiltonian is cutoff at very large photon energies (ultraviolet cutoff) and regularized at very small photon energies (infrared regularization). The infrared regularization is then removed, and the rep...
We consider a spin-$\frac12$ electron in a translation-invariant model of non-relativistic Quantum Electrodynamics (QED). Let $H(\vp,\sig)$ denote the fiber Hamiltonian corresponding to the conserved total momentum $\vp\in\R^3$ of the Pauli electron and the photon field, regularized by a fixed ultraviolet cutoff in the interaction term, and an infr...
This work addresses the problem of infrared mass renormalization for a scalar electron in a translation-invariant model of non-relativistic QED. We assume that the interaction of the electron with the quantized electromagnetic field comprises a fixed ultraviolet regularization and an infrared regularization parametrized by $\sigma>0$. For the value...
We study a class of Schr\"odinger operators on $\Z^2$ with a random potential decaying as $|x|^{-\dex}$, $0<\dex\leq\frac12$, in the limit of small disorder strength $\lambda$. For the critical exponent $\dex=\frac12$, we prove that the localization length of eigenfunctions is bounded below by $2^{\lambda^{-\frac14+\eta}}$, while for $0<\dex<\frac1...
We study the macroscopic scaling and weak coupling limit for a random Schrödinger equation on
\(\mathbb{Z}^3\). We prove that the Wigner transforms of a large class of “macroscopic” solutions converge in r
th mean to solutions of a linear Boltzmann equation, for any 1 ≤ r < ∞. This extends previous results where convergence in expectation was estab...
We study the macroscopic scaling and weak coupling limit for a random Schroedinger equation on Z^3. We prove that the Wigner transforms of a large class of "macroscopic" solutions converge in r-th mean to solutions of a linear Boltzmann equation, for any finite value of r in R_+. This extends previous results where convergence in expectation was es...
We report on recent work, [1], concerning lower bounds on the localization length of eigenfunctions in the three-dimensional Anderson model at weak disorders, that uses an extension of methods developed by L. Erdös and H.-T. Yau. Our results are similar to those obtained by C. Shubin, W. Schlag and T. Wolff, [8], for dimensions one and two. Further...
We examine the binding conditions for atoms in non-relativistic QED, and
prove that removing one electron from an atom requires a positive energy. As an
application, we establish the existence of a ground state for the Helium atom.
We explore a particular approach to the analysis of dynamical and geometrical properties of autonomous, Pfaffian non-holonomic systems in classical mechanics. The method is based on the construction of a certain auxiliary constrained Hamiltonian system, which comprises the non-holonomic mechanical system as a dynamical subsystem on an invariant man...
A new variant of the isospectral Feshbach map defined on operators in Hilbert space is presented. It is constructed with the help of a smooth partition of unity, instead of projections, and is therefore called smooth Feshbach map. It is an effective tool in spectral and singular perturbation theory. As an illustration of its power, a novel operator...
We prove lower bounds on the localization length of eigenfunctions in the three-dimensional Anderson model at weak disorders. Our results are similar to those obtained by Schlag, Shubin and Wolff for dimensions one and two. We prove that with probability one, most eigenfunctions have localization lengths bounded from below by $O(\frac{\lambda^{-2}}...
We consider a Pauli–Fierz Hamiltonian for a particle coupled to a photon field. We discuss the effects of the increase of the binding energy and enhanced binding through coupling to a photon field, and prove that both effects are the results of the existence of the ground state of the self-energy operator with total momentum P = 0. © 2003 American...
We examine the binding conditions for atoms in non-relativistic QED, and prove that removing one electron from an atom requires a positive energy. As an application, we establish the existence of a ground state for the Helium atom.
We explore a particular approach to the analysis of dynamical and geometrical properties of autonomous, Pfaffian non-holonomic systems in classical mechanics. The method is based on the construction of a certain auxiliary constrained Hamiltonian system, which comprises the non-holonomic mechanical system as a dynamical subsystem on an invariant man...
We study the classical decay of unstable scalar solitons in noncommutative field theory in 2+1 dimensions. This can, but does not have to, be viewed as a toy model for the decay of D-branes in string theory. In the limit that the noncommutativity parameter \theta is infinite, the gradient term is absent, there are no propagating modes and the solit...
We consider a Pauli-Fierz Hamiltonian for a particle coupled to a photon field. We discuss the effects of the increase of the binding energy and enhanced binding through coupling to a photon field, and prove that both effects are the results of the existence of the ground state of the self-energy operator with total momentum $P = 0$.
We consider the infrared problem in a model of a freely propagating, nonrelativistic charged particle of mass 1 in interaction with the quantized electromagnetic field. The hamiltonian of the system is regularized by an infrared cutoff $\ssig\ll 1$ and an ultraviolet cutoff $\Lambda\sim 1$ in the interaction term, in units of the mass of the charge...
We approach the analysis of dynamical and geometrical properties of nonholonomic mechanical systems from the discussion of a more general class of auxiliary constrained Hamiltonian systems. The latter is constructed in a manner that it comprises the mechanical system as a dynamical subsystem, which is confined to an invariant manifold. In certain a...
This is the second part of the notes to the course on quantum theory of large systems of non-relativistic matter taught by J. Fr\"{o}hlich at the 1994 Les Houches summer school. It is devoted to a sketchy exposition of some of the beautiful and important, recent results of J.Feldman and E.Trubowitz, and J. Feldman, H. Kn\"{o}rrer, D. Lehmann, J. Ma...
