This document discusses local properties and differential forms of smooth maps and tangent bundles. It defines the derivative of a smooth map as a linear map between tangent spaces. It shows that if a smooth map is bijective and all its derivatives are injective, then it is a diffeomorphism. It also examines vector fields on manifolds and establishes an isomorphism between the module of vector fields and the module of derivations on the ring of smooth functions. Finally, it introduces the concept of orbits of vector fields and establishes existence and uniqueness theorems for solutions to differential equations defined by vector fields.
A vector is a quantity with both magnitude and direction. There are two main operations on vectors: vector addition and scalar multiplication. Vector addition involves placing the tail of one vector at the head of another and drawing the third side of the resulting triangle or parallelogram. Scalar multiplication scales the length of a vector without changing its direction. Vectors can be represented using Cartesian components, where the magnitude and direction of a vector are given by its x, y, and z values relative to a set of perpendicular axes.
This document discusses different definitions of dimension, focusing on topological dimension and Hausdorff dimension. It provides examples of fractals like the Sierpinski triangle and Koch curve that challenge classical dimension definitions. While topological dimension aligns with intuition, the Hausdorff dimension considers how content scales with measurement. The document explains how fractals like the Sierpinski triangle have topological dimension of 1 but area of 0, requiring a fractional Hausdorff dimension to adequately measure their content. This provides the conceptual basis for Mandelbrot's definition of fractals as sets where the Hausdorff dimension exceeds the topological dimension.
Analysis and algebra on differentiable manifoldsSpringer
This chapter discusses tensor fields and differential forms on manifolds. It provides definitions of tensor fields, differential forms, vector bundles, and the exterior derivative. It also introduces the Lie derivative and interior product. The chapter contains examples of vector bundles like the Möbius strip. It aims to make the reader proficient with computations involving vector fields, differential forms, and other concepts. Problems are included to help develop skills in computing integral distributions and differential ideals.
The document discusses various topics in three dimensional geometry, including:
1. Coordinate systems define three perpendicular coordinate axes dividing space into eight parts.
2. The distance between two points P and Q is calculated as the square root of the sum of the squares of the differences between their x, y, and z coordinates.
3. Section formulae give the coordinates of points dividing or projecting onto lines between two given points.
Black hole formation by incoming electromagnetic radiationXequeMateShannon
1. The document describes a known solution to Einstein's field equations that represents the formation of non-spherical black holes through the collapse of pure electromagnetic monochromatic radiation.
2. The solution describes a metric with electromagnetic radiation modeled as a null electromagnetic field. The radiation can be linearly, circularly, or elliptically polarized depending on the choice of functions.
3. The metric can describe the formation of black holes in anti-de Sitter space, the destruction of naked singularities, or the evaporation of white holes through the ingoing or outgoing flow of electromagnetic radiation.
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.
This section provides an introduction to graphs and graph theory. Key points include:
- Graphs consist of vertices and edges that connect the vertices. They can be directed or undirected.
- Common terminology is introduced, such as adjacent vertices, neighborhoods, degrees of vertices, and handshaking theorem.
- Different types of graphs are discussed, including multigraphs, pseudographs, and directed graphs.
- Examples of graph models are given for computer networks, social networks, information networks, transportation networks, and software design. Graphs can be used to model many real-world systems and applications.
A vector is a quantity with both magnitude and direction. There are two main operations on vectors: vector addition and scalar multiplication. Vector addition involves placing the tail of one vector at the head of another and drawing the third side of the resulting triangle or parallelogram. Scalar multiplication scales the length of a vector without changing its direction. Vectors can be represented using Cartesian components, where the magnitude and direction of a vector are given by its x, y, and z values relative to a set of perpendicular axes.
This document discusses different definitions of dimension, focusing on topological dimension and Hausdorff dimension. It provides examples of fractals like the Sierpinski triangle and Koch curve that challenge classical dimension definitions. While topological dimension aligns with intuition, the Hausdorff dimension considers how content scales with measurement. The document explains how fractals like the Sierpinski triangle have topological dimension of 1 but area of 0, requiring a fractional Hausdorff dimension to adequately measure their content. This provides the conceptual basis for Mandelbrot's definition of fractals as sets where the Hausdorff dimension exceeds the topological dimension.
Analysis and algebra on differentiable manifoldsSpringer
This chapter discusses tensor fields and differential forms on manifolds. It provides definitions of tensor fields, differential forms, vector bundles, and the exterior derivative. It also introduces the Lie derivative and interior product. The chapter contains examples of vector bundles like the Möbius strip. It aims to make the reader proficient with computations involving vector fields, differential forms, and other concepts. Problems are included to help develop skills in computing integral distributions and differential ideals.
The document discusses various topics in three dimensional geometry, including:
1. Coordinate systems define three perpendicular coordinate axes dividing space into eight parts.
2. The distance between two points P and Q is calculated as the square root of the sum of the squares of the differences between their x, y, and z coordinates.
3. Section formulae give the coordinates of points dividing or projecting onto lines between two given points.
Black hole formation by incoming electromagnetic radiationXequeMateShannon
1. The document describes a known solution to Einstein's field equations that represents the formation of non-spherical black holes through the collapse of pure electromagnetic monochromatic radiation.
2. The solution describes a metric with electromagnetic radiation modeled as a null electromagnetic field. The radiation can be linearly, circularly, or elliptically polarized depending on the choice of functions.
3. The metric can describe the formation of black holes in anti-de Sitter space, the destruction of naked singularities, or the evaporation of white holes through the ingoing or outgoing flow of electromagnetic radiation.
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.
This section provides an introduction to graphs and graph theory. Key points include:
- Graphs consist of vertices and edges that connect the vertices. They can be directed or undirected.
- Common terminology is introduced, such as adjacent vertices, neighborhoods, degrees of vertices, and handshaking theorem.
- Different types of graphs are discussed, including multigraphs, pseudographs, and directed graphs.
- Examples of graph models are given for computer networks, social networks, information networks, transportation networks, and software design. Graphs can be used to model many real-world systems and applications.
This document provides an introduction to topological graph theory. It begins with definitions of basic graph theory concepts from a topological perspective, such as representing graphs with curved arcs instead of straight lines. It then discusses graph drawings, incidence matrices, vertex valence, and graph maps/isomorphisms. Important classes of graphs are introduced, such as trees, paths, cycles, and complete graphs. The document aims to introduce preliminary concepts of topological graph theory to students in a simple manner.
This document discusses various forms of equations for lines including point-slope form, two-point form, slope-intercept form, intercept form, and normal form. It provides the definitions and step-by-step processes for deriving the equation of a line given certain characteristics like two points on the line, the slope and a point, the slope and y-intercept, x and y-intercepts, or the length of the perpendicular from the origin and the angle it makes with the x-axis.
On algorithmic problems concerning graphs of higher degree of symmetrygraphhoc
Since the ancient determination of the five platonic solids the study of symmetry and regularity has always
been one of the most fascinating aspects of mathematics. One intriguing phenomenon of studies in graph
theory is the fact that quite often arithmetic regularity properties of a graph imply the existence of many
symmetries, i.e. large automorphism group G. In some important special situation higher degree of
regularity means that G is an automorphism group of finite geometry. For example, a glance through the
list of distance regular graphs of diameter d < 3 reveals the fact that most of them are connected with
classical Lie geometry. Theory of distance regular graphs is an important part of algebraic combinatorics
and its applications such as coding theory, communication networks, and block design. An important tool
for investigation of such graphs is their spectra, which is the set of eigenvalues of adjacency matrix of a
graph. Let G be a finite simple group of Lie type and X be the set homogeneous elements of the associated
geometry. The complexity of computing the adjacency matrices of a graph Gr on the vertices X such that
Aut GR = G depends very much on the description of the geometry with which one starts. For example, we
can represent the geometry as the totality of 1 cosets of parabolic subgroups 2 chains of embedded
subspaces (case of linear groups), or totally isotropic subspaces (case of the remaining classical groups), 3
special subspaces of minimal module for G which are defined in terms of a G invariant multilinear form.
The aim of this research is to develop an effective method for generation of graphs connected with classical
geometry and evaluation of its spectra, which is the set of eigenvalues of adjacency matrix of a graph. The
main approach is to avoid manual drawing and to calculate graph layout automatically according to its
formal structure. This is a simple task in a case of a tree like graph with a strict hierarchy of entities but it
becomes more complicated for graphs of geometrical nature. There are two main reasons for the
investigations of spectra: (1) very often spectra carry much more useful information about the graph than a
corresponding list of entities and relationships (2) graphs with special spectra, satisfying so called
Ramanujan property or simply Ramanujan graphs (by name of Indian genius mathematician) are important
for real life applications (see [13]). There is a motivated suspicion that among geometrical graphs one
could find some new Ramanujan graphs.
ON ALGORITHMIC PROBLEMS CONCERNING GRAPHS OF HIGHER DEGREE OF SYMMETRYFransiskeran
Since the ancient determination of the five platonic solids the study of symmetry and regularity has always
been one of the most fascinating aspects of mathematics. One intriguing phenomenon of studies in graph
theory is the fact that quite often arithmetic regularity properties of a graph imply the existence of many
symmetries, i.e. large automorphism group G. In some important special situation higher degree of
regularity means that G is an automorphism group of finite geometry. For example, a glance through the
list of distance regular graphs of diameter d < 3 reveals the fact that most of them are connected with
classical Lie geometry. Theory of distance regular graphs is an important part of algebraic combinatorics
and its applications such as coding theory, communication networks, and block design. An important tool
for investigation of such graphs is their spectra, which is the set of eigenvalues of adjacency matrix of a
graph. Let G be a finite simple group of Lie type and X be the set homogeneous elements of the associated
geometry.
The document summarizes the spectral theory and principle of limiting absorption for the elasticity operator in infinite, homogeneous, anisotropic media. It defines the elasticity operator and establishes its properties, including that it is self-adjoint. It derives the wave equation for elastic media from Hooke's law and the equations of motion. It shows plane wave solutions to the wave equation and establishes Christoffel's equation.
The document discusses direction cosines and ratios of lines, and equations to represent lines and planes in space. It provides equations in both vector and Cartesian form to define lines passing through points or parallel to vectors, lines passing through two points, the angle between lines, and the shortest distance between lines. It also provides equations to define planes passing through points or vectors, through three non-collinear points, perpendicular to a vector, and the distance from a plane to a point.
Analysis and algebra on differentiable manifoldsSpringer
This chapter discusses integration on manifolds. It defines orientation of manifolds and orientation-preserving maps between manifolds. It presents Stokes' theorem and Green's theorem, which relate integrals over boundaries to integrals of differential forms. It introduces de Rham cohomology groups, which classify closed forms modulo exact forms. Examples are given of calculating the orientability and cohomology of manifolds like the cylinder, Möbius strip, and real projective plane.
Computer graphics are pictures and movies created using computers - usually referring to image data created by a computer specifically with help from specialized graphical hardware and software. It is a vast and recent area in computer science.The phrase was coined by computer graphics researchers Verne Hudson and William Fetter of Boeing in 1960. Another name for the field is computer-generated imagery, or simply CGI.
Important topics in computer graphics include user interface design, sprite graphics, vector graphics, 3D modeling, shaders, GPU design, and computer vision, among others. The overall methodology depends heavily on the underlying sciences of geometry, optics, and physics. Computer graphics is responsible for displaying art and image data effectively and beautifully to the user, and processing image data received from the physical world. The interaction and understanding of computers and interpretation of data has been made easier because of computer graphics. Computer graphic development has had a significant impact on many types of media and has revolutionized animation, movies, advertising, video games, and graphic design generally.
The document discusses various graph theory topics including isomorphism, cut sets, labeled graphs, and Hamiltonian circuits. It defines isomorphism as two graphs being structurally identical with a one-to-one correspondence between their vertices and edges. Cut sets are edges whose removal would disconnect a connected graph. Labeled graphs assign labels or weights to their vertices and/or edges. A Hamiltonian circuit is a closed walk that visits each vertex exactly once.
The document discusses group theory and its applications in physics. It begins by introducing symmetry groups that are important in physics, including translations, rotations, and Lorentz transformations. It then discusses the use of group theory in formulating fundamental forces and the Standard Model of particle physics. The document provides definitions of group theory concepts like groups, operations, identity, and inverse. It explains how group theory provides a mathematical framework for describing physical symmetries.
Proportional and decentralized rule mcst gamesvinnief
This document presents two cost allocation rules - the proportional rule and the decentralized rule - for minimum cost spanning extension problems. The proportional rule allocates costs proportionally based on an algorithm that constructs a minimum cost spanning extension while simultaneously allocating link costs. The decentralized rule is based on an algorithm that constructs the extension in a decentralized manner, with links being added one by one and their costs immediately allocated. Both rules are shown to be refinements of the irreducible core defined for these problems. The proportional rule is then characterized axiomatically.
Vershinin 2013_about presentations of braid groups and theirPanagiote Ligouras
In the paper we give a survey of rather new notions and results which generalize
classical ones in the theory of braids. Among such notions are various inverse monoids of
partial braids. We also observe presentations different from standard Artin presentation for
generalizations of braids. Namely, we consider presentations with small number of generators,
Sergiescu graph-presentations and Birman-Ko-Lee presentation. The work of V. V. Chaynikov
on the word and conjugacy problems for the singular braid monoid in Birman-Ko-Lee generators
is described as well.
This document provides definitions and concepts related to graph theory. It begins with a brief history of graph theory and then defines basic concepts such as graphs, nodes, edges, adjacency, incidence, isomorphism, subgraphs, walks, trails, paths, connectedness, trees, and spanning trees. It also introduces different types of graphs including null graphs, cycle graphs, path graphs, complete graphs, bipartite graphs, and complete bipartite graphs. Finally, it discusses how vector spaces can be associated with graphs and defines the properties of cycle and cutset spaces.
This document provides a summary of a master's thesis in mathematics. The thesis studies generalisations of the fact that every Riesz homomorphism between spaces of continuous functions on compact Hausdorff spaces can be written as a composition multiplication operator. Specifically, the thesis examines Riesz homomorphisms between spaces of extended continuous functions on extremally disconnected spaces, known as Maeda-Ogasawara spaces. It proves that every Riesz homomorphism on such a space has a generalized composition multiplication form involving a continuous map between the underlying spaces. The thesis also applies these results to spaces of measurable functions and explores Riesz homomorphisms between spaces of continuous functions with values in a Banach lattice.
This document defines key concepts related to straight lines, including their various forms of equations, slope, intercepts, and how to calculate them. It discusses the general, standard, point-slope, and intercept forms of linear equations. It also explains how to find the slope between two points using the slope formula, and how to determine the x-intercept and y-intercept from a line's equation. Examples are provided to illustrate these concepts.
We make use of the conformal compactification of Minkowski spacetime M# to explore a way of describing general, nonlinear Maxwell fields with conformal symmetry. We distinguish the inverse Minkowski spacetime [M#]−1 obtained via conformal inversion, so as to discuss a doubled compactified spacetime on which Maxwell fields may be defined. Identifying M# with the projective light cone in (4+2)-dimensional spacetime, we write two independent conformal-invariant functionals of the 6-dimensional Maxwellian field strength tensors - one bilinear, the other trilinear in the field strengths -- which are to enter general nonlinear constitutive equations. We also make some remarks regarding the dimensional reduction procedure as we consider its generalization from linear to general nonlinear theories.
Cap 2_Nouredine Zettili - Quantum Mechanics.pdfOñatnom Adnara
This document discusses mathematical tools used in quantum mechanics. It introduces Hilbert spaces and their properties including linear vector spaces, scalar products, completeness, and bases. Vectors in a Hilbert space belong to an abstract vector space and satisfy properties like orthonormality. Operators acting on these vectors are linear. Examples of finite and infinite dimensional Hilbert spaces are given for the 3D Euclidean space and space of complex functions. Linear independence of sets of vectors and functions is also discussed.
Common Fixed Point Theorem for Weakly Compatible Maps in Intuitionistic Fuzzy...inventionjournals
In this paper, we prove some common fixed point theorem for weakly compatible maps in intuitionistic fuzzy metric space for two, four and six self mapping.
Presentation on Greater Chicago Food Depository from "Acting Up - Using Theater & Technology for Social Change," from DePaul's School for New Learning Distance Education Program, Winter 2010.
Este documento habla sobre la higiene industrial y sus objetivos como reconocer y evaluar los riesgos en el ambiente laboral, eliminar enfermedades profesionales, prevenir empeoramiento de enfermedades, y mantener la salud de los trabajadores. Explica los factores de riesgo físicos como ruido, vibraciones, calor/frío y radiaciones, y los factores de riesgo ergonómicos como carga física estática por posturas prolongadas y carga dinámica. También cubre cómo las posturas de trabajo inade
This document provides an introduction to topological graph theory. It begins with definitions of basic graph theory concepts from a topological perspective, such as representing graphs with curved arcs instead of straight lines. It then discusses graph drawings, incidence matrices, vertex valence, and graph maps/isomorphisms. Important classes of graphs are introduced, such as trees, paths, cycles, and complete graphs. The document aims to introduce preliminary concepts of topological graph theory to students in a simple manner.
This document discusses various forms of equations for lines including point-slope form, two-point form, slope-intercept form, intercept form, and normal form. It provides the definitions and step-by-step processes for deriving the equation of a line given certain characteristics like two points on the line, the slope and a point, the slope and y-intercept, x and y-intercepts, or the length of the perpendicular from the origin and the angle it makes with the x-axis.
On algorithmic problems concerning graphs of higher degree of symmetrygraphhoc
Since the ancient determination of the five platonic solids the study of symmetry and regularity has always
been one of the most fascinating aspects of mathematics. One intriguing phenomenon of studies in graph
theory is the fact that quite often arithmetic regularity properties of a graph imply the existence of many
symmetries, i.e. large automorphism group G. In some important special situation higher degree of
regularity means that G is an automorphism group of finite geometry. For example, a glance through the
list of distance regular graphs of diameter d < 3 reveals the fact that most of them are connected with
classical Lie geometry. Theory of distance regular graphs is an important part of algebraic combinatorics
and its applications such as coding theory, communication networks, and block design. An important tool
for investigation of such graphs is their spectra, which is the set of eigenvalues of adjacency matrix of a
graph. Let G be a finite simple group of Lie type and X be the set homogeneous elements of the associated
geometry. The complexity of computing the adjacency matrices of a graph Gr on the vertices X such that
Aut GR = G depends very much on the description of the geometry with which one starts. For example, we
can represent the geometry as the totality of 1 cosets of parabolic subgroups 2 chains of embedded
subspaces (case of linear groups), or totally isotropic subspaces (case of the remaining classical groups), 3
special subspaces of minimal module for G which are defined in terms of a G invariant multilinear form.
The aim of this research is to develop an effective method for generation of graphs connected with classical
geometry and evaluation of its spectra, which is the set of eigenvalues of adjacency matrix of a graph. The
main approach is to avoid manual drawing and to calculate graph layout automatically according to its
formal structure. This is a simple task in a case of a tree like graph with a strict hierarchy of entities but it
becomes more complicated for graphs of geometrical nature. There are two main reasons for the
investigations of spectra: (1) very often spectra carry much more useful information about the graph than a
corresponding list of entities and relationships (2) graphs with special spectra, satisfying so called
Ramanujan property or simply Ramanujan graphs (by name of Indian genius mathematician) are important
for real life applications (see [13]). There is a motivated suspicion that among geometrical graphs one
could find some new Ramanujan graphs.
ON ALGORITHMIC PROBLEMS CONCERNING GRAPHS OF HIGHER DEGREE OF SYMMETRYFransiskeran
Since the ancient determination of the five platonic solids the study of symmetry and regularity has always
been one of the most fascinating aspects of mathematics. One intriguing phenomenon of studies in graph
theory is the fact that quite often arithmetic regularity properties of a graph imply the existence of many
symmetries, i.e. large automorphism group G. In some important special situation higher degree of
regularity means that G is an automorphism group of finite geometry. For example, a glance through the
list of distance regular graphs of diameter d < 3 reveals the fact that most of them are connected with
classical Lie geometry. Theory of distance regular graphs is an important part of algebraic combinatorics
and its applications such as coding theory, communication networks, and block design. An important tool
for investigation of such graphs is their spectra, which is the set of eigenvalues of adjacency matrix of a
graph. Let G be a finite simple group of Lie type and X be the set homogeneous elements of the associated
geometry.
The document summarizes the spectral theory and principle of limiting absorption for the elasticity operator in infinite, homogeneous, anisotropic media. It defines the elasticity operator and establishes its properties, including that it is self-adjoint. It derives the wave equation for elastic media from Hooke's law and the equations of motion. It shows plane wave solutions to the wave equation and establishes Christoffel's equation.
The document discusses direction cosines and ratios of lines, and equations to represent lines and planes in space. It provides equations in both vector and Cartesian form to define lines passing through points or parallel to vectors, lines passing through two points, the angle between lines, and the shortest distance between lines. It also provides equations to define planes passing through points or vectors, through three non-collinear points, perpendicular to a vector, and the distance from a plane to a point.
Analysis and algebra on differentiable manifoldsSpringer
This chapter discusses integration on manifolds. It defines orientation of manifolds and orientation-preserving maps between manifolds. It presents Stokes' theorem and Green's theorem, which relate integrals over boundaries to integrals of differential forms. It introduces de Rham cohomology groups, which classify closed forms modulo exact forms. Examples are given of calculating the orientability and cohomology of manifolds like the cylinder, Möbius strip, and real projective plane.
Computer graphics are pictures and movies created using computers - usually referring to image data created by a computer specifically with help from specialized graphical hardware and software. It is a vast and recent area in computer science.The phrase was coined by computer graphics researchers Verne Hudson and William Fetter of Boeing in 1960. Another name for the field is computer-generated imagery, or simply CGI.
Important topics in computer graphics include user interface design, sprite graphics, vector graphics, 3D modeling, shaders, GPU design, and computer vision, among others. The overall methodology depends heavily on the underlying sciences of geometry, optics, and physics. Computer graphics is responsible for displaying art and image data effectively and beautifully to the user, and processing image data received from the physical world. The interaction and understanding of computers and interpretation of data has been made easier because of computer graphics. Computer graphic development has had a significant impact on many types of media and has revolutionized animation, movies, advertising, video games, and graphic design generally.
The document discusses various graph theory topics including isomorphism, cut sets, labeled graphs, and Hamiltonian circuits. It defines isomorphism as two graphs being structurally identical with a one-to-one correspondence between their vertices and edges. Cut sets are edges whose removal would disconnect a connected graph. Labeled graphs assign labels or weights to their vertices and/or edges. A Hamiltonian circuit is a closed walk that visits each vertex exactly once.
The document discusses group theory and its applications in physics. It begins by introducing symmetry groups that are important in physics, including translations, rotations, and Lorentz transformations. It then discusses the use of group theory in formulating fundamental forces and the Standard Model of particle physics. The document provides definitions of group theory concepts like groups, operations, identity, and inverse. It explains how group theory provides a mathematical framework for describing physical symmetries.
Proportional and decentralized rule mcst gamesvinnief
This document presents two cost allocation rules - the proportional rule and the decentralized rule - for minimum cost spanning extension problems. The proportional rule allocates costs proportionally based on an algorithm that constructs a minimum cost spanning extension while simultaneously allocating link costs. The decentralized rule is based on an algorithm that constructs the extension in a decentralized manner, with links being added one by one and their costs immediately allocated. Both rules are shown to be refinements of the irreducible core defined for these problems. The proportional rule is then characterized axiomatically.
Vershinin 2013_about presentations of braid groups and theirPanagiote Ligouras
In the paper we give a survey of rather new notions and results which generalize
classical ones in the theory of braids. Among such notions are various inverse monoids of
partial braids. We also observe presentations different from standard Artin presentation for
generalizations of braids. Namely, we consider presentations with small number of generators,
Sergiescu graph-presentations and Birman-Ko-Lee presentation. The work of V. V. Chaynikov
on the word and conjugacy problems for the singular braid monoid in Birman-Ko-Lee generators
is described as well.
This document provides definitions and concepts related to graph theory. It begins with a brief history of graph theory and then defines basic concepts such as graphs, nodes, edges, adjacency, incidence, isomorphism, subgraphs, walks, trails, paths, connectedness, trees, and spanning trees. It also introduces different types of graphs including null graphs, cycle graphs, path graphs, complete graphs, bipartite graphs, and complete bipartite graphs. Finally, it discusses how vector spaces can be associated with graphs and defines the properties of cycle and cutset spaces.
This document provides a summary of a master's thesis in mathematics. The thesis studies generalisations of the fact that every Riesz homomorphism between spaces of continuous functions on compact Hausdorff spaces can be written as a composition multiplication operator. Specifically, the thesis examines Riesz homomorphisms between spaces of extended continuous functions on extremally disconnected spaces, known as Maeda-Ogasawara spaces. It proves that every Riesz homomorphism on such a space has a generalized composition multiplication form involving a continuous map between the underlying spaces. The thesis also applies these results to spaces of measurable functions and explores Riesz homomorphisms between spaces of continuous functions with values in a Banach lattice.
This document defines key concepts related to straight lines, including their various forms of equations, slope, intercepts, and how to calculate them. It discusses the general, standard, point-slope, and intercept forms of linear equations. It also explains how to find the slope between two points using the slope formula, and how to determine the x-intercept and y-intercept from a line's equation. Examples are provided to illustrate these concepts.
We make use of the conformal compactification of Minkowski spacetime M# to explore a way of describing general, nonlinear Maxwell fields with conformal symmetry. We distinguish the inverse Minkowski spacetime [M#]−1 obtained via conformal inversion, so as to discuss a doubled compactified spacetime on which Maxwell fields may be defined. Identifying M# with the projective light cone in (4+2)-dimensional spacetime, we write two independent conformal-invariant functionals of the 6-dimensional Maxwellian field strength tensors - one bilinear, the other trilinear in the field strengths -- which are to enter general nonlinear constitutive equations. We also make some remarks regarding the dimensional reduction procedure as we consider its generalization from linear to general nonlinear theories.
Cap 2_Nouredine Zettili - Quantum Mechanics.pdfOñatnom Adnara
This document discusses mathematical tools used in quantum mechanics. It introduces Hilbert spaces and their properties including linear vector spaces, scalar products, completeness, and bases. Vectors in a Hilbert space belong to an abstract vector space and satisfy properties like orthonormality. Operators acting on these vectors are linear. Examples of finite and infinite dimensional Hilbert spaces are given for the 3D Euclidean space and space of complex functions. Linear independence of sets of vectors and functions is also discussed.
Common Fixed Point Theorem for Weakly Compatible Maps in Intuitionistic Fuzzy...inventionjournals
In this paper, we prove some common fixed point theorem for weakly compatible maps in intuitionistic fuzzy metric space for two, four and six self mapping.
Presentation on Greater Chicago Food Depository from "Acting Up - Using Theater & Technology for Social Change," from DePaul's School for New Learning Distance Education Program, Winter 2010.
Este documento habla sobre la higiene industrial y sus objetivos como reconocer y evaluar los riesgos en el ambiente laboral, eliminar enfermedades profesionales, prevenir empeoramiento de enfermedades, y mantener la salud de los trabajadores. Explica los factores de riesgo físicos como ruido, vibraciones, calor/frío y radiaciones, y los factores de riesgo ergonómicos como carga física estática por posturas prolongadas y carga dinámica. También cubre cómo las posturas de trabajo inade
La institución educativa Alberto Lleras Camargo sede Camelias busca implementar estrategias metodológicas efectivas para superar las dificultades en los procesos de lectura y escritura en los grados de transición a tercero. Estas estrategias incluyen estimular el proceso de lectoescritura a través del método global y el juego dramático, así como promover el compromiso de los padres y generar nuevas actividades que despierten el interés de los estudiantes.
The document discusses the Black-Scholes option pricing model (BSOPM) and its assumptions. It outlines the derivation of the BSOPM, specifying the process the stock price follows and constructing a riskless portfolio to hedge the option. It then presents the BSOPM formula and defines the terms. The document also discusses the standard normal distribution curve used in the formula and provides an example calculation. Finally, it covers extensions such as dividends, the relationship between the BSOPM and binomial option pricing model, and estimating volatility.
Valsts darba inspekcijas Sadarbības un attīstības nodaļas vadītājas Lindas Matisānes prezentācija Pasaules darba aizsardzības dienai veltītā konferencē
Este documento describe los pasos para elaborar manjar de leche, incluyendo los materiales e insumos necesarios como leche, azúcar, bicarbonato y gelatina. Explica el procedimiento de colocar la leche en una paila y cocinarla a fuego lento mientras se agregan los otros ingredientes y se remueve con una paleta de madera. Finalmente, una vez evaporada la leche en un 50%, se añade la esencia de vainilla y se envasa el producto terminado.
1) The document describes a math mini project on bi-linear mapping conducted by students at Vivekanand Education Society's Institute of Technology.
2) It provides background information on bi-linear mapping, including that it is a function that is linear in each of its arguments and combines elements from two vector spaces into a third vector space. Matrix multiplication is provided as an example.
3) The document outlines several uses of bi-linear mapping, including its important applications in cryptography through the use of pairings over elliptic curves. It also discusses the history and future scope of bi-linear mapping.
This document provides an introduction and overview of matchings in graph theory. It begins with definitions of basic matching concepts like maximum matchings, perfect matchings, alternating paths, and augmenting paths. It then presents Hall's Marriage Theorem, which provides a necessary and sufficient condition for a bipartite graph to have a perfect matching. The document also includes proofs of theorems on maximum matchings and corollaries like every k-regular bipartite graph having a perfect matching. The overall summary is that this document covers fundamental definitions and results regarding matchings in graph theory.
Operator’s Differential geometry with Riemannian Manifoldsinventionjournals
This document discusses differential geometry concepts related to Riemannian manifolds. It begins with definitions of topological manifolds, smooth manifolds, tangent vectors, and Riemannian metrics. It then defines what makes a smooth manifold into a Riemannian manifold by endowing it with a Riemannian metric tensor. Finally, it proves that there exists a unique torsion-free and metric-compatible connection on any Riemannian manifold called the Levi-Civita connection.
The document discusses vector fields and line integrals. Some key points:
- A vector field associates a vector with each point in a region, such as a velocity vector field showing wind patterns.
- Line integrals generalize the idea of integration over an interval to integration over a curve. The line integral of a function f over a curve C is defined as the limit of Riemann sums that multiply f by the length of curve segments.
- Line integrals can be used to calculate properties like work done by a force field or circulation in fluid flow. Their value does not depend on the parametrization of the curve C.
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dSolution The concept of Derivative is at th.pdftheaksmart2011
Dry Ice., is a manufacturer of air conditioners that has seen its demand grow significantly. The
companyanticipates nationwide demand for the next year to be 180,000 units in the South,
120,000 units inin the Midwest, 110,000 units in the East, and 100,000 units in the West.
Managers at DryIce are designingthe manufacturing network and have selected four potential
sites-- New York, Atlanta, Chicago, and San DiegoPlants could have a capacity of either 200,000
or 400,000 units. The annual fixed costs are at the four locations areshown in the Table, along
with the cost of producing and shipping an air conditioner to each of the four markets.Where
should DryIce build its factories and how large should they be?Dry Ice., is a manufacturer of air
conditioners that has seen its demand grow significantly. The companyanticipates nationwide
demand for the next year to be 180,000 units in the South, 120,000 units inin the Midwest,
110,000 units in the East, and 100,000 units in the West. Managers at DryIce are designingthe
manufacturing network and have selected four potential sites-- New York, Atlanta, Chicago,
and San DiegoPlants could have a capacity of either 200,000 or 400,000 units. The annual fixed
costs are at the four locations areshown in the Table, along with the cost of producing and
shipping an air conditioner to each of the four markets.Where should DryIce build its factories
and how large should they be?
Solution
If the fixed cost is not taken into consideration then,
Dry Ice has the maximum demand in the south region as: 180,000 units
The company should thus build a plant size of 400,000 units (maximum possible) in order to
satisfy the demand of the regions and earn economies of scale.
The site of the company should be chosen from the factors like: proximity to the markets,
perishable or nonperishable goods, nearness to the warehouse, suppliers convenient etc..
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Local properties and differential forms of smooth map and tangent bundle
1. Mathematical Theory and Modeling
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.13, 2013
www.iiste.org
Local Properties and Differential Forms of Smooth Map and
Tangent Bundle
Md. Shafiul Alam 1 , Khondokar M. Ahmed 2 & Bijan Krishna Saha 3
1
Department of Mathematics, University of Barisal, Barisal, Bangladesh
2
3
Department of Mathematics, University of Dhaka, Dhaka, Bangladesh
Department of Mathematics, University of Barisal, Barisal, Bangladesh
1
Corresponding author: Email: shafiulmt@gmail.com, Tel: +88-01819357123
Abstract
In this paper, some basic properties of differential forms of smooth map and tangent bundle are
d e fi n ed b y
is t he d eri va tive of at a,
developed. The li n ear m ap
where
is a smooth map and
. If
is a smooth bijective ma p and if the maps
are all injective, then is a diffeomorphism. Finally, it is shown that if X is a
vector field on a manifold M then there is a radial neighbourhood
gh
of
in
and a
smooth map
such that
and
, where
is given
by
.
Keywords: Smooth map, manifold, tangent bundle, isomorphism, diffeomorphism, vector field.
1. Introduction
S. Kobayashi1 and K. Nomizu1 first worked on differential forms of smooth map and tangent bundle. Later H.
Flanders2, F. Warner3, M. Spivak4 and J. W. Milnor5 also worked on differential forms of smooth map and
tangent bundle. A number of significant results were obtained by S. Cairns 6, W. Greub9, E. Stamm9, R. G.
Swan10, R. L. Bishop12, R. J. Crittenden12 and others.
Let M be a smooth manifold and let S (M) be the ring of smooth functions on M. A tangent vector of
M at a point
is a linear map S (M)
such that
S (M).
under the linear operations
The tangent vectors form a real vector space
S (M).
is called the tangent space of M at a.
Let E be an n-dimensional real vector space. Let
be an open subset of E and let
define a linear isomorphism
. If
is a smooth map of
sp ace F, then the classic al d erivative o f at a is the linear map
Moreover, in the special case
. We shall
into a second vector
given by
we have the product formula
S (O).
S (O)
given by
This shows that the linear map
of O at a. Hence, we have a canonical linear map
is a tangent vector
given by
.
2. The Derivative of a Smooth Map
Let
be a smooth map and let
. The li nea r map
is ca ll ed t h e d er i va tive of at a. It will be denoted by
,
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d efi ned b y
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S (N),
If
map
is a second smooth map, then
we have
and
Lemma 1 . Let
to
map from
Proof. Let
.
. Moreover, for the identity
. In particular, if
is a diffeomorphism, then
are inverse linear isomorphisms.
, then
.
and the corre spondence
be a smooth map.
S (N),
.
defines a linear
S (N)
induces a homomorphism
S (M) given by
is a linear map from S (N) to . Moreover,
S (N)
and so
. Clearly
is linear. Therefore, the lemma is proved.
be a basis for
Proposition 1. Let
(a)
the functions
the functions
S ( ).
Then
S ( ) are given by
(b)
and let
S ( ) satisfy
Proof. By the fundamental theorem of calculus we have
Thus the theorem follows, with
Proposition 2. The canonical linear map
.
is an isomorphism.
Proof. First we consider the case
. We show first that
is surjective. Let
and let
S
where the
satisfy conditions (a) and (b) of Proposition 1.
( ). Write
Since maps the constant function
into zero, so
Since the functions are independent of f, we can write
surjective. To show that
is injective, let be any linear function in . Then for
Now suppose
. Then
. Hence
and so
is injective.
Finally, let be any open subset of E and let
identity map and we have the commutative diagram
be the inclusion map. Then
102
. Thus
,
is
.
is the
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is a linear isomorphism, the canonical linear map
Since
completes the proof.
. Then dim
Corollary 1. Let M be a smooth manifold and let
Proof. Let
we find dim
is an isomorphism, which
be a chart for M such that
= dim
= dim = dim .
= dim
m
. Using the result of Lemma 1 and Proposition 2
be a smooth map
Proposition 3. The derivative of a constant map is zero. Conversely, let
Then, if M is connected, is a constant map.
such that
Proof. Assume that
function given by
follows that each
is the constant map
where
. Hence, for
.
S ( ) is the constant
. It
. Then, for
,
is a smooth map satisfying
and let
be
Conversely, assume that
connected. Then, given two points
and , there exists a smooth curve
such
and
. Consider the map
.
We have
that
.
Now using an atlas for N we see that
must be constant. In particular
. Thus is a constant map and the proposition is proved.
and so
3. Local Properties of Smooth Maps
be a smooth map. Then
is called a local diffeomorphism at a point
is a linear isomorphism. If is a local diffeomorphism for all points
called a local diffeomorphism of M into N.
if the map
, it is
Theorem 1. Let
point. Then
be a given
Let
be a smoo th map where d im M = n and dim N = r. Let
(a)
If is a local diffeomorphism at a, there are neighbourhoods U of a and V of b such that
maps U diffeomorphically onto V.
(b)
If
is injective, there are neighbourhoods U of a, V of b, and
such that
a diffeomorphism
(c)
If
of 0 in
.
is surjective, there are neighbourhoods U of a, V of b, and W of 0 in
a diffeomorphism
the projection.
and
such that
where
, and
is
. In part (a), then, we are
Proof. By using charts we may reduce to the case
assuming that
is an isomorphism, and the conclusion is the inverse function theorem.
For part (b), we choose a subspace E of
given by
such that
, and consider the map
. Then
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.
It follows that
i s i n j e c t i v e a n d t h u s a n i s o m o r p h i s m (r = dim Im
+ dim E = n +
dim E ). Thus part (a) implies the existence of neighborhoods U of a, V of b, and
of 0 in E such that
is a diffeomorphism. Clearly,
such that
Finally, for part (c), we choose a subspace E of
projection induced by this decomposition and define
. Let
. It follows easily that
Then
Hence there are neighborhoods U of a, V of b and W of
diffeomorphism. Hence the proof of the theorem is complete.
be the
by
is a linear isomorphism.
such that
is a smooth bijective map and if the maps
Proposition 4. If
all injective, then is a diffeomorphism.
is a
are
is inj ective, we have
. Now we show that r
Proof. Let d im M = n , dim N = r. Since
= n. In fact, according to Theorem 1, part (b), for every
there are neighborhoods
of a, V of
and
of
together with a diffeomorphism
such that the diagram
commutes (i denotes the inclusion map opposite 0).
of
such that each
is compact and contained
Choose a countable open covering
in some
. Since is surjective, it follows that
. Now assume that r > n. Then the
diagram implies that no
contains an open set. Thus, N could not be Hausdorff. This contradiction
shows that
. Since
, is a local diffeomorphism. On the other hand, is bijective. Since it
is a local diffeomorphism, Theorem 1 implies that its inverse is smooth. Thus is a diffeomorphism and the
theorem is proved.
Definition 1. A q uotient manifold o f a manifold M is a manifold N together with a smooth map
such that and each linear map
is surjective and thus dim M dim
N.
Lemma 2. Let
make N into a quotient manifold of M. Then the map
is open.
, there is a neighbourhood U of a such that the restriction
Proof. It is sufficient to show that, for each
of to U is open. This follows at once from part (c) of Theorem 1.
4. Submanifolds
Let M be a manifold. An embedded manifold is a pair
, where N is a second manifold and
is a smooth map such that the derivative
is injective. In particular, since the maps
are injective, it follows that dim N dim M. Given an embedded manifold
, the subset
may be considered as a bijective map
. A submanifold of a
manifold M is an embedded manifold
such that
is a homeomorphism, when
is
given the topology induced b y the topology of M. If N is a subset o f M and is the inclusion map,
we say simply that N is a submanifold of M.
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Proposition 5. Let
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be a submanifold of M. Assume that Q is a smooth manifold and
is a commutative diagram of maps. Then
is smooth if and only if
is smooth.
Proof. If is smooth then clearly is smooth. Conversely, assume that is smooth. Fix a point a
Q and set
. Since
is injective, there are neighbourhoods U, V of b in N and M respectively,
such that
.
and there is a smooth map
Since N is a submanifold of , the map is continuous. Hence there is a neighbourhood W of a such that
. Then
, where
denote the restrictions of
to W. It follows that
and so is smooth in W; thus is a smooth map. Hence the proposition is proved.
5. Vector Fields
A vector field X on a manifold M is a cross-section in the tangent bundle
. Thus a vector field X
a tangent vector
such that the map
so obtained is smooth.
assigns to every point
The vector fields on M form a module over the ring S ( ) which will be denoted by
.
onto the S ( )-module of derivations in the
Theorem 2. There is a canonical isomorphism of
algebra S ( ), i.e., Der S ( ).
Proof. Let
be a vector field. For each
S ( ), we define a function
.
is smooth. To see this we may assume that
is smooth. Hence every vector field X on M determines a map
S( )
Obviously,
S ( ) given by
.
is a derivation in the algebra S ( ). The assignment
Der S ( ).
We show now that
is an isomorphism. Suppose
on M by
. But then
, for some
defines a homomorphism
. Then
S ( ).
This implies that
Then, for every point
; i.e. X = 0. To prove that
, determines the vector
is surjective, let
, given by
be any derivation in S ( ).
S ( ).
. T o sho w that this map is s mo o th, fix a point
. Using a
construct vector fields
and smooth functions
on M such that
, (V is some neighbourhood of a). Then
the vectors
form a basis for
. Hence, for each
, there is a unique
system of numbers
such that
. Applying
to
we obtain
. Hence
. Since the
are smooth
functions on M, this equation shows that X is smooth in V; i.e. X is a vector field. Finally, it follows from the
definition that
. Thus is surjective, which completes the proof.
We d efine
chart, it is
easy
by
to
6. Differential Equations
Let X be a vector field on a manifold M. An orbit for X is a smooth map
105
where
is some
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open interval such that
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.
Co n sid er t h e p ro d u ct m an i fo ld
. W e cal l a s ub se t
or
where is an open interval on
union and finite intersection of radial sets is again radial.
and
Proposition 6. Let X be a vector field on M. Fix
of X such that
. Moreover, if
an orbit
some
then
.
Proof. For the first statement of the proposition we may assume
stand ard P icard existenc e theo rem.
radial if for each
,
containing the point 0. The
. Then there is an interval and
are orbits for X which agree at
. In this case it is the
for which
is both closed and
To prove the second part we show that the set of
open, and hence all of . It is obviously closed. To show that it is open we may assume
and
then we apply the Picard uniqueness theorem, which completes the proof.
Theorem 3. Let X be a vector field on a manifold M. Then there is a radial neighbourhood
in
and a smooth map
such that
of
and
where
is given by
. Moreover,
is uniquely determined by X.
Proof. Let
be an atlas for M. The Picard existence theorem implies our theorem for each
. Hence there are radial neighbourhoods
of
in
and there are smooth maps
and
.
such that
. Then
is a radial neighbourhood of
in
. Moreover,
is a
Now set
; if
, then
is an interval
radial neighbourhood of
containing . Clearly,
are orbits of X agreeing at 0, and so by Proposition 6 they
agree in . It follows that they agree in
. Thus the
defines a smooth map
which
follows from Proposition 6. Thus the theorem is proved.
has the desired properties. The uniqueness of
7. Conclusions
Differential forms are among the fundamental analytic objects treated in this paper. The
derivative of a smooth map appears as a bundle map between the corresponding tangent
bundles. They are the cross-sections in the exterior algebra bundle of the dual of the tangent
bundle and they form a graded anticommutative algebra. Vector fields on a manifold has been
introduced as cross-sections in the tangent bundle. The module of vector fields is canonically
isomorphic to the module of derivations in the ring of smooth functions.
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107
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