C. T. J. Dodson

A User’s Guide to Algebraic Topology

Preface. Introduction and Overview. 1. Basics of Extension and Lifting Problems. 2. Up to Homotopy is Good Enough. 3. Homotopy Group Theory. 4. Homology and Cohomology Theories. 5. Examples in Homology and Cohomology. 6. Sheaf and Spectral Theories. 7. Bundle Theory. 8. Obstruction Theory. 9. Applications. A: Algebra. B: Topology. C: Manifolds and Bundles. D: Tables of Homotopy Groups. E: Computational Algebraic Topology. Bibliography. Index.

Modeling a class of stochastic porous media

This note extends E. H. Lloyds model of pore structure in random fibre networks to a large class of stochastic fibre networks containing the random model as a special case. The key to the generalization is the substitution of a family of gamma distributions for the negative exponential family used for intercrossing distances on fibres. This allows closed expressions to be obtained for the variance and mean of the equivalent pore size distributions in a planar array of line elements representing fibres. The analytical details have been made available in a Mathematica notebook, via the World Wide Web. The result has application in modeling the forming of nonwoven textiles and paper from fibre suspensions, and in modeling their void structures and transmission of fluids.

Spatial Statistics of Stochastic Fiber Networks

From the known statistics of fiber-fiber contacts in random fiber networks, an analytic estimate is obtained for the variance of local porosity in random fiber suspensions and evolving filtrate networks. The variance of local porosity, and hence the distribution of projected areal density, seem to depend on fiber geometry only through the cube of mean diameter. Also, the coefficient of variation of local flow rate perpendicular to the plane of the pad is, to a first approximation, independent of the mode of flow. Analytic estimates are obtained also for the effect of fiber clumping on the variance of local porosity of pads for small inspection zones.

Experiments in Mathematics Using Maple

I Pre-Calculus Mathematics.- 1 Introduction to Maple.- 1.1 Documentation.- 1.2 First Maple Session.- 1.2.1 Basic Operations.- 1.2.2 Solving Equations.- 1.3 More Commands.- 1.3.1 Calling Function Packages.- 1.3.2 Isolating Variables.- 1.3.3 Variable Names.- 1.3.4 Syntax.- 1.3.5 Assigning Variables.- 1.3.6 Ditto.- 1.3.7 Notation of Mathematical Expressions.- 1.3.8 Commonly Used Commands.- 1.4 Plots.- 1.4.1 Options.- 1.4.2 Text.- 1.4.3 Implicit Plots.- 1.4.4 3D Plots.- 1.4.5 Animations.- 1.5 Preparing Worksheets for Printing.- 2 Functions.- 2.1 Relations and Functions.- 2.2 Multiplication by a Real Number.- 2.3 Addition and Subtraction of Functions.- 2.4 Multiplication and Division of Functions.- 2.5 Composition of Functions.- 2.6 Inverse Functions.- 3 Quadratic Functions.- 3.1 Quadratic Functions.- 3.2 Parabola.- 3.3 Vertex.- 3.4 Maximum and Minimum Values.- 3.5 Applications of Quadratic Functions.- 4 Solving Quadratic Equations.- 4.1 Roots of Quadratic Equations.- 4.2 Nature of Roots of Quadratic Equations.- 4.3 Quadratic Equations in Other Variables.- 4.4 Linear-Quadratic Systems.- 4.5 Slope of a Tangent to a Parabola.- 5 Polynomial Functions.- 5.1 Polynomial Functions.- 5.2 Division of Polynomials.- 5.3 Product and Sum of Roots.- 5.4 Related Roots.- 5.5 Roots of Higher Order Polynomials.- 6 Exponential Functions.- 6.1 Properties of Exponentials.- 6.2 Scientific Notation.- 6.3 Table of Values.- 6.4 Exponential Growth and Decay.- 7 Logarithmic Functions.- 7.1 Properties of Logarithms.- 7.3 pH Scale.- 7.4 Simple Interest.- 7.5 Compound Interest.- 7.6 Equivalent Rates.- 8 Circular Functions.- 8.1 Primary Trigonometric Functions.- 8.2 Reciprocal Trigonometric Functions.- 8.3 Inverse Circular Functions.- 8.4 Transformations.- 8.5 Addition of Circular Functions.- 8.6 Simple Harmonic Motion.- 9 Trigonometry.- 9.1 Basic Trigonometry.- 9.2 Trigonometric Identities.- 9.3 Trigonometric Equations.- 9.4 Power Series Expansions.- 9.5 Right-Angled Triangles.- 9.6 Law of Sines.- 9.7 Law of Cosines.- 9.8 Vectors.- 9.8.1 Dot Product.- 9.9 Bisector of a Triangle.- 10 Similar Figures.- 10.1 Similar Figures.- 10.1.1 Similar Triangles.- 10.2 Length of a Perpendicular.- 10.3 Areas of Similar Triangles.- 10.4 Dilatations and Similar Figures.- 11 Circles and Spheres.- 11.1 Circle.- 11.1.1 Intersection of a Line and a Circle.- 11.1.2 Tangent to a Circle.- 11.1.3 Arc Length.- 11.1.4 Area Bounded by a Circle.- 11.2 Sphere.- 11.2.1 Intersection of a Plane and a Sphere.- 11.2.2 Volume and Surface Area.- 12 Loci.- 12.1 Locus.- 12.2 Equations and Inequations of a Locus.- 12.3 Circles Associated with a Triangle.- 12.3.1 Circumcircle and Circumcentre.- 12.3.2 Inscribed Circle and Incentre.- 12.3.3 Bisectors of Interior Angles.- 12.3.4 Exscribed Circles and E-Centres.- 12.3.5 Centroid of a Triangle.- 12.3.6 Orthocentre of a Triangle.- 12.4 Equations of Loci After a Transformation.- 12.4.1 Translation.- 12.4.2 Stretch.- 12.4.3 Reflection.- 12.4.4 Dilatation.- 12.5 Simulations.- 13 Sequences and Series.- 13.1 Sequences.- 13.1.1 Arithmetic Sequences.- 13.1.2 Geometric Sequences.- 13.2 Series.- 13.2.1 Arithmetic Series.- 13.2.2 Geometric Series.- 14 Statistics and Probability.- 14.1 Organizing and Presenting Data.- 14.1.1 Plots.- 14.1.2 Mean, Median, Mode, and Standard Deviation.- 14.2 Probability of Events.- 14.2.1 Binomial Theorem.- II Beginning Calculus.- 15 Secants and Tangents.- 15.1 Slope of a Line.- 15.2 Slope of a Secant.- 15.3 Slope of a Tangent.- 15.4 Equation of the Tangent to a Curve.- 16 Sequences and Limits.- 16.1 Sequences.- 16.2 Limit of an Infinite Sequence.- 16.3 Sum of an Infinite Geometric Series.- 16.4 Limit of a Function.- 16.5 Rules for Limits.- 16.5.1 Constants Rule.- 16.5.2 Sum Rule.- 16.5.3 Product Rule.- 16.5.4 Quotient Rule.- 16.5.5 Exponential Rule.- 16.5.6 Inequality Rule.- 16.5.7 Sandwich Rule.- 16.5.8 Squeeze Rule.- 16.6 Continuous Functions.- 16.7 Limits Involving

Stochastic texture image estimators for local spatial anisotropy and its variability

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Bivariate Normal Thickness-Density Structure in Real Near-Planar Stochastic Fiber Networks

.- 17 Derivatives of Functions.- 17.1 Derivative.- 17.2 Differentiating from First Principles.- 17.3 Rules of Differentiation.- 17.3.1 Derivative of a Constant.- 17.3.2 Power Rule.- 17.3.3 Derivative of a Sum of Functions.- 17.3.4 Chain Rule.- 17.3.5 Product Rule.- 17.3.6 Quotient Rule.- 17.4 Higher Order Derivatives.- 17.5 Notation.- 17.6 Implicit Differentiation.- 18 Functions and Graphs.- 18.1 Plotting Functions.- 18.2 X- and Y-Intercepts.- 18.3 Asymptotes.- 18.3.1 Vertical Asymptotes.- 18.3.2 Horizontal Asymptotes.- 18.3.3 Oblique Asymptotes.- 18.4 Symmetry.- 18.5 Increasing and Decreasing Functions.- 18.6 Concavity.- 18.7 Relative Maxima and Minima.- 18.8 Inflection Point.- 18.9 Plot of f(x).- 19 Rates.- 19.1 Position, Velocity, and Acceleration.- 19.2 Rate of Change.- 19.3 Related Rates of Change.- 20 Integration.- 20.1 Approximation of the Area Under a Curve.- 20.2 Definite Integral.- 20.3 Indefinite Integral.- 20.4 Fundamental Theorem of Calculus.- 20.5 Rules of Integration.- 20.5.1 Sum of Functions.- 20.5.2 Function Times a Constant.- 20.5.3 Power Rule.- 20.6 Area Between Curves.- 20.7 Differential Equations.- 20.8 Integration by Parts.- 20.9 Partial Fractions.- 20.10 Numerical Integration.- 20.10.1 Trapezoidal Rule.- 20.10.2 Simpsons Rule.- 21 Trigonometry.- 21.1 Compound Angles.- 21.2 Graphs of Trigonometric Functions.- 21.3 Derivative of Sine Function.- 21.4 Derivatives of Trigonometric Functions.- 21.5 Maxima and Minima.- 21.6 Integrals of Trigonometric Functions.- 21.7 Areas Defined by Trigonometric Functions.- 22 Exponents and Logarithms.- 22.1 Base for Natural Logarithm.- 22.2 Exponential Growth and Decay.- 22.3 Derivatives.- 22.4 Integrals.- 23 Polar Coordinates.- 23.1 Converting Coordinates.- 23.2 Circles in Polar Coordinates.- 23.3 Area in Polar Coordinates.- 23.3.1 Sector of a Circle.- 23.3.2 Region Enclosed by a Curve.- Appendices.- A Solutions to Part I Exercises.- A.1 Functions.- A.2 Quadratics.- A.3 Solving Quadratics.- A.4 Polynomials.- A.5 Exponential Functions.- A.6 Logarithmic Functions.- A.8 Trigonometry.- A.9 Similar Figures.- A.10 Circles and Spheres.- A.11 Loci.- A.12 Sequences and Series.- A.13 Statistics and Probability.- B Solutions to Part II Exercises.- B.1 Secants and Tangents.- B.2 Sequences and Limits.- B.3 Derivatives of Functions.- B.4 Functions and Graphs.- B.5 Rates.- B.6 Integration.- B.7 Trigonometry.- B.8 Exponents and Logarithms.- B.9 Polar Coordinates.- C.1 angles.- C.2 showAntiderivative.

Spatial Statistics and Information Geometry for Parametric Statistical Models of Galaxy Clustering

Some operators from the theory of texture image analysis are used to obtain a new method for the evaluation of anisotropy in planar stochastic structures. A local gradient function provides information on local anisotropy and its variability, from 2-D density images for foil ma- terials like polymer sheets, nonwoven textiles and paper. Such images can be captured by radiography or light transmission; results are re- ported for a range of paper structures. The method has potential for on-line application to monitoring and control of anisotropy and its vari- ability, as well as local density itself, in continuous manufacturing pro- cesses.

On the bundle of principal connections and the stability of b-incompleteness of manifolds

A new image analysis technique is proposed for the evaluation of local anisotropy and its variability in stochastic texture images. It utilizes the gradient function to provide information on local anisotropy, from two-dimensional (2-D) density images for foil materials like polymer sheets, nonwoven textiles, and paper. Such images can be captured by radiography or light-transmission; results are reported for a range of paper structures, and show that the proposed technique is more robust to unfavorable imaging conditions than other approaches. The method has potential for on-line application to monitoring and control of anisotropy and its variability, as well as local density itself, in continuous manufacturing processes.