Journal of Mathematical Analysis and Applications 2 , , Integral Equations and Operator Theory 67 4 , , Numerical functional analysis and optimization 31 1 , , Numerical functional analysis and optimization 27 , , Integral Equations and Operator Theory 38 1 , , International Workshop on Intelligent Virtual Agents, , Journal of Artificial Societies and Social Simulation 17 3 , 7 , Ayuda Privacidad Condiciones. Abstract: Random constraint satisfaction problems encode many interesting questions in the study of random graphs such as the chromatic and independence numbers.

Ideas from statistical physics provide a detailed description of phase transitions and properties of these models. We will discuss the one step replica symmetry breaking transition that many such models undergo and the Satisfiability Threshold for the random K-SAT model. Abstract: Wave propagation in random media can be studied by multiscale and stochastic analysis. We review some recent advances and their applications. We study the associated moment equations and describe the propagation of coherent and incoherent waves. We quantify the scintillation of the wave and the fluctuations of the Wigner distribution.

These results make it possible to introduce and characterize correlation-based imaging methods. Abstract: We present an overview of scalable load balancing algorithms which provide favorable delay performance in large-scale systems, and yet only require minimal implementation overhead. The supermarket model is a dynamic counterpart of the classical balls-and-bins setup where balls must be sequentially distributed across bins.

However, a nominal implementation of the JSQ policy involves a prohibitive communication burden in large-scale deployments. Stochastic coupling techniques play an instrumental role in establishing the asymptotic optimality and universality properties, and augmentations of the coupling constructions allow these properties to be extended to infinite-server settings and network scenarios. We additionally show how the communication overhead can be reduced yet further by the so-called Join-the-Idle-Queue JIQ scheme, leveraging memory at the dispatcher to keep track of idle servers.

## VIAF ID: 114119373 (Personal)

Abstract: It is not difficult to find stories of a crisis in modern science, either in the popular press or in the scientific literature. There are likely multiple sources for this crisis. It could be argued that this misuse of statistical methods is caused by a shift in how data is used in 21st century science compared to its use in the midth century which presumed scientists had formal statistical hypotheses before collecting data.

With the advent of sophisticated statistical software available to anybody this paradigm has been shifted to one in which scientists collect data first and ask questions later. Abstract: Random matrix theory has played an important role in recent work on statistical network analysis. In this paper, we review recent results on regimes of concentration of random graphs around their expectation, showing that dense graphs concentrate and sparse graphs concentrate after regularization. We also review relevant network models that may be of interest to probabilists considering directions for new random matrix theory developments, and random matrix theory tools that may be of interest to statisticians looking to prove properties of network algorithms.

Applications of concentration results to the problem of community detection in networks are discussed in detail. Abstract: The last twenty-or-so years have seen spectacular progress in our understanding of the fine spectral properties of large-dimensional random matrices. These results have also shown light on the behavior of various statistical estimators used in multivariate statistics. In this short note, we will describe new strands of results, which show that intuition and techniques built on the theory of random matrices and concentration of measure ideas shed new light and bring to the fore new ideas about an arguably even more important set of statistical tools, namely M-estimators and certain bootstrap methods.

Limit shapes occur in, for example, random integer partitions or in random interface models such as the dimer model. Typically limit shapes can be described by some variational formula based on a large deviations estimate. We discuss limit shapes for certain 2-dimensional interface models, and explain how their underlying analytic structure is related to a conjectural in some cases conformal invariance property for the models.

Abstract: Modern data analysis challenges require building complex statistical models with massive numbers of parameters. It is nowadays commonplace to learn models with millions of parameters by using iterative optimization algorithms. What are typical properties of the estimated models?

In some cases, the high-dimensional limit of a statistical estimator is analogous to the thermodynamic limit of a certain disordered statistical mechanics system. Building on mathematical ideas from the mean-field theory of disordered systems, exact asymptotics can be computed for high-dimensional statistical learning problems. This theory suggests new practical algorithms and new procedures for statistical inference. Also, it leads to intriguing conjectures about the fundamental computational limits for statistical estimation.

Abstract: Statistical inference from large-scale data can benefit from sources of heterogeneity. We discuss recent progress of the mathematical formalization and theory for exploiting heterogeneity towards predictive stability and causal inference in high-dimensional models. The topic is directly motivated by a broad range of applications and we will show an illustration from molecular biology with gene knock out experiments.

Combinatorics Combinatorial structures. Enumeration: exact and asymptotic. Graph theory. Probabilistic and extremal combinatorics. Designs and finite geometries. Relations with linear algebra, representation theory and commutative algebra. Topological and analytical techniques in combinatorics. Combinatorial geometry. Combinatorial number theory.

Additive combinatorics. Polyhedral combinatorics and combinatorial optimization. Connections with sections 1, 2, 3, 4, 7, 9, 12, Abstract: The nonnegative Grassmannian is a cell complex with rich geometric, algebraic, and combinatorial structures. Its study involves interesting combinatorial objects, such as positroids and plabic graphs.

Remarkably, the same combinatorial structures appeared in many other areas of mathematics and physics, e. We discuss new ways to think about these structures. In particular, we identify plabic graphs and more general Grassmannian graphs with polyhedral subdivisions induced by 2-dimensional projections of hypersimplices. This implies a close relationship between the positive Grassmannian and the theory of fiber polytopes and the generalized Baues problem. This suggests natural extensions of objects related to the positive Grassmannian. Abstract: We call simple graphs with a linear order on the vertices ordered graphs.

This is a survey on the ongoing research in the extremal theory of ordered graphs with an emphasis on open problems. The worst-case complexity of the problem has recently been brought down from moderately exponential to quasipolynomial time. Arguably, the GI problem boils down to filling the gap between symmetry , a global property of objects, and regularity , a local concept.

Recent progress on the problem relies on a combination of the asymptotic theory of permutation groups and asymptotic properties of highly regular combinatorial structures called coherent configurations. Abstract: Why do natural and interesting sequences often turn out to be log-concave? We illustrate with several examples from combinatorics.

Abstract: In this survey we describe a recently-developed technique for bounding the number and controlling the typical structure of finite objects with forbidden substructures. We attempt to convey to the reader a general high-level overview of the method, focusing on a small number of illustrative applications in areas such as extremal graph theory, Ramsey theory, additive combinatorics, and discrete geometry, and avoiding technical details as much as possible. Abstract: We survey results on counting graphs with given degree sequence, focusing on asymptotic results, and mentioning some of the applications of these results.

The main recent development is the proof of a conjecture that facilitates access to the degree sequence of a random graph via a model incorporating independent binomial random variables. The basic method used in the proof was to examine the changes in the counting function when the degrees are perturbed. We compare with several previous uses of this type of method. Abstract: The so-called graph limit theory is an emerging diverse subject at the meeting point of many different areas of mathematics.

It enables us to view finite graphs as approximations of often more perfect infinite objects. In this survey paper we tell the story of some of the fundamental ideas in structural limit theories and how these ideas led to a general algebraic approach the nilspace approach to higher order Fourier analysis. Abstract: We give a broad survey of recent results in enumerative combinatorics and their complexity aspects.

Abstract: We survey some aspects of the perfect matching problem in hypergraphs, with particular emphasis on structural characterisation of the existence problem in dense hypergraphs and the existence of designs. Mathematical Aspects of Computer Science Complexity theory and design and analysis of algorithms. Formal languages. Computational learning. Algorithmic game theory. Coding theory. Semantics and verification of programs. Symbolic computation.

Quantum computing. Computational geometry, computer vision. Connections with sections 1, 2, 3, 4, 12, 13, At a high level, this framework casts the graph problem at hand as a convex optimization task and then applies to it an appropriate method from the continuous optimization toolkit. We survey how this new approach led to the first in decades progress on the maximum flow problem and then briefly sketch the challenges that still remain. The complexity of an algorithm is measured by the number of queries that it makes. Query complexity is widely used for studying quantum algorithms, for two reasons.

First, it includes many of the known quantum algorithms including Grover's quantum search and a key subroutine of Shor's factoring algorithm. Second, one can prove lower bounds on the query complexity, bounding the possible quantum advantage. In the last few years, there have been major advances on several longstanding problems in the query complexity.

We give an overview of unconditional average-case lower bounds for this problem and its colored variant in a few important restricted classes of Boolean circuits. Abstract: Efficient verification of computation, also known as delegation of computation , is one of the most fundamental notions in computer science, and in particular it lies at the heart of the P vs.

NP question. This article contains a high level overview of the evolution of proofs in computer science, and shows how this evolution is instrumental to solving the problem of delegating computation. We highlight a curious connection between the problem of delegating computation and the notion of no-signaling strategies from quantum physics.

High dimensional estimation problems can be formulated as system of polynomial equalities and inequalities, and thus give rise to natural probability distributions over polynomial systems. Sum of squares proofs not only provide a powerful framework to reason about polynomial systems, but they are constructive in that there exist efficient algorithms to search for sum-of-squares proofs. The efficiency of these algorithms degrade exponentially in the degree of the sum-of-squares proofs.

Understanding and characterizing the power of sum-of-squares proofs for estimation problems has been a subject of intense study in recent years. On one hand, there is a growing body of work utilizing sum-of-squares proofs for recovering solutions to polynomial systems whenever the system is feasible. On the other hand, a broad technique referred to as pseudocalibration has been developed towards showing lower bounds on degree of sum-of-squares proofs.

Finally, the existence of sum-of-squares refutations of a polynomial system has been shown to be intimately connected to the spectrum of associated low-degree matrix valued functions. This talk will survey some of the major developments in the area. The problem has a wide range of applications in machine learning, computer vision, databases and other fields. Over the last two decades many efficient solutions to this problem were developed. In this article we survey these developments, as well as their connections to questions in geometric functional analysis and combinatorial geometry.

This approach has led to the discovery of many meaningful relationships between problems, and to equivalence classes. Research on SETH-based lower bounds has flourished in particular in recent years showing that the classical algorithms are optimal for problems such as Approximate Diameter, Edit Distance, Frechet Distance and Longest Common Subsequence.

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This paper surveys the current progress in this area, and highlights some exciting new developments. Abstract: How many basic operations addition, subtraction, multiplication and division operations are required to compute a given multivariate polynomial? In this talk, we will describe a paradigm for obtaining lower bounds on the number of operations required and apply it to some restricted classes of arithmetic computation. Numerical Analysis and Scientific Computing Design of numerical algorithms and analysis of their accuracy, stability, convergence and complexity.

Approximation theory. Applied and computational aspects of harmonic analysis. Numerical solution of algebraic, functional, stochastic, differential, and integral equations. Connections with sections 8, 9, 10, 12, 14, 16, We briefly recall the theory of Dal Maso—LeFloch—Murat to define weak solutions of nonconservative systems and how it has been used to establish the notion of path-conservative schemes.

Next, a family of high order finite volume methods combining a reconstruction operator and a first order path-conservative scheme is described. Then, the well-balanced property of the proposed methods is analyzed. Finally, some challenging examples on tsunami modeling are shown. Abstract: This lecture serves as an invitation to further studies on nonlocal models, their mathematics, computation, and applications.

We sample our recent attempts in the development of a systematic mathematical framework for nonlocal models, including basic elements of nonlocal vector calculus, well-posedness of nonlocal variational problems, coupling to local models, convergence and compatibility of numerical approximations, and applications to nonlocal mechanics and diffusion. We also draw connections with traditional models and other relevant mathematical subjects. Abstract: The sedimentation of a suspension is a unit operation widely used in mineral processing, chemical engineering, wastewater treatment, and other industrial applications.

Mathematical models that describe these processes and may be employed for simulation, design and control are usually given as nonlinear, time-dependent partial differential equations that in one space dimension include strongly degenerate convection-diffusion-reaction equations with discontinuous coefficients, and in two or more dimensions, coupled flow-transport problems.

These models incorporate non-standard properties that have motivated original research in applied mathematics and numerical analysis. This contribution summarizes recent advances, and presents original numerical results, for three different topics of research: a novel method of flux identification for a scalar conservation law from observation of curved shock trajectories that can be observed in sedimentation in a cone; a new description of continuous sedimentation with reactions including transport and reactions of biological components; and the numerical solution of a multi-dimensional sedimentation-consolidation system by an augmented mixed-primal method, including an a posteriori error estimation.

Abstract: Kinetic modeling and computation face the challenges of multiple scales and uncertainties. Developing efficient multiscale computational methods, and quantifying uncertainties arising in their collision kernels or scattering coefficients, initial or boundary data, forcing terms, geometry, etc. In this article we will report our recent progress in the study of multiscale kinetic equations with uncertainties modelled by random inputs.

We first study the mathematical properties of uncertain kinetic equations, including their regularity and long-time behavior in the random space, and sensitivity of their solutions with respect to the input and scaling parameters. Using the hypocoercivity of kinetic operators, we provide a general framework to study these mathematical properties for general class of linear and nonlinear kinetic equations in various asymptotic regimes.

Abstract: In this paper, we review a set of fast and spectrally accurate methods for rapid evaluation of three dimensional electrostatic and Stokes potentials. The algorithms use the so-called Ewald decomposition and are FFT-based, which makes them naturally most efficient for the triply periodic case. Two key ideas have allowed efficient extension of these Spectral Ewald SE methods to problems with periodicity in only one or two dimensions: an adaptive 3D FFT that apply different upsampling rates locally combined with a new method for FFT based solutions of free space harmonic and biharmonic problems.

The latter approach is also used to extend to the free space case, with no periodicity. For the non-radial kernels of Stokes flow, the structure of their Fourier transform is exploited to extend the applicability from the radial harmonic and biharmonic kernels. A window function is convolved with the point charges to assign values on the FTT grid. Spectral accuracy is attained with a variable number of points in the support of the window function, tuning a shape parameter according to this choice.

A new window function, recently introduced in the context of a non-uniform FFT algorithm, allows for further reduction in the computational time as compared to the truncated Gaussians previously used in the SE method. Abstract: For centuries, many important theories and models of physical phenomena have been characterized by partial differential equations. But numerical methods for approximating such equations have only appeared over the last half century with the emergence of computers.

Principal among these methods are finite elements. Today major challenges remain with the advent of modern computer architectures and the need for massively parallel algorithms. Traditionally the assembling of finite element matrices and the computation of many a posteriori error estimators is obtained by local operators and thus regarded as cheap and of optimal order complexity.

However optimal order complexity is not necessarily equivalent to short run-times, and memory traffic may slow down the execution considerably. Here we discuss several ingredients, such as discretization and solver, for efficient approximations of coupled multi-physics problems. Surrogate finite element operators allow for a fast on-the-fly computation of the stiffness matrix entries in a matrix free setting. A variational crime analysis then yields two-scale a priori estimates.

To balance the dominating components, the scheme is enriched by an adaptive steering based on a hierarchical decomposition of the residual. Several numerical examples illustrate the need for a performance aware numerical analysis. Abstract: In recent years there has been very substantial growth in stochastic modelling in many application areas, and this has led to much greater use of Monte Carlo methods to estimate expected values of output quantities from stochastic simulation.

However, such calculations can be expensive when the cost of individual stochastic simulations is very high. Multilevel Monte Carlo greatly reduces the computational cost by performing most simulations with low accuracy at a correspondingly low cost, with relatively few being performed at high accuracy and a high cost. This article reviews the key ideas behind the multilevel Monte Carlo method. Some applications are discussed to illustrate the flexibility and generality of the approach, and the challenges in its numerical analysis.

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Abstract: Quasicrystals are one kind of fascinating aperiodic structures, and give a strong impact on material science, solid state chemistry, condensed matter physics and soft matters. The theory of quasicrystals, included in aperiodic order, has grown rapidly in mathematical and physical areas over the past few decades.

Many scientific problems have been explored with the efforts of physicists and mathematicians. However, there are still lots of open problems which might to be solved by the close collaboration of physicists, mathematicians and computational mathematicians. In this article, we would like to bridge the physical quasicrystals and mathematical quasicrystals from the perspective of numerical mathematics. Abstract: A large variety of efficient numerical methods, of the finite volume, finite difference and DG type, have been developed for approximating hyperbolic systems of conservation laws.

However, very few rigorous convergence results for these methods are available. We survey the state of the art on this crucial question of numerical analysis by summarizing classical results of convergence to entropy solutions for scalar conservation laws. Very recent results on convergence of ensemble Monte Carlo methods to the measure-valued and statistical solutions of multi-dimensional systems of conservation laws are also presented. Abstract: In this article, we overview recent developments of modern computational methods for the approximate solution of phase-field problems.

The main difficulty for developing a numerical method for phase field equations is a severe stability restriction on the time step due to nonlinearity and high order differential terms. It is known that the phase field models satisfy a nonlinear stability relationship called gradient stability, usually expressed as a time-decreasing free-energy functional. This property has been used recently to derive numerical schemes that inherit the gradient stability. The first part of the article will discuss implicit-explicit time discretizations which satisfy the energy stability.

The second part is to discuss time-adaptive strategies for solving the phase-field problems, which is motivated by the observation that the energy functionals decay with time smoothly except at a few critical time levels. The classical operator-splitting method is a useful tool in time discrtization.

### Duplicate citations

In the final part, we will provide some preliminary results using operator-splitting approach. Control Theory and Optimization Minimization problems. Controllability, observability, stability. Stochastic systems and control. Optimal control. Optimal design, shape design. Linear, non-linear, integer, and stochastic programming. Inverse problems.

Connections with sections 9, 10, 12, 15, Privat and E. Zuazua, concerning the problem of optimizing the shape and location of sensors and actuators for systems whose evolution is driven by a linear partial differential equation. This problem is frequently encountered in applications where one wants to optimally design sensors in order to maximize the quality of the reconstruction of solutions by using only partial observations, or to optimally design actuators in order to control a given process with minimal efforts. For example, we model and solve the following informal question: what is the optimal shape and location of a thermometer?

Note that we want to optimize not only the placement but also the shape of the observation or control subdomain over the class of all possible measurable subsets of the domain having a prescribed Lebesgue measure. By probabilistic considerations we model this optimal design problem as the one of maximizing a spectral functional interpreted as a randomized observability constant, which models optimal observabnility for random initial data. Solving this problem strongly depends on the operator in the PDE model and requires fine knowledge on the asymptotic properties of eigenfunctions of that operator.

For parabolic equations like heat, Stokes or anomalous diffusion equations, we prove the existence and uniqueness of a best domain, proved to be regular enough, and whose algorithmic construction depends in general on a finite number of modes. We describe this methodology and outline some of its applications in various domains. Abstract: We consider inverse problems for hyperbolic equations and systems and the solutions of these problems based on the focusing of waves.

Several inverse problems for linear equations can be solved using control theory. When the coefficients of the modelling equation are unknown, the construction of the point sources requires solving blind control problems. For non-linear equations we consider a new artificial point source method that applies the non-linear interaction of waves to create microlocal points sources inside the unknown medium.

The novel feature of this method is that it utilizes the non-linearity as a tool in imaging, instead of considering it as a difficult perturbation of the system. To demonstrate the method, we consider the non-linear wave equation and the coupled Einstein and scalar field equations. Abstract: The realization that many nondifferentiable functions exhibit some form of structured nonsmoothness has been atracting the efforts of many researchers in the last decades. Identifying theoretically and computationally certain manifolds where a nonsmooth function behaves smoothly poses challenges for the nonsmooth optimization community.

We review a sequence of milestones in the area that led to the development of algorithms of the bundle type that can track the region of smoothnes and mimic a Newton algorithm to converge with superlinear speed. The new generation of bundle methods is sufficiently versatile to deal with structured objective functions, even when the available information is inexact. Abstract: We establish or refute the optimality of inexact second-order methods for unconstrained nonconvex optimization from the point of view of worst-case evaluation complexity, improving and generalizing our previous results.

To this aim, we consider a new general class of inexact second-order algorithms for unconstrained optimization that includes regularization and trust-region variations of Newton's method as well as of their linesearch variants. These examples provide lower bounds on the worst-case evaluation complexity of methods in our class when applied to smooth problems satisfying the relevant assumptions. Abstract: Efficient representations of convex sets are of crucial importance for many algorithms that work with them.

It is well-known that sometimes, a complicated convex set can be expressed as the projection of a much simpler set in higher dimensions called a lift of the original set. This is a brief survey of recent developments in the topic of lifts of convex sets. Our focus will be on lifts that arise from affine slices of real positive semidefinite cones known as psd or spectrahedral lifts. The main result is that projection representations of a convex set are controlled by factorizations, through closed convex cones, of an operator that comes from the convex set.

This leads to several research directions and results that lie at the intersection of convex geometry, combinatorics, real algebraic geometry, optimization, computer science and more. Mathematics in Science and Technology Mathematics and its applications to physical sciences, engineering sciences, life sciences, social and economic sciences, and technology.

Mathematics in interdisciplinary research. The interplay of mathematical modeling, mathematical analysis, and scientific computation, and its impact on the understanding of scientific phenomena and on the solution of real life problems. Connections with sections 9, 10, 11, 12, 14, 15, Abstract: Single-particle cryo-electron microscopy cryo-EM has recently joined X-ray crystallography and NMR spectroscopy as a high-resolution structural method for biological macromolecules.

This paper focuses on the mathematical principles underlying existing algorithms for structure determination using single particle cryo-EM. Abstract: This paper is a review article on semi-supervised and unsupervised graph models for classification using similarity graphs and for community detection in networks.

The paper reviews graph-based variational models built on graph cut metrics. The equivalence between the graph mincut problem and total variation minimization on the graph for an assignment function allows one to cast graph-cut variational problems in the language of total variation minimization, thus creating a parallel between low dimensional data science problems in Euclidean space e. The connection paves the way for new algorithms for data science that have a similar structure to well-known computational methods for nonlinear partial differential equations.

This paper focuses on a class of methods build around diffuse interface models e. Semi-supervised learning with a small amount of training data can be carried out in this framework with diverse applications ranging from hyperspectral pixel classification to identifying activity in police body worn video. It can also be extended to the context of uncertainty quantification with Gaussian noise models.

The problem of community detection in networks also has a graph-cut structure and algorithms are presented for the use of threshold dynamics for modularity optimization. With efficient methods, this allows for the use of network modularity for unsupervised machine learning problems with unknown number of classes. Abstract: In this paper, we beginning by reviewing a certain number of mathematical challenges posed by the modelling of collective dynamics and self-organization.

Then, we focus on two specific problems, first, the derivation of fluid equations from particle dynamics of collective motion and second, the study of phase transitions and the stability of the associated equilibria. Abstract: We present a mathematical view of the structure of matter based on the invariance of the classical equations of physics. Abstract: I will report on recent developments in a class of algorithms, known as threshold dynamics, for computing the motion of interfaces by mean curvature.

These algorithms try to generate the desired interfacial motion just by alternating two very simple operations: Convolution, and thresholding. They can be extended to the multi-phase setting of networks of surfaces, and to motion by weighted anisotropic mean curvature, while maintaining the simplicity of the original version.

These extensions are relevant in applications such as materials science, where they allow large scale simulation of models for microstructure evolution in polycrystals. Mathematics Education and Popularization of Mathematics Range of research and key issues in mathematics education, from elementary school to higher education. Modern developments in effective popularization of mathematics, from publications, to museums, to online communication.

Connections with sections 17 and My goal is to highlight some of their differences. How will I proceed? I could proceed by giving a definition, T, of the term theory and by choosing some differentiating criteria such as c1, c2, etc. Theories, then, could be distinguished in terms of whether or not they include the criteria c1, c2, etc.

However, in this article I will take a different path. In the first part I will focus on a few well-known theories in Mathematics Education and discuss their differences in terms of their theoretical stances. In the last part of the article, I will comment on a sociocultural emergent trend. Abstract: This paper presents a line of research in didactics of mathematics developed during the past decade within the Anthropological Theory of the Didactic around what we call study and research paths SRPs.

SRPs are initially proposed as a study format based on the inquiry of open questions, which can be implemented at all educational levels, from pre-school to university, including teacher education and professional development. Additionally, they provide a general schema for analysing any kind of teaching and learning process, by especially pointing out the more or less explicit questions that lead the study process and the way new knowledge is built or introduced to elaborate answers to these questions.

Current research on SRPs focuses on their didactic ecology , defined as the set of conditions required to generally implement SRPs at different educational levels, together with the constraints that hinder their development and dissemination. History of Mathematics Historical studies of all of the mathematical sciences in all periods and all cultural settings.

## Jean-Christophe Wallet/HomePage/Research interest

Ivasyshen Anatoly N. Myroslav L. Guoan Bi Yonghong Zeng. Approximate Solution of Operator Equations. Krasnosel'skii G. Vainikko R. Zabreyko Ya. Ruticki V. Different Faces of Geometry International Mathematical. Introduction to Analysis of the Infinite Book I.

Leonhard Euler J. Hyperspherical Harmonics Applications in Quantum Theory. Topics in Fixed Point Theory. Infinite Processes Background to Analysis. Mathematical Logic. Carmona J. Mechanics, Boundary Layers and Function Spaces. Diarmuid O'Mathuna. Wong H. Leong H. Encounter with Mathematics. Edwin Hewitt Kenneth Allen Ross. Dajun Guo V.