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Books > Science & Mathematics > Mathematics > Topology > General

Tight Polyhedral Submanifolds and Tight Triangulations (Paperback, 1995 ed.): Wolfgang Kuhnel Tight Polyhedral Submanifolds and Tight Triangulations (Paperback, 1995 ed.)
Wolfgang Kuhnel
R1,066 Discovery Miles 10 660 Ships in 18 - 22 working days

This volume is an introduction and a monograph about tight polyhedra. The treatment of the 2-dimensional case is self- contained and fairly elementary. It would be suitable also for undergraduate seminars. Particular emphasis is given to the interplay of various special disciplines, such as geometry, elementary topology, combinatorics and convex polytopes in a way not found in other books. A typical result relates tight submanifolds to combinatorial properties of their convex hulls. The chapters on higher dimensions generalize the 2-dimensional case using concepts from combinatorics and topology, such as combinatorial Morse theory. A number of open problems is discussed.

Geometric Measure Theory (Paperback, Reprint of the 1st ed. Berlin, Heidelberg, New York 1969): Herbert Federer Geometric Measure Theory (Paperback, Reprint of the 1st ed. Berlin, Heidelberg, New York 1969)
Herbert Federer
R1,824 Discovery Miles 18 240 Ships in 18 - 22 working days

"This book is a major treatise in mathematics and is essential in the working library of the modern analyst." (Bulletin of the London Mathematical Society)

Difference Spaces and Invariant Linear Forms (Paperback, 1994 ed.): Rodney Nillsen Difference Spaces and Invariant Linear Forms (Paperback, 1994 ed.)
Rodney Nillsen
R1,104 Discovery Miles 11 040 Ships in 18 - 22 working days

Difference spaces arise by taking sums of finite or fractional differences. Linear forms which vanish identically on such a space are invariant in a corresponding sense. The difference spaces of L2 (Rn) are Hilbert spaces whose functions are characterized by the behaviour of their Fourier transforms near, e.g., the origin. One aim is to establish connections between these spaces and differential operators, singular integral operators and wavelets. Another aim is to discuss aspects of these ideas which emphasise invariant linear forms on locally compact groups. The work primarily presents new results, but does so from a clear, accessible and unified viewpoint, which emphasises connections with related work.

Dynamical Systems I - Ordinary Differential Equations and Smooth Dynamical Systems (Hardcover, 1st ed 1988. 2nd printing 1994):... Dynamical Systems I - Ordinary Differential Equations and Smooth Dynamical Systems (Hardcover, 1st ed 1988. 2nd printing 1994)
D.V. Anosov; D.V. Anosov; Translated by E.R. Dawson, D. O'Shea; Edited by V. I. Arnol'd; …
R2,669 Discovery Miles 26 690 Ships in 18 - 22 working days

From the reviews: "The reading is very easy and pleasant for the non-mathematician, which is really noteworthy. The two chapters enunciate the basic principles of the field, ... indicate connections with other fields of mathematics and sketch the motivation behind the various concepts which are introduced.... What is particularly pleasant is the fact that the authors are quite successful in giving to the reader the feeling behind the demonstrations which are sketched. Another point to notice is the existence of an annotated extended bibliography and a very complete index. This really enhances the value of this book and puts it at the level of a particularly interesting reference tool. I thus strongly recommend to buy this very interesting and stimulating book." "Journal de Physique"

Handbook of Splines (Hardcover): Gheorghe Micula, Sanda Micula Handbook of Splines (Hardcover)
Gheorghe Micula, Sanda Micula
R3,657 Discovery Miles 36 570 Ships in 18 - 22 working days

The numerous publications on spline theory during recent decades show the importance of its development on modern applied mathematics. The purpose of this text is to give an approach to the theory of spline functions, from the introduction of the phrase "spline" by I.J. Schoenbergin 1946 to the newest theories of spline-wavelets or spline-fractals, emphasizing the significance of the relationship between the general theory and its applications. In addition, this volume provides material on spline function theory, as well as an examination of basic methods in spline functions. The authors have complemented the work with a reference section to stimulate further study.

The Space of Dynamical Systems with the C0-Topology (Paperback, 1994 ed.): Sergei Yu Pilyugin The Space of Dynamical Systems with the C0-Topology (Paperback, 1994 ed.)
Sergei Yu Pilyugin
R1,103 Discovery Miles 11 030 Ships in 18 - 22 working days

This book is an introduction to main methods and principal results in the theory of Co(remark: o is upper index!!)-small perturbations of dynamical systems. It is the first comprehensive treatment of this topic. In particular, Co(upper index!)-generic properties of dynamical systems, topological stability, perturbations of attractors, limit sets of domains are discussed. The book contains some new results (Lipschitz shadowing of pseudotrajectories in structurally stable diffeomorphisms for instance). The aim of the author was to simplify and to "visualize" some basic proofs, so the main part of the book is accessible to graduate students in pure and applied mathematics. The book will also be a basic reference for researchers in various fields of dynamical systems and their applications, especially for those who study attractors or pseudotrajectories generated by numerical methods.

Geometric Theory of Semilinear Parabolic Equations (Paperback, 1st ed. 1981. 3rd printing 1993): Daniel Henry Geometric Theory of Semilinear Parabolic Equations (Paperback, 1st ed. 1981. 3rd printing 1993)
Daniel Henry
R2,091 Discovery Miles 20 910 Ships in 18 - 22 working days
Symbolic Dynamics and Hyperbolic Groups (Paperback, 1993 ed.): Michel Coornaert, Athanase Papadopoulos Symbolic Dynamics and Hyperbolic Groups (Paperback, 1993 ed.)
Michel Coornaert, Athanase Papadopoulos
R1,072 Discovery Miles 10 720 Ships in 18 - 22 working days

Gromov's theory of hyperbolic groups have had a big impact in combinatorial group theory and has deep connections with many branches of mathematics suchdifferential geometry, representation theory, ergodic theory and dynamical systems. This book is an elaboration on some ideas of Gromov on hyperbolic spaces and hyperbolic groups in relation with symbolic dynamics. Particular attention is paid to the dynamical system defined by the action of a hyperbolic group on its boundary. The boundary is most oftenchaotic both as a topological space and as a dynamical system, and a description of this boundary and the action is given in terms of subshifts of finite type. The book is self-contained and includes two introductory chapters, one on Gromov's hyperbolic geometry and the other one on symbolic dynamics. It is intended for students and researchers in geometry and in dynamical systems, and can be used asthe basis for a graduate course on these subjects.

Topological Methods for Ordinary Differential Equations - Lectures given at the 1st Session of the Centro Internazionale... Topological Methods for Ordinary Differential Equations - Lectures given at the 1st Session of the Centro Internazionale Matematico Estivo (C.I.M.E.) held in Montecatini Terme, Italy, June 24-July 2, 1991 (Paperback, 1993 ed.)
Massimo Furi; Patrick Fitzpatrick; Edited by Pietro Zecca; Mario Martelli, Jean Mawhin, …
R1,795 Discovery Miles 17 950 Ships in 18 - 22 working days

The volume contains the texts of four courses, given by the authors at a summer school that sought to present the state of the art in the growing field of topological methods in the theory of o.d.e. (in finite and infinitedimension), and to provide a forum for discussion of the wide variety of mathematical tools which are involved. The topics covered range from the extensions of the Lefschetz fixed point and the fixed point index on ANR's, to the theory of parity of one-parameter families of Fredholm operators, and from the theory of coincidence degree for mappings on Banach spaces to homotopy methods for continuation principles. CONTENTS: P. Fitzpatrick: The parity as an invariant for detecting bifurcation of the zeroes of one parameter families of nonlinear Fredholm maps.- M. Martelli: Continuation principles and boundary value problems.- J. Mawhin: Topological degree and boundary value problems for nonlinear differential equations.- R.D. Nussbaum: The fixed point index and fixed point theorems.

Global Analysis - Studies and Applications V (Paperback, 1992 ed.): Yuri G. Borisovich, Yuri E. Gliklikh Global Analysis - Studies and Applications V (Paperback, 1992 ed.)
Yuri G. Borisovich, Yuri E. Gliklikh
R1,608 Discovery Miles 16 080 Ships in 18 - 22 working days

This volume (a sequel to LNM 1108, 1214, 1334 and 1453) continues the presentation to English speaking readers of the Voronezh University press series on Global Analysis and Its Applications. The papers are selected fromtwo Russian issues entitled "Algebraic questions of Analysis and Topology" and "Nonlinear Operators in Global Analysis." CONTENTS: YuE. Gliklikh: Stochastic analysis, groups of diffeomorphisms and Lagrangian description of viscous incompressible fluid.- A.Ya. Helemskii: From topological homology: algebras with different properties of homological triviality.- V.V. Lychagin, L.V. Zil'bergleit: Duality in stable Spencer cohomologies.- O.R. Musin: On some problems of computational geometry and topology.- V.E. Nazaikinskii, B.Yu. Sternin, V.E.Shatalov: Introduction to Maslov's operational method (non-commutative analysis and differential equations).- Yu.B. Rudyak: The problem of realization of homology classes from Poincare up to the present.- V.G. Zvyagin, N.M. Ratiner: Oriented degree of Fredholm maps of non-negativeindex and its applications to global bifurcation of solutions.- A.A. Bolibruch: Fuchsian systems with reducible monodromy and the Riemann-Hilbert problem.- I.V. Bronstein, A.Ya. Kopanskii: Finitely smooth normal forms of vector fields in the vicinity of a rest point.- B.D. Gel'man: Generalized degree of multi-valued mappings.- G.N. Khimshiashvili: On Fredholmian aspects of linear transmission problems.- A.S. Mishchenko: Stationary solutions of nonlinear stochastic equations.- B.Yu. Sternin, V.E. Shatalov: Continuation of solutions to elliptic equations and localisation of singularities.- V.G. Zvyagin, V.T. Dmitrienko: Properness of nonlinear elliptic differential operators in H-lder spaces.

Geometric Topology: Recent Developments - Lectures given on the 1st Session of the Centro Internazionale Matematico Estivo... Geometric Topology: Recent Developments - Lectures given on the 1st Session of the Centro Internazionale Matematico Estivo (C.I.M.E.) held at Monteca- tini Terme, Italy, June 4-12, 1990 (Paperback, 1991 ed.)
Jeff Cheeger; Edited by Paolo de Bartolomeis; Mikhail Gromov; Edited by Franco Tricerri; Christian Okonek, …
R1,104 Discovery Miles 11 040 Ships in 18 - 22 working days

Geometric Topology can be defined to be the investigation of global properties of a further structure (e.g. differentiable, Riemannian, complex, algebraic etc.) one can impose on a topological manifold. At the C.I.M.E. session in Montecatini, in 1990, three courses of lectures were given onrecent developments in this subject which is nowadays emerging as one of themost fascinating and promising fields of contemporary mathematics. The notesof these courses are collected in this volume and can be described as: 1) the geometry and the rigidity of discrete subgroups in Lie groups especially in the case of lattices in semi-simple groups; 2) the study of the critical points of the distance function and its appication to the understanding of the topology of Riemannian manifolds; 3) the theory of moduli space of instantons as a tool for studying the geometry of low-dimensional manifolds. CONTENTS: J. Cheeger: Critical Points of Distance Functions and Applications to Geometry.- M. Gromov, P. Pansu, Rigidity of Lattices: An Introduction.- Chr. Okonek: Instanton Invariants and Algebraic Surfaces.

Geometric Modelling (Paperback, Softcover reprint of the original 1st ed. 1993): Gerald Farin, H. Hagen, H. Noltemeier, W.... Geometric Modelling (Paperback, Softcover reprint of the original 1st ed. 1993)
Gerald Farin, H. Hagen, H. Noltemeier, W. Knoedel
R1,436 Discovery Miles 14 360 Ships in 18 - 22 working days

In this volume experts from university and industry are presenting new technologies for solving industrial problems as well as important and practicable impulses for new research. The following topics are treated: - solid modelling - geometry processing - feature modelling - product modelling - surfaces over arbitrary topologies - blending methods - scattered data algorithms - smooting and fairing algorithms - NURBS 21 articles are giving a state-of-the-art survey of the relevant problems and issues in the rapidly growing area of geometric modelling.

Axioms and Hulls (Paperback, 1992 ed.): Donald E. Knuth Axioms and Hulls (Paperback, 1992 ed.)
Donald E. Knuth
R1,408 Discovery Miles 14 080 Ships in 18 - 22 working days

One way to advance the science of computational geometry is to make a comprehensive study of fundamental operations that are used in many different algorithms. This monograph attempts such an investigation in the case of two basic predicates: the counterclockwise relation pqr, which states that the circle through points (p, q, r) is traversed counterclockwise when we encounter the points in cyclic order p, q, r, p, ...; and the incircle relation pqrs, which states that s lies inside that circle if pqr is true, or outside that circle if pqr is false. The author, Donald Knuth, is one of the greatest computer scientists of our time. A few years ago, he and some of his students were looking at amap that pinpointed the locations of about 100 cities. They asked, "Which ofthese cities are neighbors of each other?" They knew intuitively that some pairs of cities were neighbors and some were not; they wanted to find a formal mathematical characterization that would match their intuition.This monograph is the result.

Lectures on Vanishing Theorems (Paperback, 1992 ed.): Esnault, Vieweg Lectures on Vanishing Theorems (Paperback, 1992 ed.)
Esnault, Vieweg
R2,408 Discovery Miles 24 080 Ships in 18 - 22 working days

Introduction M. Kodaira's vanishing theorem, saying that the inverse of an ample invert ible sheaf on a projective complex manifold X has no cohomology below the dimension of X and its generalization, due to Y. Akizuki and S. Nakano, have been proven originally by methods from differential geometry ([39J and [1]). Even if, due to J.P. Serre's GAGA-theorems [56J and base change for field extensions the algebraic analogue was obtained for projective manifolds over a field k of characteristic p = 0, for a long time no algebraic proof was known and no generalization to p > 0, except for certain lower dimensional manifolds. Worse, counterexamples due to M. Raynaud [52J showed that in characteristic p > 0 some additional assumptions were needed. This was the state of the art until P. Deligne and 1. Illusie [12J proved the degeneration of the Hodge to de Rham spectral sequence for projective manifolds X defined over a field k of characteristic p > 0 and liftable to the second Witt vectors W2(k). Standard degeneration arguments allow to deduce the degeneration of the Hodge to de Rham spectral sequence in characteristic zero, as well, a re sult which again could only be obtained by analytic and differential geometric methods beforehand. As a corollary of their methods M. Raynaud (loc. cit.) gave an easy proof of Kodaira vanishing in all characteristics, provided that X lifts to W2(k).

Robust Stabilization in the Gap-topology (Paperback): Luc C. G. J. M. Habets Robust Stabilization in the Gap-topology (Paperback)
Luc C. G. J. M. Habets
R1,377 Discovery Miles 13 770 Ships in 18 - 22 working days

The author presents a topological approach to the problem of robustness of dynamic feedback control. First the gap-topology is introduced as a distance measure between systems. In this topology, stability of the closed loop system is a robust property. Furthermore, it is possible to solve the problem of optimally robust control in this setting. The book can be divided into two parts. The first chapters form an introduction to the topological approach towards robust stabilization. Although of theoretical nature, only general mathematical knowledge is required from the reader. The second part is devoted to compensator design. Several algorithms for computing an optimally robust controller in the gap-topology are presented and worked out. Therefore we hope that the book will not only be of interest to theoreticians, but that also practitioners will benefit from it.

Topics in Set Theory - Lebesgue Measurability, Large Cardinals, Forcing Axioms, Rho-functions (Paperback, 1991 ed.): Mohamed... Topics in Set Theory - Lebesgue Measurability, Large Cardinals, Forcing Axioms, Rho-functions (Paperback, 1991 ed.)
Mohamed Bekkali
R1,066 Discovery Miles 10 660 Ships in 18 - 22 working days

During the Fall Semester of 1987, Stevo Todorcevic gave a series of lectures at the University of Colorado. These notes of the course, taken by the author, give a novel and fast exposition of four chapters of Set Theory. The first two chapters are about the connection between large cardinals and Lebesque measure. The third is on forcing axioms such as Martin's axiom or the Proper Forcing Axiom. The fourth chapter looks at the method of minimal walks and p-functions and their applications. The book is addressed to researchers and graduate students interested in Set Theory, Set-Theoretic Topology and Measure Theory.

Stable Homotopy Groups of Spheres - A Computer-Assisted Approach (Paperback, 1990 ed.): Stanley O. Kochman Stable Homotopy Groups of Spheres - A Computer-Assisted Approach (Paperback, 1990 ed.)
Stanley O. Kochman
R1,532 Discovery Miles 15 320 Ships in 18 - 22 working days

A central problem in algebraic topology is the calculation of the values of the stable homotopy groups of spheres +*S. In this book, a new method for this is developed based upon the analysis of the Atiyah-Hirzebruch spectral sequence. After the tools for this analysis are developed, these methods are applied to compute inductively the first 64 stable stems, a substantial improvement over the previously known 45. Much of this computation is algorithmic and is done by computer. As an application, an element of degree 62 of Kervaire invariant one is shown to have order two. This book will be useful to algebraic topologists and graduate students with a knowledge of basic homotopy theory and Brown-Peterson homology; for its methods, as a reference on the structure of the first 64 stable stems and for the tables depicting the behavior of the Atiyah-Hirzebruch and classical Adams spectral sequences through degree 64.

Homotopy Theory and Related Topics - Proceedings of the International Conference held at Kinosaki, Japan, August 19-24, 1988... Homotopy Theory and Related Topics - Proceedings of the International Conference held at Kinosaki, Japan, August 19-24, 1988 (Paperback, 1990 ed.)
Mamoru Mimura
R1,248 Discovery Miles 12 480 Ships in 18 - 22 working days
Minimal Projections in Banach Spaces - Problems of Existence and Uniqueness and their Application (Paperback, 1990 ed.):... Minimal Projections in Banach Spaces - Problems of Existence and Uniqueness and their Application (Paperback, 1990 ed.)
Wlodzimierz Odyniec, Grzegorz Lewicki
R1,090 Discovery Miles 10 900 Ships in 18 - 22 working days

The book is devoted to two natural problems, the existence and unicity of minimal projections in Banach space. Connections are established between the latter and unicity in mathematical programming problems and also with the problem of the characterization of Hilbert spaces. The book also contains a Kolmogorov type criterion for minimal projections and detailed descriptions of the Fourier operators. Presenting both new results and problems for further investigations, this book is addressed to researchers and graduate students interested in geometric functional analysis and to applications.

The Shape of Space (Paperback, 3rd edition): Jeffrey R. Weeks The Shape of Space (Paperback, 3rd edition)
Jeffrey R. Weeks
R1,596 Discovery Miles 15 960 Ships in 9 - 17 working days

The Shape of Space, Third Edition maintains the standard of excellence set by the previous editions. This lighthearted textbook covers the basic geometry and topology of two- and three-dimensional spaces-stretching students' minds as they learn to visualize new possibilities for the shape of our universe. Written by a master expositor, leading researcher in the field, and MacArthur Fellow, its informal exposition and engaging exercises appeal to an exceptionally broad audience, from liberal arts students to math undergraduate and graduate students looking for a clear intuitive understanding to supplement more formal texts, and even to laypeople seeking an entertaining self-study book to expand their understanding of space. Features of the Third Edition: Full-color figures throughout "Picture proofs" have replaced algebraic proofs Simpler handles-and-crosscaps approach to surfaces Updated discussion of cosmological applications Intuitive examples missing from many college and graduate school curricula About the Author: Jeffrey R. Weeks is a freelance geometer living in Canton, New York. With support from the U.S. National Science Foundation, the MacArthur Foundation and several science museums, his work spans pure mathematics, applications in cosmology and-closest to his heart-exposition for the general public.

The Geometry of Iterated Loop Spaces (Paperback, 1st ed. 1972. 2nd printing 1989): J.P. May The Geometry of Iterated Loop Spaces (Paperback, 1st ed. 1972. 2nd printing 1989)
J.P. May
R870 Discovery Miles 8 700 Ships in 18 - 22 working days
Topology and Analysis - The Atiyah-Singer Index Formula and Gauge-Theoretic Physics (Paperback, Softcover reprint of the... Topology and Analysis - The Atiyah-Singer Index Formula and Gauge-Theoretic Physics (Paperback, Softcover reprint of the original 1st ed. 1985)
D.D. Bleecker; B. Booss; Translated by A Mader; D.D. Bleecker
R2,472 Discovery Miles 24 720 Ships in 18 - 22 working days

The Motivation. With intensified use of mathematical ideas, the methods and techniques of the various sciences and those for the solution of practical problems demand of the mathematician not only greater readi ness for extra-mathematical applications but also more comprehensive orientations within mathematics. In applications, it is frequently less important to draw the most far-reaching conclusions from a single mathe matical idea than to cover a subject or problem area tentatively by a proper "variety" of mathematical theories. To do this the mathematician must be familiar with the shared as weIl as specific features of differ ent mathematical approaches, and must have experience with their inter connections. The Atiyah-Singer Index Formula, "one of the deepest and hardest results in mathematics," "probably has wider ramifications in topology and analysis than any other single result" (F. Hirzebruch) and offers perhaps a particularly fitting example for such an introduction to "Mathematics" In spi te of i ts difficulty and immensely rich interrela tions, the realm of the Index Formula can be delimited, and thus its ideas and methods can be made accessible to students in their middle * semesters. In fact, the Atiyah-Singer Index Formula has become progressively "easier" and "more transparent" over the years. The discovery of deeper and more comprehensive applications (see Chapter 111. 4) brought with it, not only a vigorous exploration of its methods particularly in the many facetted and always new presentations of the material by M. F."

Conformal Geometry (Paperback, 1988 Ed.): Ravi S. Kulkarni, Ulrich Pinkall Conformal Geometry (Paperback, 1988 Ed.)
Ravi S. Kulkarni, Ulrich Pinkall
R1,622 Discovery Miles 16 220 Ships in 18 - 22 working days

The contributions in this volume summarize parts of a seminar on conformal geometry which was held at the Max-Planck-Institut fur Mathematik in Bonn during the academic year 1985/86. The intention of this seminar was to study conformal structures on mani folds from various viewpoints. The motivation to publish seminar notes grew out of the fact that in spite of the basic importance of this field to many topics of current interest (low-dimensional topology, analysis on manifolds . . . ) there seems to be no coherent introduction to conformal geometry in the literature. We have tried to make the material presented in this book self-contained, so it should be accessible to students with some background in differential geometry. Moreover, we hope that it will be useful as a reference and as a source of inspiration for further research. Ravi Kulkarni/Ulrich Pinkall Conformal Structures and Mobius Structures Ravi S. Kulkarni* Contents 0 Introduction 2 1 Conformal Structures 4 2 Conformal Change of a Metric, Mobius Structures 8 3 Liouville's Theorem 12 n 4 The GroupsM(n) andM(E ) 13 5 Connection with Hyperbol ic Geometry 16 6 Constructions of Mobius Manifolds 21 7 Development and Holonomy 31 8 Ideal Boundary, Classification of Mobius Structures 35 * Partially supported by the Max-Planck-Institut fur Mathematik, Bonn, and an NSF grant. 2 O Introduction (0. 1) Historically, the stereographic projection and the Mercator projection must have appeared to mathematicians very startling."

Hodge Cycles, Motives, and Shimura Varieties (English, French, Paperback, 1st Corrected ed. 1982. Corr. 2nd printing 0): Pierre... Hodge Cycles, Motives, and Shimura Varieties (English, French, Paperback, 1st Corrected ed. 1982. Corr. 2nd printing 0)
Pierre Deligne, James S. Milne, Arthur Ogus, Kuang-Yen Shih
R2,244 Discovery Miles 22 440 Ships in 18 - 22 working days

This volume collects six related articles. The first is the notes (written by J.S. Milne) of a major part of the seminar "Periodes des Int grales Abeliennes" given by P. Deligne at I'.B.E.S., 1978-79. The second article was written for this volume (by P. Deligne and J.S. Milne) and is largely based on: N Saavedra Rivano, Categories tannakiennes, Lecture Notes in Math. 265, Springer, Heidelberg 1972. The third article is a slight expansion of part of: J.S. Milne and Kuang-yen Shih, Sh ura varieties: conjugates and the action of complex conjugation 154 pp. (Unpublished manuscript, October 1979). The fourth article is based on a letter from P. De1igne to R. Langlands, dated 10th April, 1979, and was revised and completed (by De1igne) in July, 1981. The fifth article is a slight revision of another section of the manuscript of Milne and Shih referred to above. The sixth article, by A. Ogus, dates from July, 1980.

Critical Point Theory and Submanifold Geometry (Paperback, 1988 ed.): Richard S. Palais, Chuu-Lian Terng Critical Point Theory and Submanifold Geometry (Paperback, 1988 ed.)
Richard S. Palais, Chuu-Lian Terng
R1,830 Discovery Miles 18 300 Ships in 18 - 22 working days
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