Contents PREFACE to The Mathematica GuideBooks CHAPTER Introduction and Orientation 0.1 Overview 0.1.1 0.1.2 0.1.3 0.1.4 0.2 Requirements 0.2.1 0.2.2 0.3 Exercises Solutions The Books Versus the Electronic Components 0.5.1 0.5.2 0.5.3 0.6 Doing Computer Mathematics Programming Paradigms Exercises and Solutions 0.4.1 0.4.2 0.5 Hardware and Software Reader Prerequisites What the GuideBooks Are and What They Are Not 0.3.1 0.3.2 0.4 Content Summaries Relation of the Four Volumes Chapter Structure Code Presentation Style Working with the Notebook Reproducibility of the Results Earlier Versions of the Notebooks Style and Design Elements 0.6.1 0.6.2 0.6.3 0.6.4 0.6.5 0.6.6 0.6.7 Text and Code Formatting References Variable Scoping, Input Numbering, and Warning Messages Graphics Notations and Symbols Units Cover Graphics 0.7 Production History 0.8 Four General Suggestions © 2004, 2005 Springer Science+Business Media, Inc Printed from THE MATHEMATICA GUIDEBOOKS @ @ References P R O G R A M M I N G CHAPTER Introduction to Mathematica 1.0 Remarks 1.1 Basics of Mathematica as a Programming Language 1.1.1 General Background In and Out Numbering † General Naming, Spelling, and Capitalization Conventions for Symbols † Options and Option Settings † Messages † Add-On Packages 1.1.2 Elementary Syntax Common Shortcuts † Parentheses, Braces, and Brackets † Comments Inside Code † Font Usage † Referring to Outputs † Functional Programming Style † “Ideal” Formatting 1.2 Introductory Examples 1.2.0 1.2.1 Remarks Numerical Computations Periodic Continued Fractions † Pisot Numbers † Fast Integer Arithmetic † Digit Sums † Numerical Integration † Numerical ODE Solving † Burridge–Knopoff Earthquake Model † Trajectories in a Random Two-Dimensional Potential † Numerical PDE Solving † Benney PDE † Sierpinski Triangle-Generating PDE † Monitoring Numerical Algorithms † Hilbert Matrices † Distances between Matrix Eigenvalues † Special Functions of Mathematical Physics † Sums and Products † Computing a High-Precision Value for Euler’s Constant g † Numerical Root-Finding † Roots of Polynomials † Jensen Disks † De Rham’s Function † Logistic Map † Built-in PseudoCompiler † Forest Fire Model † Iterated Digit Sums † Modeling a Sinai Billiard 1.2.2 Graphics Gibbs Phenomena † Fourier Series of Products of Discontinuous Functions † Dirichlet Function † Counting Digits † Apollonius Circles † Generalized Weierstrass Function † 3D Plots † Plotting Parametrized Surfaces † Plotting Implicitly Defined Surfaces † Graphics-Objects as Mathematica Expressions † Kepler Tiling † Fractal Post Sign † Polyhedral Flowers † Gauss Map Animation † Random Polyehdra © 2004, 2005 Springer Science+Business Media, Inc THE MATHEMATICA GUIDEBOOKS to PROGRAMMING—GRAPHICS—NUMERICS—SYMBOLICS 1.2.3 Symbolic Calculations Differentiation † Integration † Symbolic Solutions of ODEs † Vandermonde Matrix † LU Decomposition of a Vandermonde Matrix † Redheffer Matrix † Symbolic Representations of Polynomial Roots † Solving Systems of Polynomials † Eliminating Variables from Polynomial Systems † Series Expansions † L’Hôspital’s Rule † Radical Expressions of Trigonometric Function Values † Prime Factorizations † Symbolic Summation † Proving Legendre’s Elliptic Integral Identity † Geometric Theorem Proofs Using Gröbner Bases † Medial Parallelograms † Inequality Solving † Symbolic Description of a Thickened Lissajous Curve † Simplifications under Assumptions † Numbers with Identical Digits in the Decimal and Continued Fraction Expansions † Conformal Map of a Square to the Unit Disk † Vortex Motion in a Rectangle † Magnetic Field of a Magnet with Air Gap † Localized Propagating Solution of the Maxwell Equation † Customized Notations † Schmidt Decomposition of a Two-Particle State 1.2.4 Programming Large Calculations † Partitioning Integers † Binary Splitting-Based Fast Factorial † Bolyai Expansion in Nested Radicals † Defining Pfaffians † Bead Sort Algorithm † Structure of Larger Programs † Making Platonic Solids from Tori † Equipotential Surfaces of a Charged Icosahedral Wireframe † Tube along a 3D Hilbert Curve 1.3 What Computer Algebra and Mathematica 5.1 Can and Cannot Do What Mathematica Does Well † What Mathematica Does Reasonably Well † What Mathematica Cannot Do † Package Proposals † What Mathematica Is and What Mathematica Not Is † Impacts of Computer Algebra † Relevant Quotes † Computer Algebra and Human Creativity † New Opportunities Opened by Computer Algebra † Computer Mathematics—The Joy Now and the Joy to Come @ @ Exercises @ @ Solutions Computing Wishes and Proposals † Computer Algebra Systems 100 Proposals for Problems to Tackle † Sources of Interesting and Challenging Problems † ISSAC Challenge Problems † 100$–100Digit Challenge © 2004, 2005 Springer Science+Business Media, Inc Printed from THE MATHEMATICA GUIDEBOOKS @ @ References CHAPTER Structure of Mathematica Expressions 2.0 Remarks 2.1 Expressions Everything Is an Expression † Hierarchical Structure of Symbolic Expressions † Formatting Possibilities † Traditional Mathematics Notation versus Computer Mathematics Notation † Typeset Forms † Heads and Arguments † Symbols † Nested Heads † Input Form and the Formatting of Programs 2.2 Simple Expressions 2.2.1 Numbers and Strings Formatting Fractions † Integers † Autosimplifications † Rational Numbers † Approximate Numbers † Real Numbers † Complex Numbers † Autonumericalization of Expressions † Strings † HighPrecision Numbers † Inputting Approximate Numbers † Inputting High-Precision Numbers † Approximate Zeros 2.2.2 Simplest Arithmetic Expressions and Functions Basic Arithmetic Operations † Reordering Summands and Factors † Precedences of Simple Operators † Algebraic Numbers † Domains of Numeric Functions † Autoevaluations of Sums, Differences, Products, Quotients, and Powers 2.2.3 Elementary Transcendental Functions Exponential and Logarithmic Functions † Trigonometric and Hyperbolic Functions † Exponential Singularities † Picard’s Theorem † Secants Iterations † Exact and Approximate Arguments † Postfix Notation † Infix Notation 2.2.4 Mathematical Constants Imaginary Unit † p † Autoevaluations of Trigonometric Functions † Base of the Natural Logarithm † Golden Ratio † Euler’s Constant g † Directed and Undirected Infinities † Indeterminate Expressions 2.2.5 Inverse Trigonometric and Hyperbolic Functions Multivalued Functions † Inverse Trigonometric Functions † Inverse Hyperbolic Functions † Complex Number Characteristics † Real and Imaginary Parts of Symbolic Expressions † Branch Points and Branch Cuts † Branch Cuts Not Found in Textbooks 2.2.6 Do Not Be Disappointed Real versus Complex Arguments † Seemingly Missing Simplifications † Principal Sheets of Multivalued Functions 2.2.7 Exact and Approximate Numbers Symbols and Constants † Numericalization to Any Number of Digits † Precision of Real Numbers † Precision of Complex Numbers 2.3 Nested Expressions 2.3.1 An Example Constructing Nested Expressions † Canonical Order † Displaying Outlines of Expressions † Displaying Nested Expressions © 2004, 2005 Springer Science+Business Media, Inc THE MATHEMATICA GUIDEBOOKS to PROGRAMMING—GRAPHICS—NUMERICS—SYMBOLICS 2.3.2 Analysis of a Nested Expression A Large Expression † Parts of Expressions † Recursive Part Extraction † Depths of Expressions † Extracting Multiple Parts † Extracting Parts Hierarchically † Locating Subexpressions in Expressions † Level Specifications † Length of Expressions † Leaves of Expressions 2.4 Manipulating Numbers 2.4.1 Parts of Fractions and Complex Numbers Rational Numbers as Raw Objects † Numerators and Denominators † Complex Numbers as Raw Objects † Real and Imaginary Parts 2.4.2 Digits of Numbers Digits of Integers † Digits of Real Numbers † Writing Numbers in Any Base † Counting Digits of Numbers † Fibonacci Chain Map Animation @ @ Overview @ @ Exercises Analyzing the Levels of an Expression † Branch Cuts of Nested Algebraic Functions † Analyzing the Branch Cut Structure of Inverse Hyperbolic Functions † “Strange” Analytic Functions @ @ Solutions Principal Roots † Analyzing a Large Expression † Levels Counted from Top and Bottom † Branch Cuts of Iz4 M Branch Cuts of z+ 1êz z- 1êz † Riemann Surface of arctanHtanHz ê 2L ê 2L Singularities † 1ê4 † Repeated Mappings of @ @ References CHAPTER Definitions and Properties of Functions 3.0 Remarks 3.1 Defining and Clearing Simple Functions 3.1.1 Defining Functions Immediate and Delayed Function Definitions † Expansion and Factorization of Polynomials † Expansion and Factorization of Trigonometric Expressions † Patterns † Nested Patterns † Patterns in Function Definitions † Recursive Definitions † Indefinite Integration † Matching Patterns † Definitions for Special Values † Functions with Several Arguments † Ordering of Definitions 3.1.2 Clearing Functions and Values Clearing Symbol Values † Clearing Function Definitions † Clearing Specific Definitions † Removing Symbols † Matching Names by Name Fragments † Metacharacters in Strings © 2004, 2005 Springer Science+Business Media, Inc Printed from THE MATHEMATICA GUIDEBOOKS 3.1.3 Applying Functions Univariate and Multivariate Functions † Prefix Notation † Postfix Notation † Infix Notation 3.2 Options and Defaults Meaning and Usage of Options † Lists as Universal Containers † Options of Functions † Plotting Simple Functions † Extracting Option Values † Setting Option Values 3.3 Attributes of Functions Meaning and Usage of Attributes † Assigning Attributes to Functions † Commutative Functions † Associative Functions † Functions Operating Naturally on Lists † Numerical Functions † Differentiation of Functions † Protected Functions † Preventing the Evaluation of Expressions † Forcing the Evaluation of Expressions 3.4 Downvalues and Upvalues Function Definitions Associated with Heads † Function Definitions Associated with Specific Arguments † Downvalues and Upvalues † Timing for Adding and Removing Definitions † Caching † Values of Symbols † Numerical Values of Symbols 3.5 Functions that Remember Their Values Caching Function Values † Multiple Assignments † Simplification of Expressions † Timings of Computations † Takeuchi Function 3.6 Functions in the l-Calculus l-Calculus † Functions as Mappings † Functions without Named Arguments † Self-Reproducing Functions † Splicing of Arguments † Sequences of Arguments † Pure Functions with Attributes † Nested Pure Functions 3.7 Repeated Application of Functions Applying Functions Repeatedly † Iterative Maps † Solving an ODE by Iterated Integration † Iterated Logarithm in the Complex Plane † Fixed Points of Maps † Fixed Point Iterations † Newton’s Method for Square Root Extraction † Basins of Attractions † Cantor Series 3.8 Functions of Functions Compositions of Functions † Applying Lists of Heads † Inverse Functions † Differentiation of Inverse Functions @ @ Overview @ @ Exercises @ @ Solutions Predicting Results of Inputs † Nice Polynomial Expansions † Laguerre Polynomials † Puzzles † Unexpected Outputs † Power Tower † Cayley Multiplication Matching Unevaluated Arguments † Equality of Pure Functions † Invalid Patterns † Counting Function Applications © 2004, 2005 Springer Science+Business Media, Inc THE MATHEMATICA GUIDEBOOKS to PROGRAMMING—GRAPHICS—NUMERICS—SYMBOLICS @ @ References CHAPTER Meta-Mathematica 4.0 Remarks 4.1 Information on Commands 4.1.1 Information on a Single Command Built-in Function Definitions as Outputs † Information about Functions † Listing of All Built-in Commands † Messages † Printing Text and Cells † Warnings and Error Messages † Wrong and “Unexpected” Inputs † Suppressing Messages † Carrying out Multiple Calculations in One Input 4.1.2 A Program that Reports on Functions Converting Strings to Expressions † Converting Expressions to Strings † String Form of Typeset Expressions 4.2 Control over Running Calculations and Resources 4.2.1 Intermezzo on Iterators Do Loops † Multiple Iterators † Possible Iterator Constructions † Iterator Step Sizes 4.2.2 Control over Running Calculations and Resources Aborting Calculations † Protecting Calculations from Aborts † Interrupting and Continuing Calculations † Collecting Data on the Fly † Time-Constrained Calculations † Memory-Constrained Calculations † Time and Memory Usage in a Session † Expressions Sharing Memory † Memory Usage of Expressions 4.3 The $-Commands 4.3.1 System-Related Commands Mathematica Versions † The Date Function † Smallest and Largest Machine Real Numbers 4.3.2 Session-Related Commands In and Out Numbering † Input History † Collecting Messages † Display of Graphics † Controlling Recursions and Iterations † Deep Recursions † Ackermann Function 4.4 Communication and Interaction with the Outside 4.4.1 Writing to Files Extracting Function Definitions † Writing Data and Definitions to Files † Reading Data and Definitions from Files † File Manipulations 4.4.2 Simple String Manipulations Concatenating Strings † Replacing Substrings † General String Manipulations † Case Sensitivity and Metacharacters † A Program that Prints Itself 4.4.3 Importing and Exporting Data and Graphics Importing and Exporting Files † Importing Web Pages † Importing From and To Strings † Making Low-Resolution JPEGs © 2004, 2005 Springer Science+Business Media, Inc Printed from THE MATHEMATICA GUIDEBOOKS 4.5 Debugging Displaying Steps of Calculations † Evaluation Histories as Expressions † Recursion versus Iteration † Interactive Inputs 4.6 Localization of Variable Names 4.6.1 Localization of Variables in Iterator Constructions Sums and Products † Scoping of Iterator Variables 4.6.2 Localization of Variables in Subprograms Scoping Constructs † Lexical Scoping † Dynamic Scoping † Local Constants † Temporary Variables † Variable Scoping in Pure Functions † Creating Unique Variables † Nonlocal Program Flow 4.6.3 Comparison of Scoping Constructs Delayed Assignments in Scoping Constructs † Temporarily Changing Built-in Functions † Variable Localization in Iterators † Scoping in Nested Pure Functions † Nesting Various Scoping Constructs † Timing Comparisons of Scoping Constructs 4.6.4 Localization of Variables in Contexts Contexts † Variables in Contexts † Searching through Contexts † Manipulating Contexts † Beginning and Ending Contexts 4.6.5 Contexts and Packages Loading Packages † General Structure of Packages † Private Contexts † Analyzing Context Changes 4.6.6 Special Contexts and Packages Developer Functions † Special Simplifiers † Bit Operations † Experimental Functions † Standard Packages 4.7 The Process of Evaluation Details of Evaluating an Expression † Analyzing Evaluation Examples † Standard Evaluation Order † Nonstandard Evaluations † Held Arguments @ @ Overview @ @ Exercises @ @ Solutions Frequently Seen Messages † Unevaluated Arguments † Predicting Results of Inputs † Analyzing Context Changes † Evaluated versus Unevaluated Expressions Shortcuts for Functions † Functions with Zero Arguments † Small Expressions that Are Large † Localization of Iterator Variables † Dynamical Context Changes † Local Values © 2004, 2005 Springer Science+Business Media, Inc THE MATHEMATICA GUIDEBOOKS to PROGRAMMING—GRAPHICS—NUMERICS—SYMBOLICS @ @ References CHAPTER Restricted Patterns and Replacement Rules 5.0 5.1 Remarks Boolean and Related Functions 5.1.1 Boolean Functions for Numbers Truth Values † Predicates † Functions Ending with Q † Numbers and Numeric Quantities † Integer and Real Numbers † Compound Numeric Quantities † Exact and Inexact Numbers † Primality † Gaussian Primes † Stating Symbolic and Verifying Numeric Inequalities † Comparisons of Numbers † Ordering Relations † Positivity 5.1.2 Boolean Functions for General Expressions Testing Expressions for Being a Polynomial † Vectors and Matrices † Mathematical Equality † Equality and Equations † Structural Equality † Identity of Expressions † Equality versus Identity † Canonical Order † Membership Tests 5.1.3 Logical Operations Boolean Operations † And, Or, Not, and Xor † Rewriting Logical Expressions † Precedences of Logical Operators 5.1.4 Control Structures Branching Constructs † The If Statement † Undecidable Conditions † While and For Loops † Prime Numbers in Arithmetic Progression 5.1.5 Piecewise Functions Piecewise Defined Functions † Canonicalization of Piecewise Functions † Composition of Piecewise Functions † Interpreting Functions as Piecewise Functions † Specifying Geometric Regions † Endpoint Distance Distribution of Random Flights 5.2 Patterns 5.2.1 Patterns for Arbitrary Variable Sequences Simple Patterns † Patterns for Multiple Arguments † Testing Patterns † Named Patterns † Trace of Products of Gamma Matrices † Shortcuts for Patterns † Avoiding Evaluation in Patterns † Literal Patterns 5.2.2 Patterns with Special Properties Optional Arguments † Default Values for Optional Arguments † Repeated Arguments † Excluding Certain Patterns † Alternative Arguments † Restricted Patterns † Pattern Tests † Conditional Patterns † Recursive Definitions † Pattern-Based Evaluation of Elliptic Integrals † Generating Tables † Selecting Elements from Lists † All Syntactically Correct Shortcuts © 2004, 2005 Springer Science+Business Media, Inc Printed from THE MATHEMATICA GUIDEBOOKS 10 5.2.3 Attributes of Functions and Pattern Matching Pattern Matching in Commutative and Associative Functions † Arguments in Any Order † Nested Functions † Automatic Use of Defaults † Analyzing Matchings and Recursions in Pattern and Attribute Combinations 5.3 Replacement Rules 5.3.1 Replacement Rules for Patterns Immediate and Delayed Rules † One-Time and Repeated Replacements † Unevaluated Replacements † Common Pattern Matching Pitfalls † Finding All Possible Replacements † Scoping in Rules † Replacements and Attributes † Modeling Function Definitions † Options and Rules † Replacing Position-Specified Parts of Expressions 5.3.2 Large Numbers of Replacement Rules Optimized Rule Application † Complexity of Optimized Rule Application 5.3.3 Programming with Rules Examples of Rule-Based Programs † Splitting Lists † Cycles of Permutations † Sorting of Complex Numbers † Cumulative Maxima † Dividing Lists † House of the Nikolaus † Polypaths † Rule-Based versus Other Programming Styles 5.4 String Patterns Strings with Pattern Elements † Patterns for Character Sequences † String-Membership Tests † Shortest and Longest Possible Matches † Overlapping Matches † Counting Characters † Replacing Characters † All Possible Replacements † Analyzing the Online Documentation † Cumulative Letter Frequencies @ @ Overview @ @ Exercises Rule-Based Expansion of Polynomials † All Possible Patterns from a Given Set of Shortcuts † Extending Built-in Functions † General Finite Difference Weights † Zeta Function Derivatives † Operator Products † q-Binomial Theorem † q-Derivative † Ordered Derivatives † Differentiating Parametrized Matrices † Ferrer Conjugates † Hermite Polynomial Recursions † Peakons † Puzzles † Catching Arguments and Their Head in Calculations † Nested Scoping @ @ Solutions Modeling Noncommutative Operations † Campbell–Baker–Hausdorff Formula † Counting Function Calls Using Side Effects † q-Deformed Pascal Triangle † Ordered Derivative † Avoiding Infinite Recursions in Pattern Matchings † Dynamically Generated Definitions © 2004, 2005 Springer Science+Business Media, Inc THE MATHEMATICA GUIDEBOOKS to PROGRAMMING—GRAPHICS—NUMERICS—SYMBOLICS 237 GluedPolygons[n_Integer?(# >= 3&), angle:α_?(Im[N[#]] === 0&), iter Integer?(# >= 0&), faceShape:(Polygon | Line), opts _Rule] := Module[{c = N[Cos[α]], s = N[Sin[α]], myUnion, Y, 9, allm, argch, makeHole, makeLine, U = #/Sqrt[#.#]&, ∂ = 10^-6}, (* a completely transitive Union *) myUnion[l_] := Union[l, SameTest -> ((Plus @@ (#.#& /@ (#1 - #2))) < ∂&)]; (* construction of next layer *) (* rotate a point *) Y[point_, rotPoint_, {dir1_, dir2_, dir3_}] := Module[{δ = point - rotPoint, parallel, normal}, parallel = δ.dir1 dir1; normal = Sqrt[#.#]&[δ - parallel]; rotPoint + c normal dir2 + s normal dir3 + parallel]; (* rotate points *) 9[l_] := Module[{dir1, dir2, dir3, first}, (* orthogonal directions *) dir1 = U[Subtract @@ Take[l, 2]]; dir2 = U[(Plus @@ l)/Length[l] - (Plus @@ Take[l, 2])/2]; dir3 = -Cross[dir1, dir2]; Map[N[Y[#, l[[1]], {dir1, dir2, dir3}]]&, l, {-2}]]; (* prepare lists *) allm[l_] := Table[RotateLeft[l, i], {i, Length[l] - 1}]; argch[l_] := Join[Reverse[Take[l, 2]], Reverse[Drop[l, 2]]]; (* make a hole in a polygon *) makeHole[l_] := With[{mp = (Plus @@ l)/Length[l], h = Append[#, First[#]]&[l]}, MapThread[Polygon[Join[#1, Reverse[#2]]]&, {Partition[h, 2, 1], Partition[mp + 0.8(# - mp)& /@ h, 2, 1]}]]; (* wireframe or polygons *) makeLine[l_] := Line[Append[l, First[l]]]; (* show graphics *) Show[Graphics3D[If[faceShape === Polygon, makeHole[#], makeLine[#]]& /@ Join[{Table[N[{Cos[ϕ], Sin[ϕ], 0}], {ϕ, 0, 2Pi - 2Pi/n, 2Pi/n}]}, (* build layer on layer *) If[iter > 0, Flatten[NestList[myUnion[argch /@ (9 /@ Flatten[Join[allm /@ #], 1])]&, Join[argch /@ (9 /@ #)]&[(* one face *) Table[Table[N[{Cos[ϕ], Sin[ϕ], 0}], {ϕ, ϕ0, ϕ0 + 2Pi - 2Pi/n, 2Pi/n}], {ϕ0, 0, 2Pi - 2Pi/n, 2Pi/n}]], iter - 1], 1], {}]]], opts]] These are the functions we are interested in interestingFunctions = {Reverse, Join, Dot, Map, Partition, Apply, Take, MapThread, Drop, Table, Part, Flatten}; To monitor the number of calls to the built-in function func, we unprotect these functions and add a new rule to it The new rule never matches (the False in the condition), but as a side effect of the test, we monitor that they were called (Unprotect[#]; counter[#] = 0; HoldPattern[#[ _]] := Null /; (counter[#] = counter[#] + 1; False))& /@ interestingFunctions; Now, we run the construction of the glued polygons GluedPolygons[5, 3Pi/4, 1, Polygon, DisplayFunction -> Identity]; Here is the actual number of calls to the functions under consideration {#, counter[#]}& /@ interestingFunctions © 2004, 2005 Springer Science+Business Media, Inc Printed from THE MATHEMATICA GUIDEBOOKS 238 As a side effect in the condition testing, we not only monitor the call itself, but we also store the arguments used to call func Here, this is implemented (Unprotect[#]; bag[#] = Bag[]; HoldPattern[#[args _]] := Null /; (bag[#] = Bag[bag[#], Bag[args]]; False))& /@ interestingFunctions; Now, we run the construction of the glued polygons again GluedPolygons[5, 3Pi/4, 1, Polygon, DisplayFunction -> Identity]; For instance, Apply was called 16 times with Plus as its first argument Count[bag[Apply], Plus, Infinity] Σ (* session summary *) TMGBs`PrintSessionSummary[] References ÷1 P Abbott The Mathematica Journal 3, n1 (1992) fi1 ÷2 L Aceto, D Trigiante Rend Circ Mat Palermo S 68, 219 (2002) fi1 ÷3 A Adler Math Intell 14, n3, 14 (1992) fi1 ÷4 A Adler, L C Washington J Number Th 52, 179 (1995) fi1 DOI-Link ÷5 Y Aharonov, L Davidovich, N Zagury Phys Rev A 48, 1687 (1993) fi1 DOI-Link ÷6 M Ahmed, J De Loera, R Hemmecke arXiv:math.CO/0201108 (2002) fi1 Get Preprint ÷7 R Albert A.-L Barabási 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Iteration Theory and its Functional Equations, SpringerVerlag, Berlin, 1985 fi1 BookLink ÷78 U Dudley Mathematical Cranks, Mathematical Association of America, Washington, 1992 fi1 BookLink ÷79 D Dumont Math Comput 33, 1293 (1979) fi1 ÷80 R V Durand, C Franck J Phys A 32, 4955 (1999) fi1 DOI-Link ÷81 M Dvornikov arXiv:hep-ph/0411101 (2004) fi1 Get Preprint ÷82 W Ebeling, T Pưshel Europhys Lett 26, 241 (1994) fi1 ÷83 A Edalat, P J Potts Electr Notes Theor Comput Sc 6, (1997) fi1 http://www.elsevier.com/gej- ng/31/29/23/31/23/61/tcs6007.ps ÷84 S B Edgar, A Hưglund arXiv:gr-qc/0105066 (2001) fi1 Get Preprint ÷85 E Elizalde arXiv:cond-mat/9906229 (1999) fi1 Get Preprint ÷86 P Erdős, V Lev, G Rauzy, C Sandor, A Sárközy Discr Math 200, 119 (1999) fi1 fi2 DOI-Link ÷87 A G Fellouris, L K Matiadou J Phys A 35, 9183 (2002) fi1 DOI-Link ÷88 E Fick Einführung in die Grundlagen der Quantenmechanik, Geest and Portig, Leipzig, 1981 ÷89 M Fiedler Lin Alg Appl 372, 325 (2003) ÷90 E Formanek J Algebra 258, 310 (2002) fi1 BookLink fi1 DOI-Link fi1 DOI-Link ÷91 M Friedman, A Kandel Fundamentals of Computer Analysis, CRC Press, Boca Raton, 1994 ÷92 C.-E Froeberg Numerical Mathematics, Addison-Wesley, Redwood City, 1985 ÷93 B R Frieden Found Phys 9, 883 (1986) fi1 DOI-Link © 2004, 2005 Springer Science+Business Media, Inc fi1 BookLink fi1 BookLink Printed from THE MATHEMATICA GUIDEBOOKS 242 ÷94 L V Furlan Das Harmoniegesetz der Statistik, Verlag für Recht und Gesellschaft AG, Basel, 1946 fi1 BookLink ÷95 S Fussy, G Grưssing, H Schwabl, A Scrinzi Phys Rev A 48, 3470 (1993) ÷96 P Gaillard, V Matveev Preprints MPI 31-2002 (2002) fi1 DOI-Link fi1 http://www.mpim-bonn.mpg.de/cgi- bin/preprint/preprint_search.pl/MPI-2002-31.ps?ps=MPI-2002-31 ÷97 S Galam Physica A 274, 132 (1999) fi1 DOI-Link ÷98 F R Gantmacher The Theory of Matrices, Chelsea, New York, 1959 fi1 BookLink (5) ÷99 S Garfunkel, C A Steen Mathematik in der Praxis, Spektrum der Wissenschaft-Verlag, Heidelberg, 1989 fi1 BookLink ÷100 R S Garibaldi arXiv:math.LA/0203276 (2002) fi1 Get Preprint ÷101 I Gelfand, S Gelfand, V Retakh, R Wilson arXiv:math.QA/0208146 (2002) fi1 Get Preprint ÷102 D V Georgievskiĭ, M V Shamolin Dokl Phys 380, 47 (2001) fi1 ÷103 H Gies arXiv:hep-th/9909500 (1999) fi1 Get Preprint ÷104 F Gobel, R P Nederpelt Am Math Monthly 78, 1097 (1971) fi1 ÷105 V V Goldman, J H J Molenkamp, J A van Hulzen in A Griewank, G F Corliss (eds.) Automatic DifferentiaÖ tion of Algorithms: Theory, Implementation, and Application, SIAM, Philadelphia, 1991 fi1 BookLink ÷106 R N Goldman in D Kirk (ed.) Graphics Gems III, Academic Press, Boston, 1992 fi1 BookLink ÷107 R Goldman IEEE Comput Graphics Appl n3, 66 (2003) fi1 ÷108 E Goles, M Morvan, H D Phan in D Krob, A A Mikhalev, A V Mikhalev (eds.) Formal Power Series and Algebraic Combinatorics, Springer-Verlag, Berlin, 2000 fi1 BookLink ÷109 G H Golub, C F van Loan Matrix Computations, Johns Hopkins University Press, Baltimore, 1989 fi1 BookLink (4) ÷110 L L Gonỗalves, L B Gonỗalves arXiv:cond-mat/0501136 (2005) Get Preprint ÷111 G A Gottwald, M Nicol Physica A 303, 387 (2002) fi1 DOI-Link ÷112 A Graham Kronecker Products and Matrix Calculus: with Applications, Ellis Horwood, Chichester, 1981 fi1 BookLink (2) ÷113 F Graner in B Dubrulle, F Graner, D Sornette (eds.) Scale Invariance and Beyond Springer-Verlag, Berlin, 1997 fi1 BookLink ÷114 F A Graybill Introduction to Matrices with Applications in Statistics, Wadsworth, Belmont, 1969 fi1 fi2 BookLink ÷115 M Gross, A Hubeli Preprint ETH 338/2000 (2000) fi1 ftp://ftp.inf.ethz.ch/pub/publications/techreports/3xx/338.abstract © 2004, 2005 Springer Science+Business Media, Inc THE MATHEMATICA GUIDEBOOKS to PROGRAMMING—GRAPHICS—NUMERICS—SYMBOLICS fi1 DOI-Link ÷116 G Grưssing Phys Lett A 131, (1988) ÷117 G Grưssing Physica D 50, 321 (1991) 243 fi1 DOI-Link ÷118 G Grưssing, A Zeilinger Physica B+C 151, 366 (1988) fi1 fi1 DOI-Link ÷119 G Grưssing, A Zeilinger Physica D 31, 70 (1988) ÷120 M G Guillemot Europhys Lett 53, 155, (2001) fi1 DOI-Link ÷121 H Guiter, M V Arapov Studies on Zipf’s law, Studienverlag Dr N Brockmeyer, Bochum, 1982 fi1 BookLink ÷122 R K Guy, J F Selfridge Am Math Monthly 80, 868 (1973) fi1 ÷123 K B Hajra, P.Sen arXiv:cond-mat/0409017 (2004) fi1 fi2 Get Preprint ÷124 M Halibard, I Kanter Physica A 249, 525 (1998) fi1 DOI-Link ÷125 A J Hanson in P S Heckbert (ed.) Graphics Gems IV, Academic Press, Boston, 1994 fi1 BookLink ÷126 J F Harris Ph D Thesis, Canterbury, 1999 fi1 ÷127 W A Harris, Jr., J P Fillmore, D R Smith SIAM Rev 43, 694 (2001) fi1 DOI-Link ÷128 R Haydock J Phys A 7, 2120 (1974) fi1 DOI-Link ÷129 R Heckmann Theor Comput Sc 279, 65, (2002) fi1 DOI-Link ÷130 E R Hedrick Ann Math 1, 49 (1899) fi1 ÷131 C J Henrich Am Math Monthly 98, 481 (1991) fi1 ÷132 H Hemme Bild der Wissenschaft n11, 178 (1987) fi1 ÷133 H Hemme Bild der Wissenschaft n10, 164 (1988) fi1 ÷134 H Hemme Bild der Wissenschaft n9, 143 (1989) fi1 ÷135 E Herlt, N Salié Spezielle Relativitätstheorie, Akademie-Verlag, Berlin, 1978 fi1 BookLink ÷136 N J Highham Math Comput 46, 537 (1986) fi1 ÷137 T P Hill Am Math Monthly 102, 322 (1995) fi1 ÷138 T P Hill Proc Am Math Soc 123, 887 (1995) fi1 ÷139 T P Hill Stat Sci 10, 354 (1995) fi1 ÷140 D E Holz, H Orland, A Zee arXiv:math-ph/0204015 (2002) fi1 fi2 Get Preprint ÷141 R Honsberger Ingenuity in Mathematics, Random House, New York, 1970 © 2004, 2005 Springer Science+Business Media, Inc fi1 BookLink Printed from THE MATHEMATICA GUIDEBOOKS 244 ÷142 R Honsberger More Mathematical Morsels, American Mathematical Society, 1991 fi1 BookLink (2) ÷143 S Humphries, C Krattenthaler arXiv:math.AC/0411061 (2004) fi1 Get Preprint ÷144 J A H Hunter, J S Madachy Mathematical Diversions, Van Nostrand, Princeton, 1963 ÷145 W Hürlimann MPS: Pure mathematics/0306006 (2003) fi1 BookLink fi1 http://www.mathpreprints.com/math/Preprint/werner.huerlimann/20030603/1/IntPowers.pdf ÷146 W Hürlimann MPS: Pure mathematics/0306013 (2003) fi1 http://www.mathpreprints.com/math/Preprint/werner.huerlimann/20030624/1/GenBenford.pdf ÷147 A Ilachinski Cellular Automata, World Scientific, Singapore, 2001 fi1 BookLink fi1 DOI-Link ÷148 D Ismailescu, R Radoičić Comput Geom 27, 257 (2004) ÷149 M Itskov ZAMM 82, 535 (2002) fi1 DOI-Link ÷150 D M Jackson, R Aleliunas Can J Math 29, 971 (1977) fi1 ÷151 B C Johnson Methol Comput Appl Prob 3, 35 (2001) ÷152 J.-M Jolion J Math Imag Vision 14, 73 (2002) fi1 fi2 DOI-Link fi1 DOI-Link ÷153 B K Jones in D Abbott, L B Kish (eds.) Unsolved Problems of Noise and Fluctuations, American Institute of Physics, Melville, 2000 fi1 fi2 BookLink ÷154 J H Jordan Am Math Monthly 71, 61 (1964) fi1 ÷155 S A Kamal Matrix Tensors Quart 31, 64 (1981) fi1 ÷156 I Kantner, D A Kessler Phys Rev Lett 74, 4559 (1995) fi1 DOI-Link ÷157 Y Kawamura Progr Theor Phys 107, 1105 (2002) fi1 ÷158 J D O’Keeffe Int J Math Edu Sci Technol 12, 541 (1981) fi1 ÷159 J S Kelly Arrow Impossibility Theorems, Academic Press, New York, 1978 fi1 BookLink ÷160 R Kerner arXiv:math-ph/0011023 (2000) fi1 Get Preprint ÷161 I Kim, G Mahler arXiv:quant-ph/9902020 (1999) fi1 Get Preprint ÷162 I Kim, G Mahler arXiv:quant-ph/9902024 (1999) fi1 Get Preprint ÷163 J B Kim, J E Dowdy J Korean Math Soc 17, 141 (1980) fi1 ÷164 D E Knuth The Art of Computer Programming, v.2, Addison-Wesley, Reading, 1969 fi1 BookLink ÷165 D E Knuth The Art of Computer Programming, v 3, Addison-Wesley, Reading, 1998 fi1 BookLink ÷166 I Kogan, A M Perelomov, G W Semenoff arXiv:math-ph/0205038 (2002) fi1 Get Preprint © 2004, 2005 Springer Science+Business Media, Inc THE MATHEMATICA GUIDEBOOKS to PROGRAMMING—GRAPHICS—NUMERICS—SYMBOLICS 245 ÷167 W Kolakoski Am Math Monthly 72, 674 (1965) fi1 ÷168 A V Kontorovich, S J Miller arXiv:math.NT/0412003 (2004) fi1 Get Preprint ÷169 K Kopferman Mathematische Aspekte der Wahlverfahren, BI, Mannheim, 1991 fi1 BookLink ÷170 Y N Kosovtsov arXiv:math-ph/0409035 (2004) fi1 Get Preprint ÷171 G Kowalewski Magische Quadrate und magische Parkette, Teubner, Leipzig, 1939 fi1 ÷172 A Kozlowski The Mathematica Journal 9, 483 (2004) fi1 ÷173 M Kraitchik Mathematical Recreations, Dover, New York, 1953 fi1 BookLink ÷174 C Krattenthaler Sém Lothar Combinat B 42q (1999) fi1 http://80www.mat.univie.ac.at.proxy2.library.uiuc.edu/~slc/wpapers/s42kratt.html ÷175 C Krattenthaler arXiv:math.CO/0503507 (2005) fi1 Get Preprint ÷176 W A Kreiner Z Naturf 58a, 618 (2003) fi1 ÷177 F D Kronewitter arXiv:math.LA/0101245 (2001) fi1 Get Preprint ÷178 H Kučera, W N Francis Computational Analysis of Present-Day American English, Brown University Press, Providence, 1970 fi1 BookLink ÷179 S Kunoff Fibon Quart 25, 365 (1987) fi1 ÷180 S Lakić, M S Petković ZAMM 78, 173 (1998) fi1 DOI-Link ÷181 A Lakshminarajan, N L Balazs Ann Phys 226, 350 (1993) fi1 DOI-Link ÷182 P Lancaster, M Tismenetsky The Theory of Matrices, Academic Press, Orlando, 1985 ÷183 C T Lang Fibon Quart 24, 349 (1986) fi1 ÷184 A Lascoux Ann Combinat 1, 91 (1997) fi1 ÷185 D H Lehmer Am Math Monthly 37, 294 (1930) fi1 ÷186 D S Lemons Am J Phys 54, 816 (1986) fi1 DOI-Link ÷187 D Lenares Proc ACRL 1999 (1999) fi1 http://www.ala.org/acrl/lenares.pdf ÷188 I E Leonard SIAM Rev 38, 507 (1996) fi1 DOI-Link ÷189 Y L Loh, S N Taraskin, S R Elliott Phys Rev E 63, 056706 (2001) fi1 DOI-Link ÷190 M Lotan Am Math Monthly 56, 535 (1948) fi1 ÷191 T.-T Lu, S.-H Shiou Comput Math Appl 43, 119 (2002) fi1 DOI-Link © 2004, 2005 Springer Science+Business Media, Inc fi1 BookLink Printed from THE MATHEMATICA GUIDEBOOKS 246 ÷192 H Lütkepohl Handbook of Matrices, John Wiley, Chichester, 1996 fi1 fi2 BookLink (2) ÷193 I Marek, K Zitny Matrix Analysis for Applied Sciences I, Teubner, Stuttgart, 1983 fi1 ÷194 R Maeder Programming in Mathematica, Addison-Wesley, Reading, 1991 fi1 BookLink (3) ÷195 R Maeder The Mathematica Journal 2, n1, 37 (1992) fi1 ÷196 H M Mahmoud Sorting, Wiley, New York, 2000 fi1 BookLink ÷197 L C Malacarne, R S Mendes Physica A 286, 391 (2000) fi1 DOI-Link ÷198 B J Malešević Univ Beograd Publ Elektrotehn Fak 7, 105 (1998) fi1 ÷199 B J Malešević Univ Beograd Publ Elektrotehn Fak 9, 29 (1998) fi1 ÷200 B J Malešević arXiv:math.CO/0409287 (2004) fi1 Get Preprint ÷201 M Marsili, Y.-C Zhang Phys Rev Lett 80, 2741 (1998) fi1 DOI-Link ÷202 H Martini in T Bisztriczky, P McMullen, R Schneider, A, Ivić Weiss Polytopes: Abstract, Convex and Computational, Kluwer, Dordrecht, 1994 fi1 BookLink ÷203 H Martini in O Giering, J Hoschek (eds.) Geometrie und ihre Anwendungen, Carl Hanser, München, 1994 fi1 BookLink ÷204 G Másson, B Shapiro Exper Math 10, 609 (2001) fi1 ÷205 C Mauduit in J.-M Gambaudo, P Hubert, P Tisseur, S Vaienti (eds.) Dynamical Systems, World Scientific, Singapore, 2000 fi1 BookLink ÷206 B M McCoy Int J Mod Phys A 14, 3921 (1999) fi1 DOI-Link ÷207 D P Mehendale arXiv:math.GM/0503578 (2005) fi1 Get Preprint ÷208 E Meissel Math Ann 2, 636 (1870) fi1 ÷209 E Meissel Math Ann 3, 523 (1870) fi1 ÷210 D A Meyer arXiv:quant-ph/0111069 (2001) fi1 Get Preprint ÷211 D Middleton An Introduction to Statistical Communication Theory, McGraw–Hill, New York, 1960 BookLink ÷212 R Miller Am Math Monthly 85, 183 (1978) fi1 ÷213 R Milson arXiv:math.CO/0003126 (2000) fi1 Get Preprint ÷214 A Miyake arXiv:quant-ph/0206111 (2002) fi1 Get Preprint ÷215 A Miyake, M Wadati arXiv:quant-ph/0212146 (2002) fi1 Get Preprint © 2004, 2005 Springer Science+Business Media, Inc fi1 THE MATHEMATICA GUIDEBOOKS to PROGRAMMING—GRAPHICS—NUMERICS—SYMBOLICS 247 ÷216 M A Montemurro Physica A 300, 567 (2001) fi1 DOI-Link ÷217 M A Montemurro in M Gell-Mann, C Tsallis Nonextensive Entropy–Interdisciplinary Applications, Oxford University Press, Oxford, 2004 fi1 BookLink (2) ÷218 A Moesner Sitzungsberichte Math.-Naturw Klasse der Bayerischen Akademie der Wissenschaften 29, 1952 (1951) fi1 ÷219 C Moler, C Van Loan SIAM Rev 45, (2003) fi1 DOI-Link ÷220 H Moritz, B Hofmann-Wellenhof Geometry, Relativity, Geodesy, Whichmann, Karlsruhe, 1993 fi1 BookLink ÷221 T Muir A Treatise on the Theory of Determinants, Dover, New York 1960 fi1 BookLink (2) ÷222 G L Naber The Geometry of Minkowski Spacetime, Springer-Verlag, New York, 1992 fi1 BookLink (2) ÷223 W Narkiewicz The Development of Prime Number Theory, Springer-Verlag, Berlin, 2000 fi1 BookLink ÷224 A Nayak, A Vishwanath arXiv:quant-ph/0010117 (2000) fi1 Get Preprint ÷225 M E J Newman arXiv:cond-mat/0011144 (2000) fi1 Get Preprint ÷226 M E J Newman Proc Natl Acad Sci USA 98, 404 (2001) fi1 DOI-Link ÷227 M E J Newman arXiv:cond-mat/0412004 (2004) fi1 Get Preprint ÷228 M J Nigrini J Am Tax Ass 18, 72 (1996) fi1 ÷229 T Nowicki Invent Math 144, 233 (2001) fi1 DOI-Link ÷230 A Odlyzko Preprint (2000) fi1 http://www.research.att.com/~amo/doc/rapid.evolution.abst ÷231 R Oldenburger Am Math Monthly 47, 25 (1940) fi1 ÷232 I Paasche Compositio Math 12, 263 (1956) fi1 ÷233 A Palazzolo Am J Phys 44, 63 (1976) fi1 DOI-Link ÷234 F Palmer (ed.) Selected Papers by J R Firth, Longman, London, 1968 fi1 BookLink ÷235 B N Parlett Lin Alg Appl 355, 85 (2002) fi1 DOI-Link ÷236 E Pascal Die Determinanten, Teubner, Leipzig, 1900 fi1 ÷237 W Pauli Theory of Relativity, Pergamon Press, New York, 1958 fi1 BookLink ÷238 M Peczarski in R Möhring, R Raman (eds.) Algorithms - ESA 2002, Springer-Verlag, Berlin, 2002 fi1 BookLink ÷239 A R Penner Am J Phys 69, 332, (2001) fi1 DOI-Link © 2004, 2005 Springer Science+Business Media, Inc Printed from THE MATHEMATICA GUIDEBOOKS 248 ÷240 R Perline Phys Rev E 54, 220 (1996) fi1 DOI-Link ÷241 L Pietronero, E Tosatti, V Tosatti, A Vespignani arXiv:cond-mat 9808305 (1998) fi1 Get Preprint ÷242 L Pietronero, E Tosatti, V Tosatti, A Vespignani Physica A 293, 297 (2001) fi1 DOI-Link ÷243 R S Pinkham Ann Math Stat 32, 1223 (1962) fi1 ÷244 J F Plebanski, M Przanowski J Math Phys 29, 2334 (1988) fi1 DOI-Link ÷245 E R Prakasan, A Kumar, A Sagar, L Mohan, S K Singh, V L Kalyane, V Kumar arXiv:physics/0308107 (2003) fi1 Get Preprint ÷246 D Prato, C Tsallis J Math Phys 41, 3278 (2000) fi1 DOI-Link ÷247 J.-C Puchta, J Spilker Math Semesterber 49, 209 (2002) fi1 DOI-Link ÷248 E J Putzer Am J Math 73, (1966) fi1 ÷249 R A Raimi Am Math Monthly 83, 521 (1976) fi1 ÷250 L Rastelli, A Sen, B Zwiebach arXiv:hep-th/0111281 (2001) fi1 Get Preprint ÷251 P N Rathie, P Zưrnig Int J Math Math Sci 60, 3827 (2003) fi1 ÷252 P Renauld New Zealand J Math 31, 73 (2002) fi1 ÷253 W Reyes Nieuw Archief Wiskunde 9, 299 (1991) fi1 ÷254 D Richards Math Mag 53, 101 (1980) fi1 ÷255 C T Ridgely Am J Phys 67, 414 (1999) fi1 DOI-Link ÷256 R F Rinehart Am Math Monthly 62, 395 (1955) fi1 ÷257 L Rodman in M Hazewinkel (ed.) Handbook of Algebra v.1, Elsevier, Amsterdam, 1996 ÷258 A Rogers, P Loly Am J Phys 72, 786 (2004) fi1 BookLink fi1 DOI-Link ÷259 W W Rouse Ball, H S M Coxeter Mathematical Recreations and Essays, University of Toronto Press, Toronto, 1974 fi1 BookLink (3) ÷260 D G Saari The Geometry of Voting, Springer-Verlag, New York, 1994 fi1 BookLink (2) ÷261 D G Saari Chaotic Elections! A Mathematicians Looks at Voting, American Mathematical Society, Providence, 2001 fi1 BookLink ÷262 D G Saari Math Mag 70, 83 (1997) fi1 ÷263 D Sandell Math Scientist 16, 78 (1991) fi1 ÷264 L San Martin, Y Oono Phys Rev E 57, 4795 (1998) fi1 DOI-Link © 2004, 2005 Springer Science+Business Media, Inc THE MATHEMATICA GUIDEBOOKS to PROGRAMMING—GRAPHICS—NUMERICS—SYMBOLICS 249 ÷265 P Schatte ZAMM 53, 553 (1973) fi1 ÷266 A Schenkel, J Zhang, Y C Zhang Fractals 1, 47 (1993) fi1 ÷267 E Scholz arXiv:math.HO/0409578 (2004) fi1 Get Preprint ÷268 C Schmoeger Lin Alg Appl 359, 169 (2003) fi1 DOI-Link ÷269 E Schmutzer Relativistische Physik, Geest and Portig, Leipzig, 1968 fi1 BookLink ÷270 H Schubert Zwưlf Geduldspiele, Gưschen, Leipzig, 1899 fi1 fi1 fi2 fi3 BookLink ÷271 R Sedgewick, P Flajolet Analysis of Algorithms, Addison Wesley, Reading, 1996 ÷272 R Sharipov arXiv:math.DG/0503332 (2005) fi1 Get Preprint ÷273 R Shaw Int J Math Edu Sci Technol 18, 803 (1987) fi1 ÷274 W Sierpinski A Selection of Problems in the Theory of Numbers, Pergamon, New York, 1964 ÷275 W Sierpinski Elementary Theory of Numbers, North Holland, Amsterdam, 1988 fi1 BookLink fi1 BookLink (2) ÷276 Z K Silagadze Complex Systems 11, 465 (1997) fi1 ÷277 Z K Silagadze arXiv:physics/9901035 (1999) fi1 Get Preprint ÷278 Z K Silagadze arXiv:hep-ph/0106235 (2001) fi1 Get Preprint ÷279 B Sing arXiv:math-ph/0207037 (2002) fi1 Get Preprint ÷280 J Skilling Phil Trans R Soc Lond 278, 15 (1975) fi1 ÷281 J Skilling in J Skilling (ed.) Maximum Entropy and Bayesian Methods, Kluwer, Dordrecht, 1989 fi1 BookLink ÷282 M A Snyder, J H Curry, A M Dougherty Phys Rev E 64, 026222 (2001) fi1 DOI-Link ÷283 E Stade Rocky Mountain J Math 29, 691 (1999) fi1 ÷284 P S Stanimirović, M B Tasić Appl Math Comput 135, 443 (2003) fi1 DOI-Link ÷285 R P Stanley Enumerative Combinatorics, Cambridge University Press, Cambridge 1999 fi1 BookLink (5) ÷286 H M Stark An Introduction to Number Theory, Markham, Chicago, 1970 fi1 BookLink ÷287 E Stensholt SIAM Rev 38, 96 (1996) fi1 ÷288 T J Stieltjes J reine angew Math 89, 343 (1880) fi1 ÷289 Y Stolov, M Idel, S Solomon arXiv:cond-mat/0008192 (2000) fi1 Get Preprint ÷290 F J Studnička Monatsh Math 10, 338 (1899) fi1 © 2004, 2005 Springer Science+Business Media, Inc Printed from THE MATHEMATICA GUIDEBOOKS 250 ÷291 Z.-W Sun Discr Math 257, 143 (2002) fi1 DOI-Link ÷292 A Taivalsaari ACM Comput Surv 28, 438 (1996) fi1 DOI-Link ÷293 S.-I Takekuma Hitotsubashi J Econom 38, 139 (1997) fi1 ÷294 J.-I Tamura in V Berthé, S Ferenczi, C Mauduit, A Siegel (eds.) Substitutions in Dynamics, Arithmetics and Combinatorics, Springer-Verlag, Berlin, 2002 fi1 BookLink ÷295 V Tapia arXiv:gr-qc/0408007 (2004) fi1 Get Preprint ÷296 A D Taylor Mathematics and Politics, Springer-Verlag, New York, 1995 fi1 BookLink (2) ÷297 A D Taylor Am Math Monthly 109, 321 (2002) fi1 ÷298 C R Tolle, J L Budzien, R A LaViolette Chaos 10, 331 (2000) fi1 DOI-Link ÷299 L N Trefethen, D Bau, III Numerical Linear Algebra, SIAM, 1997 fi1 BookLink (2) ÷300 G Troll, P beim Graben Phys Rev E 57, 1347 (1998) fi1 DOI-Link ÷301 M Trott The Mathematica GuideBook for Graphics, Springer-Verlag, New York, 2004 fi1 fi2 fi3 fi4 fi5 fi6 fi7 BookLink ÷302 M Trott The Mathematica GuideBook for Numerics, Springer-Verlag, New York, 2005 fi6 fi7 fi8 fi9 fi10 fi11 fi12 fi1 fi2 fi3 fi4 fi5 BookLink ÷303 M Trott The Mathematica GuideBook for Symbolics, Springer-Verlag, New York, 2005 fi6 fi7 fi8 fi9 fi10 fi11 fi12 fi13 fi14 fi15 fi16 fi17 ÷304 B Tsaban arXiv:math.NA/0204028 (2003) fi1 fi2 fi3 fi4 fi5 BookLink fi1 Get Preprint ÷305 C Tsallis arXiv:cond-mat/9903356 (1999) fi1 Get Preprint ÷306 C Tsallis, M P de Albuquerque arXiv:cond-mat/9903433 (1999) fi1 Get Preprint ÷307 C Tsallis Anais Acad Brasil Ciências 74, 393 (2002) fi1 ÷308 L U Uko Math Scientist 18, 67 (1993) fi1 ÷309 C Van den Broeck, J M R Parrondo Phys Rev Lett 71, 2355 (1993) fi1 DOI-Link ÷310 I Vardi The Mathematica Journal 1, n3, 53 (1991) fi1 ÷311 R Vein, P Dale Determinants and Their Applications in Mathematical Physics, Springer-Verlag, New York, 1999 fi1 BookLink ÷312 G Venkatasubbiah Math Student 7, 101 (1940) fi1 ÷313 P Vignolo, A Minguzzi, M P Tosi Phys Rev Lett 85, 2850 (2000) © 2004, 2005 Springer Science+Business Media, Inc fi1 DOI-Link THE MATHEMATICA GUIDEBOOKS to PROGRAMMING—GRAPHICS—NUMERICS—SYMBOLICS 251 ÷314 D Wagner The Mathematica Journal 6, n1, 54 (1996) fi1 ÷315 Y H Wang, L Tang, Y S Lou Math Scientist 24, 96 (1999) fi1 ÷316 D S Watkins SIAM Rev 34, 427 (1982) fi1 ÷317 J J Wavrik Comput Sc J Moldova 4, (1996) fi1 ÷318 S Weinberg The Quantum Theory of Fields v.1, Cambridge University Press, Cambridge, 1996 fi1 fi2 BookLink ÷319 A Weinmann J Lond Math Soc 35, 265 (1960) fi1 ÷320 T West Comput Phys Commun 77, 286 (1993) fi1 DOI-Link ÷321 H Weyl The Theory of Groups and Quantum Mechanics, Dover, New York, 1931 ÷322 R Wheeldon, S Counsell arXiv:cs.SE/0305037 (2003) fi1 BookLink fi1 Get Preprint ÷323 D V Widder Trans Am Math Soc 30, 126 (1928) fi1 ÷324 J H Wilkinson The Algebraic Eigenvalue Problem, Oxford, Clarendon, 1965 ÷325 F Wille Humor in der Mathematik, Vandenhoeck & Ruprecht, Gưttingen, 1987 fi1 BookLink (2) fi1 BookLink ÷326 D Withoff The Mathematica Journal 4, n2, 56 (1994) fi1 ÷327 S Wolfram Rev Mod Phys 55, 601 (1983) fi1 DOI-Link ÷328 H Wolkowicz, G P H Styan Lin Alg 29, 471 (1980) fi1 ÷329 D R Woodall Math Intell 8, n4, 36 (1986) fi1 ÷330 G Xin arXiv:math.CO/0409468 (2004) fi1 Get Preprint ÷331 S Y Yan Number Theory for Computing, Springer-Verlag, Berlin, 2000 fi1 BookLink (2) ÷332 A C.-C Yang, C.-K Peng, H.-W Yien, A L Goldberger Physica A 329, 473 (2003) fi1 DOI-Link ÷333 C Yu, H Song arXiv:math-ph/0412060 (2004) fi1 Get Preprint ÷334 ZEIT magazin 4.9.1992 page 68 LOGELEI VON ZWEISTEIN (1992) fi1 ÷335 D Zeitlin Am Math Monthly 65, 345 (1958) fi1 ÷336 Y Z Zhang Special Relativity and Its Experimental Tests, World Scientific, Singapore, 1997 fi1 BookLink ÷337 L Zhipeng, C Lin, W Huajia arXiv:math.ST/0408057 (2004) fi1 fi2 Get Preprint ÷338 G K Zipf Human Behavior and the Principle of Least Effort, Addison-Wesley, Cambridge, 1949 BookLink © 2004, 2005 Springer Science+Business Media, Inc fi1 ... and Mathematica 5.1 Can and Cannot Do What Mathematica Does Well † What Mathematica Does Reasonably Well † What Mathematica Cannot Do † Package Proposals † What Mathematica Is and What Mathematica. ..Printed from THE MATHEMATICA GUIDEBOOKS @ @ References P R O G R A M M I N G CHAPTER Introduction to Mathematica 1.0 Remarks 1.1 Basics of Mathematica as a Programming Language 1.1.1... Expressions † Formatting Possibilities † Traditional Mathematics Notation versus Computer Mathematics Notation † Typeset Forms † Heads and Arguments † Symbols † Nested Heads † Input Form and the Formatting