Search result: Catalogue data in Spring Semester 2021

Mathematics Bachelor Information
Electives
Selection: Algebra, Number Thy, Topology, Discrete Mathematics, Logic
NumberTitleTypeECTSHoursLecturers
401-3058-00LCombinatorics IW4 credits2GN. Hungerbühler
AbstractThe course Combinatorics I and II is an introduction into the field of enumerative combinatorics.
ObjectiveUpon completion of the course, students are able to classify combinatorial problems and to apply adequate techniques to solve them.
ContentContents of the lectures Combinatorics I and II: congruence transformation of the plane, symmetry groups of geometric figures, Euler's function, Cayley graphs, formal power series, permutation groups, cycles, Bunside's lemma, cycle index, Polya's theorems, applications to graph theory and isomers.
Prerequisites / NoticeRecognition of credits as an elective course in the Mathematics Bachelor's or Master's Programmes is only possible if you have not received credits for the course unit 401-3052-00L Combinatorics (which was for the last time taught in the spring semester 2008).
401-3109-65LProbabilistic Number Theory Information W8 credits4GE. Kowalski
AbstractThe course presents some results of probabilistic number theory in a unified manner, including distribution properties of the number of prime divisors of integers, probabilistic properties of the zeta function and statistical distribution of exponential sums.
ObjectiveThe goal of the course is to present some results of probabilistic number theory in a unified manner.
ContentThe main concepts will be presented in parallel with the proof of a few main theorems:
(1) the Erdős-Wintner and Erdős-Kac theorems concerning the distribution of values of arithmetic functions;
(2) the distribution of values of the Riemann zeta function, including Selberg's central limit theorem for the Riemann zeta function on the critical line;
(3) the Chebychev bias for primes in arithmetic progressions;
(4) functional limit theorems for the paths of partial sums of families of exponential sums.
Lecture notesThe lecture notes for the class are available at

https://www.math.ethz.ch/~kowalski/probabilistic-number-theory.pdf
Prerequisites / NoticePrerequisites: Complex analysis, measure and integral, and at least the basic language of probability theory (the main concepts, such as convergence in law, will be recalled).
Some knowledge of number theory is useful but the main results will also be summarized.
401-3362-21LSpectral Theory of Eisenstein Series Information W4 credits2VP. D. Nelson
AbstractWe plan to discuss the basic theory of Eisenstein series and the spectral decomposition of the space of automorphic forms, with focus on the groups GL(2) and GL(n).
Objective
Prerequisites / NoticeSome familiarity with basics on Lie groups and functional analysis would be helpful, and some prior exposure to modular forms or homogeneous spaces may provide useful motivation.
401-4116-12LLectures on Drinfeld Modules Information W6 credits3VR. Pink
AbstractDrinfeld modules: Basic theory, analytic uniformization, moduli spaces, good/bad/semistable reduction, Tate modules, Galois representations, endomorphism rings, etc.
Objective
ContentA central role in the arithmetic of fields of positive characteristic p is played by the Frobenius map x ---> x^p. The theory of Drinfeld modules exploits this map in a systematic fashion. Drinfeld modules of rank 1 can be viewed as analogues of the multiplicative group and are used in the class field theory of global function fields. Drinfeld modules of arbitrary rank possess a rich theory which has many aspects in common with that of elliptic curves, including analytic uniformization, moduli spaces, good/bad/semistable reduction, Tate modules, Galois representations.

A full understanding of Drinfeld modules requires some knowledge in the arithmetic of function fields and, for comparison, the arithmetic of elliptic curves, which cannot all be presented in the framework of this course. Relevant results from these areas will be presented only cursorily when they are needed, but a fair amount of the theory can be developed without them.
LiteratureDrinfeld, V. G.: Elliptic modules (Russian), Mat. Sbornik 94 (1974), 594--627, translated in Math. USSR Sbornik 23 (1974), 561--592.

Deligne, P., Husemöller, D: Survey of Drinfeld modules, Contemp. Math. 67, 1987, 25-91.

Goss, D.: Basic structures in function field arithmetic. Springer-Verlag, 1996.

Drinfeld modules, modular schemes and applications. Proceedings of the workshop held in Alden-Biesen, September 9¿14, 1996. Edited by E.-U. Gekeler, M. van der Put, M. Reversat and J. Van Geel. World Scientific Publishing Co., Inc., River Edge, NJ, 1997.

Thakur, Dinesh S.: Function field arithmetic. World Scientific Publishing Co., Inc., River Edge, NJ, 2004.

Further literature will be indicated during the course
Selection: Geometry
NumberTitleTypeECTSHoursLecturers
401-4118-21LSpectral Theory of Hyperbolic Surfaces Information W4 credits2VC. Burrin
AbstractThe Laplacian plays a prominent role in many parts of mathematics. On a flat surface like the torus, understanding its spectrum is the topic of Fourier analysis, whose 19th century development allowed to solve the heat and wave equations. On the sphere, one studies spherical harmonics. In this course, we will study the spectrum of hyperbolic surfaces and its Maass forms (eigenfunctions).
ObjectiveWe will start from scratch, with an overview of hyperbolic geometry and harmonic analysis on the hyperbolic plane. The objectives are to prove the spectral theorem and Selberg's trace formula, and explore applications in geometry and number theory.
ContentTentative syllabus:
Hyperbolic geometry (the hyperbolic plane and Fuchsian groups)
Construction of arithmetic hyperbolic surfaces
Harmonic analysis on the hyperbolic plane
The spectral theorem
Selberg's trace formula
Applications in geometry (isoperimetric inequalities, geodesic length spectrum)
and number theory (links to the Riemann zeta function and Riemann hypothesis)

Possible further topics (if time permits):
Eisenstein series
Explicit constructions of Maass forms (after Maass)
A special case of the Jacquet-Langlands correspondence (after the exposition of Bergeron, see references)
LiteratureNicolas Bergeron, The Spectrum of Hyperbolic Surfaces, Springer Universitext 2011.
Armand Borel, Automorphic forms on SL(2,R), Cambridge University Press 1997.
Peter Buser, Geometry and spectra of compact Riemann surfaces, Birkhäuser 1992.
Henryk Iwaniec, Spectral methods of automorphic forms. Graduate studies in mathematics, AMS 2002.
Prerequisites / NoticeKnowledge of the material covered in the first two years of bachelor studies is assumed. Prior knowledge of differential geometry, functional analysis, or Riemann surfaces is not required.
401-4206-17LGroups Acting on Trees Information W6 credits3GB. Brück
AbstractAs a main theme, we will see how an action of a group on a tree enables us to break the group into smaller pieces, and thus gain better understanding of its structure.
ObjectiveLearn basics of Bass-Serre theory; get to know concepts from geometric group theory.
ContentAs a mathematical object, a tree is a graph without any loops. It turns out that if a group acts on such an object, the algebraic structure of the group has a nice description in terms of the combinatorics of the graph. In particular, groups acting on trees can be decomposed in a certain way into simpler pieces.These decompositions can be described combinatorially, but are closely related to concepts from topology such as fundamental groups and covering spaces.

This interplay between (elementary) concepts of algebra, combinatorics and geometry/topology is typical for geometric group theory. The course can also serve as an introduction to basic concepts of this field.

Topics that will be covered in the lecture include:
- Trees and their automorphisms
- Different characterisations of free groups
- Amalgamated products and HNN extensions
- Graphs of groups
- Kurosh's theorem on subgroups of free (amalgamated) products
LiteratureJ.-P. Serre, Trees. (Translated from the French by John Stillwell). Springer-Verlag, 1980. ISBN 3-540-10103-9

O. Bogopolski. Introduction to group theory. EMS Textbooks in Mathematics. European Mathematical Society (EMS), Zürich, 2008. x+177 pp. ISBN: 978-3-03719-041-8

C. T. C. Wall. The geometry of abstract groups and their splittings. Revista Matemática Complutense vol. 16(2003), no. 1, pp. 5-101
Prerequisites / NoticeBasic knowledge of group theory; being familiar with fundamental groups (e.g. the Seifert-van-Kampen Theorem) and covering theory is definitely helpful, although not strictly necessary.
In particular, the standard material of the first two years of the Mathematics Bachelor is sufficient.
401-3056-00LFinite Geometries I
Does not take place this semester.
W4 credits2GN. Hungerbühler
AbstractFinite geometries I, II: Finite geometries combine aspects of geometry, discrete mathematics and the algebra of finite fields. In particular, we will construct models of axioms of incidence and investigate closing theorems. Applications include test design in statistics, block design, and the construction of orthogonal Latin squares.
ObjectiveFinite geometries I, II: Students will be able to construct and analyse models of finite geometries. They are familiar with closing theorems of the axioms of incidence and are able to design statistical tests by using the theory of finite geometries. They are able to construct orthogonal Latin squares and know the basic elements of the theory of block design.
ContentFinite geometries I, II: finite fields, rings of polynomials, finite affine planes, axioms of incidence, Euler's thirty-six officers problem, design of statistical tests, orthogonal Latin squares, transformation of finite planes, closing theorems of Desargues and Pappus-Pascal, hierarchy of closing theorems, finite coordinate planes, division rings, finite projective planes, duality principle, finite Moebius planes, error correcting codes, block design
Literature- Max Jeger, Endliche Geometrien, ETH Skript 1988

- Albrecht Beutelspacher: Einführung in die endliche Geometrie I,II. Bibliographisches Institut 1983

- Margaret Lynn Batten: Combinatorics of Finite Geometries. Cambridge University Press

- Dembowski: Finite Geometries.
401-3574-61LIntroduction to Knot Theory Information
Does not take place this semester.
W6 credits3G
AbstractIntroduction to the mathematical theory of knots. We will discuss some elementary topics in knot theory and we will repeatedly centre on how this knowledge can be used in secondary school.
ObjectiveThe aim of this lecture course is to give an introduction to knot theory. In the course we will discuss the definition of a knot and what is meant by equivalence. The focus of the course will be on knot invariants. We will consider various knot invariants amongst which we will also find the so called knot polynomials. In doing so we will again and again show how this knowledge can be transferred down to secondary school.
ContentDefinition of a knot and of equivalent knots.
Definition of a knot invariant and some elementary examples.
Various operations on knots.
Knot polynomials (Jones, ev. Alexander.....)
LiteratureAn extensive bibliography will be handed out in the course.
Prerequisites / NoticePrerequisites are some elementary knowledge of algebra and topology.
Selection: Analysis
NumberTitleTypeECTSHoursLecturers
401-3378-19LEntropy in Dynamics Information W8 credits4GM. Einsiedler
AbstractDefinition and basic property of measure theoretic dynamical entropy (elementary and conditionally). Ergodic theorem for entropy. Topological entropy and variational principle. Measures of maximal entropy. Equidistribution of periodic points. Measure rigidity for commuting maps on the circle group.
ObjectiveThe course will lead to a firm understanding of measure theoretic dynamical entropy and its applications within dynamics. We will start with the basic properties of (conditional) entropy, relate it to the question of effective coding techniques, discuss and prove the Shannon-McMillan-Breiman theorem that is also known as the ergodic theorem for entropy. Moreover, we will discuss a topological counter part and relate this topological entropy to the measure theoretic entropy by the variational principle. We will use these methods to classify certain natural homogeneous measures, prove equidistribution of periodic points on compact quotients of hyperbolic surfaces, and establish a measure rigidity theorem for commuting maps on the circle group.
Lecture notesEntropy book under construction, available online under
https://tbward0.wixsite.com/books/entropy
Prerequisites / NoticeNo prior knowledge of dynamical systems will be assumed but measure theory will be assumed and very important. Doctoral students are welcome to attend the course for 2KP.
Selection: Numerical Analysis
NumberTitleTypeECTSHoursLecturers
401-3426-21LTime-Frequency AnalysisW4 credits2GR. Alaifari
AbstractThis course gives a basic introduction to time-frequency analysis from the viewpoint of applied harmonic analysis.
ObjectiveBy the end of the course students should be familiar with the concept of the short-time Fourier transform, the Bargmann transform, quadratic time-frequency representations (ambiguity function and Wigner distribution), Gabor frames and modulation spaces. The connection and comparison to time-scale representations will also be subject of this course.
ContentTime-frequency analysis lies at the heart of many applications in signal processing and aims at capturing time and frequency information simultaneously (as opposed to the classical Fourier transform). This course gives a basic introduction that starts with studying the short-time Fourier transform and the special role of the Gauss window. We will visit quadratic representations and then focus on discrete time-frequency representations, where Gabor frames will be introduced. Later, we aim at a more quantitative analysis of time-frequency information through modulation spaces. At the end, we touch on wavelets (time-scale representation) as a counterpart to the short-time Fourier transform.
LiteratureGröchenig, K. (2001). Foundations of time-frequency analysis. Springer Science & Business Media.
Prerequisites / NoticeFunctional analysis, Fourier analysis, complex analysis, operator theory
Selection: Probability Theory, Statistics
NumberTitleTypeECTSHoursLecturers
401-6102-00LMultivariate Statistics
Does not take place this semester.
W4 credits2Gnot available
AbstractMultivariate Statistics deals with joint distributions of several random variables. This course introduces the basic concepts and provides an overview over classical and modern methods of multivariate statistics. We will consider the theory behind the methods as well as their applications.
ObjectiveAfter the course, you should be able to:
- describe the various methods and the concepts and theory behind them
- identify adequate methods for a given statistical problem
- use the statistical software "R" to efficiently apply these methods
- interpret the output of these methods
ContentVisualization / Principal component analysis / Multidimensional scaling / The multivariate Normal distribution / Factor analysis / Supervised learning / Cluster analysis
Lecture notesNone
LiteratureThe course will be based on class notes and books that are available electronically via the ETH library.
Prerequisites / NoticeTarget audience: This course is the more theoretical version of "Applied Multivariate Statistics" (401-0102-00L) and is targeted at students with a math background.

Prerequisite: A basic course in probability and statistics.

Note: The courses 401-0102-00L and 401-6102-00L are mutually exclusive. You may register for at most one of these two course units.
401-4626-00LAdvanced Statistical Modelling: Mixed Models
Does not take place this semester.
W4 credits2VM. Mächler
AbstractMixed Models = (*| generalized| non-) linear Mixed-effects Models, extend traditional regression models by adding "random effect" terms.

In applications, such models are called "hierarchical models", "repeated measures" or "split plot designs". Mixed models are widely used and appropriate in an aera of complex data measured from living creatures from biology to human sciences.
Objective- Becoming aware how mixed models are more realistic and more powerful in many cases than traditional ("fixed-effects only") regression models.

- Learning to fit such models to data correctly, critically interpreting results for such model fits, and hence learning to work the creative cycle of responsible statistical data analysis:
"fit -> interpret & diagnose -> modify the fit -> interpret & ...."

- Becoming aware of computational and methodological limitations of these models, even when using state-of-the art software.
ContentThe lecture will build on various examples, use R and notably the `lme4` package, to illustrate concepts. The relevant R scripts are made available online.

Inference (significance of factors, confidence intervals) will focus on the more realistic *un*balanced situation where classical (ANOVA, sum of squares etc) methods are known to be deficient. Hence, Maximum Likelihood (ML) and its variant, "REML", will be used for estimation and inference.
Lecture notesWe will work with an unfinished book proposal from Prof Douglas Bates, Wisconsin, USA which itself is a mixture of theory and worked R code examples.

These lecture notes and all R scripts are made available from
https://github.com/mmaechler/MEMo
Literature(see web page and lecture notes)
Prerequisites / Notice- We assume a good working knowledge about multiple linear regression ("the general linear model') and an intermediate (not beginner's) knowledge about model based statistics (estimation, confidence intervals,..).

Typically this means at least two classes of (math based) statistics, say
1. Intro to probability and statistics
2. (Applied) regression including Matrix-Vector notation Y = X b + E

- Basic (1 semester) "Matrix calculus" / linear algebra is also assumed.

- If familiarity with [R](https://www.r-project.org/) is not given, it should be acquired during the course (by the student on own initiative).
401-4627-00LEmpirical Process Theory and ApplicationsW4 credits2VS. van de Geer
AbstractEmpirical process theory provides a rich toolbox for studying the properties of empirical risk minimizers, such as least squares and maximum likelihood estimators, support vector machines, etc.
Objective
ContentIn this series of lectures, we will start with considering exponential inequalities, including concentration inequalities, for the deviation of averages from their mean. We furthermore present some notions from approximation theory, because this enables us to assess the modulus of continuity of empirical processes. We introduce e.g., Vapnik Chervonenkis dimension: a combinatorial concept (from learning theory) of the "size" of a collection of sets or functions. As statistical applications, we study consistency and exponential inequalities for empirical risk minimizers, and asymptotic normality in semi-parametric models. We moreover examine regularization and model selection.
401-4637-67LOn Hypothesis TestingW4 credits2VF. Balabdaoui
AbstractThis course is a review of the main results in decision theory.
ObjectiveThe goal of this course is to present a review for the most fundamental results in statistical testing. This entails reviewing the Neyman-Pearson Lemma for simple hypotheses and the Karlin-Rubin Theorem for monotone likelihood ratio parametric families. The students will also encounter the important concept of p-values and their use in some multiple testing situations. Further methods for constructing tests will be also presented including likelihood ratio and chi-square tests. Some non-parametric tests will be reviewed such as the Kolmogorov goodness-of-fit test and the two sample Wilcoxon rank test. The most important theoretical results will reproved and also illustrated via different examples. Four sessions of exercises will be scheduled (the students will be handed in an exercise sheet a week before discussing solutions in class).
Literature- Statistical Inference (Casella & Berger)
- Testing Statistical Hypotheses (Lehmann and Romano)
Selection: Financial and Insurance Mathematics
NumberTitleTypeECTSHoursLecturers
401-3888-00LIntroduction to Mathematical Finance Information
A related course is 401-3913-01L Mathematical Foundations for Finance (3V+2U, 4 ECTS credits). Although both courses can be taken independently of each other, only one will be recognised for credits in the Bachelor and Master degree. In other words, it is not allowed to earn credit points with one for the Bachelor and with the other for the Master degree.
W10 credits4V + 1UD. Possamaï
AbstractThis is an introductory course on the mathematics for investment, hedging, portfolio management, asset pricing and financial derivatives in discrete-time financial markets. We discuss arbitrage, completeness, risk-neutral pricing and utility maximisation. We prove the fundamental theorem of asset pricing and the hedging duality theorems, and also study convex duality in utility maximization.
ObjectiveThis is an introductory course on the mathematics for investment, hedging, portfolio management, asset pricing and financial derivatives in discrete-time financial markets. We discuss arbitrage, completeness, risk-neutral pricing and utility maximisation, and maybe other topics. We prove the fundamental theorem of asset pricing and the hedging duality theorems in discrete time, and also study convex duality in utility maximization.
ContentThis course focuses on discrete-time financial markets. It presumes a knowledge of measure-theoretic probability theory (as taught e.g. in the course "Probability Theory"). The course is offered every year in the Spring semester.

This course is the first of a sequence of two courses on mathematical finance. The second course "Mathematical Finance" (MF II), 401-4889-00, focuses on continuous-time models. It is advisable that the present course, MF I, is taken prior to MF II.

For an overview of courses offered in the area of mathematical finance, see Link.
Lecture notesThe course is based on different parts from different textbooks as well as on original research literature. Lecture notes will not be available.
LiteratureLiterature:

Michael U. Dothan, "Prices in Financial Markets", Oxford University Press

Hans Föllmer and Alexander Schied, "Stochastic Finance: An Introduction in Discrete Time", de Gruyter

Marek Capinski and Ekkehard Kopp, "Discrete Models of Financial Markets", Cambridge University Press

Robert J. Elliott and P. Ekkehard Kopp, "Mathematics of Financial Markets", Springer
Prerequisites / NoticeA related course is "Mathematical Foundations for Finance" (MFF), 401-3913-01. Although both courses can be taken independently of each other, only one will be given credit points for the Bachelor and the Master degree. In other words, it is also not possible to earn credit points with one for the Bachelor and with the other for the Master degree.

This course is the first of a sequence of two courses on mathematical finance. The second course "Mathematical Finance" (MF II), 401-4889-00, focuses on continuous-time models. It is advisable that the present course, MF I, is taken prior to MF II.

For an overview of courses offered in the area of mathematical finance, see Link.
401-3629-00LQuantitative Risk Management Information W4 credits2V + 1UP. Cheridito
AbstractThis course introduces methods from probability theory and statistics that can be used to model financial risks. Topics addressed include loss distributions, risk measures, extreme value theory, multivariate models, copulas, dependence structures and operational risk.
ObjectiveThe goal is to learn the most important methods from probability theory and statistics used in financial risk modeling.
Content1. Introduction
2. Basic Concepts in Risk Management
3. Empirical Properties of Financial Data
4. Financial Time Series
5. Extreme Value Theory
6. Multivariate Models
7. Copulas and Dependence
8. Operational Risk
Lecture notesCourse material is available on https://people.math.ethz.ch/~patrickc/qrm
LiteratureQuantitative Risk Management: Concepts, Techniques and Tools
AJ McNeil, R Frey and P Embrechts
Princeton University Press, Princeton, 2015 (Revised Edition)
http://press.princeton.edu/titles/10496.html
Prerequisites / NoticeThe course corresponds to the Risk Management requirement for the SAA ("Aktuar SAV Ausbildung") as well as for the Master of Science UZH-ETH in Quantitative Finance.
401-3923-00LSelected Topics in Life Insurance MathematicsW4 credits2VM. Koller
AbstractStochastic Models for Life insurance
1) Markov chains
2) Stochastic Processes for demography and interest rates
3) Cash flow streams and reserves
4) Mathematical Reserves and Thiele's differential equation
5) Theorem of Hattendorff
6) Unit linked policies
Objective
401-3917-00LStochastic Loss Reserving MethodsW4 credits2VR. Dahms
AbstractLoss Reserving is one of the central topics in non-life insurance. Mathematicians and actuaries need to estimate adequate reserves for liabilities caused by claims. These claims reserves have influence all financial statements, future premiums and solvency margins. We present the stochastics behind various methods that are used in practice to calculate those loss reserves.
ObjectiveOur goal is to present the stochastics behind various methods that are used in prctice to estimate claim reserves. These methods enable us to set adequate reserves for liabilities caused by claims and to determine prediction errors of these predictions.
ContentWe will present the following stochastic claims reserving methods/models:
- Stochastic Chain-Ladder Method
- Bayesian Methods, Bornhuetter-Ferguson Method, Credibility Methods
- Distributional Models
- Linear Stochastic Reserving Models, with and without inflation
- Bootstrap Methods
- Claims Development Result (solvency view)
- Coupling of portfolios
LiteratureM. V. Wüthrich, M. Merz, Stochastic Claims Reserving Methods in Insurance, Wiley 2008.
Prerequisites / NoticeThe exams ONLY take place during the official ETH examination periods.

This course will be held in English and counts towards the diploma "Aktuar SAV".
For the latter, see details under www.actuaries.ch.

Basic knowledge in probability theory is assumed, in particular conditional expectations.
401-3956-00LEconomic Theory of Financial Markets Restricted registration - show details W4 credits2VM. V. Wüthrich
AbstractThis lecture provides an introduction to the economic theory of financial markets. It presents the basic financial and economic concepts to insurance mathematicians and actuaries.
ObjectiveThis lecture aims at providing the fundamental financial and economic concepts to insurance mathematicians and actuaries. It focuses on portfolio theory, cash flow valuation and deflator techniques.
ContentWe treat the following topics:
- Fundamental concepts in economics
- Portfolio theory
- Mean variance analysis, capital asset pricing model
- Arbitrage pricing theory
- Cash flow theory
- Valuation principles
- Stochastic discounting, deflator techniques
- Interest rate modeling
- Utility theory
Prerequisites / NoticeThe exams ONLY take place during the official ETH examination period.

This course will be held in English and counts towards the diploma of "Aktuar SAV". For the latter, see details under www.actuaries.ch.

Knowledge in probability theory, stochastic processes and statistics is assumed.
401-3936-00LData Analytics for Non-Life Insurance Pricing Restricted registration - show details W4 credits2VC. M. Buser, M. V. Wüthrich
AbstractWe study statistical methods in supervised learning for non-life insurance pricing such as generalized linear models, generalized additive models, Bayesian models, neural networks, classification and regression trees, random forests and gradient boosting machines.
ObjectiveThe student is familiar with classical actuarial pricing methods as well as with modern machine learning methods for insurance pricing and prediction.
ContentWe present the following chapters:
- generalized linear models (GLMs)
- generalized additive models (GAMs)
- neural networks
- credibility theory
- classification and regression trees (CARTs)
- bagging, random forests and boosting
Lecture notesThe lecture notes are available from:
https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2870308
Prerequisites / NoticeThis course will be held in English and counts towards the diploma of "Aktuar SAV".
For the latter, see details under www.actuaries.ch

Good knowledge in probability theory, stochastic processes and statistics is assumed.
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