The Concept of Dimension
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The document discusses topological dimension and fractal dimension. It provides examples of objects with different dimensions, such as a line being 1-dimensional, a plane being 2-dimensional, and the Cantor set having a fractal dimension of 0.6309. Fractals are self-similar objects that can have non-integer dimensions calculated using the logarithmic formula D = log(N)/log(1/r), where N is the number of parts and r is the scaling ratio. The Koch snowflake is used to illustrate this, with N=4, r=1/3, giving a fractal dimension of 1.26.
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This document discusses dimensions in mathematics and physics. It begins by explaining one-dimensional, two-dimensional, and three-dimensional objects like lines, squares, cubes, and tesseracts. It then discusses higher dimensions posited by theories like string theory and M-theory. Key definitions of dimension discussed include topological dimension, Hausdorff dimension, and covering dimension. In physics, it discusses the three spatial dimensions and time as the fourth dimension, as well as theories proposing additional curled up dimensions to explain phenomena.
Chap6_Sec1.ppt
The document discusses using integrals to calculate the area between two curves. It provides examples of finding areas bounded above and below by functions, including cases where the boundary curves intersect and cases where graphical methods are needed to find approximate intersection points. The key formula given is that the area between curves y=f(x) and y=g(x) from x=a to x=b is the integral from a to b of f(x)-g(x) dx. Examples are worked out demonstrating the application of this formula.
Lecture3
This document provides a summary of a lecture on graphs and algorithms. It discusses Hall's Theorem and provides a proof using induction. It then gives several examples of how Hall's Theorem can be applied, such as scheduling tasks among machines. The document also discusses maximum flows, the Konig-Egervary Theorem relating matchings and vertex covers, and Menger's Theorem relating disjoint paths and minimum cuts.
Section 1.3 -- The Coordinate Plane
The coordinate plane is formed by intersecting two number lines, called the x-axis and y-axis, at their zero points. The point of intersection is called the origin. To graph an inequality in two variables, graph the boundary curve and shade the region where the inequality is true. The distance formula can be used to find the distance between two points by treating it as the hypotenuse of a right triangle formed by the differences in the x and y coordinates. The midpoint formula finds the point halfway between two points by averaging the x and y coordinates. A circle is defined as all points equidistant from a center point, where the distance from the center is the radius. The standard form of a circle equation relates the
Lar calc10 ch04_sec2
The document discusses methods for approximating and calculating the area of plane regions. It introduces sigma notation for writing sums and explores using rectangles to approximate the area of a region bounded by a graph and the x-axis. Upper and lower sums are defined by using maximum and minimum heights of a function over subintervals, and the area is defined as the limit of the Riemann sums as the subintervals approach zero width. Examples demonstrate finding upper and lower sums and calculating the exact area of a region using limits.
Lesson 11 plane areas area by integration
This document discusses calculating the area under a curve using integration. It begins by approximating the area under an irregular shape using squares and rectangles. It then introduces defining the area A as a limit of approximating rectangles as their width approaches 0. This is written as the integral from a to b of f(x) dx, where f(x) is the curve. Examples are given of setting up definite integrals to calculate the areas under curves and between two curves. Steps for determining the area of a plane figure using integration are also provided.
MA101-Lecturenotes(2019-20)-Module 13 (1).pdf
This document defines scalar and vector fields and provides examples of each. It also discusses line integrals, which integrate a scalar or vector field over a curve. Line integrals of vector fields can be used to calculate work done by a force field or fluid flow along a curve. The line integral is path independent for conservative vector fields, which can be expressed as the gradient of a potential function.
MA101-Lecturenotes(2019-20)-Module 13 (1).pdf
- A scalar field assigns a scalar value to each point in a region of space, like temperature. A vector field assigns a vector, like fluid velocity.
- Line integrals calculate the work done or fluid flow along a path by integrating the dot product of the field and the path's tangent vector.
- Conservative vector fields have a potential function whose derivatives give the field. Line integrals of conservative fields depend only on the endpoints and not the path taken.
Newton cotes integration method
This presentation is a part of Computer Oriented Numerical Method . Newton-Cotes formulas are an extremely useful and straightforward family of numerical integration techniques.
Beginning direct3d gameprogrammingmath01_primer_20160324_jintaeks
This document provides an introduction to mathematics concepts for direct3D game programming, including:
- 2D rotation formulas for rotating a point about an angle
- Definitions of domain, codomain, exponentiation, factorial, square root, and cubic root
- Explanations of inverse functions, Pythagorean theorem, summation notation, and practice problems for various concepts
- Overviews of the Cartesian coordinate system, trigonometric functions, and how to implement them in programs
L 4 4
The document discusses the Fundamental Theorem of Calculus, which states that differentiation and integration are inverse operations. It provides examples of using the theorem to evaluate definite integrals and find the average value of functions. The Second Fundamental Theorem of Calculus describes integrals as accumulation functions and relates the integral and derivative. The Net Change Theorem applies the Fundamental Theorem to relate the net change in a function to its integral over an interval. Examples demonstrate using these theorems to solve problems in physics, chemistry, and particle motion.
AndersSchreiber_ThesisFinal
- The author derives the Schwarzschild metric in D spatial dimensions and one time dimension to investigate the effects of a fractal deviation from three dimensions.
- Using the metric, the author considers phenomena like perihelion precession, bending of light, and gravitational redshift to set bounds on the parameter δ that quantifies the deviation from three dimensions.
- The best upper bound found is |δ| < 5.0 × 10-9, from measurements of perihelion precession in binary pulsar systems. This is consistent with zero deviation from three dimensions.
An Analysis and Study of Iteration Procedures
In computational mathematics, an iterative method is a scientific technique that utilizes an underlying speculation to produce a grouping of improving rough answers for a class of issues, where the n th estimate is gotten from the past ones. A particular execution of an iterative method, including the end criteria, is a calculation of the iterative method. An iterative method is called joined if the relating grouping meets for given starting approximations. A scientifically thorough combination investigation of an iterative method is typically performed notwithstanding, heuristic based iterative methods are additionally normal. This Research provides a survey of iteration procedures that have been used to obtain fixed points for maps satisfying a variety of contractive conditions. Dr. R. B. Singh | Shivani Tomar ""An Analysis and Study of Iteration Procedures"" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-4 , June 2019, URL: https://www.ijtsrd.com/papers/ijtsrd23715.pdf
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Visualization of general defined space data
A new algorithm is presented which determines the dimensionality and signature of a measured space. The
algorithm generalizes the Map Maker’s algorithm
from 2D to n dimensions and works the same for 2D
measured spaces as the Map Maker’s algorithm but with better efficiency. The difficulty of generalizing the
geometric approach of the Map Maker’s algorithm from 2D to 3D and then to higher dimensions is
avo
ided by using this new approach. The new algorithm preserves all distances of the distance matrix and
also leads to a method for building the curved space as a subset of the N
-
1 dimensional embedding space.
This algorithm has direct application to Scientif
ic Visualization for data viewing and searching based on
Computational Geometry.
3 d scaling and translation in homogeneous coordinates
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Representing Graphs by Touching Domains
This document discusses representing graphs by touching domains, specifically:
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- It defines key terms like planar graphs, 3-connected graphs, and dual graphs.
- The proof of Koebe's theorem involves showing that there exists a function mapping radii to a defect vector such that there is a fixed point where the defect is zero, implying tangency between circles. This is done by showing the function maps a convex set to itself.
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Syllabus
Chapter-1
Introduction to objective, scope and outcome the subject
Chapter 2
Introduction: Scope and Specialization of Civil Engineering, Role of civil Engineer in Society, Impact of infrastructural development on economy of country.
Chapter 3
Surveying: Object Principles & Types of Surveying; Site Plans, Plans & Maps; Scales & Unit of different Measurements.
Linear Measurements: Instruments used. Linear Measurement by Tape, Ranging out Survey Lines and overcoming Obstructions; Measurements on sloping ground; Tape corrections, conventional symbols. Angular Measurements: Instruments used; Introduction to Compass Surveying, Bearings and Longitude & Latitude of a Line, Introduction to total station.
Levelling: Instrument used Object of levelling, Methods of levelling in brief, and Contour maps.
Chapter 4
Buildings: Selection of site for Buildings, Layout of Building Plan, Types of buildings, Plinth area, carpet area, floor space index, Introduction to building byelaws, concept of sun light & ventilation. Components of Buildings & their functions, Basic concept of R.C.C., Introduction to types of foundation
Chapter 5
Transportation: Introduction to Transportation Engineering; Traffic and Road Safety: Types and Characteristics of Various Modes of Transportation; Various Road Traffic Signs, Causes of Accidents and Road Safety Measures.
Chapter 6
Environmental Engineering: Environmental Pollution, Environmental Acts and Regulations, Functional Concepts of Ecology, Basics of Species, Biodiversity, Ecosystem, Hydrological Cycle; Chemical Cycles: Carbon, Nitrogen & Phosphorus; Energy Flow in Ecosystems.
Water Pollution: Water Quality standards, Introduction to Treatment & Disposal of Waste Water. Reuse and Saving of Water, Rain Water Harvesting. Solid Waste Management: Classification of Solid Waste, Collection, Transportation and Disposal of Solid. Recycling of Solid Waste: Energy Recovery, Sanitary Landfill, On-Site Sanitation. Air & Noise Pollution: Primary and Secondary air pollutants, Harmful effects of Air Pollution, Control of Air Pollution. . Noise Pollution Harmful Effects of noise pollution, control of noise pollution, Global warming & Climate Change, Ozone depletion, Greenhouse effect
Text Books:
1. Palancharmy, Basic Civil Engineering, McGraw Hill publishers.
2. Satheesh Gopi, Basic Civil Engineering, Pearson Publishers.
3. Ketki Rangwala Dalal, Essentials of Civil Engineering, Charotar Publishing House.
4. BCP, Surveying volume 1
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