Time Value of Money and Present Value Essay Example

Time Value of Money and Present Value Paper Date: 14/11/2012 52. Annuities: You are saving for the college education of your two children. They are two years apart in age; one will begin college 15 years from today and the other will begin 17 years from today. You estimate your children’s college expenses to be $23,000 per year per child, payable at the beginning of each school year. The annual interest rate is 5. 5 percent. How much money must you deposit in account each year to fund your children’s education? Your deposits begin one year from today. You will make your last deposit when your oldest child enters college. Assume four years of college Solution: Cost of 1 year at university = 23,000 N=4 I=5. 5% PMT=23,000 CPT PV = 80,618. 45 For the first child the PV = 80,618. 45/ (1. 055) ^14 = $38,097. 81 For the second child the PV = 80,618. 45/ (1. 055) ^16 = $34,229. 07 Therefore the total cost today of your children’s college expense will be the addition of the 2 = $72,326. 88 This is the present value of my annual savings, which are an annuity, so to get the amount I am supposed to save each year would be: PV=72,326. 88 N=15 I=5. 5 CPT PMT = 7,205. 6 57. Calculating Annuity Values: Bilbo Baggins wants to save money to meet three objectives. First, he would like to be able to retire 30 years from now with retirement income of $25,000 per month for 20 years, with the first payment received 30 years and 1 month from now. Second, he would like to purchase a cabin in Rivendell in 10 years at an estimated cost of $350,000. Third, after he passes on at the end of the 20 years of withdrawals, he would like to leave an inheritance of $750,000 to his nephew Frodo. He can afford to save $2,100 per month for the next 10 years. If he can earn an 11 percent EAR before he retires and an 8 percent EAR after he retires, how much will he have to save each month in years 11 through 30? Solution: We will write a custom essay sample on Time Value of Money and Present Value specifically for you for only $16.38 $13.9/page Order now We will write a custom essay sample on Time Value of Money and Present Value specifically for you FOR ONLY $16.38 $13.9/page Hire Writer We will write a custom essay sample on Time Value of Money and Present Value specifically for you FOR ONLY $16.38 $13.9/page Hire Writer First we get the FV of the 2,100 savings over 10 years Bilbo Baggins can afford to save $2,100 dollars per month for the next 10 years therefore at 10 years he would have saved: PMT = 2,100 I = 10. 48 / 12 = 0. 873 N = 10 x 12 = 120 CPT FV = $442,201. 15 So after 10 years he would be able to purchase his yacht at the price of $350,000, and he would be left with a balance of $92,201. 15 This $92,201. 15 will be our current PV at year 10. At year 30, the year when Bilbo retires, the $92,201. 15 would become 92,201. 15*(1. 11) ^20 = $620,283. 23 Second we have to find out how much the inheritance of 750,000 would be at year 30: 750,000/1. 8^20= $160,911. 16 Third In order for him to be able to withdraw a sum of 25,000 per month for the next 20 years after his retirement, we should now calculate this annuity’s present value: N= 20 x 12 = 240 I= 7. 72 / 12 = 0. 643 PMT= 25,000 CPT PV = $3,052,135. 26 Adding up the PV’s of the $750,000 and the annuity, we will get $3,213,046. 32 We will subtract the future value at year 30 of the $92,201. 15 ($620,283. 23) which we saved at year 10 from $3,213,046. 32 to get $2,592,763. 09 We are now left with an annuity that pays $2,592,763. 09 at year 30, and a time period of 20 years (yr11-30) To calculate the yearly PMT, we have FV= $2,592,763. 09 I= 10. 48 / 12 = 0. 873 N= 20 x 12 = 240 CPT PMT = 3,207. 33 Therefore the monthly PMT Bilbo would have to save each month through years 11-30 would be = $3,207. 33 34. Valuing bonds: Mallory Corporation has two different bonds, currently outstanding. Bond M has a face value of $20,000 and matures in twenty years. The bond makes no payments for the first six years, then pays $1,200 every 6 months over the subsequent eight years, and finally pays $1,500 every 6 months over the last years. Bond N also has a face value of $20,000 and a maturity of 20 years; it makes n coupon payments over the life of the bond. If the required return on both these bonds is 10% compounded semiannually, what is the current price of bond M? Of bond N? Solution: The price of a bond is equal to PV of expected future cash flows Bond M: Face value 20,000 Present value of 20,000 = 20,000/ (1. 05) ^40 = $2,840. 91 First we need to get the present value of the annuity for the 1,500 semiannual PMTs at year 14 Present Value of Annuity = $13,295 $13,295 becomes $3,391 at year 0 We then get the annuity of the 1,200 semiannual PMTs at year 6, and then at Present Value $13,005 at year 6 with a PV of $7,242 at year 0 The sum of the 3 PV’s gives us the value of the bond ,841 + 3,391 + 7,242 = $13,474 Bond N Face value 20,000 Present value of 20,000 = 20,000/ (1. 05) ^40 = $2,840. 91 38. Non-constant growth: Storico Co. just paid a dividend of aud 3. 5 per share. The company will increase its dividend by 20% next year, and will then reduce its dividend growth rate by 5% per year, until it reaches the industry average o f 5% industry average growth, after which the company will keep a constant growth rate forever. If the required return on Storico stock is 13%, what will a share of stock sell for today? Solution  : D0 = $3. 5 D1= 3. 5*1. 2= $4. 2 D2= 4. 2*1. 15= $4. 3 D3=4. 83*1. 1= $5. 31 D4=5. 31*1. 05= $5. 58 Since the first 4 periods are different we get the PV of each one alone, then as of the 4th year we get the perpetuity of the rest, and sum them up to get the final NPV We now get the PV of each Dividend PV D1 = 4. 2/ (1. 13) = $3. 72 PV D2 = 4. 83/ (1. 13) ^2 = $3. 78 PV D3 = 5. 31/ (1. 13) ^3 = $3. 68 So the PVs of D1+D2+D3 = $11. 18 NPV of perpetuity at constant growth = 5. 58(0. 08) / (1. 13) ^3 = 69. 75 / (1. 13) ^3 = $48. 34 NPV perpetuity + NPV dividends = NPV price of stock today 48. 34 + 11. 18 = $59. 52

3. List 2 key enviromental issues for each of the Essays - Biology

3. List 2 key enviromental issues for each of the Essays - Biology 3. List 2 key enviromental issues for each of the following region and describe about how are the issues changing: The wilderness North These areas have important economic values in their trees, animals, scenery, and other natural resources. Resources exploitation involves significant trade-offs as these ecosystems are sensitive to insults and take a long time to repair damage caused by exploitation. Many short-term political and economic decisions have failed to look at long-term environment implications. Native peoples in these areas are sensitive to changes in land use or government policy that would force changes in traditional ways of life. Increasingly sophisticated in negotiations. The Agricultural Middle Middle of North America is dominated by intensive agriculture. Original, natural ecosystems have been replaced by managed agriculture Tremendous economic value. Mostly private land- large economic risks. Major non-point pollution source Soil erosion and groundwater contamination Fertilizer and Pesticides The forested West Government and commercial timber companies own large sections of land. Historically, much f this timber has been sold at a loss. In 1993, USFS directed to stop below-cost timber sales. Timber officials claim access to public land is necessary to remain in business and support the economy; conservationists argue ecological and intangible values outweigh economic value. Northern Spotted owl has become a symbol of conflict between logging and preservation. The Dry West Characterized by area where rainfall is inadequate for agriculture, but adequate enough to allow livestock production. Because much of western U.S. is of low economic value, most is still controlled by U.S government. Encourages use by providing cheap water for livestock and irrigation, cheap grazing fees, and access for industrial development. As cities grow, conflict arises between urban dwellers and ranchers and farmers. Increased demand will result in shortages and resulting trade-off decisions Low population density areas tend towards wilderness character. Economic livestock and wildness preservation. Great Lakes and Industrial Northeast: Great Lakes and Northeast are dominated by large metropolitan complexes with large, complex resource demands. Many older cities have declined, leaving behind abandoned sites and environmental problems. One of the greatest problems is water contamination from toxic material Bioaccumulation in food chain. Fish Advisories The South Microcosm of all other regions. Extremely rapid population growth in some areas such as coastal regions Pockets of extreme poverty.

According to material Coursework Example | Topics and Well Written Essays - 750 words

According to material - Coursework Example That it was written at a time when the Catholic doctrines were still held in high esteem (during the Middle Age) clearly portrays this. Perhaps the death of Ackermann’s wife is symbolic of the fall of the Roman Empire and Roman Catholic Church that paved way for transformation of the church in England (Parker 145). According to Luther, God, and not the church or its agents could only offer salvation. He opposed the selling of indulges (pieces of paper) that supposedly got people to heaven. His open disagreement with the Roman Catholic Church saw him write the 95 theses, where he explained his dissatisfaction with the church, and his eventual championing for the Protestant Church (Parker 91). Tears of the Fatherhood talks about the toll the thirty years of war had on Germany. It hurt the economy. Property was destroyed, lives lost, and cities ruined. Major roads were closed and for a long time Germany felt the effects of the war. Social amenities such as schools, hospitals and recreation centers were reduced to rubble. The war had excesses such as raping of young women and girls by the enemy soldiers (Parker p182). The poem uses poetic devices such as symbolism. Nature is used such as the blocked river; the river is blocked with corpses. The reader gets the picture and one is therefore able to understand how grave the effects of the war were. The title Tears of the Fatherhood depicts the mourning by the German citizens who start all over to revive the economy. However, the memories of the war will live with them, just as a father mourns the death of a child. In summary, the Communist Manifesto argued that capitalism lead to classification in the society that creates social conflicts. As such, they asserted that capitalism was unstable. The communists intended to stage a revolution and see equitable distribution of resources in the society.

Research Methods Coursework Example | Topics and Well Written Essays - 1000 words

Research Methods - Coursework Example There are many situations that would drive a researcher to want to conduct a historical research. One of those situations is the urge to understand a certain culture. The full length understanding of a culture (for examples its education or religion) requires that information about the past events of that culture be examined in order to understand its present and even predict what will happen to the future. This is possible and reliable since historical research focuses on patterns that happened in the past therefore making it easier to compare with the current patterns. Understanding of the past of a culture may is also necessary in order to compare it with the history of other cultures. Part 2 Explain the difference between external and internal evidence and give examples of each. Internal evidence in a research is determined by the absence of any confounds and it ensures that the results of the researcher are according to the procedures specified and no deviations have been made. Internal evidence is mostly used to determine causal effects and relationships in a research and this therefore means that it is bound to have many threats to it. External evidence on the other hand describes the amount of supportive information that can be acquired from other sources or other previous research to justify the current research. It can also be defined widely to include the extent the research results can be generalized to other settings other than the one currently the research is on. Part 3 Provide an explanation and an example of the following descriptive research designs: 1. Observation studies: These are done through the researcher just merely observing the subjects of the research and not manipulating them. The researcher then records the observations while they are taking place to avoid missing any details and the analysis will be carried out of the recorded information from the observation. The observation may take place for a long or short time depending on th e research objectives. This is common when dealing with research about animals where there are communication barriers. 2. Correlation research: This is a form of descriptive research design where a relationship is sought and established between the variables in a research. The relationships between the variables (if any) are used to further understand and justify the research design. An example of this type of research is in social research like poverty where there are several variables which seem interrelated. 3. Developmental designs: (there are three different types) These are research designs used to examine human relationships and interaction throughout their development and the time each developmental design takes varies. The first is the cross-sectional design where a researcher researches on different subjects with different characteristics but within a single time period that is usually specified for example researching about different age-related subjects who have the same characteristics. The other is the longitudinal design where the time period for the research is not specified and it involves studying the same subjects over and over again for a long period of time. This is true for example in medical research. Lastly, the third developmental design is the cross- sequential design. In this design, the research subjects are tested on a cross-sectional basis (ensuring the differences in traits) but repeatedly for long

What's the news Essay Example | Topics and Well Written Essays - 250 words

What's the news - Essay Example The reading further stipulates considerations that journalists should have in mind when collecting news these are: timeliness, proximity, prominence and consequences that their news might have. The reading makes sense as it enlightens those pursuing journalism career with the elements they ought to consider to make their profession successful. The reading highlights important fundamentals to be considered which are timeliness, proximity, prominence and consequences of particular news (Harrison 73). In case a journalist considers these elements when collecting news, they will automatically provide the best news to their readers that will make them yearn for more news from the same paper. For instance a paper has news that meets all the requirements above will make their readers have enthusiasm for their news thus their paper will definitely sell and in turn provide its editors and reporters with constant income that will improve their living standards (Harrison

Design of Spatial Decoupling Scheme

Design of Spatial Decoupling Scheme Design of Spatial Decoupling Scheme using Singular Value Decomposition for Multi-User Systems Abstract In this paper, we present the use of a polynomial singular value decomposition (PSVD) algorithm to examine a spatial decoupling based block transmission design for multiuser systems. This algorithm facilitates joint and optimal decomposition of matrices arising inherently in multiuser systems. Spatial decoupling allows complex multichannel problems of suitable dimensionality to be spectrally diagonalized by computing a reduced-order memoryless matrix through the use of the coordinated transmit precoding and receiver equalization matrices. A primary application of spatial decoupling based system can be useful in discrete multitone (DMT) systems to combat the induced crosstalk interference, as well as in OFDM with intersymbol interference. We present here simulation-based performance analysis results to justify the use of PSVD for the proposed algorithm. Index Terms-polynomial singular value decomposition, paraunitary systems, MIMO system. INTRODUCTION Block transmission based systems allows parallel, ideally noninterfering, virtual communication channels between multiuser channels. Minimally spatial decoupling channels are needed whenever more than two transmitting channels are communicate simultaneously. The channel of our interest here, is the multiple input multiple output channels, consisting of multiple MIMO capable source terminals and multiple capable destinations. This scenario arises, obviously, in multi-user channels. Since certain phases of relaying involves broadcasting, it also appears in MIMO relaying contexts. The phrase MIMO broadcast channel is frequently used in a loose sense in the literature, to include point-to-multipoint unicast (i.e. private) channels carrying different messages from a single source to each of the multiple destinations (e.g. in multi-user MIMO). Its use in this paper is more specific, and denotes the presence of at least one common virtual broadcast channel from the source to the destinations. The use of iterative and non-iterative spatial decoupling techniques in multiuser systems to achieve independent channels has been investigated, for instance in [1]-[9]. Their use for MIMO broadcasting, which requires common multipoint-to-multipoint MIMO channels is not much attractive, given the fact that the total number of private and common channels is limited by the number of antennas the source has. Wherever each receiver of a broadcast channel conveys what it receives orthogonally to the same destination, as in the case of pre-and post-processing block transmission, the whole system can be envisaged as a single point-to-point MIMO channel. Block transmission techniques have been demonstrated for point-to-point MIMO channels to benefit the system complexities. Other advantages includes: (i) channel interference is removed by creating $K$ independent subchannels; (ii) paraunitarity of precoder allows to control transmit power; (iii) paraunitarity of equalizer does not amplify the channel noise; (iv) spatial redundancy can be achieved by discarding the weakest subchannels. Though the technique outperform the conventional signal coding but had its own demerits.   Amongst many, it shown in cite{Ta2005,Ta2007} that an appropriate additional amount of additive samples  still require individual processing, e.g. per- tone equalisation, to remove ISI, and   the receiver does not exploit the case of structured noise. However, the choice of optimal relay gains, although known for certain cases (e.g. [10], [11]), is not straightforward with this approach. Since the individual equalization have no non-iterative means of decoding the signals, this approach cannot be used with decode-and-forward (DF), and code-and-forward (CF) relay processing schemes. The use of zero-forcing at the destination has been examined [12], [13] as a mean of coordinated beamforming, since it does not require transmitter processing. The scheme scales to any number of destinations, but requires each destination to have no less antennas than the source. Although not used as commonly as the singular value decomposition (SVD), generalized singular value decomposition (GSVD) [14, Thm. 8.7.4] is not unheard of in the wireless literature. It has been used in multi-user MIMO transmission [15], [16], MIMO secrecy communication [17], [18], and MIMO relaying [19]. Reference [19] uses GSVD in dual-hop AF relaying with arbitrary number of relays. Since it employs zero-forcing at the relay for the forward channel, its use of GSVD appears almost similar to the use of SVD in [1]. Despite GSVD being the natural generalization of SVD for two matrices, we are yet to see in the literature, a generalization of SVD-based beamforming to GSVD-based beamforming. Although the purpose and the use is somewhat different, the reference [17, p.1] appears to be the first to hint the possible use of GSVD for beamforming. In present work, we illustrate how GSVD can be used for coordinated beamforming in source-to-2 destination MIMO broadcasting; thus in AF, DF and CF MIMO relaying. We also present comparative, simulation-based performance analysis results to justify GSVD-based beamforming. The paper is organized as follows: Section II presents the mathematical framework, highlighting how and under which constraints GSVD can be used for beamforming. Section III examines how GSVD-based beamforming can be applied in certain simple MIMO and MIMO relaying configurations. Performance analysis is conducted in section IV on one of these applications. Section V concludes with some final remarks. Notations: Given a matrix A and a vector v, (i) A(i, j)  gives the ith element on the jth column of A; (ii) v(i)  {ˆ y1 }R(r+1,r+s) = ˜Π£{x }R(r+1,r+s) + _ UHn1 _ R(r+1,r+s) ,   {ˆ y2 }R(pà ¢Ã‹â€ Ã¢â‚¬â„¢t+r+1,pà ¢Ã‹â€ Ã¢â‚¬â„¢t+r+s) = ˜Άº{x }R(r+1,r+s) + _ VHn2 _ R(pà ¢Ã‹â€ Ã¢â‚¬â„¢t+r+1,pà ¢Ã‹â€ Ã¢â‚¬â„¢t+r+s) , {ˆ y1 }R(1,r) = {x }R(1,r) + _ UHn1 _ R(1,r) , {ˆ y2 }R(pà ¢Ã‹â€ Ã¢â‚¬â„¢t+r+s+1,p) = {x }R(r+s+1,t) + _ VHn2 _ R(pà ¢Ã‹â€ Ã¢â‚¬â„¢t+r+s+1,p) . (1)  gives the element of v at the ith position. {A}R(n) and  {A}C(n) denote the sub-matrices consisting respectively of the  first n rows, and the first n columns of A. Let {A}R(m,n)  denote the sub-matrix consisting of the rows m through n  of A. The expression A = diag (a1, . . . , an) indicates that  A is rectangular diagonal; and that first n elements on its  main diagonal are a1, . . . , an. rank (A) gives the rank of  A. The operators ( à £Ã†â€™Ã‚ » )H, and ( à £Ã†â€™Ã‚ »)à ¢Ã‹â€ Ã¢â‚¬â„¢1 denote respectively the  conjugate transpose and the matrix inversion. C mÃÆ'-n is the  space spanned by mÃÆ'-n matrices containing possibly complex  elements. The channel between the wireless terminals T1 and  T2 in a MIMO system is designated T1 à ¢Ã¢â‚¬  Ã¢â‚¬â„¢T2.   II. MATHEMATICAL FRAMEWORK Let us examine GSVD to see how it can be used for  beamforming. There are two major variants of GSVD in the  literature (e.g. [20] vs. [21]). We use them both here to  elaborate the notion of GSVD-based beamforming. A. GSVD Van Loan definition Let us first look at GSVD as initially proposed by Van Loan [20, Thm. 2]. Definition 1: Consider two matrices, H à ¢Ã‹â€ Ã‹â€ C mÃÆ'-n with  m à ¢Ã¢â‚¬ °Ã‚ ¥n, and G à ¢Ã‹â€ Ã‹â€ C pÃÆ'-n, having the same number n of  columns. Let q = min (p, n). H and G can be jointly  decomposed as H = UÃŽÂ £Q, G = VΆºQ (2) where (i) U à ¢Ã‹â€ Ã‹â€ C mÃÆ'-m,V à ¢Ã‹â€ Ã‹â€ C pÃÆ'-p are unitary, (ii) Q à ¢Ã‹â€ Ã‹â€  C nÃÆ'-n non-singular, and (iii) ÃŽÂ £= diag (à Ã†â€™1, . . . , à Ã†â€™n) à ¢Ã‹â€ Ã‹â€  C mÃÆ'-n, à Ã†â€™i à ¢Ã¢â‚¬ °Ã‚ ¥0; Άº= diag (ÃŽÂ »1, . . . , ÃŽÂ »q) à ¢Ã‹â€ Ã‹â€ C pÃÆ'-n, ÃŽÂ »i à ¢Ã¢â‚¬ °Ã‚ ¥0. As a crude example, suppose that G and H above represent  channel matrices of MIMO subsystems S à ¢Ã¢â‚¬  Ã¢â‚¬â„¢D1 and S à ¢Ã¢â‚¬  Ã¢â‚¬â„¢D2  having a common source S. Assume perfect channel-stateinformation  (CSI) on G and H at all S,D1, and D2. With  a transmit precoding matrix Qà ¢Ã‹â€ Ã¢â‚¬â„¢1, and receiver reconstruction  matrices UH,VH we get q non-interfering virtual broadcast channels. The invertible factor Q in (2) facilitates jointprecoding  for the MIMO subsystems; while the factors U,V   allow receiver reconstruction without noise enhancement. Diagonal  elements 1 through q of ÃŽÂ £,Άºrepresent the gains  of these virtual channels. Since Q is non-unitary, precoding  would cause the instantaneous transmit power to fluctuate. This is a drawback not present in SVD-based beamforming. Transmit signal should be normalized to maintain the average  total transmit power at the desired level. This is the essence of GSVD-based beamforming for  a single source and two destinations. As would be shown  in Section III, this three-terminal configuration appears in  various MIMO subsystems making GSVD-based beamforming  applicable. B. GSVD Paige and Saunders definition Before moving on to applications, let us appreciate GSVDbased  beamforming in a more general sense, through another  form of GSVD proposed by Paige and Saunders [21, (3.1)]. This version of GSVD relaxes the constraint m à ¢Ã¢â‚¬ °Ã‚ ¥n present  in (2). Definition 2: Consider two matrices, H à ¢Ã‹â€ Ã‹â€ C mÃÆ'-n and  G à ¢Ã‹â€ Ã‹â€ C pÃÆ'-n, having the same number n of columns. Let CH = _ HH,GH _ à ¢Ã‹â€ Ã‹â€ C nÃÆ'-(m+p), t = rank(C), r = t à ¢Ã‹â€ Ã¢â‚¬â„¢rank (G) and s = rank(H) + rank (G) à ¢Ã‹â€ Ã¢â‚¬â„¢t. H and G can be jointly decomposed as H = U (ÃŽÂ £ 01 )Q = UÃŽÂ £{Q}R(t) , G = V (Άº 02 )Q = VΆº{Q}R(t) , (3) where (i) U à ¢Ã‹â€ Ã‹â€ C mÃÆ'-m,V à ¢Ã‹â€ Ã‹â€ C pÃÆ'-p are unitary, (ii) Q à ¢Ã‹â€ Ã‹â€ C nÃÆ'-n non-singular, (iii) 01 à ¢Ã‹â€ Ã‹â€ C mÃÆ'-(nà ¢Ã‹â€ Ã¢â‚¬â„¢t), 02 à ¢Ã‹â€ Ã‹â€  C pÃÆ'-(nà ¢Ã‹â€ Ã¢â‚¬â„¢t) zero matrices, and (iv) ÃŽÂ £Ãƒ ¢Ã‹â€ Ã‹â€ C mÃÆ'-t,ΆºÃƒ ¢Ã‹â€ Ã‹â€  C pÃÆ'-t have structures ÃŽÂ £_ à ¢Ã… ½Ã¢â‚¬ º à ¢Ã… ½Ã‚  IH ˜Π£ 0H à ¢Ã… ½Ã… ¾ à ¢Ã… ½Ã‚   and Άº_ à ¢Ã… ½Ã¢â‚¬ º à ¢Ã… ½Ã‚  0G ˜Άº IG à ¢Ã… ½Ã… ¾ à ¢Ã… ½Ã‚  . IH à ¢Ã‹â€ Ã‹â€ C rÃÆ'-r and IG à ¢Ã‹â€ Ã‹â€ C (tà ¢Ã‹â€ Ã¢â‚¬â„¢rà ¢Ã‹â€ Ã¢â‚¬â„¢s)ÃÆ'-(tà ¢Ã‹â€ Ã¢â‚¬â„¢rà ¢Ã‹â€ Ã¢â‚¬â„¢s) are identity  matrices. 0H à ¢Ã‹â€ Ã‹â€ C (mà ¢Ã‹â€ Ã¢â‚¬â„¢rà ¢Ã‹â€ Ã¢â‚¬â„¢s)ÃÆ'-(tà ¢Ã‹â€ Ã¢â‚¬â„¢rà ¢Ã‹â€ Ã¢â‚¬â„¢s), and 0G à ¢Ã‹â€ Ã‹â€  C (pà ¢Ã‹â€ Ã¢â‚¬â„¢t+r)ÃÆ'-r are zero matrices possibly having no  rows or no columns. ˜Π£= diag (à Ã†â€™1, . . . , à Ã†â€™s) ,˜Άº= diag (ÃŽÂ »1, . . . , ÃŽÂ »s) à ¢Ã‹â€ Ã‹â€ C sÃÆ'-s such that 1 > à Ã†â€™1 à ¢Ã¢â‚¬ °Ã‚ ¥. . . à ¢Ã¢â‚¬ °Ã‚ ¥ à Ã†â€™s > 0, and à Ã†â€™2 i + ÃŽÂ »2i = 1 for i à ¢Ã‹â€ Ã‹â€  {1, . . . , s}. Let us examine (3) in the MIMO context. It is not difficult  to see that a common transmit precoding matrix _ Qà ¢Ã‹â€ Ã¢â‚¬â„¢1 _ C(t) and receiver reconstruction matrices UH,VH would jointly  diagonalize the channels represented by H and G.  For broadcasting, only the columns (r+1) through (r +s)  of ÃŽÂ £and Άºare of interest. Nevertheless, other (t à ¢Ã‹â€ Ã¢â‚¬â„¢s)  columns, when they are present, may be used by the source  S to privately communicate with the destinations D1 and configuration # common channels # private channels S à ¢Ã¢â‚¬  Ã¢â‚¬â„¢ {D1,D2} S à ¢Ã¢â‚¬  Ã¢â‚¬â„¢D1 S à ¢Ã¢â‚¬  Ã¢â‚¬â„¢D2 m > n,p à ¢Ã¢â‚¬ °Ã‚ ¤n p n à ¢Ã‹â€ Ã¢â‚¬â„¢p 0 m à ¢Ã¢â‚¬ °Ã‚ ¤n, p > n m 0 n à ¢Ã‹â€ Ã¢â‚¬â„¢m m à ¢Ã¢â‚¬ °Ã‚ ¥n, p à ¢Ã¢â‚¬ °Ã‚ ¥n n 0 0 m + p à ¢Ã‹â€ Ã¢â‚¬â„¢n n à ¢Ã‹â€ Ã¢â‚¬â„¢p n à ¢Ã‹â€ Ã¢â‚¬â„¢m (m + p) > n n à ¢Ã¢â‚¬ °Ã‚ ¥(m + p) 0 m p TABLE I NUMBERS OF COMMON CHANNELS AND PRIVATE CHANNELS FOR  DIFFERENT CONFIGURATIONS D2. It is worthwhile to compare this fact with [22], and  appreciate the similarity and the conflicting objectives GSVDbased  beamforming for broadcasting has with MIMO secrecy  communication. Thus we can get ˆ y1 à ¢Ã‹â€ Ã‹â€ C mÃÆ'-1, ˆ y2 à ¢Ã‹â€ Ã‹â€ C pÃÆ'-1 as in (1) at  the detector input, when x à ¢Ã‹â€ Ã‹â€ C tÃÆ'-1 is the symbol vector  transmitted. It can also be observed from (1) that the private  channels always have unit gains; while the gains of common  channels are smaller. Since, à Ã†â€™is are in descending order, while the ÃŽÂ »is ascend  with i, selecting a subset of the available s broadcast channels  (say k à ¢Ã¢â‚¬ °Ã‚ ¤s channels) is somewhat challenging. This highlights  the need to further our intuition on GSVD. C. GSVD-based beamforming Any two MIMO subsystems having a common source  and channel matrices H and G can be effectively reduced,  depending on their ranks, to a set of common (broadcast) and  private (unicast) virtual channels. The requirement for having  common channels is rank (H) + rank (G) > rank (C) where C = _ HH,GH _ H. When the matrices have full rank, which is the case with  most MIMO channels (key-hole channels being an exception),  this requirement boils down to having m +p > n . Table I  indicates how the numbers of common channels and private  channels vary in full-rank MIMO channels. It can be noted  that the cases (m > n,p à ¢Ã¢â‚¬ °Ã‚ ¤n) and (m à ¢Ã¢â‚¬ °Ã‚ ¥n, p à ¢Ã¢â‚¬ °Ã‚ ¥n)  correspond to the form of GSVD discussed in the Subsection II-A. Further, the case n à ¢Ã¢â‚¬ °Ã‚ ¥(m + p) which produces only  private channels with unit gains, can be seen identical to zero  forcing at the transmitter. Thus, GSVD-based beamforming is  also a generalization of zero-forcing. Based on Table I, it can be concluded that the full-rank  min (n,m + p) of the combined channel always gets split  between the common and private channels. D. MATLAB implementation A general discussion on the computation of GSVD is found  in [23]. Let us focus here on what it needs for simulation:  namely its implementation in the MATLAB computational  environment, which extends [14, Thm. 8.7.4] and appears as  less restrictive as [21]. The command [V, U, X, Lambda, Sigma] = gsvd(G, H);  gives1 a decomposition similar to (3). Its main deviations  from (3) are,   1Reverse order of arguments in and out of gsvd function should be noted. ) ) D1 y1 , r1 S x ,w ( ( ) ) D2 y2 , r2 _ H1 __ n1 _ __ H2 n2 Fig. 1. Source-to-2 destination MIMO broadcast system  Ãƒ ¢Ã¢â€š ¬Ã‚ ¢ QH = X à ¢Ã‹â€ Ã‹â€ C nÃÆ'-t is not square when t . Precoding  for such cases would require the use of the pseudo-inverse  operator. à ¢Ã¢â€š ¬Ã‚ ¢ ÃŽÂ £has the same block structure as in (3). But the structure  of Άºhas the block 0G shifted to its bottom as follows: Άº_ à ¢Ã… ½Ã¢â‚¬ º à ¢Ã… ½Ã‚  ˜Άº IG 0G à ¢Ã… ½Ã… ¾ à ¢Ã… ½Ã‚  . This can be remedied by appropriately interchanging the  rows of Άºand the columns of V. However, restructuring  ÃƒÅ½Ã¢â‚¬ ºis not a necessity, since the column position of the  block ˜Άºwithin Άºis what matters in joint precoding.   Following MATLAB code snippet for example jointly  diagonalizes H,G to obtain the s common channels (3)  would have given. MATLAB code % channel matrices H = (randn(m,n)+i*randn(m,n))/sqrt(2); G = (randn(p,n)+i*randn(p,n))/sqrt(2); % D1, D2: diagonalized channels [V,U,X,Lambda,Sigma] = gsvd(G,H); w = X*inv(X*X); C = [H G]; t = rank(C); r = t rank(G); s = rank(H)+rank(G)-t; D1 = U(:,r+1:r+s)*H*w(:,r+1:r+s); D2 = V(:,1:s)*G*w(:,r+1:r+s); III. APPLICATIONS Let us look at some of the possible applications of GSVDbased beamforming. We assume the Van Loan form of GSVD  for simplicity, having taken for granted that the dimensions  are such that the constraints hold true. Nevertheless, the Paige  and Saunders form should be usable as well. A. Source-to-2 destination MIMO broadcast system   Consider the MIMO broadcast system shown in Fig. 1,  where the source S broadcasts to destinations D1 and D2.  MIMO subsystems S à ¢Ã¢â‚¬  Ã¢â‚¬â„¢D1 and S à ¢Ã¢â‚¬  Ã¢â‚¬â„¢D2 are modeled  to have channel matrices H1 ,H2 and additive complex   Gaussian noise vectors n1 , n2. Let x = [x1, . . . , xn]T ) ) R1 y1 , F1 ( ( S x ,w ( ( ) ) D y3 ,r1 y4 ,r2 ) ) R2 y2 , F2 ( ( _ ___ H3 _ n3 H1 ___ n1 _ ___ H2 n2 _ H4 ___ n4 Fig. 2. MIMO relay system with two 2-hop-branches  be the signal vector desired to be transmitted over n à ¢Ã¢â‚¬ °Ã‚ ¤Ã‚  min (rank (H1 ) , rank (H2 )) virtual-channels. The source  employs a precoding matrix w. The input y1 , y2 and output ˆ y1 , ˆ y2 at the receiver filters   r1 , r2 at D1 and D2 are given by y1 = H1wx + n1 ; ˆ y1 = r1 y1 , y2 = H2wx + n2 ; ˆ y2 = r2 y2 . Applying GSVD we get H1 = U1 ÃŽÂ £1 V and H2 = U2 ÃŽÂ £2V. Choose the precoding matrix w = ÃŽÂ ± _ Và ¢Ã‹â€ Ã¢â‚¬â„¢1 _ C(n) ; and receiver reconstruction matrices r1 = _ U1 H _ R(n) _ , r2 = U2 H _ R(n) . The constant ÃŽÂ ± normalizes the total average transmit power. Then we get, ˆ y1(i) = ÃŽÂ ±ÃƒÅ½Ã‚ £1(i, i) x(i) + Ëœn1(i) , ˆ y2(i) = ÃŽÂ ±ÃƒÅ½Ã‚ £2(i, i) x(i) + Ëœn2(i), ià ¢Ã‹â€ Ã‹â€  {1 . . . n}, where Ëœn1 , Ëœn2 have the same noise distributions as n1 , n2 .  B. MIMO relay system with two 2-hop-branches (3 time-slots) Fig. 2 shows a simple MIMO AF relay system where a  source S communicates a symbol vector x with a destination  D via two relays R1 and R2. MIMO channels S à ¢Ã¢â‚¬  Ã¢â‚¬â„¢R1, S à ¢Ã¢â‚¬  Ã¢â‚¬â„¢ R2, R1 à ¢Ã¢â‚¬  Ã¢â‚¬â„¢D and R2 à ¢Ã¢â‚¬  Ã¢â‚¬â„¢D are denoted: Hi , i à ¢Ã‹â€ Ã‹â€  {1, 2, 3, 4}. Corresponding channel outputs and additive complex Gaussian  noise vectors are yi , ni for i à ¢Ã‹â€ Ã‹â€  {1, 2, 3, 4}. Assume relay  operations to be linear, and modeled as matrices F1 and F2 . Assume orthogonal time-slots for transmission. The source  S uses w as the precoding matrix. Destination D uses  different reconstruction matrices r1 , r2 during the time slots  2 and 3. Then we have: Time slot 1: y1 = H1wx + n1 , y2 = H2wx + n2 Time slot 2: y3 = H3 F1 y1 + n3 Time slot 3: y4 = H4 F2 y2 + n4 Let ˆ y = r1 y3 +r2 y4 be the input to the detector. Suppose n à ¢Ã¢â‚¬ °Ã‚ ¤min i (rank (Hi )) virtual-channels are in use. ) ) R y1 , F ( ( S x ,w ( ( ) ) D y2 ,r1 y3 ,r2 _ ___ H3 _ n3 H1 ___ n1 H2 _ n2 Fig. 3. MIMO relay system having a direct path and a relayed path  Applying GSVD on the broadcast channel matrices we get H1 = U1 ÃŽÂ £1 Q and H2 = U2 ÃŽÂ £2 Q. Through SVD we  obtain H3 = V1 Άº1 R1 H and H4 = V2 Άº2 R2 H. Choose w = ÃŽÂ ± _ Qà ¢Ã‹â€ Ã¢â‚¬â„¢1 _ C(n) ; F1 = R1U1 H; F2 = R2U2 H; r1 = _ V1 H _ R(n) ; r2 = _ V2 H _ R(n) . The constant ÃŽÂ ± normalizes  the total average transmit power. Then we get

A Respectable Trade: Slavery :: Market Systems England Essays

A Respectable Trade: Slavery Many economic systems are revealed in A Respectable Trade: Slavery, Feudalism, Self-Employment, and Capitalism. England in 1788 was entering a period of economic transition. Viewing this finite period in A Respectable Trade allows us, as economists, to dissect the different market systems prevalent during that time. Slavery is the market system most focused on in A Respectable Trade. Josiah's "respectable trade" involves trading sugar, cocoa, coffee and cotton in Africa for captured Negro men, women and children. He then ships these "slaves" to the Caribbean, where he sells them. He makes all of his money in the sale of these people. While Josiah and Sarah Cole have been involved in the slave trade for many years, in 1788 they have just begun to experience the immediate effects of slaves in their lives. Josiah has determined that he will make more money if he ships some slaves to England to train as house slaves. He has married Frances so that she will train and teach them while they live with the Cole's in England. Josiah, Sarah, and Frances are learning the techniques of the slave master. As the film progresses, Josiah becomes more crass and unfeeling toward the slaves, seeing them solely as property. When the slaves first arrive, he feels awkward and anxious about harming them. He knows that he should punish them and lord over them, but he is more comfortable allowing Bates to reprimand and beat the slaves. He allows his customer to rape the slave girl, but he is uncomfortable doing so and does not want to watch. However, at the end of the movie, he stands over Bates while he severely beats Matthew, watching closely with no remorse. Holding human beings as property by chaining them and locking them in the house, controlling their lives and fates by selling them and forcing them to work, Josiah Cole has become a cruel slave master. Frances has a chief role in the slave system. Marrying Josiah, she becomes a teacher and a manager of the slaves in her home. She teaches them English, manners, and proper ways to serve their masters so they may become a more successful sale for Josiah. She does not do this because she desires his success, but because she is held in marriage in a feudal contract. Francis, a young woman without significant funds, without supportive family, and without an acceptable job, has few options in life.