Topology is a branch of mathematics developed in modern times to study continuous phenomena. Chinese names originated from Greek τ ο π ο λ ο γ? Transliteration of α. Topology, originally meaning landform, was introduced by scientists in the middle of19th century. At that time, I mainly studied some geometric problems arising from the needs of mathematical analysis. So far, topology mainly studies the invariant properties and invariants of topological space under topological transformation.
Branch discipline
Point set topology is also called general topology.
Combinatorial topology
algebraic topology
Differential topology
Geometric topology
analysis situs
Topology is an important and basic branch of mathematics. At first, it is a branch of geometry, which studies the properties of geometric figures that remain unchanged under continuous deformation (the so-called continuous deformation, figuratively speaking, allows deformation such as expansion and distortion, but does not allow cutting and bonding); Now it has developed into a branch of mathematics that studies continuous phenomena. Because of the diversity of continuous expressions and research methods in mathematics, topology is divided into several branches with different research objects and methods. In the gestation stage of topology, at the end of 19, there have been two directions: point set topology and combinatorial topology. Now, the former has evolved into a general topology, while the latter has become an algebraic topology. Then differential topology and geometric topology appeared one after another.
Mathematically, the problem of the Seven Bridges in Konigsberg, the polyhedral euler theorem and the four-color problem are all important problems in the development history of topology.
Konigsberg (now Kaliningrad, Russia) is the capital of East Prussia, and the Pledgel River runs through it. /kloc-in the 0/8th century, seven bridges were built on this river, connecting the two islands in the middle of the river with the riverbank. People often walk on it in their spare time. One day, someone asked: can we just walk on each bridge once and finally return to the original position? This question looks very simple and interesting, attracting everyone. Many people are trying various methods, but no one can do it. It seems that it is not so easy to get a clear and ideal answer.
1736, someone found the great mathematician Euler with this question. After some thinking, Euler quickly gave the answer in a unique way. Euler first simplified the problem. He regarded two small islands and the river bank as four points respectively, and seven bridges as connecting lines between these four points. Then the question is simplified to, can you draw this figure with one stroke? After further analysis, Euler came to the conclusion that it is impossible to walk every bridge and finally return to its original position. And gives the conditions that all the drawings can be drawn. This is the pioneer of topology.
In the history of topology, there is also a famous and important theorem about polyhedron, which is also related to Euler. The content of this theorem is: if the number of vertices, edges and faces of a convex polyhedron is V, then they always have such a relationship: f+v-e=2.
According to the euler theorem of polyhedron, we can get an interesting fact: there are only five regular polyhedrons. They are regular tetrahedron, regular hexahedron, regular octahedron, regular dodecahedron and regular icosahedron.
The famous "four-color problem" is also related to the development of topology. The four-color problem, also known as the four-color conjecture, is one of the three major mathematical problems in the modern world.
The four-color conjecture was put forward by Britain. 1852, when Francis guthrie, who graduated from London University, came to a scientific research unit to do map coloring, he found an interesting phenomenon: "It seems that every map can be colored with four colors, so countries with the same border will be colored with different colors."
1872, Kelly, the most famous mathematician in Britain at that time, formally put forward this question to the London Mathematical Society, so the four-color conjecture became a concern of the world mathematical community. Many first-class mathematicians in the world have participated in the great battle of four-color conjecture. During the two years from 1878 to 1880, Kemp and Taylor, two famous lawyers and mathematicians, respectively submitted papers to prove the four-color conjecture and announced that they had proved the four-color theorem. But later mathematician Hurwood pointed out that Kemp's proof and his own accurate calculation were wrong. Soon, Taylor's proof was also denied. As a result, people began to realize that this seemingly simple topic is actually a difficult problem comparable to Fermat's conjecture.
Since the 20th century, scientists have basically proved the four-color conjecture according to Kemp's idea. After the emergence of electronic computers, the process of proving the four-color conjecture has been greatly accelerated due to the rapid improvement of calculation speed and the emergence of man-machine dialogue. 1976, American mathematicians Appel and Harken spent 1200 hours on two different computers at the University of Illinois in the United States, made 1000 billion judgments, and finally completed the proof of the four-color theorem. However, many mathematicians are not satisfied with the achievements made by computers. They think there should be a simple and clear written proof method.
The above examples are all related to geometric figures, but these problems are different from traditional geometry, but some new geometric concepts. These are pioneers of "topology".
What is topology?
The English name of Topology is topology, and the literal translation is geography, which is similar to topography and geomorphology. In the early days of China, it was translated into situational geometry, continuous geometry and geometry under one-to-one continuous transformation group. But these translations are not easy to understand, and the unified mathematical terminology from 65438 to 0956 recognizes it as topology and transliteration.
Topology is a branch of geometry, but this geometry is different from the usual plane geometry and solid geometry. Usually, the research object of plane geometry or solid geometry is the positional relationship between points, lines and surfaces and their measurement properties. Topology has nothing to do with the measurement properties and quantitative relations of the length, size, area and volume of the research object.
For example, in the usual plane geometry, if one figure on the plane moves to another figure, if they are completely coincident, then the two figures are said to be conformal. However, the graph studied in topology changes in motion, regardless of its size or shape. In topology, there is no element that cannot be bent, and the size and shape of each figure can be changed. For example, when Euler solved the problem of the Seven Bridges in Konigsberg, he did not consider its size and shape, but only the number of points and lines. These are the starting points of topological thinking.
What are the topological properties? Firstly, we introduce topological equivalence, which is an easy-to-understand topological property.
Topology does not discuss the concept of congruence between two graphs, but the concept of topological equivalence. For example, circles, squares and triangles are all equivalent graphs under topological transformation, although their shapes and sizes are different. The three things on the left are topologically equivalent, in other words, from the perspective of topology, they are exactly the same.
Select some points on a sphere and connect them with disjoint lines, so that the sphere is divided into many blocks by these lines. Under topological transformation, the number of points, lines and blocks is still the same as the original number, which is topological equivalence. Generally speaking, for a closed surface with arbitrary shape, as long as the surface is not torn or cut, its transformation is topological change, and topological equivalence exists.
It should be pointed out that torus does not have this property. For example, if the torus is cut as shown on the left, it will not be divided into many pieces, but just become a curved bucket. In this case, we say that a sphere cannot be topologically torus. So sphere and torus are topologically different surfaces.
The combination relationship and order relationship between points and lines on a straight line remain unchanged under topological transformation, which is a topological property. In topology, the closed properties of curves and surfaces are also topological properties.
The plane and surface we usually talk about usually have two sides, just like a piece of paper has two sides. But the German mathematician Mobius (1790 ~ 1868) discovered Mobius surface in 1858. This surface cannot be painted with different colors on both sides.
There are many invariants and invariants of topological transformation, which are not introduced here.
After the establishment of topology, it has also developed rapidly due to the development needs of other mathematical disciplines. Especially after Riemann founded Riemann geometry, he took the concept of topology as the basis of analytic function theory, which further promoted the progress of topology.
Since the twentieth century, set theory has been introduced into topology, which has opened up a new look for topology. Topology becomes the corresponding concept of arbitrary point set. Some problems that need to be accurately described in topology can be discussed by sets.
Because a large number of natural phenomena are continuous, topology has the possibility of extensive contact with various practical things. Through the study of topology, we can make clear the set structure of space and grasp the functional relationship between spaces. Since 1930s, mathematicians have made more in-depth research on topology and put forward many new concepts. Such as uniform structure, abstract distance, approximate space, etc. There is a branch of mathematics called differential geometry, which uses differential tools to study the bending of lines and surfaces near a point, and topology studies the global relationship of surfaces. Therefore, there should be some essential connection between these two disciplines. 1945, China mathematician Chen Shengshen established the connection between algebraic topology and differential geometry, which promoted the development of global geometry.
Until today, topology has been divided into two branches in theory. One branch focuses on analytical methods, called point set topology, or analytical topology. Another branch focuses on algebraic methods, called algebraic topology. Now, these two branches have a unified trend.
Topology was originally called situation analysis, which was put forward by G.W. Leibniz in 1679. The word topology (Chinese transliteration) was put forward by J.B. Listing in 1847, which comes from the Greek position, situation and knowledge.
Starting from 185 1, B. Riemann proposed that in order to learn functions and integrals, it is necessary to learn situational analysis. From then on, a systematic study of topology began.
The founder of combinatorial topology is h poincare. He introduced topological problems in his work of analysis and mechanics, especially in the study of complex functions and single values of curves determined by differential equations. He discussed the topological classification of three-dimensional manifolds and put forward the famous Poincare conjecture.
Another source of topology is the rigor of analysis. The strict definition of real numbers promoted G Cantor to systematically study point sets in Euclidean space from 1873, and obtained many topological concepts. Such as: gathering points, open sets, connectivity, etc. Under the influence of point set theory, the concept of universal function (that is, function) appeared in the analysis. Take a function set as a geometric object, discuss its limit, and finally lead to the concept of abstract space.
Some basic examples of topological problems:
The problem of the seventh bridge in Konigsberg (one shot problem). How can a pedestrian cross seven bridges, and each bridge only passes once? This18th century intellectual game was simplified by L. Euler as a question of whether the network drawn with thin lines can be drawn with one stroke, and then he proved it impossible. Whether a pen can draw a network has nothing to do with the length of a line, but only depends on the connection between points and lines. Imagine a network made of soft and elastic materials. After it is bent and stretched, whether it can be drawn in one stroke will not change.
Euler polyhedron formula and surface classification. Euler found that no matter what shape of convex polyhedron, there is always such a relationship among the number of vertices, the number of edges and the number of faces. It can be proved that there are only five regular polyhedrons. If a polyhedron is not convex but has a frame shape (Figure 33), there will always be a polyhedron regardless of the shape of the frame. This shows that convex shape and frame shape have more essential differences than long and short curves. Generally speaking, there is a hole in the shape of a frame.
Under continuous deformation, the surface of convex body can become spherical, and the surface of frame can become torus (tire tread). Neither of them can be transformed into each other through continuous deformation (Figure 34). How many different types of sealing surfaces are there under continuous deformation? How to identify them? This is the main problem of topology research in the second half of 19 century.
Knotting problem. A closed curve that does not intersect itself in space will be knotted. Ask whether a knot can be untied (that is, whether it can be transformed into an oblate circle) or whether two knots can be transformed into each other (as shown in Figure 35, whether two trefoil knot can be transformed into each other). At the same time, it is far from easy to give strict proof.
Wiring problem (embedding problem). Can a complex network be laid on a plane without crossing? This problem naturally occurs when making printed circuits. Figure 36. On the left, a diagonal can be placed on the plane by moving it out of the square. However, no matter how you move it, the two pictures in Figure 37 cannot be placed on the same plane. 1930 k？ Kuratowski proved that whether a network can be embedded in a plane depends on whether it does not contain one of these two graphs.
These examples show that there are some properties of geometric figures that cannot be studied by traditional geometric methods. These properties have nothing to do with length and angle, but they show the characteristics of the overall structure of the figure. This property is the so-called topological property of graphs.
Origin of topology
Geometric topology is a branch of mathematics formed in19th century, which belongs to the category of geometry. Some contents about topology appeared as early as the eighteenth century. Some isolated problems were discovered at that time, which later played an important role in the formation of topology.
Mathematically, the problem of the Seven Bridges in Konigsberg, the polyhedral euler theorem and the four-color problem are all important problems in the development history of topology.
Konigsberg (now Kaliningrad, Russia) is the capital of East Prussia, and the Pledgel River runs through it. /kloc-in the 0/8th century, seven bridges were built on this river, connecting the two islands in the middle of the river with the riverbank. People often walk on it in their spare time. One day, someone asked: can we just walk on each bridge once and finally return to the original position? This question seems simple and interesting, attracting everyone. Many people are trying various methods, but no one can do it. It seems that it is not so easy to get a clear and ideal answer.
1736, someone found the great mathematician Euler with this question. After some thinking, Euler quickly gave the answer in a unique way. Euler first simplified the problem. He regarded two small islands and the river bank as four points respectively, and seven bridges as connecting lines between these four points. Then the question is simplified to, can you draw this figure with one stroke? After further analysis, Euler came to the conclusion that it is impossible to walk every bridge and finally return to its original position. And gives the conditions that all the drawings can be drawn. This is the pioneer of topology.
In the history of topology, there is also a famous and important theorem about polyhedron, which is also related to Euler. The content of this theorem is: if the number of vertices, edges and faces of a convex polyhedron is V, then they always have such a relationship: f+v-e=2.
According to the euler theorem of polyhedron, we can get an interesting fact: there are only five regular polyhedrons. They are regular tetrahedron, regular hexahedron, regular octahedron, regular dodecahedron and regular icosahedron.
The famous "four-color problem" is also related to the development of topology. The four-color problem, also known as the four-color conjecture, is one of the three major mathematical problems in the modern world.
The four-color conjecture was put forward by Britain. 1852, when Francis guthrie, who graduated from London University, came to a scientific research unit to do map coloring, he found an interesting phenomenon: "It seems that every map can be colored with four colors, so countries with the same border will be colored with different colors."
1872, Kelly, the most famous mathematician in Britain at that time, formally put forward this question to the London Mathematical Society, so the four-color conjecture became a concern of the world mathematical community. Many first-class mathematicians in the world have participated in the great battle of four-color conjecture. During the two years from 1878 to 1880, Kemp and Taylor, two famous lawyers and mathematicians, respectively submitted papers to prove the four-color conjecture and announced that they had proved the four-color theorem. But later mathematician Hurwood pointed out that Kemp's proof and his own accurate calculation were wrong. Soon, Taylor's proof was also denied. As a result, people began to realize that this seemingly simple topic is actually a difficult problem comparable to Fermat's conjecture.
Since the 20th century, scientists have basically proved the four-color conjecture according to Kemp's idea. After the emergence of electronic computers, the process of proving the four-color conjecture has been greatly accelerated due to the rapid improvement of calculation speed and the emergence of man-machine dialogue. 1976, American mathematicians Appel and Harken spent 1200 hours on two different computers at the University of Illinois in the United States, made 1000 billion judgments, and finally completed the proof of the four-color theorem. However, many mathematicians are not satisfied with the achievements made by computers. They think there should be a simple and clear written proof method.
The above examples are all related to geometric figures, but these problems are different from traditional geometry, but some new geometric concepts. These are pioneers of "topology".
What is topology?
The English name of Topology is topology, and the literal translation is geography, which is similar to topography and geomorphology. In China's early years, it was translated into situational geometry, continuous geometry and geometry under one-to-one continuous transformation group. But these translations are not easy to understand, and the unified mathematical terminology from 65438 to 0956 recognizes it as topology and transliteration.
Topology is a branch of geometry, but this geometry is different from the usual plane geometry and solid geometry. Usually, the research object of plane geometry or solid geometry is the positional relationship between points, lines and surfaces and their measurement properties. Topology has nothing to do with the measurement properties and quantitative relations of the length, size, area and volume of the research object.
For example, in the usual plane geometry, if one figure on the plane moves to another figure, if they are completely coincident, then the two figures are said to be conformal. However, the graph studied in topology changes in motion, regardless of its size or shape. In topology, there is no element that cannot be bent, and the size and shape of each figure can be changed. For example, when Euler solved the problem of the Seven Bridges in Konigsberg, he did not consider its size and shape, but only the number of points and lines. These are the starting points of topological thinking.
What are the topological properties? Firstly, we introduce topological equivalence, which is an easy-to-understand topological property.
Topology does not discuss the concept of congruence between two graphs, but the concept of topological equivalence. For example, circles, squares and triangles are all equivalent graphs under topological transformation, although their shapes and sizes are different. The three things on the left are topologically equivalent, in other words, from the perspective of topology, they are exactly the same.
Select some points on a sphere and connect them with disjoint lines, so that the sphere is divided into many blocks by these lines. Under topological transformation, the number of points, lines and blocks is still the same as the original number, which is topological equivalence. Generally speaking, for a closed surface with arbitrary shape, as long as the surface is not torn or cut, its transformation is topological change, and topological equivalence exists.
Origin of topology
Geometric topology is a branch of mathematics formed in19th century, which belongs to the category of geometry. Some contents about topology appeared as early as the eighteenth century. Some isolated problems were discovered at that time, which later played an important role in the formation of topology.
Mathematically, the problem of the Seven Bridges in Konigsberg, the polyhedral euler theorem and the four-color problem are all important problems in the development history of topology.
Konigsberg (now Kaliningrad, Russia) is the capital of East Prussia, and the Pledgel River runs through it. /kloc-in the 0/8th century, seven bridges were built on this river, connecting the two islands in the middle of the river with the riverbank. People often walk on it in their spare time. One day, someone asked: can we just walk on each bridge once and finally return to the original position? This question looks very simple and interesting, attracting everyone. Many people are trying various methods, but no one can do it. It seems that it is not so easy to get a clear and ideal answer.
1736, someone found the great mathematician Euler with this question. After some thinking, Euler quickly gave the answer in a unique way. Euler first simplified the problem. He regarded two small islands and the river bank as four points respectively, and seven bridges as connecting lines between these four points. Then the question is simplified to, can you draw this figure with one stroke? After further analysis, Euler came to the conclusion that it is impossible to walk every bridge and finally return to its original position. And gives the conditions that all the drawings can be drawn. This is the pioneer of topology.
In the history of topology, there is also a famous and important theorem about polyhedron, which is also related to Euler. The content of this theorem is: if the number of vertices, edges and faces of a convex polyhedron is V, then they always have such a relationship: f+v-e=2.
According to the euler theorem of polyhedron, we can get an interesting fact: there are only five regular polyhedrons. They are regular tetrahedron, regular hexahedron, regular octahedron, regular dodecahedron and regular icosahedron.
The famous "four-color problem" is also related to the development of topology. The four-color problem, also known as the four-color conjecture, is one of the three major mathematical problems in the modern world.
The four-color conjecture was put forward by Britain. 1852, when Francis guthrie, who graduated from London University, came to a scientific research unit to do map coloring, he found an interesting phenomenon: "It seems that every map can be colored with four colors, so countries with the same border will be colored with different colors."
1872, Kelly, the most famous mathematician in Britain at that time, formally put forward this question to the London Mathematical Society, so the four-color conjecture became a concern of the world mathematical community. Many first-class mathematicians in the world have participated in the great battle of four-color conjecture. During the two years from 1878 to 1880, Kemp and Taylor, two famous lawyers and mathematicians, respectively submitted papers to prove the four-color conjecture and announced that they had proved the four-color theorem. But later mathematician Hurwood pointed out that Kemp's proof and his own accurate calculation were wrong. Soon, Taylor's proof was also denied. As a result, people began to realize that this seemingly simple topic is actually a difficult problem comparable to Fermat's conjecture.
Since the 20th century, scientists have basically proved the four-color conjecture according to Kemp's idea. After the emergence of electronic computers, the process of proving the four-color conjecture has been greatly accelerated due to the rapid improvement of calculation speed and the emergence of man-machine dialogue. 1976, American mathematicians Appel and Harken spent 1200 hours on two different computers at the University of Illinois in the United States, made 1000 billion judgments, and finally completed the proof of the four-color theorem. However, many mathematicians are not satisfied with the achievements made by computers. They think there should be a simple and clear written proof method.
The above examples are all related to geometric figures, but these problems are different from traditional geometry, but some new geometric concepts. These are pioneers of "topology".
What is topology?
The English name of Topology is topology, and the literal translation is geography, which is similar to topography and geomorphology. In China's early years, it was translated into situational geometry, continuous geometry and geometry under one-to-one continuous transformation group. But these translations are not easy to understand, and the unified mathematical terminology from 65438 to 0956 recognizes it as topology and transliteration.
Topology is a branch of geometry, but this geometry is different from the usual plane geometry and solid geometry. Usually, the research object of plane geometry or solid geometry is the positional relationship between points, lines and surfaces and their measurement properties. Topology has nothing to do with the measurement properties and quantitative relations of the length, size, area and volume of the research object.
For example, in the usual plane geometry, if one figure on the plane moves to another figure, if they are completely coincident, then the two figures are said to be conformal. However, the graph studied in topology changes in motion, regardless of its size or shape. In topology, there is no element that cannot be bent, and the size and shape of each figure can be changed. For example, when Euler solved the problem of the Seven Bridges in Konigsberg, he did not consider its size and shape, but only the number of points and lines. These are the starting points of topological thinking.
What are the topological properties? Firstly, we introduce topological equivalence, which is an easy-to-understand topological property.
Topology does not discuss the concept of congruence between two graphs, but the concept of topological equivalence. For example, circles, squares and triangles are all equivalent graphs under topological transformation, although their shapes and sizes are different. The three things on the left are topologically equivalent, in other words, from the perspective of topology, they are exactly the same.
Select some points on a sphere and connect them with disjoint lines, so that the sphere is divided into many blocks by these lines. Under topological transformation, the number of points, lines and blocks is still the same as the original number, which is topological equivalence. Generally speaking, for a closed surface with arbitrary shape, as long as the surface is not torn or cut, its transformation is topological change, and topological equivalence exists.
It should be pointed out that torus does not have this property. For example, if the torus is cut as shown on the left, it will not be divided into many pieces, but just become a curved bucket. In this case, we say that a sphere cannot be topologically torus. So sphere and torus are topologically different surfaces.
The combination relationship and order relationship between points and lines on a straight line remain unchanged under topological transformation, which is a topological property. In topology, the closed properties of curves and surfaces are also topological properties.
The plane and surface we usually talk about usually have two sides, just like a piece of paper has two sides. But the German mathematician Mobius (1790 ~ 1868) discovered Mobius surface in 1858. This surface cannot be painted with different colors on both sides.
There are many invariants and invariants of topological transformation, which are not introduced here.
After the establishment of topology, it has also developed rapidly due to the development needs of other mathematical disciplines. Especially after Riemann founded Riemann geometry, he took the concept of topology as the basis of analytic function theory, which further promoted the progress of topology.
Since the twentieth century, set theory has been introduced into topology, which has opened up a new look for topology. Topology becomes the corresponding concept of arbitrary point set. Some problems that need to be accurately described in topology can be discussed by sets.
Because a large number of natural phenomena are continuous, topology has the possibility of extensive contact with various practical things. Through the study of topology, we can make clear the set structure of space and grasp the functional relationship between spaces. Since 1930s, mathematicians have made more in-depth research on topology and put forward many new concepts. Such as uniform structure, abstract distance, approximate space, etc. There is a branch of mathematics called differential geometry, which uses differential tools to study the bending of lines and surfaces near a point, and topology studies the global relationship of surfaces. Therefore, there should be some essential connection between these two disciplines. 1945, China mathematician Chen Shengshen established the connection between algebraic topology and differential geometry, which promoted the development of global geometry.
Until today, topology has been divided into two branches in theory. One branch focuses on analytical methods, called point set topology, or analytical topology. Another branch focuses on algebraic methods, called algebraic topology. Now, these two branches have a unified trend.
Topology is widely used in functional analysis, Lie group theory, differential geometry, differential equations and many other branches of mathematics.
References:
/Maths/Maths _ Branch/Topology _ total.htm It should be pointed out that torus does not have this property. For example, if the torus is cut as shown on the left, it will not be divided into many pieces, but just become a curved bucket. In this case, we say that a sphere cannot be topologically torus. So sphere and torus are topologically different surfaces.
The combination relationship and order relationship between points and lines on a straight line remain unchanged under topological transformation, which is a topological property. In topology, the closed properties of curves and surfaces are also topological properties.
The plane and surface we usually talk about usually have two sides, just like a piece of paper has two sides. But the German mathematician Mobius (1790 ~ 1868) discovered Mobius surface in 1858. This surface cannot be painted with different colors on both sides.
There are many invariants and invariants of topological transformation, which are not introduced here.
After the establishment of topology, it has also developed rapidly due to the development needs of other mathematical disciplines. Especially after Riemann founded Riemann geometry, he took the concept of topology as the basis of analytic function theory, which further promoted the progress of topology.
Since the twentieth century, set theory has been introduced into topology, which has opened up a new look for topology. Topology becomes the corresponding concept of arbitrary point set. Some problems that need to be accurately described in topology can be discussed by sets.
Because a large number of natural phenomena are continuous, topology has the possibility of extensive contact with various practical things. Through the study of topology, we can make clear the set structure of space and grasp the functional relationship between spaces. Since 1930s, mathematicians have made more in-depth research on topology and put forward many new concepts. Such as uniform structure, abstract distance, approximate space, etc. There is a branch of mathematics called differential geometry, which uses differential tools to study the bending of lines and surfaces near a point, and topology studies the global relationship of surfaces. Therefore, there should be some essential connection between these two disciplines. 1945, China mathematician Chen Shengshen established the connection between algebraic topology and differential geometry, which promoted the development of global geometry.
Until today, topology has been divided into two branches in theory. One branch focuses on analytical methods, called point set topology, or analytical topology. Another branch focuses on algebraic methods, called algebraic topology. Now, these two branches have a unified trend.
Topology in functional analysis