Taxation Essay Example | Topics and Well Written Essays - 500 words - 2

Taxation - Essay Example However, this can only be possible if the public participate in funding the state in order to purchase necessary weapons and also employ the disciplinary forces. The benefit enjoyed by the public in terms of services due to their tax contribution is immeasurable. It is; therefore, appropriate to suggest that Chancellor Osborne is right when he relays the state’s perceptive of tax evasion. The Shaban Mahamood of Shadow exchequer secretary, comments on the state of living being higher than the standard expectation. From his comments, it is clear that, Shaban expects the state to make the cost of living to that level where each of the citizens living United Kingdom can afford. He has held the state at ransom for not playing its role in reducing the cost of living to its general public. Shaban’s comments clearly indicate that the state then has all the rights to demand and even place criminal charges to people evading taxation. The reason behind the support of the state is that, it will only be in a position to provide the so said services if the public plays its rightful role of paying taxes (Davidson 2011). offshore has been compared to a criminal act. The state has even gone further to formulate policies that strengths the taxman. In addition, the people who willingly provide the state with information regarding the evasion of tax by others will be rewarded heftily. As though this is not enough, the UK state has promised to cooperate with the rest of countries that are fighting tax evasion. In the new criminal act targeting the tax evaders, the state has included harsher punishment, longer jail terms and double payment of the evaded taxes. Furthermore, the state continues to quote the amount of recovered taxes in the past two state budget years. As was of rubberstamping the essence of paying taxes, the politician has also used the issue in their campaigns. It is clear from

Identify Leadership Style Essay Example for Free

Identify Leadership Style Essay As a leader, it sometimes takes a step back to look at oneself to realize what type of leader you are to understand your strengths and weaknesses, so you can lead an effective team to success. After lots of leadership quizs and readings, my leadership style is participative (Democratic) with an emphasis on the consensus view. â€Å"A participative leader seeks to involve other people in the process, possibly including subordinates, peers, superiors and other stakeholders. Often, however, as it is within the managers whim to give or deny control to his or her subordinates, most participative activity is within the immediate team (Tannenbaum and Schmitt, 1958). † Some of the characteristics of the participative leader and that of myself are: creating an atmosphere were creativity is wanted and rewarded, engaging in the group decision making process and while keeping the final say over decisions, allowing members to share ideas and thoughts. A1a: Two Strengths Using the participative leadership style gives me the most advantages when working and making decisions in a group. The main two strengths of my leadership style (participative) is: Empowerment and Better Team Decisions.  Empowerment: as defined by Dictionary.com, â€Å"v.to give power or authority; to authorize, especially by official means. To enable or permit (Dictionary.com, 2014).† How does it feel when you feel like you have a say in a group or team? You feel valued. I enjoy allowing my team to feel valued and employ their decisions to make for the best decision possible. The feeling that group members get from participating makes the final decision accepted much more. Personally, when team members have input in the final goal, they feel responsible and take it much more seriously. Better Team Decisions: Since I dont know all the answers, isnt it better crucial to have lots of people helping find the answers. In my leadership style, since we have lots of people brain-storming on ideas, it makes for better team decisions. When a tough decision is ahead, its critical to have others helping make decisions. Employees input is invaluable in a group participating setting. When I allow others to help make decisions, the quality of those decisions will only increase. A1b: Two Weaknesses With this effective leadership style brings some cons. One of the cons of this style of leadership is time restrictions. When the team is liberated into making decisions, it takes awhile to come to great group decision. This type of leadership allows lots of discussions and hearing everyones opinions, which in-turn slows down the whole process. This style is a slower process, but the pro of making the best decision outweighs the con. The second con for this leadership style is that there can usually only be one decision and we as a team/group can not use everyones opinion or decision. This can cause some team members to become upset or heated, but can be avoided with a good leader leading the discussions. When allowing this style to be in a group, the group must come to a consensus in agreeing on the best decision and all parties must agree with whatever outcome is chosen. A2: Compare against two other styles Lets look at two other leadership styles and compare them to the participative leadership style. Authoritative (autocratic) which is a style that has clear expectations and usually makes decisions with little input from the team/group. In this style, the group makes fast decisions and the leader is usually the most knowledgeable team-member in the group Some similarities are that when the  team is in a bind or time is no longer of essence, the team leader must make an autocratic decision for the betterment of the group. The contributions of the team will be much lower than that of my participative style. Delegative (Laissez-Faire) is the opposite of the autocratic style in that there is little or no team guidance and the team is left to make all the decisions. This style is the least productive style because it offers no structure. The group is more likely to demand a lot from the leader and usually will not work independently. This style is used when all team members are over-qualified in the area of expertise, but also leads to lack of motivation for the group. The participating leadership style that I am apart of uses both the autocratic and the delegative style aspects to achieve maximum group potential. Participative leadership is concerned about the group and that the group has input. A3: Understanding to be more effective To be a leader you must have a mix of skills, behaviors, value sets and knowledge. To be a more effective leader one must understand their abilities and understand others and their strengths/weaknesses. This understanding includes an apprehension for other leadership styles. An impelling leader, understanding certain situations and leadership styles, can carry out different styles to lead different groups and teams of people. A great leader will supply assistance and encouragement to a group based on its needs and maintain the group by using its knowledge of different leadership styles. An effective leader can not accomplish success exclusive of a team. Those teams will consist of lots of team members and different leaders. An effective leader will be able to understand other leadership styles to support and assist those leaders. â€Å"The advantage to understanding your leadership style is that you understand your strengths and weaknesses. You can be proactive and more effective as a leader by strategically using your strengths and counteracting your weaker areas. Your style defines your values and perspective, and being aware of it will aid your communication those you work with. As the saying goes, knowledge is power. You can empower yourself and move forward in your career or interest by exercising this knowledge (Raines, 2011).† When knowing yourself and your leadership style it helps you in the workplace. Knowing your work environment and workplace challenges helps to address appropriate solutions using your leadership style. By knowing your style and the areas around you, it helps you make sound strategies for solutions to the problems and you can succeed by steering the strengths of the leadership way. Being a leader means that you must be able to know your strengths and weaknesses and the situations of the group and when to incorporate your style into the situation. Leadership is not a thing that fits all situations. The more leadership styles you are accustomed to, the more negotiable you will be able to be in the group. A4: Two problems in workplace due to different leadership Problem 1: An employee is always late for workplace. The schedule that the employee is on is one of rotating shifts (day and night). This involves different managers with different leadership styles. These different leadership styles has allowed this employee to continue its bad behaviors and continue to be late. Problem 2: Personality Clash. Your style is to talk out problems and solutions (democratic leadership style) while a manager you work with is direct and to the point (autocratic style), and doesnt want to hear about the solutions but just wants the problems fixed. â€Å"Spray gun in tablet bed has stopped working, most of the team wants to find root cause, but one of the managers wants a solution now and doesnt want to understand what caused  the problem.† A4a: Two ways to overcome each problem Problem 1: Late employee. The employee that is late knows that if LF (Laissez-Faire) manager is working that he allows an unstructured work environment. In other words, see nothing, hear nothing, speak nothing and all is correct in the work place. He has unclear expectations for this late employee and is more of people pleas-er that tackling the problem. This manager is fine if the employee is late as long as it does not effect production. This leadership style also allows him to play favorites. He has different rules for different people. As long as production is not affected, whats to bother? If AA (Autocratic) is manager, this employee knows that she can not be late to work, as is not for fear of consequences of her actions. They know that action will be taken against them if they are late. This causes a trickle down effect for employees, and their team morale. Since not all leadership team-members are on the same plane, things like this are always happening. We must fix this problem, before it gets out of hand. Fix. LF manager should first follow the rules and make the employee accountable for her actions. If that employee is late, they should have consequences like everyone else. If that does not work, then managers of this team should be aware of all times that employees are arriving and leaving work. During a monthly meeting, management could review all badge in/outs to make sure times are consistent with when employees are arriving and leaving. At these meetings, employees are questioned and held accountable for their tardiness. If this is not a viable option, management should have one manager that is responsible for all tardies/lates to work. This would be a dedicated job for the manager and take the responsibility off of the other managers and  help alleviate the leadership style differences between managers. Another quick fixer, would be to discuss this problem with LF manager and hold him accountable for allowing the employee to get away with tardies/lates. This one on one meeting would help to show him that there are consequences to his actions and that all employees need to be accountable for their actions. Problem 2: Spray gun. One way to fix this problem is to invite and have the said manager attend and hold discussions within the group to help him see that not all problems have immediate solutions. Allowing him on the team, helps him develop his leadership style and see there is not one leadership style that encompasses all problems. The leader must find it valuable for them to be on the team and to give input. Showing them that they are valued goes a long way to helping them develop as great leaders. Have a one on one with the manager to discuss reasons for and against having this meeting. If both parties that are having trouble understanding meet, then they are more apt to understand where each other is coming from. If this doesnt work, we can always call an upper level management manager to help diffuse the situation. Another solution would be coaching. Maybe this manager/leader is insecure/uncertain about work environment or doesnt want to make decisions because its not my job. Maybe this manager lacks creativity or innovation. Whatever the issue that they are dealing with, a great effective synergistic leader would see this as an opportunistic time to coach the manager in these areas to help alleviate stress, uncertainty and encourage teamwork, trust, expectations and cohesion amongst teams. A5a: Three advantages that increase productivity on these advantages â€Å"Taking a team from ordinary to extraordinary means understanding and  embracing the differences (Bennicasa, 2012).† The joint vision enables all groups to have meaningful communications—business synergy. Lets take a company that is going through a major brand transformation. The reasons not to change are abundant. The company was reaching goals and employee satisfaction was high. The company would lose brand identification through the proposed changes from upper management. The company would be dysfunctional because the whole team was not on board for the transformation. The change was a challenge from all parties involved. The bottom line came when management and the team members realized that those concerns did not compare to the importance of providing the best product and customer service to the customers. The shared vision had to be recognized from all that was serving through the transformation. The important factor was understanding that it was not about us anymore, but about the companys greater good and we had to embrace it. Meetings were held, discussions were had about working together, being a better team, how to better serve our customers, and enabling team diversity. We had to see the team synergy to be able to grow and become a better productive team. New logos and mottos and so forth did not create team synergy, but leadership had to add value to the team and show all the team members that they were required to make a better team—that is business leadership synergy. Once team synergy was found, this allowed the team members to be accountable for their actions and this included the quality of the new products. In return, this created a better product for the customer. A second advantage of having a synergy leadership style could be used when different team members are not included in a process that affects them. When a team of engineers and managers are designing a new stream line approach to the production floor to help move along the packaging process of â€Å"MMs†. A synergy leadership approach to the stream line approach would be  to include the actual workers from the packaging floor. These employees would be able to discuss best practices for help in designing the new approach. This would speed up the meeting process and would speed up implementation of new processes because the packaging team members would be included in the process. Including these team members would allow for less trial and error and more time being productive in having the new packaging approach in a workable state. Having a synergistic team is just as important than the actual packaging process because it stream lines the meeting, allows team-members to have certain expectations, keeps the meetings focused on the problems and develops and enhances rich collaborative discussions that are now easy to have—business success. The third advantage of leadership synergy would be having all upper management in a participative meeting discussing the new vision for the company. After long discussions the team has selected the vision for the company. The team then selects an authoritative figure to disburse the new vision information to the whole company group—because its not up for debate anymore. Knowing when to use different leadership roles is very important. In this model, a decision that has the absolute consensus of all upper management, will appear when all team-members is seeing the discussion directly through the same glasses – which includes the mission, the vision and the teams goals. Without these glasses, cooperative judgments can be challenging to accomplish. The hard part of making decisions and agreeing upon them in a multi-leadership style team –is that there is no common goal – changes in leadership styles will strangle the discussions, as more thoughts will only add to the difficulty in making a decision. Looking through the same lenses of the glasses permits a variety diversity into the debate and allows the group to succeed. Once we see everything through the same lens, then the upper management group  can be more productive in delivering a vision in a timely manner. This synergy of upper management leaders then starts trickling down to management teams throughout the plant and then onto regular teams and committees in the plant. This trickle down affect makes teams more productive, more efficient by allowing the meetings to flow better and to free up people to not be in meetings (to be on the floor being productive making the products). References Bennicasa, Robyn. (2012) 6 Leadership Styles, and When You Should Use Them. Retrieved from http://www.fastcompany.com/1838481/6-leadership-styles-and when-youshould-use-them. Dictionary.com. 2014. Empowerment Definition. Retrieved from http://dictionary. reference.com/browse_empowerment Raines, Stephanie. 2011. The Advantages of Knowing Your Leadership Style. Retrieved from http://smallbusiness.chron.com/advantages-knowing leadership-style18924.html. Tannenbaum, A.S. and Schmitt, W.H. (1958). How to choose a leadership pattern. Harvard Business Review, 36, March-April, 95-101.

The Algorithm of Gaussian Elimination

The Algorithm of Gaussian Elimination In linear algebra, Gaussian elimination is an algorithm for solving systems of linear equations, finding the rank of a matrix, and calculating the inverse of an invertible square matrix. Gaussian elimination is named after German mathematician and scientist Carl Friedrich Gauss. GAUSS / JORDAN (G / J) is a method to find the inverse of the matrices using elementary operations on the matrices.To find the rank of a matrix we use gauss Jordan elimination metod but we use gauss Jordan method in case we have to find only the inverse of the invertible matrix. Algorithm overview Algorithm of gauss Jordan method is simple. We have to make the matrix an identity matrix using elementary operation on it. It is firstly written in the form of AI=A We will firstly write the upper equation and then perform elementary operation the right hand side matrix matrix and simultaneously on identity matrix to obtain following matrix. I=A A-1 The process of Gaussian elimination has two parts. The first part (Forward Elimination) reduces a given system to either triangular or echelon form, or results in a degenerate equation with no solution, indicating the system has no solution. This is accomplished through the use of elementary row operations. The second step uses back substitution to find the solution of the system above. Stated equivalently for matrices, the first part reduces a matrix to row echelon form using elementary row operations while the second reduces it to reduced row echelon form, or row canonical form. Another point of view, which turns out to be very useful to analyze the algorithm, is that Gaussian elimination computes a matrix decomposition. The three elementary row operations used in the Gaussian elimination (multiplying rows, switching rows, and adding multiples of rows to other rows) amount to multiplying the original matrix with invertible matrices from the left. The first part of the algorithm computes an LU decomposition, while the second part writes the original matrix as the product of a uniquely determined invertible matrix and a uniquely determined reduced row-echelon matrix. Gaussian elimination In linear algebra, Gaussian elimination is an algorithm for solving systems of linear equations, finding the rank of a matrix, and calculating the inverse of an invertible square matrix. Gaussian elimination is named after German mathematician and scientist Carl Friedrich Gauss, which makes it an example of Stiglers law. Elementary row operations are used to reduce a matrix to row echelon form. Gauss-Jordan elimination, an extension of this algorithm, reduces the matrix further to reduced row echelon form. Gaussian elimination alone is sufficient for many applications, and is cheaper than the -Jordan version. History The method of Gaussian elimination appears in Chapter Eight, Rectangular Arrays, of the important Chinese mathematical text Jiuzhang suanshu or The Nine Chapters on the Mathematical Art. Its use is illustrated in eighteen problems, with two to five equations. The first reference to the book by this title is dated to 179 CE, but parts of it were written as early as approximately 150 BCE. It was commented on by Liu Hui in the 3rd century. The method in Europe stems from the notes of Isaac Newton.In 1670, he wrote that all the algebra books known to him lacked a lesson for solving simultaneous equations, which Newton then supplied. Cambridge University eventually published the notes as Arithmetica Universalis in 1707 long after Newton left academic life. The notes were widely imitated, which made (what is now called) Gaussian elimination a standard lesson in algebra textbooks by the end of the 18th century. Carl Friedrich Gauss in 1810 devised a notation for symmetric elimination that was adopted in the 19th century by professional hand computers to solve the normal equations of least-squares problems. The algorithm that is taught in high school was named for Gauss only in the 1950s as a result of confusion over the history of the subject Algorithm overview The process of Gaussian elimination has two parts. The first part (Forward Elimination) reduces a given system to either triangular or echelon form, or results in a degenerate equation with no solution, indicating the system has no solution. This is accomplished through the use of elementary row operations. The second step uses back substitution to find the solution of the system above. Stated equivalently for matrices, the first part reduces a matrix to row echelon form using elementary row operations while the second reduces it to reduced row echelon form, or row canonical form. Another point of view, which turns out to be very useful to analyze the algorithm, is that Gaussian elimination computes a matrix decomposition. The three elementary row operations used in the Gaussian elimination (multiplying rows, switching rows, and adding multiples of rows to other rows) amount to multiplying the original matrix with invertible matrices from the left. The first part of the algorithm computes an LU decomposition, while the second part writes the original matrix as the product of a uniquely determined invertible matrix and a uniquely determined reduced row-echelon matrix. Example Suppose the goal is to find and describe the solution(s), if any, of the following system of linear equations: The algorithm is as follows: eliminate x from all equations below L1, and then eliminate y from all equations below L2. This will put the system into triangular form. Then, using back-substitution, each unknown can be solved for. In the example, x is eliminated from L2 by adding to L2. x is then eliminated from L3 by adding L1 to L3. Formally: The result is: Now y is eliminated from L3 by adding 4L2 to L3: The result is: This result is a system of linear equations in triangular form, and so the first part of the algorithm is complete. The last part, back-substitution, consists of solving for the knowns in reverse order. It can thus be seen that Then, z can be substituted into L2, which can then be solved to obtain Next, z and y can be substituted into L1, which can be solved to obtain The system is solved. Some systems cannot be reduced to triangular form, yet still have at least one valid solution: for example, if y had not occurred in L2 and L3 after the first step above, the algorithm would have been unable to reduce the system to triangular form. However, it would still have reduced the system to echelon form. In this case, the system does not have a unique solution, as it contains at least one free variable. The solution set can then be expressed parametrically (that is, in terms of the free variables, so that if values for the free variables are chosen, a solution will be generated). In practice, one does not usually deal with the systems in terms of equations but instead makes use of the augmented matrix (which is also suitable for computer manipulations). For example: Therefore, the Gaussian Elimination algorithm applied to the augmented matrix begins with: which, at the end of the first part(Gaussian elimination, zeros only under the leading 1) of the algorithm, looks like this: That is, it is in row echelon form. At the end of the algorithm, if the Gauss-Jordan elimination(zeros under and above the leading 1) is applied: That is, it is in reduced row echelon form, or row canonical form. Example of Gauss Elimination method!!! (To solve System of Linear Equations) One simple example of G/J row operations is offered immediately above the pivoting reference; an example is below: Below is a system of equations which we will solve using G/J step 1 Below is the 1st augmented matrix :pivot on the 1 encircled in red Row operations for the 1st pivoting are named below Next we pivot on the number 5in the 2-2 position, encircled below Below is the result of performing P1 on the element in the 2-2 position. Next we must perform P2 Row operations of P2 are below The result of the 2nd pivoting is below. Now pivot on -7 encircled in red Using P1 below we change -7to 1 Below is the result of performing P1 on -7 in the 3-3 position. Next we must perform P2 Row operations of P2 are below The result of the third (and last) pivoting is below with 33 ISM matrix in blue Step [3] of G/J Re-writing the final matrix as equations gives the solution to the original system Other applications Finding the inverse of a matrix Suppose A is a matrix and you need to calculate its inverse. The identity matrix is augmented to the right of A, forming a matrix (the block matrix B = [A,I]). Through application of elementary row operations and the Gaussian elimination algorithm, the left block of B can be reduced to the identity matrix I, which leaves A 1 in the right block of B. If the algorithm is unable to reduce A to triangular form, then A is not invertible. General algorithm to compute ranks and bases The Gaussian elimination algorithm can be applied to any matrix A. If we get stuck in a given column, we move to the next column. In this way, for example, some matrices can be transformed to a matrix that has a reduced row echelon form like (the *s are arbitrary entries). This echelon matrix T contains a wealth of information about A: the rank of A is 5 since there are 5 non-zero rows in T; the vector space spanned by the columns of A has a basis consisting of the first, third, fourth, seventh and ninth column of A (the columns of the ones in T), and the *s tell you how the other columns of A can be written as linear combinations of the basis columns. Analysis Gaussian elimination to solve a system of n equations for n unknowns requires n(n+1) / 2 divisions, (2n3 + 3n2 5n)/6 multiplications, and (2n3 + 3n2 5n)/6 subtractions,[3] for a total of approximately 2n3 / 3 operations. So it has a complexity of . This algorithm can be used on a computer for systems with thousands of equations and unknowns. However, the cost becomes prohibitive for systems with millions of equations. These large systems are generally solved using iterative methods. Specific methods exist for systems whose coefficients follow a regular pattern (see system of linear equations). The Gaussian elimination can be performed over any field. Gaussian elimination is numerically stable for diagonally dominant or positive-definite matrices. For general matrices, Gaussian elimination is usually considered to be stable in practice if you usepartial pivoting as described below, even though there are examples for which it is unstable. Gauss-Jordan elimination In linear algebra, Gauss-Jordan elimination is an algorithm for getting matrices in reduced row echelon form using elementary row operations. It is variation of Gaussian elimination. Gaussian elimination places zeros below each pivot in the matrix, starting with the top row and working downwards. Matrices containing zeros below each pivot are said to be in row echelon form. Gauss-Jordan elimination goes a step further by placing zeros above and below each pivot; such matrices are said to be in reduced row echelon form. Every matrix has a reduced row echelon form, and Gauss-Jordan elimination is guaranteed to find it. It is named after Carl Friedrich Gauss and Wilhelm Jordan because it is a variation of Gaussian elimination as Jordan described in 1887. However, the method also appears in an article by Clasen published in the same year. Jordan and Clasen probably discovered Gauss-Jordan elimination independently.[1] Computer sciences complexity theory shows Gauss-Jordan elimination to have a time complexity of O(n3) for an n by n matrix (using Big O Notation. This result means it is efficiently solvable for most practical purposes. As a result, it is often used in computer software for a diverse set of applications. However, it is often an unnecessary step past Gaussian elimination. Gaussian elimination shares Gauss-Jordons time complexity of O(n3) but is generally faster. Therefore, in cases in which achieving reduced row echelon form over row echelon form is unnecessary, Gaussian elimination is typically preferred.[citation needed] Application to finding inverses If Gauss-Jordan elimination is applied on a square matrix, it can be used to calculate the matrixs inverse. This can be done by augmenting the square matrix with the identity matrix of the same dimensions and applying the following matrix operations: If the original square matrix, A, is given by the following expression: Then, after augmenting by the identity, the following is obtained: By performing elementary row operations on the [AI] matrix until it reaches reduced row echelon form, the following is the final result: The matrix augmentation can now be undone, which gives the following: A matrix is non-singular (meaning that it has an inverse matrix) if and only if the identity matrix can be obtained using only elementary row operations. Example of Gauss Jordan method!!! (To Simply Find Inverse of a Matrix) If the original square matrix, A, is given by the following expression: Then, after augmenting by the identity, the following is obtained: By performing elementary row operations on the [AI] matrix until it reaches reduced row echelon form, the following is the final result: The matrix augmentation can now be undone, which gives the following:

The Algorithm of Gaussian Elimination

The Algorithm of Gaussian Elimination In linear algebra, Gaussian elimination is an algorithm for solving systems of linear equations, finding the rank of a matrix, and calculating the inverse of an invertible square matrix. Gaussian elimination is named after German mathematician and scientist Carl Friedrich Gauss. GAUSS / JORDAN (G / J) is a method to find the inverse of the matrices using elementary operations on the matrices.To find the rank of a matrix we use gauss Jordan elimination metod but we use gauss Jordan method in case we have to find only the inverse of the invertible matrix. Algorithm overview Algorithm of gauss Jordan method is simple. We have to make the matrix an identity matrix using elementary operation on it. It is firstly written in the form of AI=A We will firstly write the upper equation and then perform elementary operation the right hand side matrix matrix and simultaneously on identity matrix to obtain following matrix. I=A A-1 The process of Gaussian elimination has two parts. The first part (Forward Elimination) reduces a given system to either triangular or echelon form, or results in a degenerate equation with no solution, indicating the system has no solution. This is accomplished through the use of elementary row operations. The second step uses back substitution to find the solution of the system above. Stated equivalently for matrices, the first part reduces a matrix to row echelon form using elementary row operations while the second reduces it to reduced row echelon form, or row canonical form. Another point of view, which turns out to be very useful to analyze the algorithm, is that Gaussian elimination computes a matrix decomposition. The three elementary row operations used in the Gaussian elimination (multiplying rows, switching rows, and adding multiples of rows to other rows) amount to multiplying the original matrix with invertible matrices from the left. The first part of the algorithm computes an LU decomposition, while the second part writes the original matrix as the product of a uniquely determined invertible matrix and a uniquely determined reduced row-echelon matrix. Gaussian elimination In linear algebra, Gaussian elimination is an algorithm for solving systems of linear equations, finding the rank of a matrix, and calculating the inverse of an invertible square matrix. Gaussian elimination is named after German mathematician and scientist Carl Friedrich Gauss, which makes it an example of Stiglers law. Elementary row operations are used to reduce a matrix to row echelon form. Gauss-Jordan elimination, an extension of this algorithm, reduces the matrix further to reduced row echelon form. Gaussian elimination alone is sufficient for many applications, and is cheaper than the -Jordan version. History The method of Gaussian elimination appears in Chapter Eight, Rectangular Arrays, of the important Chinese mathematical text Jiuzhang suanshu or The Nine Chapters on the Mathematical Art. Its use is illustrated in eighteen problems, with two to five equations. The first reference to the book by this title is dated to 179 CE, but parts of it were written as early as approximately 150 BCE. It was commented on by Liu Hui in the 3rd century. The method in Europe stems from the notes of Isaac Newton.In 1670, he wrote that all the algebra books known to him lacked a lesson for solving simultaneous equations, which Newton then supplied. Cambridge University eventually published the notes as Arithmetica Universalis in 1707 long after Newton left academic life. The notes were widely imitated, which made (what is now called) Gaussian elimination a standard lesson in algebra textbooks by the end of the 18th century. Carl Friedrich Gauss in 1810 devised a notation for symmetric elimination that was adopted in the 19th century by professional hand computers to solve the normal equations of least-squares problems. The algorithm that is taught in high school was named for Gauss only in the 1950s as a result of confusion over the history of the subject Algorithm overview The process of Gaussian elimination has two parts. The first part (Forward Elimination) reduces a given system to either triangular or echelon form, or results in a degenerate equation with no solution, indicating the system has no solution. This is accomplished through the use of elementary row operations. The second step uses back substitution to find the solution of the system above. Stated equivalently for matrices, the first part reduces a matrix to row echelon form using elementary row operations while the second reduces it to reduced row echelon form, or row canonical form. Another point of view, which turns out to be very useful to analyze the algorithm, is that Gaussian elimination computes a matrix decomposition. The three elementary row operations used in the Gaussian elimination (multiplying rows, switching rows, and adding multiples of rows to other rows) amount to multiplying the original matrix with invertible matrices from the left. The first part of the algorithm computes an LU decomposition, while the second part writes the original matrix as the product of a uniquely determined invertible matrix and a uniquely determined reduced row-echelon matrix. Example Suppose the goal is to find and describe the solution(s), if any, of the following system of linear equations: The algorithm is as follows: eliminate x from all equations below L1, and then eliminate y from all equations below L2. This will put the system into triangular form. Then, using back-substitution, each unknown can be solved for. In the example, x is eliminated from L2 by adding to L2. x is then eliminated from L3 by adding L1 to L3. Formally: The result is: Now y is eliminated from L3 by adding 4L2 to L3: The result is: This result is a system of linear equations in triangular form, and so the first part of the algorithm is complete. The last part, back-substitution, consists of solving for the knowns in reverse order. It can thus be seen that Then, z can be substituted into L2, which can then be solved to obtain Next, z and y can be substituted into L1, which can be solved to obtain The system is solved. Some systems cannot be reduced to triangular form, yet still have at least one valid solution: for example, if y had not occurred in L2 and L3 after the first step above, the algorithm would have been unable to reduce the system to triangular form. However, it would still have reduced the system to echelon form. In this case, the system does not have a unique solution, as it contains at least one free variable. The solution set can then be expressed parametrically (that is, in terms of the free variables, so that if values for the free variables are chosen, a solution will be generated). In practice, one does not usually deal with the systems in terms of equations but instead makes use of the augmented matrix (which is also suitable for computer manipulations). For example: Therefore, the Gaussian Elimination algorithm applied to the augmented matrix begins with: which, at the end of the first part(Gaussian elimination, zeros only under the leading 1) of the algorithm, looks like this: That is, it is in row echelon form. At the end of the algorithm, if the Gauss-Jordan elimination(zeros under and above the leading 1) is applied: That is, it is in reduced row echelon form, or row canonical form. Example of Gauss Elimination method!!! (To solve System of Linear Equations) One simple example of G/J row operations is offered immediately above the pivoting reference; an example is below: Below is a system of equations which we will solve using G/J step 1 Below is the 1st augmented matrix :pivot on the 1 encircled in red Row operations for the 1st pivoting are named below Next we pivot on the number 5in the 2-2 position, encircled below Below is the result of performing P1 on the element in the 2-2 position. Next we must perform P2 Row operations of P2 are below The result of the 2nd pivoting is below. Now pivot on -7 encircled in red Using P1 below we change -7to 1 Below is the result of performing P1 on -7 in the 3-3 position. Next we must perform P2 Row operations of P2 are below The result of the third (and last) pivoting is below with 33 ISM matrix in blue Step [3] of G/J Re-writing the final matrix as equations gives the solution to the original system Other applications Finding the inverse of a matrix Suppose A is a matrix and you need to calculate its inverse. The identity matrix is augmented to the right of A, forming a matrix (the block matrix B = [A,I]). Through application of elementary row operations and the Gaussian elimination algorithm, the left block of B can be reduced to the identity matrix I, which leaves A 1 in the right block of B. If the algorithm is unable to reduce A to triangular form, then A is not invertible. General algorithm to compute ranks and bases The Gaussian elimination algorithm can be applied to any matrix A. If we get stuck in a given column, we move to the next column. In this way, for example, some matrices can be transformed to a matrix that has a reduced row echelon form like (the *s are arbitrary entries). This echelon matrix T contains a wealth of information about A: the rank of A is 5 since there are 5 non-zero rows in T; the vector space spanned by the columns of A has a basis consisting of the first, third, fourth, seventh and ninth column of A (the columns of the ones in T), and the *s tell you how the other columns of A can be written as linear combinations of the basis columns. Analysis Gaussian elimination to solve a system of n equations for n unknowns requires n(n+1) / 2 divisions, (2n3 + 3n2 5n)/6 multiplications, and (2n3 + 3n2 5n)/6 subtractions,[3] for a total of approximately 2n3 / 3 operations. So it has a complexity of . This algorithm can be used on a computer for systems with thousands of equations and unknowns. However, the cost becomes prohibitive for systems with millions of equations. These large systems are generally solved using iterative methods. Specific methods exist for systems whose coefficients follow a regular pattern (see system of linear equations). The Gaussian elimination can be performed over any field. Gaussian elimination is numerically stable for diagonally dominant or positive-definite matrices. For general matrices, Gaussian elimination is usually considered to be stable in practice if you usepartial pivoting as described below, even though there are examples for which it is unstable. Gauss-Jordan elimination In linear algebra, Gauss-Jordan elimination is an algorithm for getting matrices in reduced row echelon form using elementary row operations. It is variation of Gaussian elimination. Gaussian elimination places zeros below each pivot in the matrix, starting with the top row and working downwards. Matrices containing zeros below each pivot are said to be in row echelon form. Gauss-Jordan elimination goes a step further by placing zeros above and below each pivot; such matrices are said to be in reduced row echelon form. Every matrix has a reduced row echelon form, and Gauss-Jordan elimination is guaranteed to find it. It is named after Carl Friedrich Gauss and Wilhelm Jordan because it is a variation of Gaussian elimination as Jordan described in 1887. However, the method also appears in an article by Clasen published in the same year. Jordan and Clasen probably discovered Gauss-Jordan elimination independently.[1] Computer sciences complexity theory shows Gauss-Jordan elimination to have a time complexity of O(n3) for an n by n matrix (using Big O Notation. This result means it is efficiently solvable for most practical purposes. As a result, it is often used in computer software for a diverse set of applications. However, it is often an unnecessary step past Gaussian elimination. Gaussian elimination shares Gauss-Jordons time complexity of O(n3) but is generally faster. Therefore, in cases in which achieving reduced row echelon form over row echelon form is unnecessary, Gaussian elimination is typically preferred.[citation needed] Application to finding inverses If Gauss-Jordan elimination is applied on a square matrix, it can be used to calculate the matrixs inverse. This can be done by augmenting the square matrix with the identity matrix of the same dimensions and applying the following matrix operations: If the original square matrix, A, is given by the following expression: Then, after augmenting by the identity, the following is obtained: By performing elementary row operations on the [AI] matrix until it reaches reduced row echelon form, the following is the final result: The matrix augmentation can now be undone, which gives the following: A matrix is non-singular (meaning that it has an inverse matrix) if and only if the identity matrix can be obtained using only elementary row operations. Example of Gauss Jordan method!!! (To Simply Find Inverse of a Matrix) If the original square matrix, A, is given by the following expression: Then, after augmenting by the identity, the following is obtained: By performing elementary row operations on the [AI] matrix until it reaches reduced row echelon form, the following is the final result: The matrix augmentation can now be undone, which gives the following:

Spinal Cord Injury Essay -- Biology Essays Research Papers

Spinal Cord Injury Spinal cord injury is a serious problem that effects close to 250,000 people in the United States with 10,000 people being injured per year . There are many things that can lead to spinal cord injury, including athletic injuries, car accidents, and recreational activities like swimming and biking. It primarily effects those between the ages of 16 and 30 and drastically effects the rest of their lives. It is a very debilitating injury that requires extensive medical care, often leaves the patients in a great deal of pain for the rest of their lives(2), and the treatment of which costs $10 billion dollars a year in the US.(facts from site 1) With all of these factors spurring research on there is a strong drive to find a cure for such a devastating injury. Spinal cord injuries can happen anywhere along the spinal cord, but the exact location of the trauma determines the effects that the injury will have. Injuries in the lower back, between the Sacrum(S1-S5) and Lumbar(L1-L5) vertebrae mainly effect the legs. Breaks in the Thoracic(T1-T12) vertebrae, located in the middle of the back, effect the torso and portions of the arms. While injuries in the spine above the shoulder blades, the Cervical(C1-C8) vertebrae, effect not only movement in the neck, but functions such as breathing, speaking, and eating. In the past, some functions have been able to be regained by some individuals, but after the initial recovery period most people see little improvement over the course of their lives. The main reason for such a poor recovery is that the nerve cells in the spinal cord do not regenerate on their own. Once the spinal cord develops, two things keep it from growing. One of which is an inhibitor protein and the ... ...en very bleak in the past, major hurdles have been overcome and science is now working on a way to help people recover from their injuries instead of teaching them how to live with them. Web Sites used in this Paper (1) Welcome to the American Paralysis Association - APACURE.COM http://www.apacure.com/mainfram.html -This site contains a short video on axon degeneration and an animated illustrations on the process of death and regeneration in nerve cells. (2) Theories on the Effects of Acupuncture on the Nervous System Emma Christensen, [deals with concepts of pain] http://serendip.brynmawr.edu/bb/neuro/neuro98/202s98-paper1/Christensen.html (3) Spinal Cord Injury - - Research Highlights http://www.nin ds.nih.gov/healinfo/disorder/sci/scispec.htm (4) Spinal Cord Injury Center http://www.med.nyu.edu/clnre s95/spincord.htm

Sex Discrimination at Walmart

Sex Discrimination at Wal-Mart OMM640 Business Ethics and Social Responsibility Dr. : David Britton May 14, 2012 Betty Dukes along with five other women filled a law suit against Wal-Mart Inc. in 2001 for discrimination against women, denying them their raises and also their promotions. Betty Dukes and the other women hope that they can stand for hundreds of thousands of other women who might have been similarly affected by this type of behavior when they were there also. Years later the ladies got the go ahead to represent 1. 6 other women in the case seeking back wages and maybe even punitive damages from Wal-Mart.This is by far the biggest class action suit against such a huge company and by these ladies getting the go ahead they have to prove to a court that Wal-Mart treated them unfairly. Wal-Mart has denied that such atrocities have ever taken place. If it is found that Wal-Mart did in fact do these things not only would their image ne tarnished, they would also end up paying b illions of dollars to these ladies and open a door for other suits to be filed against them for whatever purpose and also sends a message saying no matter how big a company you are you are still going to be held accountable for the way in which you treat people.That is why laws were established for this precise purpose. Wal-Mart was hoping to have the case dismissed and have all the women file separately which would be easier for them because all of the women filing together would stand to gain millions from them. The financial impact a law suit of this magnitude might have on Wal-Mart would be loss of business, loss of millions of dollars paid to the women and the majority of shoppers that frequent Wal-Mart are women and if they are found to be discriminating against women of course other women will not shop there.Them they would have to end up letting go a Lot of people if they don’t have the money to sustain them; companies that supply to them would pull out, and with the criticism they would get would probably destroy the company. Wal-Mart has maintained that they did not discriminate against Betty and the other women that filed claim against, but I believe that Wal-Mart knew exactly what was going on and just chose to ignore Betty when she was making her complaints. They began to take her seriously when she filed the suit.Some of the moral complaints the women were suing Wal-Mart for was for statistical disparities such female workers were less likely to be paid the proper wages as the men; they were denied promotion in a timely fashion different to those of the men (Boatright, 2009, pg. 199). There was a lot of bias involved with all of this; it even has a hint of gender stereotyping. I believe the moral complaints were justified because Betty made complaints to managers and higher up and because she did this they began to treat her differently and demoted her for not opening a cash register with a penny but for talking.When it came to promotions opportunities they were not made known, and those that she wanted they kept telling her they were filled and hired male counter parts that were not fit for the position. What made Wal-Mart managers determine what part of the store you worked was based on your gender and that was wrong, not because they were women meant they could not do what was needed. The women by far were more experienced than the men and the stayed longer on the job also.It was shown based on the records of the trial and brought up in the news that this massive company had so discrepancies that they had to correct as it pertained to the way they treated women. They would first have to start off improving their employment policies and what they are looking for in individual. They have to be an equal opportunity employer and employ People based on their skills and experience on a particular job. Women need to be promoted into management programs just as men do, not showing favoritism to men over women.They have to allow people to work anywhere in the store as long as they can do the job and not putting them in whatever department they think they should be in. Develop a program that teaches all employees about diversity in the work place and place emphasis on respecting women and treating them fairly. Starting an affirmative program would be to the benefit of the company, they have to learn that women are priced just as highly as men and the way they. For women with kids it is hard for them to work as they would because they have no child care and this is something that Wal-Mart must put in place.Proper compensation for the work being done and it should be equal pay for both parties involved. Training programs for employees that want to advancement and how to work with each other. In conclusion I would have to say it would have been a good luck for the women to win that discrimination law suit against Wal-Mart Inc. I am actually surprised at the outcome but I understand it. None the less Bett y Duke allowed the world to see Wal-Mart for who they really are and allowed them to see that they were treating women unfairly and they now have the opportunity to reevaluate the way they do things.Women have the right to be treated fairly and equally and also have the right to be paid and promoted just as any man. The law suit also opened up the eyes of all the other companies that were watching with a keen interest. References: Boatright, J. (2009). Ethics and the conduct of business (6th Ed) Upper Saddle River, NJ Prentice Hall Emily Friedman, April 16, 2010, Appeals Court Rules Wal-Mart Sex Discrimination Case Can Go to Trial Retrieved May 14, 2014 from http://abcnews. go. com/WN/Business/wal-mart-sex-discrimination-case-trial/story? d=10480510 Wal-Mart sex-bias case could have wide impact Retrieved May 14, 2012 from http://www. msnbc. msn. com/id/42250811/ns/business-careers/t/wal-mart-sex-bias-case-could-have-wide-impact/ Wal-Mart Wins Request in Bias Case Retrieved May 15, 2 012 from http://www. blackchristiannews. com/news/2009/02/wal-mart-wins-request-in-bias-case. html Betty. V Goliath Retrieved May 15, 2012 from http://walmartwatch. com/wp-content/blogs. dir/2/files/pdf/dukes_backgrounder. pdf

Free Essays on Black Arts Movement

An Examination of The Black Arts Movement In a 1968 essay, "The Black Arts Movement", Larry Neal proclaimed the Black Arts Movement was â€Å"the aesthetic and spiritual sister of the Black Power concept†. The Black Arts movement, usually referred to as a 1960s movement, solidified in 1965 and broke apart around 1975. The movement’s major players were Amiri Baraka/Leroi Jones, Adrienne Kennedy, Ron Karenga, Larry Neal, Sonia Sanchez, and many more Black artists at this particular time in American history. This Black intellectual revolution examined and targeted many assumptions in the artistic world, specifically the role of the text, the timelessness of art, the responsibility of artists to their communities, and the significance of oral forms in the struggles of Black folk . This paper will explore several concepts promoted by the Black Arts Movement, in particular, cultural nationalism, the Black Aesthetic and the role of the artist in the community. Cultural nationalism was founded on the belief that blacks and whites have separate values, histories, intellectual traditions and lifestyles; therefore, in reality, there are two separate Americas. Cultural nationalism was often expressed as an abstract and aesthetic return to the motherland (rarely an actual return) and a recognition of traditional African roots, that biased education and stereotyped representations in the mass media had torn from the souls of African-Americans. Ron Karenga, one of the most prominent voices of the cultural nationalist, states, in his essay â€Å"Black Cultural Nationalists†, â€Å"Let our art remind us of our distaste for the enemy, our love for each other, and our commitment to the revolutionary struggle that will be fought with the rhythmic reality of a permanent revolution". The goal of the cultural nationalist was the realization of a Black community based on a common descent and language. Moreover, the cultural nationalis... Free Essays on Black Arts Movement Free Essays on Black Arts Movement An Examination of The Black Arts Movement In a 1968 essay, "The Black Arts Movement", Larry Neal proclaimed the Black Arts Movement was â€Å"the aesthetic and spiritual sister of the Black Power concept†. The Black Arts movement, usually referred to as a 1960s movement, solidified in 1965 and broke apart around 1975. The movement’s major players were Amiri Baraka/Leroi Jones, Adrienne Kennedy, Ron Karenga, Larry Neal, Sonia Sanchez, and many more Black artists at this particular time in American history. This Black intellectual revolution examined and targeted many assumptions in the artistic world, specifically the role of the text, the timelessness of art, the responsibility of artists to their communities, and the significance of oral forms in the struggles of Black folk . This paper will explore several concepts promoted by the Black Arts Movement, in particular, cultural nationalism, the Black Aesthetic and the role of the artist in the community. Cultural nationalism was founded on the belief that blacks and whites have separate values, histories, intellectual traditions and lifestyles; therefore, in reality, there are two separate Americas. Cultural nationalism was often expressed as an abstract and aesthetic return to the motherland (rarely an actual return) and a recognition of traditional African roots, that biased education and stereotyped representations in the mass media had torn from the souls of African-Americans. Ron Karenga, one of the most prominent voices of the cultural nationalist, states, in his essay â€Å"Black Cultural Nationalists†, â€Å"Let our art remind us of our distaste for the enemy, our love for each other, and our commitment to the revolutionary struggle that will be fought with the rhythmic reality of a permanent revolution". The goal of the cultural nationalist was the realization of a Black community based on a common descent and language. Moreover, the cultural nationalis...