Out, Out by Robert Frost - 782 Words

Robert Frosts poem â€Å"Out, Out,† paints a strange and bizarre death image to readers; A young boys death due to a carnivorous chainsaw who sought blood, slicing the boys hand off. Robert makes readers understand why he would paint such a tragic accident with various narrative elements, such as personification, many signs of imagery, emotions, and perceptions throughout the story. Also, Frost references William Shakespeare’s work, â€Å"Macbeth.† This gives readers who have read Macbeth before, an idea of what’s to come in the end of the poem, the feeling of sadness and death. This analysis will show the main theme of the boys death, who died doing the work of a man. Robert Frost starts out by showing readers the year and location; 1945, New England, Vermont. This is the period where World War I occurs, however, Frost does not describe the land destroyed and covered with dead bodies, instead Vermont is a place where exists; â€Å"Five Mountain ranges on e behind the other under the sunset far into Vermont.† This makes readers understand that this place is located in a beautiful place out in the wilderness away from war. The location is very significant to the story, I have done a bit of research about Frosts life; Frost was forced to return to America due to the war in 1915, a war that would have killed hundreds of innocent little boys. With this being said, the reader now understands why the boy in the poem is located in a place like Vermont in a time of war, and why the boy in theShow MoreRelatedOut Out by Robert Frost548 Words   |  2 PagesRobert Frost is the author of Out Out--, â€Å"Stopping by Woods on a Snowy Evening, and Nothing Gold can Stay. His literary work communicates deep meaning through the use of metaphoric language and deception. Being raised most of his life on a farm; his works perceive the natural life of a normal person while out in nature. â€Å"Frost believes that the emphasis on everyday life al lows him to communicate with his readers more clearly; they can empathize with the struggles and emotions that are expressed inRead MoreAnalysis Of Out, Out By Robert Frost727 Words   |  3 PagesIn Robert Frost’s poem â€Å"Out, Out† an overwhelming theme of agony can be sensed as Frost incorporates his personal experiences with loss and his views on society into the narrative of this literary work. Frost uses the depiction of innocence through a young boy who suffers a fatal accident to metaphorically embed his personal struggles with the death of his two children into the poem. The section of the poem that will be analyzed is the final ten lines (25-34). The significance of this section inRead MoreAnalysis Of Out, Out By Robert Frost780 Words   |  4 PagesOut, Out and the Responsibilities of age Responsibilities may not seem very harmful, but not adhering to these responsibilities can lead to dire consequences. Said responsibilites are much more prominent in the teenage years of life. For example: driving, getting offered drugs and/or alcohol, and intercourse are all situations that teenagers might find themselves in. These situations may not be inherently bad, but because teenagers are new drivers, too young and inexperienced to properly care forRead MoreOut, Out by Robert Frost Essay836 Words   |  4 PagesOut, Out Out, Out, by Robert Frost is a gruesomely graphic and emotional poem about the tragic end of a young boys life. It is a powerful expression about the fragility of life and the fact that death can come at any time. Death is always devastating, but it is even more so when the victim is just a young boy. The fact that the boys death came right before he could Call it a day (750) leads one to think the tragedy might have been avoided and there by forces the reader to think, WhatRead MorePoem, Birches And Out, Out By Robert Frost1116 Words   |  5 PagesTheme, Figurative Speech and Tones in â€Å"Birches† and â€Å"Out, Out† by Robert Frost Robert Frost was born in 1874 in San Francisco. Descended from the New Englanders generations, his parents, make Robert Frost is much associated with New England. In addition, most of his poems were well-known as a reflection from New England life. Despite that, he was a kind of subtle poet and generally recognized as a private man. Moreover, his appearance at the inauguration of John F. Kennedy to recite â€Å"The Gift Outright†Read More Analysis of Out, Out by Robert Frost Essay591 Words   |  3 PagesAnalysis of Out, Out by Robert Frost Robert Frost tells a disturbing story in Out, Out, --, in which a little boy loses his life. The title of the poem leaves the reader to substitute the last word of the title, which some would assume would be out because of the repetition. The title is referring to the boy exiting the living world. Frost drags the readers mind into the poem with the imagistic description of the tools and atmosphere the little boy is surrounded by. Frost describesRead More Appeal of Robert Frosts Out Out Essay1055 Words   |  5 Pagesinteresting and appealing poems is Robert Frost’s â€Å"Out, Out†. The poem has the ability to make the reader visualize an event in vivid detail without making it into a short story. The poem depicts a very dramatic scene and makes it seem as if the reader is really there. Poems are generally thought to be about love and feelings, but some poems can actually be like a short story; these are called narrative poems, which means that they tell a story. The poem â€Å"Out, Out† is a great example of a narrativeRead MoreLooking Out By Robert Frost2826 Words   |  12 Pageshas found its thought and the thought has found words.† (Robert Frost) Expressing emotions is a very important thing that we do in our lives, everyone also has a different way they express emotions. In the quote by Robert Frost he says that he expresses his emotions through writing poetry. The way people express emotions is very important in their lives, it can improve or destroy many interpersonal relationships. In the book â€Å" Looking Out, Looking in† it talks about how people express emotions toRead MoreDisabled by Wilfred Owen and Out, Out by Robert Frost1516 Words   |  6 PagesCompare how the theme of loss is communicated in the poems â€Å"Disabled† by Wilfred Owen and ‘Out, Out –‘by Robert Frost In both of the poems â€Å"Out, Out’’ and ‘’Disabled’ ’has a similar theme of loss and is shown throughout each poem. Both of the poem deals with the subject of physical loss. The characters of these poems both experience losses from an accident. They create an effect, where the audience will show empathy to the two poems. In order to create this outstanding effect, they both used similarRead More A Comparison of The death of a hired man and Out, Out- by Robert Frost1199 Words   |  5 PagesA Comparison of The death of a hired man and Out, Out- by Robert Frost Robert frost was born in Vermont in 1874 and died in 1963. Robert Frost was a farmer and lived in Vermont, USA. Both poems The death of a hired man and Out, Out- are set on a farm in Vermont which is probably because of where Robert Frost lived and worked. I will know begin to discuss the similarities. As I said previously both poems are set in a farm enviroment. The poem The death of a hired man is probably set

Using Math to Find a Way to Beat the Game A Knight´s Tour

I chose this specific topic because I was interested to know if there was an actual mathematical way of beating the game called â€Å"A Knight’s Tour†. Whenl was first introduced to this game in my research for an internal assessment topic, I was rather intrigued by its goal that was required for each player to attempt to achieve. I thought that the game wouldn’t be too terribly hard because I knew fluently how the knight moved and thought that I would be able to move it in certain patterns across the board that would allow me to complete the knight’s tour. Although the goal of the game seemed easily obtainable at first glance, as I continued to play the game and take myself on a lnight’s tour, I came to conclusion that it would be nearly impossible to beat without having an algorithm to base each move off of. With each attempt at the game, I try different movements that might help to beat â€Å"a Knight’s Tour† but always end up landi ng on the same square more than once because I run out of moves, or failing to beat the game in the required 63 moves or less. After coming as close as 69 moves at the end of the tour, l decided to research the game more thoroughly and attempt to find a solution to the game without cheating, but using math based algorithms to help guide each rnovet The Rules ef A Knighfis Tour The rules seem simple at first because the player is supposed to move their knight piece in any direction, starting in the top lett corner, and make the knight land in each squareShow MoreRelatedCase Study148348 Words   |  594 Pagesresold, hired out or otherwise disposed of by way of trade in any form of binding or cover other than that in which it is published, without the prior consent of the Publishers. 2  © Pearson Education Limited 2011 Contents Acknowledgements Introduction Using this Manual Planning Your Approach Designing the Teaching Scheme A Guide to Using the Work Assignments A Guide to Using the Case Studies Strategy Lenses The Exploring Strategy Website A Guide to Using the Video Material Exploring Strategy Teachers’Read More_x000C_Introduction to Statistics and Data Analysis355457 Words   |  1422 Pageseditorial board for Statistics: A Guide to the Unknown, 4th edition. 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Cameron UNIVERSITY OF MICHIGAN Prentice Hall Boston Columbus Indianapolis New York San Francisco Upper Saddle River Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montreal Toronto Delhi Mexico City Sao Paulo Sydney Hong Kong Seoul SingaporeRead MoreStephen P. Robbins Timothy A. Judge (2011) Organizational Behaviour 15th Edition New Jersey: Prentice Hall393164 Words   |  1573 Pagesand permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. To obtain permission(s) to use material from this work, please submit a written request to Pearson Education, Inc., Permissions Department, One Lake Street, Upper Saddle River, New Jersey 07458, or you may fax your request to 201-236-3290. Many of the designations by manufacturersRead MoreProject Mgmt296381 Words   |  1186 Pagestheir wives (Kevin and Dawn, Robert and Sally) and their children (Ryan, Carly, C onnor and Lauren). C.F.G. â€Å"We must not cease from exploration and the end of all exploring will be to arrive where we begin and to know the place for the first time.† T. S. Eliot To Ann whose love and support has brought out the best in me. And, to our girls Mary, Rachel, and Tor-Tor for the joy and pride they give me. Finally, to my muse, Neil, for the faith and inspiration he instills. E.W.L Preface Since youRead MoreContemporary Issues in Management Accounting211377 Words   |  846 Pages978–0–19–928336–1 (Pbk.) 1 3 5 7 9 10 8 6 4 2 3 FOREWORD ‘ Michael Bromwich is an exemplar of all that is good about the British tradition of academic accounting. Serious in intent, he has striven both to illuminate practice and to provide ways of improving it. 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Computational Efficiency of Polar Free Essays

string(127) " the n component multivariate normal with D = I , just take the components of Y to be independent univariate standard normals\." Lecture Notes on Monte Carlo Methods Fall Semester, 2005 Courant Institute of Mathematical Sciences, NYU Jonathan Goodman, goodman@cims. nyu. edu Chapter 2: Simple Sampling of Gaussians. We will write a custom essay sample on Computational Efficiency of Polar or any similar topic only for you Order Now created August 26, 2005 Generating univariate or multivariate Gaussian random variables is simple and fast. There should be no reason ever to use approximate methods based, for example, on the Central limit theorem. 1 Box Muller It would be nice to get a standard normal from a standard uniform by inverting the distribution function, but there is no closed form formula for this distribution 2 x unction N (x) = P (X x) = v1 ? e? x /2 dx . The Box Muller method is a 2 brilliant trick to overcome this by producing two independent standard normals from two independent uniforms. It is based on the familiar trick for calculating ? 2 e? x I= /2 dx . This cannot be calculated by â€Å"integration† – the inde? nite integral does not have an algebraic expression in terms of elementary functions (exponentials, logs, trig functions). However, ? 2 e? x I2 = ? /2 e? y dx 2 ? /2 ? 2 e? (x dy = +y 2 )/2 dxdy . The last integral can be calculated using polar coordinates x = r cos (? ), y = r sin(? with area element dxdy = rdrd? , so that 2? I2 = r = 0? e? r 2 /2 rdrd? = 2? r = 0? e? r 2 /2 rdr . ? =0 Unlike the original x integral, this r integral is elementary. The substitution s = r2 /2 gives ds = rdr and ? e? s ds = 2? . I 2 = 2? s=0 The Box Muller algorithm is a probabilistic interpretation of this trick. If (X, Y ) is a pair of independent standard normals, then the probability density is a product: 2 2 1 1 ? (x2 +y2 )/2 1 e . f (x, y ) = v e? x /2  · v e? y /2 = 2? 2? 2? 1 Since this density is radially symmetric, it is natural to consider the polar coordinate random variables (R, ? de? ned by 0 ? ? 2? and X = R cos(? ), and Y = R sin(? ). Clearly ? is uniformly distributed in the interval [0, 2? ] and may be sampled using ? = 2? U1 . Unlike the original distribution function N (x), there is a simple expression for the R distribution function: 2? r G(R) = P (R ? r) = r =0 ?=0 r 1 ? r 2 /2 e rdrd? = 2? e? r 2 /2 rdr . r =0 The same change of variable r 2 /2 = s, r dr = ds (so that r = r when s = r2 /2) allows us to calculate r 2 /2 e? s dx = 1 ? e? r G(r) = 2 /2 . s=0 Therefore, we may sample R by solving the distribution function equation1 G(R) = 1 ? e? R 2 /2 = 1 ? U2 , whose solution is R = ? 2 ln(U2 ). Altogether, the Box Muller method takes independent standard uniform random variables U1 and U2 and produces independent standard normals X and Y using the formulas ? = 2? U1 , R = ?2 ln(U2 ) , X = R cos(? ) , Y = R sin(? ) . (1) It may seem odd that X and Y in (13) are independent given that they use the same R and ?. Not only does our algebra shows that this is true, but we can test the independence computationally, and it will be con? rmed. Part of this method was generating a point â€Å"at random† on the unit circle. We suggested doing this by choosing ? niformly in the interval [0, 2? ] then taking the point on the circle to be (cos(? ), sin(? )). This has the possible drawback that the computer must evaluate the sine and cosine functions. Another way to do this2 is to choose a point uniformly in the 2 ? 2 square ? 1 ? x ? 1, 1 ? y ? 1 then rejecting it if it falls outside the unit circle. The ? rst accepted point will be uniformly distributed in the unit disk x2 + y 2 ? 1, so its angle will be random and uniformly distributed. The ? nal step is to get a point on the unit circle x2 + y 2 = 1 by dividing by the length. The methods have equal accuracy (both are exact in exact arithmetic). What distinguishes them is computer performance (a topic discussed more in a later lecture, hopefully). The rejection method, with an acceptance probability ? ? 4 78%, seems e? cient, but rejection can break the instruction pipeline and slow a computation by a factor of ten. Also, the square root needed to compute 1 Recall that 1 ? U2 is a standard uniform if U2 is. for example, in the dubious book Numerical Recipies. 2 Suggested, 2 the length may not be faster to evaluate than sine and cosine. Moreover, the rejection method uses two uniforms while the ? method uses just one. The method can be reversed to solve another sampling problem, generating a random point on the â€Å"unit spnere† in Rn . If we generate n independent standard normals, then the vector X = (X1 , . . . , Xn ) has all angles equally n likely (because the probability density is f (x) = v1 ? exp(? (x2 + ·  ·  ·+x2 )/2), n 1 2 which is radially symmetric. Therefore X/ X is uniformly distributed on the unit sphere, as desired. 1. 1 Other methods for univariate normals The Box Muller method is elegant and reasonably fast and is ? ne for casual omputations, but it may not be the best method for hard core users. Many software packages have native standard normal random number generators, which (if they are any good) use expertly optimized methods. There is very fast and accurate software on the web for directly inverting the normal distribution function N (x). This is particularly important for qua si Monte Carlo, which substitutes equidistributed sequences for random sequences (see a later lecture). 2 Multivariate normals An n component multivariate normal, X , is characterized by its mean  µ = E [X ] and its covariance matrix C = E [(X ?  µ)(X ?  µ)t ]. We discuss the problem of generating such an X with mean zero, since we achieve mean  µ by adding  µ to a mean zero multivariate normal. The key to generating such an X is the fact that if Y is an m component mean zero multivariate normal with covariance D and X = AY , then X is a mean zero multivariate normal with covariance t C = E X X t = E AY (AY ) = AE Y Y t At = ADAt . We know how to sample the n component multivariate normal with D = I , just take the components of Y to be independent univariate standard normals. You read "Computational Efficiency of Polar" in category "Essay examples" The formula X = AY will produce the desired covariance matrix if we ? nd A with AAt = C . A simple way to do this in practice is to use the Choleski decomposition from numerical linear algebra. This is a simple algorithm that produces a lower triangular matrix, L, so that LLt = C . It works for any positive de? nite C . In physical applications it is common that one has not C but its inverse, H . This would happen, for example, if X had the Gibbs-Boltzmann distribution with kT = 1 (it’s easy to change this) and energy 1 X t HX , and probability 2 1 density Z exp(? 1 X t HX ). In large scale physical problems it may be impracti2 cal to calculate and store the covariance matrix C = H ? though the Choleski factorization H = LLt is available. Note that3 H ? 1 = L? t L? 1 , so the choice 3 It is traditional to write L? t for the transpose of L? 1 , which also is the inverse of Lt . 3 A = L? t works. Computing X = L? t Y is the same as solving for X in the equation Y = Lt X , which is the process of back substitution in numerical linear algebra. In some applications one knows the eigenvectors of C (which also are the eigenvectors of H ), and the corresponding eigenvalues. These (either the eigenvectors or the eigenvectors and eigenvalues) sometimes are called principal com2 ponents. Let qj be the eigenvectors, normalized to be orthonormal, and ? j the corresponding eigenvalues of C , so that 2 Cqj = ? j qj , t qj qk = ? jk . t Denote the qj component of X by Zj = qj X . This is a linear function of X and t therefore Gaussian with mean zero. It’s variance (note: Zj = Zj = X t qj ) is 2 t t t 2 E [Zj ] = E [Zj  · Zj ] = qj E [XX t ]qj = qj Cqj = ? j . A similar calculation shows that Zj and Zk are uncorrelated and hence (as components of a multivariate normal) independent. Therefore, we can generate Yj as independent standard normals and sample the Zj using Zj = ? j Yj . (2) After that, we can get an X using Zj qj . X= (3) j =1 We restate this in matrix terms. Let Q be the orthogonal matrix whose columns are the orthonormal eigenvectors of C , and let ? 2 be the diagonal ma2 trix with ? j in the (j, j ) diagonal position. The eigenvalue/eigenvector relations are CQ = Q? 2 , Qt Q = I = QQt . (4) The multivariate normal vector Z = Qt X then has covariance m atrix E [ZZ t ] = E [Qt XX t Q] = Qt CQ = ? 2 . This says that the Zj , the components of Z , are 2 independent univariate normals with variances ? j . Therefore, we may sample Z by choosing its components by (14) and then reconstruct X by X = QZ , which s the same as (15). Alternatively, we can calculate, using (17) that t C = Q? 2 Qt = Q Qt = (Q? ) (Q? ) . Therefore A = Q? satis? es AAt = C and X = AY = Q? Y = QZ has covariance C if the components of Y are independent standard univariate normals or 2 the components of Z are independent univariate normals with variance ? j . 3 Brownian motion examples We illustrate these ideas for various kids of Brownian motion. Let X (t) be a Brownian motion path. Choose a ? nal time t and a time step ? t = T /n. The 4 observation times will be tj = j ? t and the observations (or observation values) will be Xj = X (tj ). These observations may be assembled into a vector X = (X1 , . . . , Xn )t . We seek to generate sample observation vectors (or observation paths). How we do this depends on the boundary conditions. The simplest case is standard Brownian motion. Specifying X (0) = 0 is a Dirichlet boundary condition at t = 0. Saying nothing about X (T ) is a free (or Neumann) condition at t = T . The joint probability density for the observation vector, f (x) = f (x1 , . . . , xn ), is found by multiplying the conditional densities. Given Xk = X (tk ), the next observation Xk+1 = X (tk + ? ) is Gaussian with mean Xk and variance ? t, so its conditional density is v 2 1 e? (xk+1 ? Xk ) /2? t . 2? ?t Multiply these together and use X0 = 0 and you ? nd (with the convention x0 = 0) f (x1 , . . . , xn ) = 3. 1 1 2? ?t n/2 exp ?1 2 ? Deltat n? 1 (xk+1 ? xk )2 . (5) k=0 The random walk method The simplest and possibly best way to generate a sample observation path, X , comes from the derivation of (1). First generate X1 = X (? t) as a mean zero v univariate normal with mean zero and variance ? t, i. e. X1 = ? tY1 . Given X1 , X2 is a univariate normal with mean X1 and variance ? , so we may v take X2 = X1 + ? tY2 , and so on. This is the random walk method. If you just want to make standard Brownian motion paths, stop here. We push on for pedigogical purposes and to develop strategies that apply to other types of Brownian motion. We describe the random walk method in terms of the matrices above, starting by identifying the matrices C and H . Examining (1) leads to ? 2 ? 1 0  ·Ã‚ ·Ã‚ · ? ? ? 1 2 ? 1 0  ·Ã‚ ·Ã‚ · ? ? .. .. .. . . . 1 ? 0 ? 1 ? H= ?. .. ?t ? . . 2 ? 1 ?. ? .. ? . ? 1 2 0  ·Ã‚ ·Ã‚ · 0 ? 1 ? 0 .? .? .? ? ? ? ? 0? ? ? ?1 ? 1 This is a tridiagonal matrix with pattern ? 1, 2, ? except at the bottom right corner. One can calculate the covariances Cjk from the random walk representation v Xk = ? t (Y1 +  ·  ·  · + Yk ) . 5 Since the Yj are independent, we have Ckk = var(Xk ) = ? t  · k  · var(Yj ) = tk , and, supposing j k , Cjk = E [Xj Xk ] = ? tE [((Y1 +  ·  ·  · + Yj ) + (Yj +1 +  ·  ·  · + Yk ))  · (Y1 +  ·  ·  · + Yj )] = 2 ?tE (Y1 +  ·  ·  · + Yj ) = tj . These combine into the familiar formula Cjk = cov(X (tj ), X (tk )) = min(tj , tk ) . This is the same as saying that the ? 1 ?1 ? ?. ?. C = ? t ? . ? ? ? 1 matrix C is 1  ·Ã‚ ·Ã‚ · 2 2  ·Ã‚ ·Ã‚ · 2 . . . 3  ·Ã‚ ·Ã‚ · . . . 2 3  ·Ã‚ ·Ã‚ · ? 1 2? ? ? 3? .? .? .? .. . (6) The random walk method for generating X may be expresses as ? ? ? Y ? X1 1 1 0  ·Ã‚ ·Ã‚ · 01 ? ? ? ?1 1 0  ·Ã‚ ·Ã‚ · 0 ? ? . ? ?.? ?.? v? ? . ? ?.? 1 0 . . ? . .? ? . ? = ? t ? 1 1 ? ? ? ? ?. . .. ? ? ? ?. . . .. ? ? ? ? 11 1  ·Ã‚ ·Ã‚ · 1 Yn Xn Thus, X = AY with ? ? 1 0  ·Ã‚ ·Ã‚ · 01 ?1 1 0  ·Ã‚ ·Ã‚ · 0 ? ? ? v? .? .? . ?1 1 1 0 .? A = ? t ? ?. . ? .. .. ?. . ? . 11 1  ·Ã‚ ·Ã‚ · 1 (7) The reader should do the matrix multiplication to check that indeed C = AAt for ( 6) and (7). Notice that H is a sparse matrix indicating short range interactions while C is full indicating long range correlations. This is true of in great number of physical applications, though it is rare to have an explicit formula for C . 6 We also can calculate the Choleski factorization of H . The reader can convince herself or himself that the Choleski factor, L, is bidiagonal, with nonzeros only on or immediately below the diagonal. However, the formulas are simpler if we reverse the order of the coordinates. Therefore we de? ne the coordinate reversed observation vector t X = (Xn , xn? 1 , . . . , Xn ) and whose covariance matrix is ? tn ? tn? 1 ? C=? . ?. . t1 tn? 1 tn? 1  ·  ·  · t1 t1 .. .  ·Ã‚ ·Ã‚ · ? ? ? , ? t1 and energy matrix ? 1 ? 1  ·Ã‚ ·Ã‚ · 0 ? 0 .? .? .? ? ? ?. ? 0? ? ? ?1 ? 2 ? ? ? 1 2 ? 1 0  ·Ã‚ ·Ã‚ · ? ? .. .. .. . . . 1 ? 0 ? 1 ? H= .. ?t ? . . ?. . 2 ? 1 ? ? .. ? . ? 1 2 0  ·Ã‚ ·Ã‚ · 0 ? 1 We seek the Choleski factorization H = LLt ? l1 0 ? m2 l2 1? L= v ? m3 ?t ? 0 ? . .. . . . with bidiagonal ?  ·Ã‚ ·Ã‚ · ? 0 ? ?. .. ? . ? .. . Multiplying out H = LLt leads to equatio ns that successively determine the lk and mk : 2 l1 l 1 m2 2 2 l1 + l 2 l 2 m3 = 1 =? l1 = 1 , = ? 1 =? m2 = ? 1 , = 2 =? l2 = 1 , = 1 =? m3 = ? 1 , etc. , The result is H = LLt with L simply ? 1 0  ·Ã‚ ·Ã‚ · ? ? 1 10 1? .. L= v ? . ?t ? ? 1 ? . .. .. . . . . 7 ? ? ? ?. ? ? The sampling algorithm using this Y = Lt X : ? ? ? 1 Yn ? Yn? 1 ? ? ? ? ?0 ? ? 1? ? ? ? ? . ?= v ? ?.? ?t ? ?.? ?. ? ? ?. . Y1 0 information is to ? nd X from Y by solving ?1 0 1 .. . ?1 .. .  ·Ã‚ ·Ã‚ ·  ·Ã‚ ·Ã‚ · .. . 0 0 Xn . ? ? Xn? 1 . . . 0 . . ?1 X1 1 ? ? ? ? ? ? ? ? ? Solving from the bottom up (back substitution), we have Y1 = Y2 = v 1 v X1 =? X1 = ? tY1 , ?t v 1 v (X2 ? X1 ) =? X2 = X1 + ? tY2 , etc. ?t This whole process turns out to give the same random walk sampling method. Had we not gone to the time reversed (X , etc. variables, we could have calculated the bidiagonal Choleski factor L numerically. This works for any problem with a tridiagonal energy matrix H and has a name in the cont rol theory/estimation literature that escapes me. In particular, it will allow to ? nd sample Brownian motion paths with other boundary conditions. 3. 2 The Brownian bridge construction The Brownian bridge construction is useful in the mathematical theory of Brownian motion. It also is the basis for the success of quasi Monte Carlo methods in ? nance. Suppose n is a power of 2: n = 2L . We will construct the observation path X through a sequence of L re? ements. First, notice that Xn is a univariate normal with mean zero and variance T , so we may take (with Yk,l being independent standard normals) v Xn = T Y1,1 . Given the value of Xn , the midoint observation, Xn/2 , is a univariate normal4 with mean 1 Xn and variance T /4, so we may take 2 Xn 2 v 1 T = Xn + Y2,1 . 2 2 At the ? rst level, we chose the endpoint value for X . We could draw a ? rst level path by connenting Xn to zero with a straight line. At the second level, or ? rst re? nement, we created a midpoint value. The seco nd level path could be piecewise linear, connecting 0 to X n to Xn . 4 We assign this and related claims below as exercises for the student. 8 The second re? nement level creates values for the â€Å"quarter points†. Given n X n , X n is a normal with mean 1 X n and variance 1 T . Similarly, X 34 is a 2 42 2 4 2 1 1T normal with mean 2 (X n + Xn ) and variance 4 2 . Therefore, we may take 2 Xn = 4 1 1 Xn + 22 2 T Y3,1 2 and n X 34 = 1 1 (X n + Xn ) + 2 2 2 T Y3,2 . 2 1 The level three path would be piecewise linear with breakpoints at 1 , 2 , and 3 . 4 4 Note that in each case we add a mean zero normal of the appropriate variance to the linear interpolation value. In the general step, we go from the level k ? 1 path to the level k paths by creating values for the midpoints of the level k ? 1 intervals. The level k observations are X j . The values with even j are known from the previous 2k? 1 level, so we need values for odd j . That is, we want to interpolate between the j = 2m value and the j = 2m + 2 value and add a mean zero normal of the appropriate variance: X (2m+1)n = 2k? 1 1 2 mn X 2k? 1 + X (2m+2)n 2 2k? 1 + 1 2(k? 2)/2 T Ym,k . 2 The reader should check that the vector of standard normals Y = (Y1,1 , Y2,1 , Y3,1 , Y3,2 , . . . t indeed has n = 2L components. The value of this method for quasi Monte Carlo comes from the fact that the most important values that determine the large scale structure of X are the ? rst components of Y . As we will see, the components of the Y vectors of quasi Monte Carlo have uneven quality, with the ? rst components being the best. 3. 3 Principle components The principle component eigenvalues and eigenve ctors for many types of Brownian motion are known in closed form. In many of these cases, the Fast Fourier Transform (FFT) algorithm leads to a reasonably fast sampling method. These FFT based methods are slower than random walk or Brownian bridge sampling for standard random walk, but they sometimes are the most e? cient for fractional Brownian motion. They may be better than Brownian bridge sampling with quasi Monte Carlo (I’m not sure about this). The eigenvectors of H are known5 to have components (qj,k is the k th component of eigenvector qj . ) qj,k = const  · sin(? j tk ) . 5 See e. g. Numerical Analysis by Eugene Isaacson and Herbert Keller. 9 (8) The n eigenvectors and eigenvalues then are determined by the allowed values of ? j , which, in turn, are determined throught the boundary conditions. We 2 2 can ? nd ? j in terms of ? j using the eigenvalue equation Hqj = ? j qj evaluated at any of the interior components 1 k n: 1 2 [? sin(? j (tk ? ?t)) + 2 sin(? j tk ) ? sin(? j (tk + ? t))] = ? j sin(? j tk ) . ?t Doing the math shown that the eigenvalue equation is satis? ed and that 2 ?j = 2 1 ? cos(? j ? t) . ?t (9) The eigenvalue equation also is satis? ed at k = 1 because the form (8) automatically satis? es the boundary condition qj,0 = 0. This is why we used the sine and not the cosine. Only special values ? j give qj,k that satisfy the eigenvalue equation at the right boundary point k = n. 10 How to cite Computational Efficiency of Polar, Essay examples

Larry Flint Essay Example For Students

Larry Flint Essay Larry FlyntInfamous pornographer and free-speech activist Larry Flynt has brought about controversy for nearly 30 years. As the editor of Hustler magazine, Flynt has publicized pornographic obscenities in several manors. By doing this, he has challenged Americas interpretation of the First Amendment, insisted that freedom of speech include obscenities and pornography, and made the anti-porn activists and feminists fight for constitutional protection from obscenity. Larry Flynt was born on November 1, 1942. Coming from a broken home, he later joined the military under false age. Flynt was discharged and after several unsuccessful jobs, went back to serve for five years on the U.S.S. Enterprise. After the Navy he moved to Dayton, Ohio where he bought a bar and turned it into a successful strip club. In the next year, Flynt opened similar clubs in 4 different Ohio cities and sent out newsletters to his clientele. Soon, Flynt set out to make his own mens magazine. In 1974 he released the first issue of Hustler magazine. Hustler was different from other pornographic magazines. It prided itself on hard core depictions of raw sex, which often included graphic nude photos of disabled, pregnant, and elderly women. One issue featured nude pictures of former First Lady Jacqueline Kennedy Onassis, while another depicted a woman being feed into a meat grinder.# Anti-porn activists and feminists started to speak out about the controversial issues behind the explicit Hustler magazine. This is a magazine bought by pubescent boys with untempered curiosity about sexual perversions, and by adults with insatiable curiosity about sexual perversions, adults who would read monthlies serializing the Marquis de Sade, if they were more literate.# Some people tend to disagree with the feminist views of such magazines as Hustler. Women have the right to be free from discrimination not only in the workplace and in the classroom but in the bedroom as well Women should not be seen as victims in their own sexual relations with men but as equally assertive partners, just as capable of experiencing sexual pleasure.# With this new hard-core, pornographic magazine out, much controversy came about on whether or not there should be any censorship to pornography. As soon as Hustler hit the stands, Flynt had been bombarded with lawsuits. He was charged with many counts of pandering obscenity and organized crime in May, 1976. The case was significant because it suggested that individual communities had the right to define obscenity.# These court cases again questioned our First Amendment rights for both sides of the controversy. Should we be constitutionally protected from obscenities? Feminists and anti-porn activists would say yes. To them, pornography is a type of media which degrades women and throws it in the faces of the public. What they are fighting for is that freedom of speech and press allow protection from obscenity in favor of the women and of the innocent eyes. Without this exploitation, published for profit, the male writer feels censored. The woman lynched naked on a tree, or restrained with ropes and a ball gag in her mouth, has what? Freedom of what?#Others would argue that freedom of speech and press should not have boundaries or limitations. They have a strong opposition to the censorship of pornography. Many feel pornography is portraying women as equals to men with their sexuality, while some feel pornography is an ex pression, like art. Larry Flynt, pornographers, and those who are anti-censorship feel as if their freedom of speech is being challenged. Flynt said, I realized that freedom of expression could never be taken for granted.# Over the years, the controversy has still been debated, but Flynt has set his sights on a recent headline. Since the Monica Lewinsky scandal, Flynt has sided with Clinton. He has offered up to $1 million to anyone able to provide evidence of illicit sexual relations with a congressman, senator or other prominent officeholder.# Flynt received over 2000 replies to his offer, in which two stories were of strong interest to him. His first story included Bob Livingston, a Republican in line for speaker of the house. Livingston resigned from his position, making a vague admission of marital infidelity, Livingston complained about having been Larry Flynted, but a full-on Flynting promises to be a far more excruciating experience one supported by embarrassingly specific evidence for which Flynt is paying nicely.# There are several opinions toward Flynts harsh tactics against politicians. Some see him as a pervert saint, sacrificing his freedom for our liberty.# The opposing believe that, there is no place in America for his despicable, brutish politics of blackmail.# Flynt has his own reasons for exploiting politicians like Livingston and defending Clinton. You can have a front-page story published about oral sex in the newspaper, but you put a photograph of two people making love out there and you can go to jail. That says a lot about a society that condones violence and condemns sex. So I think the problem is not just the politicians, its the country as a whole. Its got to come to grips with sexuality.# Nearly 30 years ago, Larry Flynt became the notorious pornography martyr, responsible for Americans questioning the meaning of the First Amendment. For years to come freedom of expression will be argued, should there or should there not be censors hip of pornography? Court case to court case might tell but, some believe that we will never agree on anything for sure and it may depend on what the case is directly ruling on. Majority rule only works if youre also considering individual rights. Because you cant have five wolves and one sheep voting on what to have for supper,# says Flynt. He believes that, Its not money, its not politicsits who controls the *censored* that controls the world.#Bibliography# Works CitedLarry Flynt, Biography.com, January 26, 2001, (January 29, 2001). .u314e526b3a4ca2a34607d77346ed5878 , .u314e526b3a4ca2a34607d77346ed5878 .postImageUrl , .u314e526b3a4ca2a34607d77346ed5878 .centered-text-area { min-height: 80px; position: relative; } .u314e526b3a4ca2a34607d77346ed5878 , .u314e526b3a4ca2a34607d77346ed5878:hover , .u314e526b3a4ca2a34607d77346ed5878:visited , .u314e526b3a4ca2a34607d77346ed5878:active { border:0!important; } .u314e526b3a4ca2a34607d77346ed5878 .clearfix:after { content: ""; display: table; clear: both; } .u314e526b3a4ca2a34607d77346ed5878 { display: block; transition: background-color 250ms; webkit-transition: background-color 250ms; width: 100%; opacity: 1; transition: opacity 250ms; webkit-transition: opacity 250ms; background-color: #95A5A6; } .u314e526b3a4ca2a34607d77346ed5878:active , .u314e526b3a4ca2a34607d77346ed5878:hover { opacity: 1; transition: opacity 250ms; webkit-transition: opacity 250ms; background-color: #2C3E50; } .u314e526b3a4ca2a34607d77346ed5878 .centered-text-area { width: 100%; position: relative ; } .u314e526b3a4ca2a34607d77346ed5878 .ctaText { border-bottom: 0 solid #fff; color: #2980B9; font-size: 16px; font-weight: bold; margin: 0; padding: 0; text-decoration: underline; } .u314e526b3a4ca2a34607d77346ed5878 .postTitle { color: #FFFFFF; font-size: 16px; font-weight: 600; margin: 0; padding: 0; width: 100%; } .u314e526b3a4ca2a34607d77346ed5878 .ctaButton { background-color: #7F8C8D!important; color: #2980B9; border: none; border-radius: 3px; box-shadow: none; font-size: 14px; font-weight: bold; line-height: 26px; moz-border-radius: 3px; text-align: center; text-decoration: none; text-shadow: none; width: 80px; min-height: 80px; background: url(https://artscolumbia.org/wp-content/plugins/intelly-related-posts/assets/images/simple-arrow.png)no-repeat; position: absolute; right: 0; top: 0; } .u314e526b3a4ca2a34607d77346ed5878:hover .ctaButton { background-color: #34495E!important; } .u314e526b3a4ca2a34607d77346ed5878 .centered-text { display: table; height: 80px; padding-left : 18px; top: 0; } .u314e526b3a4ca2a34607d77346ed5878 .u314e526b3a4ca2a34607d77346ed5878-content { display: table-cell; margin: 0; padding: 0; padding-right: 108px; position: relative; vertical-align: middle; width: 100%; } .u314e526b3a4ca2a34607d77346ed5878:after { content: ""; display: block; clear: both; } READ: African Americans In The South Essay Buckley Jr., William F., The Honored Guest, National Review, June 14, 1999, vol. 51, issue 11, 58. Strossen, Nadine, The Perils of Pornophobia, in Conversations, ed. John Selzer (Massachusetts: Allyn Bacon, 2000) 595. Dworkin, Andrea , Reply to John Irving, in Conversations, ed. John Selzer (Massachusetts: Allyn Bacon, 2000) 592. Richardson, John H. , Larry Flynt, Esquire, March 1999, 116. Strauss, Neil, Checking in with Larry Flynt, Rolling Stone, February 18, 1999, 41. Human Sexuality