Mathematical happenings Essay

   In 1881, john Venn, a British philosopher and logician introduced the Venn diagrams which are up-to-date used in many mathematical fields such as statistics, set theory, logic, computer science and in probability analysis. John Venn was born in Hull, Yorkshire in 1934 and died on 4th April 1923 at Cambridge in England. He is recognized as the third greatest mathematician, narrowly after Sir Isaac Newton and Leonard Euler. Venn diagrams are also known as set diagrams which are defined as drawings showing all hypothetical, logical and possible relationships amongst groups of things or between finite collections of sets. To construct a Venn diagram, simple closed curves are collected and drawn in a plane. They function on a principle that, sets or classes of items with similar or logical relations should appear in the same region within such diagrams.

The diagrams first of all recognize and leave a room for the possibility of relations between the sets and then specifies the status of the given or actual relation as either null or not. Venn diagrams in most times comprise of overlapping circles, each circle representing a specified set of objects. A diagram consisting of two circles is said to be a two set Venn diagram. For more than three sets, other shapes other than circles can be employed. The intersection or the overlapping area is said to represent items sharing some characteristics. If a Venn diagram for n items is to be presented, then it must have 2n zones that are possible hypothetically and also correspond to a combination of inclusion or exclusion in the respective component sets (http://www-groups.dcs.st-and.ac.uk/history/Chronology/index.html).

In April 22, 1811, Ludwig Otto Hessian, a Germany mathematician introduced the Hessian determinant which he published on a paper entitled cubic and quadratic curves. This is a function used in determining minimum, maximum and turning point for different surfaces. It is also used in applied calculus when finding the second derivative as the turning point for curves. Hessian determinant is applied in Hessian matrix, a square matrix of second order level of partial derivatives. In case of three dimensional aspects, Hessian matrix undergoes a reformulation and applies the so called eigenvalues. Eigen values are negative at local maximum and positive at local minimum. If at least one of the eigenvalues of the Hessian determinant is zero, the function is degenerate. If however at least one is negative or positive and none is zero, the function has a saddle point.

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These determinants are widely applied in large scale optimization problems to be solved with Newton type style.

The Hessian determinant has more visibility in recent texts within the calculus of variations. Hesse’s contributions in the calculus field are believed as the major doorstep in functional analysis. Hesse is also recognized to have initiated a major shift of focus from algorithmic approach towards the variation calculus by emphasizing on analytical characterization. This analysis came after developing the theory of algebraic functions and the invariants theory. In critical analysis, if derivative of a function is zero at some given point, the function is said to have a stationary or critical point and therefore the Hessian determinant is said to be discriminant in that case. The Hessian matrix is diverse and can be applied in image processing for expressing the image processing operators. It cam also be applied in computer vision in scale space and blob detector (http://www-groups.dcs.st-and.ac.uk/history/Chronology/index.html)

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Reference:

Index for the Chronology. Retrieved on 4th November 2008 from http://www-groups.dcs.st-and.ac.uk/history/Chronology/index.html

 

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