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Activation Energy (Ea) Chemistry Definition

Activation Energy (Ea) Chemistry Definition Activation energy is the  minimum amount of energy required to initiate a reaction. It is the height of the potential energy barrier between the potential energy minima of the reactants and products. Activation energy  is denoted by Ea and typically has units of kilojoules per mole (kJ/mol) or kilocalories per mole (kcal/mol). The term activation energy was introduced by the Swedish scientist Svante Arrhenius in 1889. The Arrhenius equation relates activation energy to the rate at which a chemical reaction proceeds: k Ae-Ea/(RT) where k is the reaction rate coefficient, A is the frequency factor for the reaction, e is the irrational number (approximately equal to 2.718), Ea is the activation energy, R is the universal gas constant, and T is the absolute temperature (Kelvin). From the Arrhenius equation, it can be seen that the rate of reaction changes according to temperature. Normally, this means a chemical reaction proceeds more quickly at a higher temperature. There are, however, a few cases of negative activation energy, where the rate of a reaction decreases with temperature. Why Is Activation Energy Needed? If you mix together two chemicals, only a small number of collisions will naturally occur between the reactant molecules to make products. This is particularly true if the molecules have low kinetic energy. So, before a significant fraction of reactants can be converted into products, the free energy of the system must be overcome. The activation energy gives the reaction that little extra push needed to get going. Even exothermic reactions require activation energy to get started. For example, a stack of wood wont start burning on its own. A lit match can provide the activation energy to start combustion. Once the chemical reaction starts, the heat released by the reaction provides the activation energy to convert more reactant into product. Sometimes a chemical reaction proceeds without adding any additional energy. In this case, the activation energy of the reaction is usually supplied by heat from the ambient temperature. Heat increases the motion of the reactant molecules, improving their odds of colliding with each other and increasing the force of the collisions. The combination makes it more likely bonds between reactant will break, allowing for the formation of products. Catalysts and Activation Energy A substance that lowers the activation energy of a chemical reaction is called a catalyst. Basically, a catalyst acts by modifying the transition state of a reaction. Catalysts are not consumed by the chemical reaction and they dont change the equilibrium constant of the reaction. Relationship Between Activation Energy and Gibbs Energy Activation energy is a term in the Arrhenius equation used to calculate the energy needed to overcome the transition state from reactants to products. The Eyring equation is another relation that describes the rate of reaction, except instead of using activation energy, it includes Gibbs energy of the transition state. The Gibbs energy of the transition state factors in both enthalpy and entropy of a reaction. Activation energy and Gibbs energy are related, but not interchangeable.

Zumba Fitness Essay Example | Topics and Well Written Essays - 1000 words

Zumba Fitness - Essay Example Zumba Fitness It has been accepted in most societies due to its flexible programs that can suit the most workers. Zumba fitness has been rated among the Worlds top ten fitness programs based on the number of participants and product uniqueness. Zumba is a dance and aerobics fitness course that was started in the 1990s by Alberto Perez (Perez, 2009). He is from Columbia and started the Zumba program due to his passion for dance and music. The Zumba fitness brand consists of various products launched since 2003. They include Zumba Stape, Zumba Toning, Zumba Sentao, Zumba Kids, Zumba Toning among others. The products cater for various people in the society despite their ages. Since Alberto is a choreographer and a dancer, the fitness program incorporates music in its sessions (Perez, 2009). Zumba uses music genres such as Salsa, Samba, Martial arts, Soca and Hip hop. The different rhythms from the music can be used to change the used approaches. Moreover, the use of different genres ensures that its customers can join the program irrespective of their music preferences. Zumba fitness programs became popular in Columbia and there was a need to cause awareness into other countries (Perez, 2009). With the help of entrepreneurs, the program became international where it has been practiced in more than 180 countries globally. Perez had business ties with the United States; he was able to market his dance into the country. According to recent research, more than 10 million people take weekly classes. Zumbas success has been caused by its use of music and use of infomercials to cause awareness.

The analysis of Young goodman Brown Essay Example | Topics and Well Written Essays - 750 words

The analysis of Young goodman Brown - Essay Example For instance, he introduces Young Goodman Brown, whose name is allegorical in several manners. In addition, this is the title that he gives this narration, which is also a significant part in this narration. There are three allegories in this protagonist’s name that have individual meanings. First, there is the name Young, which seems to suggest the innocence of his character. However, this innocence is later corrupted by several factors that surround this protagonist. The second name is Goodman and is equally significant in an allegorical manner. Careful analysis of this narration reveals that there are few righteous deeds in this narration hence some irony (Neary 445). Consequently, the wife of the protagonist has an allegorical name: Faith. This name suggests hope for several significant things in the narration. In addition, the protagonist needs faith in order to survive the journey he takes through the forest. Consequently, Faith acts as a guardian to the protagonist throughout the narration. Further, Faith is his wife hence presenting a physical guide in the material world, which they live in. In particular, she pleads with the protagonist not to take the evil journey, which would later compromise his spirituality (Hoffman 98). The forest represents all the evil in the narration by Hawthorne because of its components. For instance, it has such tings as trees and wild animals that are synonymous of evil. According to Xian-chun(56) the forest is similar to the mind of the protagonist who struggles to fight evil. In addition, his mind has similar uncertainties to the ones, which people experience in any forest. Traditionally, forests are dark because of the overlapping of trees and other plantations. Similarly, Brown’s mind is dark since it has. Evidently, this forest is ruled by the devil like Brown’s mind, which compromises his spiritual actions

Renaissance and Baroque Artists and Musicians Assignment

Renaissance and Baroque Artists and Musicians - Assignment Example Leonardo Da Vinci, who is one of the greatest celebrated artists of the Renaissance, was also a sculpture, architect, scientist and engineer. He was born in a small town of Vinci, near Florence. During 1940’s his family settled in Florence and he was given â€Å"the best education that Florence, the intellectual and artistic center of Italy, could offer†. His influences from other artists are not very defined, yet, his sound interest in science and â€Å"his in-depth study of human anatomy aided him in mastering the realist art form† (Leonardo Da Vinci's Life). While all other works of arts seem to be static arts, Vinci always tried to create movements in his works and all his works are considered examples of accuracy and perfection. His exposure to his father’s scholarly texts and his apprentice under Andrea del Verrochio in Florence boosted his talents. His most famous works included â€Å"The Last Supper, The Mona Lisa and Vitruvian Man: The Proportion s of the Human Figure†. He died in Cloux, France and legends reveal that â€Å"King Francis was at his side when he died, cradling Leonardo's head in his arms. prestigious St. Michael's School in Là ¼neburg and â€Å"it is almost certain that while in Là ¼neburg, young Bach would have visited the Johanniskirche and heard (and possibly played) the church's famous organ†.. Among many others, his major works that he is famous for includes â€Å"the Brandenburg concertos, the Goldberg Variations, the Partitas, The Well-Tempered Clavier, the Mass in B Minor, the St Matthew Passion†.

Comparison Of Rate Of Convergence Of Iterative Methods Philosophy Essay

Comparison Of Rate Of Convergence Of Iterative Methods Philosophy Essay The term iterative method refers to a wide range of techniques that use successive approximations to obtain more accurate solutions to a linear system at each step In numerical analysis it attempts to solve a problem by finding successive  approximations  to the solution starting from an initial guess. This approach is in contrast to  direct methods which attempt to solve the problem by a finite sequence of operations, and, in the absence of  rounding errors, would deliver an exact solution Iterative methods are usually the only choice for non linear equations. However, iterative methods are often useful even for linear problems involving a large number of variables (sometimes of the order of millions), where direct methods would be prohibitively expensive (and in some cases impossible) even with the best available computing power. Stationary methods are older, simpler to understand and implement, but usually not as effective Stationary iterative method are the iterative methods that performs in each iteration the same operations on the current iteration vectors.Stationary iterative methods solve a linear system with an  operator  approximating the original one; and based on a measurement of the error in the result, form a correction equation for which this process is repeated. While these methods are simple to derive, implement, and analyze, convergence is only guaranteed for a limited class of matrices. Examples of stationary iterative methods are the Jacobi method,gauss seidel method  and the  successive overrelaxation method. The Nonstationary methods are based on the idea of sequences of orthogonal vectors Nonstationary methods are a relatively recent development; their analysis is usually harder to understand, but they can be highly effective These are the Iterative method that has iteration-dependent coefficients.It include Dense matrix: Matrix for which the number of zero elements is too small to warrant specialized algorithms. Sparse matrix: Matrix for which the number of zero elements is large enough that algorithms avoiding operations on zero elements pay off. Matrices derived from partial differential equations typically have a number of nonzero elements that is proportional to the matrix size, while the total number of matrix elements is the square of the matrix size. The rate at which an iterative method converges depends greatly on the spectrum of the coefficient matrix. Hence, iterative methods usually involve a second matrix that transforms the coefficient matrix into one with a more favorable spectrum. The transformation matrix is called a  preconditioner. A good preconditioner improves the convergence of the iterative method, sufficiently to overcome the extra cost of constructing and applying the preconditioner. Indeed, without a preconditioner the iterative method may even fail to converge. Rate of Convergence In  numerical analysis, the speed at which a  convergent sequence  approaches its limit is called the  rate of convergence. Although strictly speaking, a limit does not give information about any finite first part of the sequence, this concept is of practical importance if we deal with a sequence of successive approximations for an  iterative method as then typically fewer iterations are needed to yield a useful approximation if the rate of convergence is higher. This may even make the difference between needing ten or a million iterations.Similar concepts are used for  discretization  methods. The solution of the discretized problem converges to the solution of the continuous problem as the grid size goes to zero, and the speed of convergence is one of the factors of the efficiency of the method. However, the terminology in this case is different from the terminology for iterative methods. The rate of convergence of an iterative method is represented by mu (ÃŽÂ ¼) and is defined as such:   Suppose the sequence{xn}  (generated by an iterative method to find an approximation to a fixed point) converges to a point  x, then   limn->[infinity] = |xn+1-x|/|xn-x|[alpha]=ÃŽÂ ¼,  where  ÃƒÅ½Ã‚ ¼Ãƒ ¢Ã¢â‚¬ °Ã‚ ¥0 and  ÃƒÅ½Ã‚ ±(alpha)=order of convergence.   In cases where  ÃƒÅ½Ã‚ ±=2 or 3 the sequence is said to have  quadratic  and  cubic convergence  respectively. However in linear cases i.e. when  ÃƒÅ½Ã‚ ±=1, for the sequence to converge  ÃƒÅ½Ã‚ ¼Ã‚  must  be in the interval (0,1). The theory behind this is that for En+1à ¢Ã¢â‚¬ °Ã‚ ¤ÃƒÅ½Ã‚ ¼En  to converge the absolute errors must decrease with each approximation, and to guarantee this, we have to set  0 In cases where  ÃƒÅ½Ã‚ ±=1 and  ÃƒÅ½Ã‚ ¼=1  and  you know it converges (since  ÃƒÅ½Ã‚ ¼=1 does not tell us if it converges or diverges) the sequence  {xn}  is said to converge  sublinearly  i.e. the order of convergence is less than one. If  ÃƒÅ½Ã‚ ¼>1 then the sequence diverges. If  ÃƒÅ½Ã‚ ¼=0 then it is said to converge  superlinearly  i.e. its order of convergence is higher than 1, in these cases you change  ÃƒÅ½Ã‚ ±Ã‚  to a higher value to find what the order of convergence is.  In cases where  ÃƒÅ½Ã‚ ¼Ã‚  is negative, the iteration diverges. Stationary iterative methods Stationary iterative methods are methods for solving a  linear system of equations. Ax=B. where  Ã‚  is a given matrix and  Ã‚  is a given vector. Stationary iterative methods can be expressed in the simple form where neither  Ã‚  nor  Ã‚  depends upon the iteration count  . The four main stationary methods are the Jacobi Method,Gauss seidel method,  successive overrelaxation method  (SOR), and   symmetric successive overrelaxation method  (SSOR). 1.Jacobi method:- The Jacobi method is based on solving for every variable locally with respect to the other variables; one iteration of the method corresponds to solving for every variable once. The resulting method is easy to understand and implement, but convergence is slow. The Jacobi method is a method of solving a  matrix equation  on a matrix that has no zeros along its main diagonal . Each diagonal element is solved for, and an approximate value plugged in. The process is then iterated until it converges. This algorithm is a stripped-down version of the Jacobi transformation  method of  matrix diagnalization. The Jacobi method is easily derived by examining each of the  Ã‚  equations in the linear system of equations  Ã‚  in isolation. If, in the  th equation solve for the value of  Ã‚  while assuming the other entries of  Ã‚  remain fixed. This gives which is the Jacobi method. In this method, the order in which the equations are examined is irrelevant, since the Jacobi method treats them independently. The definition of the Jacobi method can be expressed with matrices  as where the matrices  ,  , and  Ã‚  represent the diagnol, strictly lower triangular, and  strictly upper triangular  parts of  , respectively Convergence:- The standard convergence condition (for any iterative method) is when the  spectral radius  of the iteration matrix à Ã‚ (D  Ãƒ ¢Ã‹â€ Ã¢â‚¬â„¢ 1R) D is diagonal component,R is the remainder. The method is guaranteed to converge if the matrix  A  is strictly or irreducibly  diagonally dominant. Strict row diagonal dominance means that for each row, the absolute value of the diagonal term is greater than the sum of absolute values of other terms: The Jacobi method sometimes converges even if these conditions are not satisfied. 2. Gauss-Seidel method:- The Gauss-Seidel method is like the Jacobi method, except that it uses updated values as soon as they are available. In general, if the Jacobi method converges, the Gauss-Seidel method will converge faster than the Jacobi method, though still relatively slowly. The Gauss-Seidel method is a technique for solving the  Ã‚  equations of the  linear system of equations  Ã‚  one at a time in sequence, and uses previously computed results as soon as they are available, There are two important characteristics of the Gauss-Seidel method should be noted. Firstly, the computations appear to be serial. Since each component of the new iterate depends upon all previously computed components, the updates cannot be done simultaneously as in the  Jacobi method. Secondly, the new iterate  Ã‚  depends upon the order in which the equations are examined. If this ordering is changed, the  components  of the new iterates (and not just their order) will also change. In terms of matrices, the definition of the Gauss-Seidel method can be expressed as where the matrices  ,  , and  Ã‚  represent the  diagonal, strictly lower triangular, and strictly upper triangular  parts of   A, respectively. The Gauss-Seidel method is applicable to strictly diagonally dominant, or symmetric positive definite matrices   A. Convergence:- Given a square system of  n  linear equations with unknown  x: The convergence properties of the Gauss-Seidel method are dependent on the matrix  A. Namely, the procedure is known to converge if either: A  is symmetric  positive definite, or A  is strictly or irreducibly  diagonally dominant. The Gauss-Seidel method sometimes converges even if these conditions are not satisfied. 3.Successive Overrelaxation method:- The successive overrelaxation method (SOR) is a method of solving a  linear system of equations  Ã‚  derived by extrapolating the  gauss-seidel method. This extrapolation takes the form of a weighted average between the previous iterate and the computed Gauss-Seidel iterate successively for each component, where  Ã‚  denotes a Gauss-Seidel iterate and  Ã‚  is the extrapolation factor. The idea is to choose a value for  Ã‚  that will accelerate the rate of convergence of the iterates to the solution. In matrix terms, the SOR algorithm can be written as where the matrices  ,  , and  Ã‚  represent the diagonal, strictly lower-triangular, and strictly upper-triangular parts of  , respectively. If  , the SOR method simplifies to the  gauss-seidel method. A theorem due to Kahan shows that SOR fails to converge if  Ã‚  is outside the interval  . In general, it is not possible to compute in advance the value of  Ã‚  that will maximize the rate of convergence of SOR. Frequently, some heuristic estimate is used, such as  Ã‚  where  Ã‚  is the mesh spacing of the discretization of the underlying physical domain. Convergence:- Successive Overrelaxation method may converge faster than Gauss-Seidel by an order of magnitude. We seek the solution to set of linear equations   In matrix terms, the successive over-relaxation (SOR) iteration can be expressed as where  ,  , and  Ã‚  represent the diagonal, lower triangular, and upper triangular parts of the coefficient matrix  ,  Ã‚  is the iteration count, and  Ã‚  is a relaxation factor. This matrix expression is not usually used to program the method, and an element-based expression is used Note that for  Ã‚  that the iteration reduces to the  gauss-seidel  iteration. As with the  Gauss seidel method, the computation may be done in place, and the iteration is continued until the changes made by an iteration are below some tolerance. The choice of relaxation factor is not necessarily easy, and depends upon the properties of the coefficient matrix. For symmetric, positive definite matrices it can be proven that  Ã‚  will lead to convergence, but we are generally interested in faster convergence rather than just convergence. 4.Symmetric Successive overrelaxation:- Symmetric Successive Overrelaxation (SSOR) has no advantage over SOR as a stand-alone iterative method; however, it is useful as a preconditioner for nonstationary methods The symmetric successive overrelaxation (SSOR) method combines two  successive overrelaxation method  (SOR) sweeps together in such a way that the resulting iteration matrix is similar to a symmetric matrix it the case that the coefficient matrix  Ã‚  of the linear system  Ã‚  is symmetric. The SSOR is a forward SOR sweep followed by a backward SOR sweep in which the  unknowns  are updated in the reverse order. The similarity of the SSOR iteration matrix to a symmetric matrix permits the application of SSOR as a preconditioner for other iterative schemes for symmetric matrices. This is the primary motivation for SSOR, since the convergence rate is usually slower than the convergence rate for SOR with optimal  .. Non-Stationary Iterative Methods:- 1.Conjugate Gradient method:- The conjugate gradient method derives its name from the fact that it generates a sequence of conjugate (or orthogonal) vectors. These vectors are the residuals of the iterates. They are also the gradients of a quadratic functional, the minimization of which is equivalent to solving the linear system. CG is an extremely effective method when the coefficient matrix is symmetric positive definite, since storage for only a limited number of vectors is required. Suppose we want to solve the following   system of linear equations Ax  =  b where the  n-by-n  matrix  A  is  symmetric  (i.e.,  AT  =  A),  positive definite  (i.e.,  xTAx  > 0 for all non-zero vectors  x  in  Rn), and  real. We denote the unique solution of this system by  x*. We say that two non-zero vectors  u  and  v  are  conjugate  (with respect to  A) if Since  A  is symmetric and positive definite, the left-hand side defines an  inner product So, two vectors are conjugate if they are orthogonal with respect to this inner product. Being conjugate is a symmetric relation: if  u  is conjugate to  v, then  v  is conjugate to  u. Convergence:- Accurate predictions of the convergence of iterative methods are difficult to make, but useful bounds can often be obtained. For the Conjugate Gradient method, the error can be bounded in terms of the spectral condition number  Ã‚  of the matrix  . ( if  Ã‚  and  Ã‚  are the largest and smallest eigenvalues of a symmetric positive definite matrix  , then the spectral condition number of  Ã‚  is  . If  Ã‚  is the exact solution of the linear system  , with symmetric positive definite matrix  , then for CG with symmetric positive definite preconditioner  , it can be shown that where  Ã‚  , and   . From this relation we see that the number of iterations to reach a relative reduction of  Ã‚  in the error is proportional to  . In some cases, practical application of the above error bound is straightforward. For example, elliptic second order partial differential equations typically give rise to coefficient matrices  Ã‚  with  Ã‚  (where  Ã‚  is the discretization mesh width), independent of the order of the finite elements or differences used, and of the number of space dimensions of the problem . Thus, without preconditioning, we expect a number of iterations proportional to  Ã‚  for the Conjugate Gradient method. Other results concerning the behavior of the Conjugate Gradient algorithm have been obtained. If the extremal eigenvalues of the matrix  Ã‚  are well separated, then one often observes so-called; that is, convergence at a rate that increases per iteration. This phenomenon is explained by the fact that CG tends to eliminate components of the error in the direction of eigenvectors associated with extremal eigenvalues first. After these have been eliminated, the method proceeds as if these eigenvalues did not exist in the given system,  i.e., the convergence rate depends on a reduced system with a smaller condition number. The effectiveness of the preconditioner in reducing the condition number and in separating extremal eigenvalues can be deduced by studying the approximated eigenvalues of the related Lanczos process. 2. Biconjugate Gradient Method-The Biconjugate Gradient method generates two CG-like sequences of vectors, one based on a system with the original coefficient matrix , and one on . Instead of orthogonalizing each sequence, they are made mutually orthogonal, or bi-orthogonal. This method, like CG, uses limited storage. It is useful when the matrix is nonsymmetric and nonsingular; however, convergence may be irregular, and there is a possibility that the method will break down. BiCG requires a multiplication with the coefficient matrix and with its transpose at each iteration. Convergence:- Few theoretical results are known about the convergence of BiCG. For symmetric positive definite systems the method delivers the same results as CG, but at twice the cost per iteration. For nonsymmetric matrices it has been shown that in phases of the process where there is significant reduction of the norm of the residual, the method is more or less comparable to full GMRES (in terms of numbers of iterations). In practice this is often confirmed, but it is also observed that the convergence behavior may be quite irregular  , and the method may even break down  . The breakdown situation due to the possible event that  Ã‚  can be circumvented by so-called look-ahead strategies. This leads to complicated codes. The other breakdown  Ã‚  situation,  , occurs when the  -decomposition fails, and can be repaired by using another decomposition. Sometimes, breakdown  Ã‚  or near-breakdown situations can be satisfactorily avoided by a restart  Ã‚  at the iteration step immediately before the breakdown step. Another possibility is to switch to a more robust method, like GMRES.  Ã‚   3. Conjugate Gradient Squared (CGS  ). The Conjugate Gradient Squared method is a variant of BiCG that applies the updating operations for the -sequence and the -sequences both to the same vectors. Ideally, this would double the convergence rate, but in practice convergence may be much more irregular than for BiCG, which may sometimes lead to unreliable results. A practical advantage is that the method does not need the multiplications with the transpose of the coefficient matrix. often one observes a speed of convergence for CGS that is about twice as fast as for BiCG, which is in agreement with the observation that the same contraction operator is applied twice. However, there is no reason that the contraction operator, even if it really reduces the initial residual  , should also reduce the once reduced vector  . This is evidenced by the often highly irregular convergence behavior of CGS  . One should be aware of the fact that local corrections to the current solution may be so large that cancelation effects occur. This may lead to a less accurate solution than suggested by the updated residual. The method tends to diverge if the starting guess is close to the solution.  Ã‚   4 Biconjugate Gradient Stabilized (Bi-CGSTAB  ). The Biconjugate Gradient Stabilized method is a variant of BiCG, like CGS, but using different updates for the -sequence in order to obtain smoother convergence than CGS. Bi-CGSTAB often converges about as fast as CGS, sometimes faster and sometimes not. CGS can be viewed as a method in which the BiCG contraction operator is applied twice. Bi-CGSTAB can be interpreted as the product of BiCG and repeatedly applied GMRES. At least locally, a residual vector is minimized  , which leads to a considerably smoother  Ã‚  convergence behavior. On the other hand, if the local GMRES step stagnates, then the Krylov subspace is not expanded, and Bi-CGSTAB will break down  . This is a breakdown situation that can occur in addition to the other breakdown possibilities in the underlying BiCG algorithm. This type of breakdown may be avoided by combining BiCG with other methods,  i.e., by selecting other values for  Ã‚   One such alternative is Bi-CGSTAB2  ; more general approaches are su ggested by Sleijpen and Fokkema. 5..Chebyshev   Iteration. The Chebyshev Iteration recursively determines polynomials with coefficients chosen to minimize the norm of the residual in a min-max sense. The coefficient matrix must be positive definite and knowledge of the extremal eigenvalues is required. This method has the advantage of requiring no inner products. Chebyshev Iteration is another method for solving nonsymmetric problems . Chebyshev Iteration avoids the computation of inner products  Ã‚  as is necessary for the other nonstationary methods. For some distributed memory architectures these inner products are a bottleneck  Ã‚  with respect to efficiency. The price one pays for avoiding inner products is that the method requires enough knowledge about the spectrum of the coefficient matrix  Ã‚  that an ellipse enveloping the spectrum can be identified  ; however this difficulty can be overcome via an adaptive construction  developed by Manteuffel  , and implemented by Ashby  . Chebyshev iteration is suitable for any non symmetric linear system for which the enveloping ellipse does not include the origin. Convergence:- In the symmetric case (where  Ã‚  and the preconditioner  Ã‚  are both symmetric) for the Chebyshev Iteration we have the same upper bound as for the Conjugate Gradient method, provided  Ã‚  and  Ã‚  are computed from  Ã‚  and  Ã‚  (the extremal eigenvalues of the preconditioned matrix  ). There is a severe penalty for overestimating or underestimating the field of values. For example, if in the symmetric case  Ã‚  is underestimated, then the method may diverge; if it is overestimated then the result may be very slow convergence. Similar statements can be made for the nonsymmetric case. This implies that one needs fairly accurate bounds on the spectrum of  Ã‚  for the method to be effective (in comparison with CG or GMRES).  Ã‚   Acceleration of convergence Many methods exist to increase the rate of convergence of a given sequence, i.e. to transform a given sequence into one converging faster to the same limit. Such techniques are in general known as series acceleration. The goal of the transformed sequence is to be much less expensive to calculate than the original sequence. One example of series acceleration is Aitkens delta -squared process.

The Odyssey Essay examples -- essays research papers

Throughout vast journeys of many heroes, no other hero had a more complex journey than Odysseus. This journey is called The Odyssey, written by Homer. It is an epic poem or story told of a hero name Odysseus on a 20-year voyage trying to get back home from the Trojan War. The great epic poem known as The Odyssey and attributed to Homer was probably first written down around the eighth century BC, but the origins of the ancient story in myth, legend, and folklore and art appear to be much older. Greek Epic Hero When you think about Greek Epic heroes, Odysseus will most likely come to mind. Odysseus is the main character in Homer's poem "The Odyssey." "The Odyssey" is a narrative poem that describes Odysseus' adventures and obstacles in his quest to return home to Ithaca, where he is king, from the Trojan War. Odysseus has been gone for two decades. All the other chieftains have returned home. However, Odysseus' whereabouts are unknown. One of the things that make Odysseus a Greek hero is that the gods favor him. In the Odyssey the gods help Odysseus many times. When Poseidon struck Odysseus' ship with lightning, Odysseus was able to survive because a goddess named Ino gave him a magical cloak. The cloak prevents Odysseus from drowning. When Circe turns Odysseus’ men into swine, Hermes the messenger god gives Odysseus a magical herb that prevents Circe from using her spells on him. Due to this Odysseus was able to gain favor in Circe's eyes and he convinced her to turn his men back to normal. Towards the end of the book, when Odysseus returns to Ithaca, Athena disguises him as a beggar. The disguise was to fool the suitors, who are eating Odysseus out of his home and are harassing his wife Penelope, and to get Odysseus access to the palace. The ancient Greeks used to say that a true hero must show respect to the gods by giving them sacrifices and praying to them. Odysseus gives sacrifices to the gods in many instances in â€Å"The Ody ssey." When Odysseus goes to Hades, the underworld, he sacrifices a ram and an ewe to the gods. When Odysseus defeats Polyphemus the Cyclops who is the son of Poseidon the god of the sea, Odysseus gives sacrifices to the gods. Lastly when Odysseus defeats those horrible suitors that outnumbered him by so much Odysseus realizes that it was the gods who helped him do this. A mere mortal could not perform the fea... ...d wars are true. Let’s take Ithaca for instances, is a real landmass in the Ionians Islands of the western part of Greece. Through the Trojan War, there is some truth to that because researches have found some parts and signs that deal with a war that took place a great time ago. They found that Troy is in what we call present-day Turkey. It is very hard to understand the book because of the depth it goes into. The translator E. V. Rieu said that â€Å"The Odyssey† was one the hardest books to translate mainly because Homer writes with emphasis on his works. But in the movie, the director Andrei Kanchalovsky said â€Å"the movie was hard to make because it was to many books that translated The Odyssey in many different ways.† In actuality, the article was the best summary of the book and half as good as the movie. Fayetteville State University By: Adrian Williams History 110 11-16-00 Book: Homer. The Odyssey. 375 Hudson Street, New York, New York 10014, USA. Penguin Books Inc, 1946. Movie: â€Å"The Odyssey†. Directed by Andrei Kanchalovsky. Starring Armand Assante, Vanessa L. Williams, and Eric Roberts. 1997. Article: http://members.nbci.com/blademaster/myths/odyssey.html

Ageing Population â€Any Resolutions?

Ageing population –any resolutions? According to the World Health Organization , a country or a city is defined as an ageing society if the number of elders who aged 65 or above exceed 7% of the whole population. With 14% of elders in population, the country would be considered as an aged society. Worse still, if the amount of elders still keep on increasing and reach 20% of the population, the country would be classified as an hyper-aged society. Recently, the ageing population has been under the spotlight all over the world.This worldwide issue has aroused public concern and surely Hong Kong is not an exception. According to the Hong Kong Population Projections published in July, 2012, there are in total 13% of elders who aged 65 or over in our population, which means that Hong Kong is going to transform into an aged society. What’s more, it is predicted that we would turn into hyper-aged society in 2026, with 23% of elderly people in population. Such shocking data ac tually did a good warning effect , which successfully remind government to take long term measures at once to tackle the problem.Or not, ageing population would lower Hong Kong’s labour force, which would then affect Hong Kong’s productivity and hence our competitiveness in the world market. Furthermore, the ageing population would also burden the health care system. To solve the problems, people from all walks of life have raised different suggestions. It is proposed to import talents from other parts of the world, and implement effective policies to convince children born here to non-local parents and their families to stay here in order to alleviate the shortage of labour force.On the other hand, the tension in existing health care system can be eased by several programmes held by the hospital authority, like the psychological support programme , which can help strengthen staff morale and address their work stress. As the above mentioned, one of the severe problems provoked by ageing population is the shortage of larbour force. According to the Summary Results of the 2011 Population Census published in February,2012, the labour force participation rate has dropped from 61. 4% in 2001 to 59. 7% in 2011.In addition, with reference to the Hong Kong Population Projections , the overall dependency ratio is predicted to rise from 352 to 712 ,which is over a double increase. Besides, it is also estimated that every three persons in the workforce will need to support one elder by 2029. Therefore, we can see that ageing population not only lower our larbour force, and hence Hong Kong’s productivity and competitiveness in international market , but it would also put more burden to the workforce, which bring great challenges to Hong Kong economy. To increase the supply of labour force in near future, it is believed there are few ways to do so.Firstly, we can introduce more talent admission schemes and seriously investigate both push and pull facto rs affecting talents to stay or to leave Hong Kong. For instance, the government are now expanding the international school system. There would be in total 5000 additional international school places in the coming four years. Furthermore, the government is planning to invite the international schools to express interest in expanding their school by using several vacant school premises. These actions definitely help import talents from other countries, but the government should note that it should not be a short term measure.The government should keep improving the education in international school and try best to assist international school to have more resources so that more school places are available for the talents and talents would stay here for better education. Or not, the students from outside Hong Kong may choose to study in other countries with better education, other than Hong Kong. Yet, there are some critics believes that it is totally not enough to solve the population problem by only those talent admission schemes.Therefore the babies born in Hong Kong to non-local parents have been seen as a simple way to alleviate the problem. But whether those babies and their families would stay here or not later on, it mainly depends on the government efforts to implement policies for that purpose and see whether the city can promise them a stable and good enough future. Compared to the talent admission schemes, it is a more effective means to meet the uprising demand of manpower as the number of mainland families who stay here is greater than those talents.This can be proved by the results of our 2011 Population Census . According to the Hong Kong Population Projections, about 77% of the Mainland women whose husband is a Hong Kong Permanent Resident, would bring their children back to Hong Kong later. And about 53% for those Mainland women whose husband is not a Hong Kong Permanent Resident would also later bring their children back to Hong Kong. Other tha n the shortage of labour force, the ageing population would also burden the health care system. More elderly people would definitely increase the demand of medical services and it would ncrease the work stress of frontline staff in hospital and the financial burden to the government. To strengthen the staff morale and ease their work stress, it is suggested the Hospital Authority can hold more relaxing measures , for instance, improving the working conditions, recruiting more full-time and part-time staff and preventing excessive overnight duties. All these are useful ways to reduce frontline workload, and it is believed it can help lower the staff turnover number, and hence help prevent the manpower shortage in public hospitals.Just like the psychological support programme which help staff deal with work stress has lower the overall turnover rate of physicians in public hospitals from 5. 2 % in 2010/11 to 4. 8% in 2011/12. And the overall turnover rate has further decreased to 4. 4 % in the first 4 months of 2012/13. All in all, it is the responsibility of the government to stop our aging situation from being worse. The government should plan detailed and long term policies to solve the aging situation. Or not, Hong Kong will be suffered badly with economic and social problems in the near future. Word count: 1035 words) REFERENCES Peng,P. (2012, February 24). The problem is not ‘non-local’ women but intermediaries. Retrieved from http://www. cdeclips. com/en/hongkong/The_problem_is_not_'non-local'_women_but_intermediaries/fullstory_72200. html Demographic Statistics Section, Census and Statistics Department. (2012,July). Hong Kong Population Projections. Retrieved from http://www. statistics. gov. hk/pub/B1120015052012XXXXB0100. pdf Pang,F. C. (2012, October 12). Letter to the Editor . Retrieved from http://www. ha. org. k/haho/ho/pad/121016chi1. pdf Li, A. (2006,August 12). Coping with old age. Retrieved from http://www. scmp. com/article/559945 /coping-old-age Census and Statistics Department. (2012, February 21). Summary Results of the 2011 Population Census. Retrieved from http://www. censtatd. gov. hk/FileManager/EN/Content_1214/presentation_summary_results. pdf HKSAR. (2012, May 30). Government releases Steering Committee on Population Policy Progress Report 2012. Retrieved from http://www. info. gov. hk/gia/general/201205/30/P201205300456. htm

Research proposal on Berlin’s Architecture

Proposed Research Topic: A situational analysis of the relationship and influence of Berlin’s past Architectural designs on the modern ones. Purposes: TeNeues (2003) argues that a situational analysis and approach should be applied on architectural designs as a practical successive progress, rather than trying to conceptualize these designs as just what the designers think off or decide to put into place. This approach will more particularly explore how the Architectural designs are a product of the past ones; as they are to a great extent influenced upon by and related to the past designs.This study will also cover the issues of the basis and sources of the model architectural designs, particularly with regard to the phenomena that has directed the architectural thinking towards this direction and the usefulness of this situational approach as it has rarely been developed and applied in Architectural literature (TeNeues, 2003). Background: I will carryout my study on the diff erent teams that are operating within the field of Architectural designing within the region of Berlin.Barkow Leibinger Architects will be the company to be used as a centre for the study as it offers the services of architecture and master planning. Additionally, this company has worked on successful projects like the Trumpf Restaurant and event space. Based on that this firm is Berlin based; it is evident from the architectural designing it has put into place that highly depicts the influence and characteristics of earlier architectural models.For this study I will evaluate and analyse their architectural designs pointing out the traits passed down from earlier designs and the influence of the earlier designs on the new ones with regard to the architectural expertise used and the materials utilised. This will be carried out in the areas of new projects for five hours a day, though I will also make enquiries regarding the areas that prove rather difficult to analyse or document abo ut on my own.The other aspect that I will utilise with reference to attaining these results will be interacting with the workers and clients to get a few views regarding the models (TeNeues, 2003). Scope: I will make use of participant-observation over a two-week period for five hours a day; typically observing the designs put into place while documenting the observation and keeping record of the views and hints given by the engineers among other workers. On other days I will vary the timing for the observation so as to attain a comparative picture of the design applications and the views of different staff.Towards attaining these I will offer structured interviews from time to time; so as to acquire information about the time of future investments, the number of participants, and the studies carried out before the operations begin (TeNeues, 2003). Theoretical framework: I will be directed normally by the interpretive point of view and more particularly by TeNeues (2003) situational analysis. The interpretive point of view mainly focuses on understanding the significance and standpoint of the choice of the design to be applied and how the choice is arrived at.I will also explore the meanings the different staff and customers have regarding these designs and the relationship they likely place between the two and the projected results. The situational perspective will further create the need that the study focuses on one or a few specific projects which are to be analysed in details; thus the most appropriate mode of studying the topic from this standpoint is a comparative analysis of the designs and the symbolic significance contained in each of them (TeNeues, 2003).Method: 1. Perform a literature evaluation on the architectural designs employed by the focus company. 2. Monitor the operations of the company through the current project five hours a day for two weeks, focusing mainly on the designs and the rationale behind the choice of the different designs and the past design influencing the choice. 3. Interview team members to clarify and provide insight into the choice of designs. I will ensure I conduct the interviews during the time of design choice and employment.However, despite the fact that the interviews will not be structured or formal; the kinds of question to be asked will include starting with broad based questions then following–up on the informants’ responses to capture their personalised meanings so as to avoid the imposition of meanings on the interviewees (TeNeues, 2003). a. Tell me about the current project b. What issues guided the choice for the utilised architectural design c.What influenced the choice of this design and what were they trying to accomplish from using the given design? e. What success expectations do you have for the project? f. How would you describe the current project in general? 4. Undertake a situational analysis of the study notes and interview notes, guided by theoretical models. 5. Write a research report that combines my understanding of the relevant theory and previous research with the results of my empirical research. Timetable: Prepare proposal by 14 JulyComplete literature review by 16 July Complete fieldwork by 18 July Complete analysis by 20 July Give presentation on 24 July Complete final report by 28 July Limitations: The tightly packed study schedule will limit the time to be allocated to this study to two weeks which may not be ideally enough to make a fully comprehensive study regarding this topic. The other limitation is that some of the officers may not be fully cooperative in providing the information required for the study (TeNeues, 2003).Delimitations: For this study I choose not to observe multiple operation units despite the fact that the comparisons might provide valuable information in order to allow for a more into depth understanding of the focus group. Additionally, I will not make use of formal or structured interviews with the vie w to minimise the chances of obtrusiveness and influencing the views of the informants (TeNeues, 2003). References TeNeues, P. (2003). Berlin and guide (Architecture and design guides), Multilingu edition. New York: TeNeues Publishers.

Comparative Analysis of the Literary Work, Lolita by Vladimir Nabokov and the Artistic Works of Balthasar Klossowski de Rola, Better Known as Balthus Essays

Comparative Analysis of the Literary Work, Lolita by Vladimir Nabokov and the Artistic Works of Balthasar Klossowski de Rola, Better Known as Balthus Essays Comparative Analysis of the Literary Work, Lolita by Vladimir Nabokov and the Artistic Works of Balthasar Klossowski de Rola, Better Known as Balthus Paper Comparative Analysis of the Literary Work, Lolita by Vladimir Nabokov and the Artistic Works of Balthasar Klossowski de Rola, Better Known as Balthus Paper Essay Topic: Lolita Comparative Analysis of the Literary work, Lolita by Vladimir Nabokov and the Artistic Works of Balthasar Klossowski de Rola, Better Known as Balthus Lolita is written as a memoir in the first person by its main character, Humbert Humbert. This is a story that could be viewed in two very different ways, two very different perspectives. One could look at it as a story of a middle age pedophile as evidenced by the quote â€Å"Humbert Humbert is without question an honest-to-God, open-and-shut sexual deviant, displaying classic ruthlessness, guile and above all attention to detail. And the other, of a middle aged man in anguish over his love for a prepubescent girl, a forbidden love. â€Å"Lolita, light of my life, fire of my loins. My sin, my soul. Lo-lee-ta: the tip of the tongue taking a trip of three steps down the palate to tap, at three, on the teeth. Lo. Lee. Ta. She was Lo, plain Lo, in the morning, standing four feet ten in one sock. She was Lola in sl acks. She was Dolly at school. She was Dolores on the dotted line. But in my arms she was always Lolita. â€Å" I chose to view this work as the later. To me it showed a middle aged man trapped in a moral dilemma. A statement from the first page of the book best says how I feel about the story. â€Å"Lolita is not about sex, but about love. Almost every page sets forth some explicit, erotic emotion or some erotic action and still it is not about sex. It is love†. Nabokov started writing Lolita in 1949 and finished in 1954. When he finished his work he had a hard time getting it published. Publishers were nervous about printing a book supposedly narrated by a pedophile. Four American publishers turned him down before he was finally able to get it published by Olympia Press of Paris and in 1958 in America. Controversy over the book only increased its readership and sales rose. To understand this controversy you need to understand the story and further understand why I considered it to be more of a tragic love interest rather than the perceived pornographic pedophile story thought of by others. The story starts out with a boyhood Humbert having his first taste of a love affair with the love of his young life, Annabel. They laugh, sneak around and display affection like young lovers do, but after three months Annabel dies of typhus and leaves a broken hearted Humbert. Though Humbert experiences relationships throughout his adult life he always feels hollow, like something is missing. Though he grows in age he still yearns for his young love. Humbert takes to fantasizing about young girls and goes as far as sitting in parks just to view the girls to get a glance of a â€Å"nymphet†. Humbert didn’t just want any young girl; she had to have certain traits that made her a â€Å"nymphet†, his â€Å"nymphet†. His criterion was a girl between nine and fourteen but as he states in his memoirs â€Å"Between those ages, are all girls nymphets? Of course not. Otherwise we, who are in the know, we lone voyagers, we nympholepts, would long have gone insane. Neither are good looks any criterion. A true nymphet is strikingly inferior to that of provisionally plain, just nice or â€Å"cute† or even â€Å"sweet â€Å"or attractive little girl. You have to be an artist, and a madman, a creature of infinite melancholy, with a bubble of hot poison in your loins in order to discern at once, by ineffable signs the little deadly demon that I deem a nymphet. † Humbert marries and tries to suppress his desire; he becomes bored with his married life and in the same time frame discovers his wife is having an affair. This leads him to divorce her and leave Paris and move to America. Once there, while looking for a place to stay he visits a home with a room to rent, the Haze home. While looking over the house Humbert is not impressed and is looking for a way to leave but during his tour of the house spots Dolores Haze, Lo, Lola, Lolita, the 12 year old daughter of the home owner, Charlotte Haze, Big Haze. He is struck; she is the nymphet that he has been searching for. Humbert agrees to rent the room, to be near his nymphet. Humbert proceeds to try to win Lo’s affection while Lo’s mother wishes to win Humberts’ affection. He ends up marrying â€Å"Big Haze†, Lo’s mother to get near her. Big Haze dies in a tragic accident soon after their wedding. This sets the stage for Humberts seduction of Lo. He ends up with his nymphet, losing her to a rival, killing the rival and dieing in jail. Ironically Lo dies soon afterward giving birth to a child. As you can see, this is more of a story of intense love rather than that of a pedophile. The situation Nobokov faced after writing Lolita is very much similar to that faced by the polish painter Balthasar Klossowski de Rola, better known as Balthus. Balthus had a difficult time getting his paintings displayed in his early days. His paintings were viewed as pornographic, almost taboo. They often depicted pubescent young girls in erotic and voyeuristic poses. He was best known for his disturbing and erotic works of girls on the verge of womanhood. One of the most notorious works from his first exhibition in Paris was The Guitar Lesson (1934), which caused controversy due to its depiction of a sexually explicit lesbian scene featuring a young girl and her teacher. As you can imagine the Guitar Lesson caused quite a controversy when it was first released. Balthus painted many pictures of young girls, most in provocative poses and situations. Balthus’ obsession with young girls parallels that of the early Humbert Humberts with the exception that he seemed to desire multiple girls and Humbert only wanted his nymphet, his Lolita. Other Balthus works that seem to parallel the plot of Lolita and the exploitation of young girls are The Golden Days, which seems to show a topless young girl sitting in a chair while her older lover seems to be building a fire in the fire place in preparation of a romantic night. One could almost envision this scene coming right out of the book. The other painting that puts you in mind of book and its controversial content is pictured here. I couldn’t locate the name but you can clearly see the relationship of this Balthus painting to the character Quilty, Humberts rival for Lo’s affection. This looks as though it were taken right out of the text, displaying Quilty and the semi nude Lo. Other samples of Balthus’ works are shown on the following page. Notice the young girls in various stages of undress, a reoccurring theme. The tie between the book and especially Humbert and the works of Balthus are very strong and if the book were in need of illustration, I’m sure Balthus would have did an excellent job as its illustrator and would have thoroughly enjoyed the job. So you would ask yourself, â€Å"Is Balthus the outstanding preserver of tradition or a mediocre painter who is only remarkable for sexual perversion and snobbery? Do his typical paintings of little girls testify to desires beyond the area of taboo or are they untouchable archetypes of purity? Balthus, who had staged his life, partly constructed his vita and surrounded himself with an aura of mystery, leaves art lovers and historians with a lot of riddles to solve. †

Scholarship Appeal Essay Example | Topics and Well Written Essays - 1250 words

Scholarship Appeal - Essay Example I eventually found the load to be highly demanding; as such, I withdrew from the class to avoid generating a low grade. Regardless of the course withdrawal, the challenging academic requirements caused my inability to meet the GPA requirements of my scholarship. I am hereby appealing to the committee to re-consider sustaining my scholarship with the firm commitment that I would work fervently to meet and maintain the required GPA. I am hereby providing the details: Other confounding events and circumstances apparently rationalized my inability to meet the GPA requirement. One of the main reasons that I attended the University of Louisville was because it was in close proximity to where my parents lived. My parents understood that I was going to College at a younger age than all my peers and they believed that I still needed a support system at least for my first full year at the University. However, because of great job opportunities in Texas for my parents, they decided that since I obtained a good GPA during my first semester, I was stable enough to be left in another state by myself. I got a job as a hostess at Griff’s restaurant to augment the financial resources I had to support living alone and studying, at the same time. With the time allocated for work, there was evidently lesser time allotted for studying and going to REACH sessions. As soon as I started to see my grades suffer detrimentally, I decided to stop the job. I gradually improved over time with my test grades but the first two test scores were way too low for me to eventually make at least a B in the class. As such, the significantly low test scores evidently caused my GPA to drop. Transportation was also a big issue for me because my parents used to accommodate my transportation requirements. With their relocation, I had no way of attending deaf events for my ASL class. ASL students were required to attend 10 hours of deaf events but I was only able to attain 4

Human Resources Development Essay Example | Topics and Well Written Essays - 1000 words - 1

Human Resources Development - Essay Example ge of branded products, â€Å"backwards expansion† strategy (saturating the target’s rural market first), innovative marketing strategy and most important of all, human resource management. Walton, from the very beginning, believed that happy and satisfied employees performed well and were responsible for happy customers. Wal-Mart’s success in human resource management is keeping their workforce of 1.3 million or more from unionizing. In addition to that, Wal-Mart has achieved a very good retention rate of their employees and has also ensured discipline and uniformity in an organization that has been growing at such a rapid pace. Wal-Mart’s distinctive human resource management policies can be listed and analyzed as follows: (1) Recruitment: In the realm of employee recruitment, Wal-Mart has targeted college students to add to their workforce.   Wal-Mart achieves this recruitment by fanning out over 80 college campuses. While they are at these colleges, they are also able to expand their demographics by looking at minority fraternities and sororities, which brings all types of people from different backgrounds, races, and genders together in the Wal-Mart family.   Having a wide variety demographic for a workforce, only serves to attract more people to seek employment with Wal-Mart because they are able to show that they have a very open hiring process. (2) Training and Development: Beyond recruitment, Wal-Mart has taken an additional step with college students by offering management training for college students while they are still in school so they are more developed and prepared upon their graduation. This program serves the purpose of making college students consider careers with Wal-Mart, and over the last few years, the program has had immense success.   The company is also highly committed towards employee development and improving their career prospects. It has a policy of recruiting more than seventy percent of its personnel in managerial