Fixed Point Developed by Space Topology
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ISSN : 2394-3580 Swadeshi Research Foundation Vol. – 6, No. - 10 August 2019 A Monthly Journal of Multidisciplinary International Peer Refereed, Review, Indexing & Impact Factor - 4.2 Research Journal Fixed point developed by space Topology Dr. V.H. Badshah Professor and Head, School of Studies in Mathematics, Vikram University, Ujjain Santoshi Alawa Research Scholar The theory of fixed point is one of the most The same definition of recursive function powerful tool of modern Mathematics. It is a rich, can be given, in by applying these results are not interesting and exciting branch of Mathematics. equivalent theorems; the Knaster–Tarski theorem Theorems Concerning the existence and properties is a much stronger result than what is used in de of Fixed points are known as fixed point theorem. notational semantics.[11] However, in light of It is a beautiful mixture of Analysis, Topology and the their intuitive meaning is the same: a Geometry. Fixed point theorems are applied in recursive function can be described as the least various fields such as Engineering, Physics, fixed point of a certain functional, mapping Chemistry, Economics, Game theory, Biology, functions to functions. Medical science, Classical analysis, Functional The above technique of iterating a function to analysis, Functional analysis, Integral equations, find a fixed point can also be used in the states Differential equations, Partial differential that any continuous strictly increasing function equations, Eigen Value problems, Boundary Value from to ordinals has one (and indeed many) fixed problems, analysis, Operator theory, Theory of points. Every on a posset has many fixed points; Non-Linear Oscillations, General and Algebraic these are the "closed elements" with respect to Topology etc. The Fixed Point theory has been the closure operator, and they are the main developed through different spaces such as reason the closure operator was defined in the Topological spaces, Metric spaces, Fuzzy Metric first place. spaces, Manger spaces, Intuitionistic fuzzy metric The theorem has applications in abstract spaces, Probabilistic metric, Fuzzy symmetric interpretation, a form of static program analysis. A spaces etc. common theme in lambda calculus is to find fixed Every involution on a finite set with an points of given lambda expressions. Every lambda odd number of elements has a fixed point; more expression has a fixed point, and a fixed-point generally, for every involution on a finite set of combinator is a "function" which takes as input a elements, the number of elements and the lambda expression and produces as output a fixed number of fixed points have the same parity. Don point of that expression.[9] An important fixed- Zagier used these observations to give a one- point combinatory is the Y combinatory used to sentence proof of Fermat's theorem on sums of give recursive definitions. In de notational two squares, by describing two involutions on the semantics of programming languages, a special same set of triples of integers, one of which can case of the Knaster–Tarski theorem is used to easily be shown to have only one fixed point and establish the semantics of recursive definitions. the other of which has a fixed point for each While the fixed-point theorem is applied to the representation of a given prime (congruent to 1 "same" function (from a logical point of view), the mod 4) as a sum of two squares. Since the first development of the theory is quite different. Let f involution has an odd number of fixed points, so be a function which maps a set S into itself; i.e. f:S does the second, and therefore there always exists → S. A fixed point of the mapping f is an element x a representation of the desired form. belonging to S such that f(x) = x. Fixed points are of SRF National & International Research Journal & Book Publication House, 320, Sewa Path, Sanjeevni Nagar, Garha, Jabalpur (M.P.) Email Id : [email protected] , www.srfresearchjournal.com , Phone- 0761-4036611, Mo. 9993332299, 9131312045 (Whatsapp) Page 1 ISSN : 2394-3580 Swadeshi Research Foundation Vol. – 6, No. - 10 August 2019 A Monthly Journal of Multidisciplinary International Peer Refereed, Review, Indexing & Impact Factor - 4.2 Research Journal interest in themselves but they also provide a way property. The annulus shown below does not have to establish the existence of a solution to a set of the fixed point property. This can easily be equations. For example, in theoretical economics, established by noting that the continuous mapping such as general equilibrium theory, there comes at which rotates the annulus has no fixed point. If point where one needs to know whether the there were no hole in the disk then the center of solution to a system of equations necessarily the disk would be the fixed point of any rotation. exists; or, more specifically, under which But the center of the disk is not part of the conditions will a solution necessarily exist. The annulus. A physical example of a fixed point of a mathematical analysis of this question usually mapping is the center of a whirlpool in a cup of tea relies on fixed point theorems. As stated when it is stirred. (The fact that the center of the previously, if f is a function which maps a set S into whirlpool moves over time is just due to the fact itself; i.e. f:S →S, a fixed point of the mapping is an that the mapping is changing over time.) There is a element x belonging to S such that f(x) = x. If the related problem involving the mapping of the system of equations for which a solution is sought points of a sphere into itself. This is a closed is of the form g(x)=0, then if the function g should surface but not a convex surface. Nevertheless any be represented as g(x)=f(x)-x. A fixed point of f is a continuous mapping of a sphere into itself does solution to g(x)=0. have a fixed point. This is illustrated by considering Below is shown an illustration of the "hair" on a coconut. Assume each hair lies Brouwer's Fixed Point Theorem for the mapping of down so the tip of a hair touch’s the sphere. the unit interval into itself. For any continuous Consider the root of the hair as being x and the mapping there is necessarily some point such that point where the tip touches the sphere as f(x). The f(x)=x. This is indicated by the point where the fact that a fixed point necessarily exists continuous function crosses the diagonal, which corresponds to the existence of a "whorl." Thus, represents x=x. There is no way to draw a You Can't Comb the Hair on a Coconut Without continuous function that starts on the vertical axis There Being a Whorl (fixed point). A very at x=0 and ends on the vertical axis at x=1 without important fixed point theorem for economic crossing or touching the diagonal. analysis is: Let point A be an arbitrary point in the disk. Point B represents the point A is mapped to. References :- A line is drawn from A to B and continued until it 1. Aage, C.T. and salunke, J.N., “On common intersects the boundary of the disk at point C. Now fixed point theorem in complete metric the mapping under consideration is the mapping space”. International Mathematical Fourm, 4, of A to C. This would be a continuous mapping of (2009), no, 3, 151-159. the interior onto the boundary. The existence of 2. Brounciari, A., “A fixed point theorem for such a mapping is a contradiction of the No mappings satisfying a general contractive Retraction Theorem so the process must break condition of integral type “, int. J. Math. Math. down somewhere. It breaks down if f(x)=x because Sci. 29 (9), 531-536 there is no unique line defined. Therefore, since 3. Grabiec M, “Fixed point in Fuzzy sets and the assumption of no fixed point leads to a systems”, 27 (1998) 385-389 contradiction of the No Retraction Theorem there 4. Goebel, K, A coincidence theorem. Bull. Acad. must be at least one fixed point. Pol. Sci., Ser. Sci. Math. Astron. Phys. 16, 733- A topological space that is such that any 735 (1968) continuous mapping of it into itself must have a 5. Poincare, H., “Analysis situs”. J. Del E. Cole fixed point is said to have the fixed point property. polytech 2 (1889), 01-123. Not all topological spaces have the fixed point SRF National & International Research Journal & Book Publication House, 320, Sewa Path, Sanjeevni Nagar, Garha, Jabalpur (M.P.) Email Id : [email protected] , www.srfresearchjournal.com , Phone- 0761-4036611, Mo. 9993332299, 9131312045 (Whatsapp) Page 2 ISSN : 2394-3580 Swadeshi Research Foundation Vol. – 6, No. - 10 August 2019 A Monthly Journal of Multidisciplinary International Peer Refereed, Review, Indexing & Impact Factor - 4.2 Research Journal Classification of Manet Attacks Dr. RB Singh (Professor), Supervisor Nidhi Malik Research Scholar, Computer Application As previously discussed, we have attack can be considered as a Black-Hole attack categorized the presently existing attacks into two when it starts dropping packets. So they can be broad categories: DATA traffic attacks and categorized under a single category. CONTROL traffic attacks. This classification is based on their common characteristics and attack goals. There are few attacks that have implications For example: Black-Hole attack drops packets on both DATA & CONTROL traffic, so they cannot every time, while Gray-Hole attack also drops be classified into these categories easily. packets but its action is based on two conditions: time or sender node.