Thursday, November 7, 2019
Free Essays on Philip Johnson
Philip Johnson The Amon Carter Museum in Fort Worth, Texas houses an impressive collection of works by Charles M. Russell and Frederic Remington and can actually be thought of as two buildings in one. The museum was originally built and designed by Philip Johnson in 1961 and forty years later Johnson was called upon again to design an addition to the building that would triple its exhibition space. Philip Johnson was born in 1906 and grew up in Cleveland, Ohio. He attended Harvard University majoring in philosophy and Greek not architecture. By the age of 26, Johnson was the founder and director of the Museum of Modern Art’s new architecture department in New York. There he cultivated an interest in architecture and co-authored the book The International Style, which introduced â€Å"Bauhaus,†a German style of architecture begun by Walter Gropius in 1918, to America (5). At the age of 34, Johnson returned to Harvard’s Graduate School of Design. After receiving his Master of Architecture in 1943, Johnson began designing his own home in New Canaan, Connecticut known as â€Å"The Glass House,†a house whose walls are made entirely of windows. Other well-known works include the Seagram’s Building and the AT&T Building both in New York. In 1979 Johnson was awarded the first Pritzker Architecture Prize for his 50 years of work and designs in the field of architecture (6). Undoubtedly, Johnson has been a great influence in architecture though some will argue that Johnson’s style is quite whimsical seeming to change as often as the wind blows (3). Johnson’s response? â€Å"After fifty years you shouldn’t be doing the same thing†(3). Though at times controversial it cannot be denied that Johnson is one of the most brilliant and renowned architects of the past century. 2 Interestingly, some of Johnson’s most acclaimed buildings are found in Texas. In Dallas alone, Johnson has designed ... Free Essays on Philip Johnson Free Essays on Philip Johnson Philip Johnson The Amon Carter Museum in Fort Worth, Texas houses an impressive collection of works by Charles M. Russell and Frederic Remington and can actually be thought of as two buildings in one. The museum was originally built and designed by Philip Johnson in 1961 and forty years later Johnson was called upon again to design an addition to the building that would triple its exhibition space. Philip Johnson was born in 1906 and grew up in Cleveland, Ohio. He attended Harvard University majoring in philosophy and Greek not architecture. By the age of 26, Johnson was the founder and director of the Museum of Modern Art’s new architecture department in New York. There he cultivated an interest in architecture and co-authored the book The International Style, which introduced â€Å"Bauhaus,†a German style of architecture begun by Walter Gropius in 1918, to America (5). At the age of 34, Johnson returned to Harvard’s Graduate School of Design. After receiving his Master of Architecture in 1943, Johnson began designing his own home in New Canaan, Connecticut known as â€Å"The Glass House,†a house whose walls are made entirely of windows. Other well-known works include the Seagram’s Building and the AT&T Building both in New York. In 1979 Johnson was awarded the first Pritzker Architecture Prize for his 50 years of work and designs in the field of architecture (6). Undoubtedly, Johnson has been a great influence in architecture though some will argue that Johnson’s style is quite whimsical seeming to change as often as the wind blows (3). Johnson’s response? â€Å"After fifty years you shouldn’t be doing the same thing†(3). Though at times controversial it cannot be denied that Johnson is one of the most brilliant and renowned architects of the past century. 2 Interestingly, some of Johnson’s most acclaimed buildings are found in Texas. In Dallas alone, Johnson has designed ...
Tuesday, November 5, 2019
ESPOSITO Surname Meaning and Origin
ESPOSITO Surname Meaning and Origin The common Italian surname Esposito was a last name commonly given to children in Italy (prior to its unification in 1861) who were abandoned or given up for adoption by their parents. The name derives from the Latin expositus, the past participle of the Latin verb exponere, which means to place outside. The Esposito surname is especially prevalent in the Naples region of Italy. Alternate Surname Spellings: ESPOSTI, ESPOSTO, ESPOSTI, DEGLI ESPOSTI, SPOSITO Surname Origin: Italian Famous People Raffaele Esposito is baker attributed with first creating the modern pizza. Genealogy Resources Contrary to what you may hear, there is no such thing as an Esposito family crest or coat of arms for the Esposito surname. Coats of arms are granted to individuals, not families, and may rightfully be used only by the uninterrupted male line descendants of the person to whom the coat of arms was originally granted. If youre interested in connecting with others who share the Esposito surname, the following resources can help: ESPOSITO Family Genealogy Forum: Free message board is focused on descendants of Esposito ancestors around the world.FamilySearch - ESPOSITO Genealogy and Family History: Explore over 350,000 digitized and transcribed historical records, as well as lineage-linked family trees for the Esposito surname.ESPOSITO Surname Mailing List: Free mailing list for researchers of the Esposito surname and its variations includes subscription details and a searchable archives of past messages.GeneaNet - Esposito Records: GeneaNet includes archival records, family trees, and other resources for individuals with the Esposito surname, with a concentration on records and families from France and other European countries.The Esposito Genealogy and Family Tree Page: Browse family trees and links to genealogical and historical records for individuals with the last name Esposito from the website of Genealogy Today. Sources: Cottle, Basil. Penguin Dictionary of Surnames. Baltimore, MD: Penguin Books, 1967.Dorward, David. Scottish Surnames. Collins Celtic (Pocket edition), 1998.Fucilla, Joseph. Our Italian Surnames. Genealogical Publishing Company, 2003.Hanks, Patrick and Flavia Hodges. A Dictionary of Surnames. Oxford University Press, 1989.Hanks, Patrick. Dictionary of American Family Names. Oxford University Press, 2003.Reaney, P.H. A Dictionary of English Surnames. Oxford University Press, 1997.Smith, Elsdon C. American Surnames. Genealogical Publishing Company, 1997.
Saturday, November 2, 2019
Economic of race and gender Assignment Example | Topics and Well Written Essays - 750 words
Economic of race and gender - Assignment Example These films could be brutal but sometimes, reality is worst than we would like it to be. The stories are just examples of what really happens in this world, whites discriminating Blacks or Asians and men acting like gods over women. What could be alarming is to know that there are also other skin colors discriminating the whites and more shockingly, women discriminating men. The films may have not shown the latter but in reality, things happen and it is not a news to break because there have been news of domestic violence also where the men are the victims. It is sad to realize that we can not escape these realities and they could happen to any of us. It is then very helpful to be educated about the issues on race and gender discrimination. Taking this class has opened my eyes and my understanding of the existence of such acts, now I know the rights of a person one needs to fight for, as well as what way of thinking one should have so as not to involve ones self in such actions. I believe that if I consider not discriminating, there would be one less racist in this world and no matter how minute its effect could be; still I could impact the world I am moving in. Diverse social positions can lay the foundations of a person’s political and economic access. For instance, a poor person can be deprived of education in his early years because he has to work in order to survive. Eventually, he will have minimal probabilities of ever going to school to get a better job in the future. Politically, he would have limited access to the rights of a regular paying worker if he ends up with menial jobs which do not give him the benefits of government employed workers. Loss of job pushes a person to self-employment particularly getting in to business because it gives advantages like being the boss of oneself, the opportunity for success and the convenience
Thursday, October 31, 2019
INTERNATIONAL STUDIES IN BUSINESS Research Paper
INTERNATIONAL STUDIES IN BUSINESS - Research Paper Example In addition to this, the study will try to discuss about the identified issues that are affecting the growth of the country. Based on these facts and issues, finally the study will recommend in favor of a particular business plan that can be run successfully in Morocco. Present Morocco’s area has been occupied since the Paleolithic times. The Maghreb was fertile than comparing to today during the Upper Paleolithic times. However, the Aterian was highly succeeded by the popular Iberomaurusian culture. This specific cultured had the similarity with the Iberian cultures. In fact the Iberomaurusian culture was highly succeeded by the popular Beaker culture in the Kingdom of Morocco. Slowly and gradually, Morocco and North Africa were drawn into the emerging Mediterranean world. It happened due to the initiatives and activities of Phoenicians. He actually made settlements between the established trading colonies during the classical period. Since then, Morocco passed several eras, such as early Islamic era, era of Berber dynasties, era of Sharifian dynasties, era of Spanish and French protectorates during the First World War, King Hassan II’s reign and King Mohammed VI’s Reign. Despite the understanding and deep reforms made by King Mohammed VI which answered maximum concerns created by international community, the demonstrators continued to raise their voice for greater reforms (Murphy, 2010). Finally, the government failed to deliver all the reforms that were promised in the year 2012. Berber and Arabic are the official languages of Morocco. Moroccan Arabic Dialects’ distinctive group is generally referred as Darija. Near about 90 percent of total population in Morocco speaks in Moroccan Arabic. On the other hand, Berber language is generally spoken in three different dialects, such as Central Atlas Tamazight, Tashelhit and Traffit. French and English are other languages that have been practiced in Morocco (Njoku,
Tuesday, October 29, 2019
Critical review of Environmental Statements Literature
Critical of Environmental Statements - Literature review Example A few of the key factors of the EMAS would include openness, periodic provision of the environmental information and transparency. These factors are also vital for the companies to build a confident relationship with the interested entities (EMAS, n.d.). Methodology In the entire evaluation process a primarily qualitative approach will be used. It would intend to evaluate the EMAS environmental statement of Park Communication based on certain benchmarks such as company performances in accordance with EMAS regulations, core indicators of company performances and the organisational goals which is expected to render an in-depth understanding of the environmental compliance practices by Park Communication. Furthermore, using statement checklist the report would present an accompanying statement of conformance that would justify the evaluation. The aspect related to the conformance of environmental statement with the prescribed guidelines will be evaluated from the EMAS environmental stat ement of Park Communications. Aims This report aims to conduct a critical evaluation of an environmental statement published in accordance with the EMAS regulation. The report would be presented to the Environmental Manager of an organisation i.e. Park Communications. Evaluating Environmental Statement Published In Accordance With the EMAS Regulation This report reviews an EMAS environmental statement of an organisation i.e. Park Communications. In this regard, the report considers the EMAS environmental statement of Park Communications, which sets the objectives for the year 2011. Park Communications is a United Kingdom based organisation which provides ‘one-point-of-contact’ services to translate, distribute and store literature of the people. The company also performs functions such as print reports and accounts, marketing literature, financial prospectuses and government publications among others (Park Communications, n.d.). It has been observed that the company has been quite committed towards the employees, customers, local environment and global environment along with taking every step by considering all these aspects. The company is aware of the fact that whatever it does would certainly have an effect upon the environment, directly as well as indirectly. In this regard, the company has taken a step further and introduced the system of EMAS in the organisation. Eco-Management and Audit Scheme (EMAS) is an initiative designed to enhance the environmental activities of companies. It further aims to identify and reward organisations to take a step further to improve its performance. According to this scheme, companies would have to present reports publicly on a regular basis that could show how they are improving their environmental activities (Park Communications, 2011). Park Communications aims at managing its business in a way that would have less effect on the environment. It has been obse
Sunday, October 27, 2019
Static Analysis of Uncertain Structures
Static Analysis of Uncertain Structures Static Analysis of Uncertain Structures Using Interval Eigenvalue Decomposition 1Mehdi Modares and 2Robert L. Mullen 1Department of Civil and Environmental Engineering Tufts University Medford, MA, 02155 2Department of Civil Engineering Case Western Reserve University Cleveland, OH, 44106 Abstract: Static analysis is an essential procedure to design a structure. Using static analysis, the structures response to the applied external forces is obtained. This response includes internal forces/moments and internal stresses that is used in the design process. However, the mechanical characteristics of the structure possess uncertainties which alter the structures response. One method to quantify the presence of these uncertainties is interval or unknown-but-bounded variables. In this work a new method is developed to obtain the bounds on structures static response using interval eigenvalue decomposition of the stiffness matrix. The bounds of eigenvalues are obtained using monotonic behavior of eigenvalues for a symmetric matrix subjected to non-negative definite perturbations. Moreover, the bounds of eigenvectors are obtained using perturbation of invariant subspaces for symmetric matrices. Comparisons with other interval finite element solution methods are presented. Using this method, it has shown that obtaining the bound on static response of an uncertain structure does not require a combinatorial or Monte-Carlo simulation procedure. Keywords: Statics, Analysis, Interval, Uncertainty  © 2008 by authors. Printed in USA. REC 2008 Modares and Mullen In design of structures, the performance of the structure must be guaranteed over its lifetime. Moreover, static analysis is a fundamental procedure for designing reliable structure that are subjected to static or quasi-static forces induced by various loading conditions and patterns. However, in current procedures for static analysis of structural systems, the existence of uncertainty in either mechanical properties of the system or the characteristics of forcing function is generally not considered. These uncertainties can be attributed to physical imperfections, modeling inaccuracies and system complexities. Although, in a design process, uncertainty is accounted for by a combination of load amplification and strength reduction factors that are based on probabilistic models of historic data, consideration of the effects of uncertainty has been removed from current static analysis of structural systems. In this work, a new method is developed to perform static analysis of a structural system in the presence of uncertainty in the systems mechanical properties as well as uncertainty in the magnitude of loads. The presence of these uncertainties is quantified using interval or unknownbut-bounded variables. This method obtains the bounds on structures static response using interval eigenvalue decomposition of the stiffness matrix. The bounds of eigenvalues are obtained using the concept of monotonic behavior of eigenvalues for a symmetric matrix subjected to non-negative definite perturbations. Furthermore, the bounds of eigenvectors are obtained using perturbation of invariant subspaces for symmetric matrices. Using this method, it has shown that obtaining the bound on static response of an uncertain structure does not require a combinatorial or MonteCarlo simulation procedure. The equation of equilibrium for a multiple degree of freedom structure is defined as a linear system of equations as:  [K]{U}={P}          (1) where, [K]is the stiffness matrix, {U}is the vector of unknown nodal displacements, and {P} is the vector of nodal forces. The solution to this system of equation is:  {U} = [K]−1{P}          (2) The concept of interval numbers has been originally applied in the error analysis associated with digital computing. Quantification of the uncertainties introduced by truncation of real numbers in numerical methods was the primary application of interval methods (Moore 1966). A real interval is a closed set defined by extreme values as (Figure 1): ~l ,zu ] ={z∈â„Å"| zl ≠¤ z ≠¤ zu} (3)  Z = [z ~ x = [a,b] Figure 1. An interval variable. In this work, the symbol (~) represents an interval quantity. One interpretation of an interval number is a random variable whose probability density function is unknown but non-zero only in the range of interval. Another interpretation of an interval number includes intervals of confidence for ÃŽ ±-cuts of fuzzy sets. The interval representation transforms the point values in the deterministic system to inclusive set values in the system with bounded uncertainty. Considering the presence of interval uncertainty in stiffness and force properties, the system of equilibrium equations, Eq.(1), is modified as an interval system of equilibrium equation as: ~~  [K]{U}={P}          (4) ~ where, [K]is the interval stiffness matrix, {U}is the vector of unknown nodal displacements, and {P} is the vector of interval nodal forces. In development of interval stiffness matrix, the physical and mathematical characteristics of the stiffness matrix must be preserves. This system of interval equations is mainly solved using computationally iterative procedures (Muhanna et al 2007) and (Neumaier and Pownuk 2007). The present method proposes a computationally efficient procedure with nearly sharp results using interval eigenvalue decomposition of stiffness matrix. While the external force can also have uncertainties, in this work only problems with interval stiffness properties are addressed. However, for functional independent variations for both stiffness matrix and external force vector, the extension of the proposed work is straightforward. 3.1. DETERMINISTIC EIGENVALUE DECOMPOSITION The deterministic symmetric stiffness matrix can be decomposed using matrix eigenvalue decomposition as:  [K] = [ÃŽ ¦][Λ][ÃŽ ¦]T          (5) where, [ÃŽ ¦] is the matrix of eigenvectors, and [Λ] is the diagonal matrix of eigenvalues. Equivalently, N  [K] =∑Î »i{à †¢i}{à †¢i}T          (6) i=1 where, the values of ÃŽ »i is the eigenvalues and the vectors{à †¢i}are their corresponding eigenvectors. Therefore, the eigenvalue decomposition of the inverse of the stiffness matrix is: equivalently, [K]−1 =[ÃŽ ¦][Λ]−1[ÃŽ ¦]T          (7) −N 1T [K] 1 =∑ {à †¢i}{à †¢i}         (8) i=1 ÃŽ »i Substituting Eq.(8) in the solution for the deterministic linear system of equation, Eq.(2), the solution for response is shown as:  {U}= ( N 1 {à †¢i}{à †¢i}T ){P}          (9) 3.2. INTERVAL EIGENVALUE DECOMPOSITION Similarly, the solution to interval system of equilibrium equations, Eq.(4), is:  {U~}= (∑N ~1 {à †¢~ }{à †¢~i}T ){P}          (10) i i=1 ÃŽ »i ~~ } are their where, the values of ÃŽ »i is the interval eigenvalues and, the vectors {à †¢i corresponding interval eigenvectors that are to be determined. 4.1. BACKGROUND The research in interval eigenvalue problem began to emerge as its applicability in science and engineering was realized. Hollot and Bartlett (1987) studied the spectra of eigenvalues of an interval matrix family which are found to depend on the spectrum of its extreme sets. Dief (1991) presented a method for computing interval eigenvalues of an interval matrix based on an assumption of invariance properties of eigenvectors. In structural dynamics, Modares and Mullen (2004) have introduced a method for the solution of the interval eigenvalue problem which determines the exact bounds of the natural frequencies of a system using Interval Finite Element formulation. 4.2. DEFINITION The eigenvalue problems for matrices containing interval values are known as the interval ~ ~ nn ) and [A] is a member of the eigenvalue problems. If [A] is an interval real matrix (A∈â„Å" ~ interval matrix ([A]∈[A]) , the interval eigenvalue problem is shown as: ~ 4.2.1. Solution for Eigenvalues The solution of interest to the real interval eigenvalue problem for bounds on each eigenvalue is ~ defined as an inclusive set of real values (ÃŽ ») such that for any member of the interval matrix, the eigenvalue solution to the problem is a member of the solution set. Therefore, the solution to the interval eigenvalue problem for each eigenvalue can be mathematically expressed as: ~l ,ÃŽ »u ]|∀[A]∈[A~]: ([A]−Î »[I]){x} = 0}              (12)  {ÃŽ »Ã¢Ë†Ë†ÃŽ »= [ÃŽ » 4.2.2. Solution for Eigenvectors: The solution of interest to the real interval eigenvalue problem for bounds on each eigenvector is defined as an inclusive set of real values of vector {~x} such that for any member of the interval matrix, the eigenvector solution to the problem is a member of the solution set. Thus, the solution to the interval eigenvalue problem for each eigenvector is: 4.3. INTERVAL STIFFNESS MATRIX The systems global stiffness can be viewed as a summation of the element contributions to the global stiffness matrix: n i=1 where [ Li ] is the element Boolean connectivity matrix and [Ki ] is the element stiffness matrix in the global coordinate system. Considering the presence of uncertainty in the stiffness properties, the non-deterministic element elastic stiffness matrix is expressed as: ~ in which, [li ,ui ] is an interval number that pre-multiplies the deterministic element stiffness matrix. This procedure preserves the physical and mathematical characteristics of the stiffness matrix. Therefore, the systems global stiffness matrix in the presence of any uncertainty is the linear summation of the contributions of non-deterministic interval element stiffness matrices: ,ui ])[Li ][Ki ][Li ] =∑ i=1i=1 in which, [Ki ] is the deterministic element elastic stiffness contribution to the global stiffness matrix. 4.4. INTERVAL EIGENVALUE PROBLEM FOR STATICS The interval eigenvalue problem for a structure with stiffness properties expressed as interval values is:  [K~]{à †¢~} = (ÃŽ »~){à †¢~} (17) Substituting Eq.(16) in Eq.(17): ]){à †¢} = (ÃŽ »){à †¢ i=1 This interval eigenvalue problem can be transformed to a pseudo-deterministic eigenvalue problem subjected to a matrix perturbation. Introducing the central and radial (perturbation) stiffness matrices as: i 1 [K~R ] =∑i=n1 (ÃŽ µi )(ui 2−li )[Ki ]  ,   Î µi =[−1,1]             (20) Using Eqs. (19,20), the non-deterministic interval eigenpair problem, Eq.(18), becomes: Hence, the determination of bounds on eigenvalues and bounds on eigenvectors of a stiffness matrix in the presence of uncertainty is mathematically interpreted as an eigenvalue problem on a ~ central stiffness matrix ([KC ]) that is subjected to a radial perturbation stiffness matrix ([KR ]). This perturbation is in fact, a linear summation of non-negative definite deterministic element stiffness contribution matrices that are scaled with bounded real numbers(ÃŽ µi ) . 5. Solution 5.1. BOUNDS ON EIGENVALUES The following concepts must be considered in order to bound the non-deterministic interval eigenvalue problem, Eq.(21). The classical linear eigenpair problem for a symmetric matrix is: with the solution of real eigenvalues (ÃŽ »1 ≠¤ÃŽ »2 ≠¤ ≠¤ÃŽ »n ) and corresponding eigenvectors ( x1, x2,, xn ). This equation can be transformed into a ratio of quadratics known as the Rayleigh quotient:  R(x) =                     (23) The Rayleigh quotient for a symmetric matrix is bounded between the smallest and the largest eigenvalues (Bellman 1960 and Strang 1976).  (24) Thus, the first eigenvalue (ÃŽ »1) can be obtained by performing an unconstrained minimization on the scalar-valued function of Rayleigh quotient: ( (25) x∈ For finding the next eigenvalues, the concept of maximin characterization can be used. This concept obtains the kth eigenvalue by imposing (k-1) constraints on the minimization of the Rayleigh quotient (Bellman 1960 and Strang 1976): ÃŽ »k = max[minR(x)]  (subject to constrains(xT zi = 0),i =1,k −1,k ≠¥ 2 ) (26) 5.1.1. Bounding the Eigenvalues for Statics Using the concepts of minimum and maximin characterizations of eigenvalues for symmetric matrices, the solution to the interval eigenvalue problem for the eigenvalues of a system with uncertainty in the stiffness characteristics (Eq.(21)) for the first eigenvalue can be shown as: n x∈Rn{x}T {x} for the next eigenvalues: ~{x}T [K~]{x}{x}T ([K ]+[K~ ]){x} 5.1.2. Deterministic Eigenvalue Problems for Bounding Eigenvalues in Statics Substituting and expanding the right-hand side terms of Eqs. (27,28): ~T [K ]{x}~ui (li +u{x} (29) Since the matrix [Ki ] is non-negative definite, the term () is non-negative. Therefore, using the monotonic behavior of eigenvalues for symmetric matrices, the upper bounds on the eigenvalues in Eqs.(19,20) are obtained by considering maximum values of interval coefficients of uncertainty (ÃŽ µ~i = [−1,1]), ((ÃŽ µi )max = 1), for all elements in the radial perturbation matrix. Similarly, the lower bounds on the eigenvalues are obtained by considering minimum values of those coefficients, ((ÃŽ µi )min =−1) , for all elements in the radial perturbation matrix. Also, it can be observed that any other element stiffness selected from the interval set will yield eigenvalues between the upper and lower bounds. This imonotonic behavior of eigenvalues can also be used for parameterization purposes. Using these concepts, the deterministic eigenvalue problems corresponding to the maximum and minimum eigenvalues are obtained (Modares and Mullen 2004) as: n n 5.2. BOUNDS ON EIGENVECTORS 5.2.1. Invariant Subspace The subspace Ç is defined to be an invariant subspace of matrix [A] if:  AÇ⊂Ç (32) Equivalently, if Ç is an invariant subspace of [A]nn and also, columns of [X1]nm form a basis forÇ, then there is a unique matrix [L1]mm such that: The matrix [L1 ] is the representation of [A] on Ç with respect to the basis [X1] and the eigenvalues of [L1] are a subset of eigenvalues of [A]. Therefore, for the invariant subspace, ({v},ÃŽ ») is an eigenpair of [L1] if and only if ({[X1]{v}},ÃŽ ») is an eigenpair of [A]. 5.2.2. Theorem of Invariant Subspaces For a real symmetric matrix [A], considering the subspace Ç with the linearly independent columns of [X1] forming a basis for Ç and the linearly independent columns of [X2] spanning the complementary subspace Ç⊥ , then, Ç is an invariant subspace of [A] iff: Therefore, invoking this condition and postulating the definition of invariant subspaces, the symmetric matrix [A] can be reduced to a diagonalized form using a unitary similarity transformation as:  [X1X2]T [A][X1X2] = à ¢Ã… ½Ã‚ ¢Ãƒ ¢Ã… ½Ã‚ ¡[X1]TT[[AA][][XX11]] à ¢Ã… ½Ã‚ £[X2] where [Li ] =[Xi ]T [A][Xi ], i =1,2. 5.2.3. Simple Invariant Subspace [X1]T [A][X2]à ¢Ã… ½Ã‚ ¤ à ¢Ã… ½Ã‚ ¡[L1] [X2]T [A][X2]à ¢Ã… ½Ã‚ ¥Ãƒ ¢Ã… ½Ã‚ ¦= à ¢Ã… ½Ã‚ ¢Ãƒ ¢Ã… ½Ã‚ £[0] [0] à ¢Ã… ½Ã‚ ¤ [L2]à ¢Ã… ½Ã‚ ¥Ãƒ ¢Ã… ½Ã‚ ¦ (35) An invariant subspace is simple if the eigenvalues of its representation [L1] are distinct from other eigenvalues of [A]. Thus, using the reduced form of [A] with respect to the unitary matrix [[X1][X2]], Ç is a simple invariant subspace if the eigenvalues of [L1] and [L2] are distinct: 5.2.4. Perturbed Eigenvector Considering the column spaces of [X1] and [X2] to span two complementary simple invariant subspaces, the perturbed orthogonal subspaces are defined as:  [Xˆ1] =[X1]+[X 2 ][P] (37)  [Xˆ2 ] =[X 2]−[X1][P]T (38) in which [P] is a matrix to be determined. Thus, each perturbed subspace is defined as a summation of the exact subspace and the contribution of the complementary subspace. Considering a symmetric perturbation[E] , the perturbed matrix is defined as: Applying the theorem of invariant subspaces for perturbed matrix and perturbed subspaces, and linearizing due to a small perturbation compared to the unperturbed matrix, Eq.(34) is rewritten as: This perturbation problem is an equation for unknown [P] in the form of a Sylvesters equation in which, the uniqueness of the solution is guaranteed by the existence of simple perturbed invariant subspaces. Finally, specializing the result for one eigenvector and solving the above equation, the perturbed eigenvector is (Stewart and Sun 1990):  {xˆ1} = {x1}+[X 2 ](ÃŽ »1[I]−[L2 ])−1[X 2 ]T [E]{x1} 5.2.5 Bounding Eigenvectors for Statics For the perturbed eigenvalue problem for statics, Eq.(21), the error matrix is: (41) ~nu [E] = [KR ] = (∑(ÃŽ µi )( i − li )[Ki ]) (42) i=12 Using the error matrix in eigenvector perturbation equation for the first eigenvector, Eq.(33) the perturbed eigenvector is: in which, {à †¢1}is the first eigenvector, (ÃŽ »1) is the first eigenvalue, [ÃŽ ¦2 ] is the matrix of remaining eigenvectors and [Λ2 ] is the diagonal matrix of remaining eigenvalues obtained from the deterministic eigenvalue problem. Eq.(30,31 and 43) is used to calculate the bounds on interval eigenvalues and interval eigenvectors in the response equation, Eq.(9). In order to attain sharper results, the functional dependency of intervals in direct interval multiplications in Eq.(9) is considered. Also, input intervals are subdivided and the union of responses of subset results is obtained. 6. Numerical Example Problem The bounds on the static response for a 2-D statically indeterminate truss with interval uncertainty present in the modulus of elasticity of each element are determined (Figure 2). The crosssectional area A, the length for horizontal and vertical members L , the Youngs moduli E for all ~ elements are E = ([0.99,1.01])E . Figure 2. The structure of 2-D truss The problem is solved using the method presented in this work. The functional dependency of intervals in the response equation is considered. A hundred-segment subdivision of input intervals is performed and the union of responses is obtained. For comparison, an exact combinatorial analysis has performed which considers lower and upper values of uncertainty for each element i.e. solving (2n = 210 =1024 ) deterministic problems. The static analysis results obtained by the present method and the brute force combination solution for the vertical displacement of the top nodes in are summarized Table (1). Lower Bound Present Method Lower Bound Combination Method Upper Bound Combination Method Upper Bound Present Method Error % U à ¢Ã… ½Ã¢â‚¬ º PL à ¢Ã… ½Ã… ¾ à ¢Ã… ½Ã…“à ¢Ã… ½Ã… ¸ à ¢Ã… ½Ã‚ AE à ¢Ã… ½Ã‚ -1.6265 -1.6244 -1.5859 -1.5838 % 0.12 Table1. Bounds on Vertical Displacement of Top Nodes The results show that the proposed robust method yields nearly sharp results in a computationally efficient manner as well as preserving the systems physics. 4.Conclusions A finite-element based method for static analysis of structural systems with interval uncertainty in mechanical properties is presented. This method proposes an interval eigenvalue decomposition of stiffness matrix. By obtaining the exact bounds on the eigenvalues and nearly sharp bounds on the eigenvectors, the proposed method is capable to obtain the nearly sharp bounds on the structures static response. Some conservative overestimation in response occurs that can be attributed to the linearization in formation of bounds of eigenvectors and also, the functional dependency of intervals in the dynamic response formulation. This method is computationally feasible and it shows that the bounds on the static response can be obtained without combinatorial or Monte-Carlo simulation procedures. This computational efficiency of the proposed method makes it attractive to introduce uncertainty into structural static analysis and design. While this methodology is shown for structural systems, its extension to various mechanics problems is straightforward. References Bellman, R. Introduction to Matrix Analysis, McGraw-Hill, New York 1960. Dief, A., Advanced Matrix theory for Scientists and Engineers, pp.262-281. Abacus Press 1991. Hollot, C. and A. Bartlett. On the eigenvalues of interval matrices, Technical Report, Department of Electrical and Computer Engineering, University of Massachusetts, Amherst, MA 1987. Modares, M. and R. L. Mullen. Free Vibration of Structures with Interval Uncertainty. 9th ASCE Specialty Conference on Probabilistic Mechanics and Structural Reliability 2004. Moore, R. E. Interval Analysis. Prentice Hall, Englewood, NJ 1966. Muhanna, R. L. and R. L. Mullen. Uncertainty in Mechanics Problems-Interval-Based Approach. Journal of Engineering Mechanics June-2001, pp.557-566 2001. Muhanna, R. L., Zhang H. and R. L. Mullen. Interval Finite Element as a Basis for Generalized Models of Uncertainty in Engineering Mechanics, Reliable Computing, Vol. 13, pp. 173-194, 2007. Neumaier, A. Interval Methods for Systems of Equations. Cambridge University Press, Cambridge 1990. Neumaier, A. and A. Pownuk. Linear Systems with Large Uncertainties, with Applications to Truss Structures, Reliable Computing, Vol. 13, pp. 149-172, 2007. Strang, G. Linear Algebra and its Applications, Massachusetts Institute of Technology, 1976. Stewart, G.W. and J. Sun. Matrix perturbation theory, Chapter 5. Academic Press, Boston, MA 1990.
Friday, October 25, 2019
The Power of Carvers Little Things Essay examples -- Carver Little Th
The Power of Carver's Little Things    To a reader unfamiliar with his work, Raymond Carver's short story, "Little Things" may seem devoid of all literary devices owning to good writing. Fortunately, these people are mistaken. With his minimalistic style, it is what Carver doesn't write that makes his work so effective. Most of Carver's short stories describe situations that many people could find themselves in and that is why his work is so appealing to readers. They are not restricted to harsh explicative details or over-dramatized language, but are allowed to create their own rationale for the actions of the characters and the consequent results.     "Little Things" begins with an explanation of the setting when Carver writes, "Cars slushed by on the streets outside, where it was getting dark. But it was getting dark on the inside too." This is the most descriptive passage in the entire story, which is only one-and-a-half pages in length, and it serves to set a mood of bleakness and animosity between the characters as well as the remainder of the piece. A scene follows in which a man is packing a suitcase and a woman is telling him she is glad he is leaving. Carver goes so far as to omit the characters' names, allowing the reader to more clearly identify with their struggle. One of the major turning points is a dozen lines into the story when the woman notices a picture of their baby and remembers it, forsaken, in the living room. The reader is compelled to ask if she had not remembered the baby at that moment would the rest of the scene have progressed in the same way? The man then follows the woman into the living room and tells her he wants the baby. This she can not allow as she turns away f... ...his own struggle with alcohol and personal strife. Regardless of the origin of the words, they force the reader to take a compelling look into his own life. Carson said that, in many ways, Carver's life was a model for all of his characters. But Carver forces readers to use their own lives as the foundation of the interpretation of the story, allowing them to relate to the characters and events themselves.  Works Cited Carson, Phillip. "Carver's Vision". (200). Online. Internet. 12 Feb. 2003. Available: world.std.com/~ptc/carver-paper.html Carver, Raymond. Where I'm calling From. "Little things." (1988, Atlantic Monthly Press). 114. Hashimoto, Hiromi. "Trying to Understand Raymond carver's Revisions." Tokai English Review. (Dec. 1995). Online. Internet. 12 Feb. 2003. Available:      people.whitman.edu/~lucetb/carver/precision.html The Power of Carver's Little Things Essay examples -- Carver Little Th The Power of Carver's Little Things    To a reader unfamiliar with his work, Raymond Carver's short story, "Little Things" may seem devoid of all literary devices owning to good writing. Fortunately, these people are mistaken. With his minimalistic style, it is what Carver doesn't write that makes his work so effective. Most of Carver's short stories describe situations that many people could find themselves in and that is why his work is so appealing to readers. They are not restricted to harsh explicative details or over-dramatized language, but are allowed to create their own rationale for the actions of the characters and the consequent results.     "Little Things" begins with an explanation of the setting when Carver writes, "Cars slushed by on the streets outside, where it was getting dark. But it was getting dark on the inside too." This is the most descriptive passage in the entire story, which is only one-and-a-half pages in length, and it serves to set a mood of bleakness and animosity between the characters as well as the remainder of the piece. A scene follows in which a man is packing a suitcase and a woman is telling him she is glad he is leaving. Carver goes so far as to omit the characters' names, allowing the reader to more clearly identify with their struggle. One of the major turning points is a dozen lines into the story when the woman notices a picture of their baby and remembers it, forsaken, in the living room. The reader is compelled to ask if she had not remembered the baby at that moment would the rest of the scene have progressed in the same way? The man then follows the woman into the living room and tells her he wants the baby. This she can not allow as she turns away f... ...his own struggle with alcohol and personal strife. Regardless of the origin of the words, they force the reader to take a compelling look into his own life. Carson said that, in many ways, Carver's life was a model for all of his characters. But Carver forces readers to use their own lives as the foundation of the interpretation of the story, allowing them to relate to the characters and events themselves.  Works Cited Carson, Phillip. "Carver's Vision". (200). Online. Internet. 12 Feb. 2003. Available: world.std.com/~ptc/carver-paper.html Carver, Raymond. Where I'm calling From. "Little things." (1988, Atlantic Monthly Press). 114. Hashimoto, Hiromi. "Trying to Understand Raymond carver's Revisions." Tokai English Review. (Dec. 1995). Online. Internet. 12 Feb. 2003. Available:      people.whitman.edu/~lucetb/carver/precision.html
Subscribe to:
Posts (Atom)