Apostila_Matemática_ColFundamental_4_8

Prévia do material em texto

�PAGE �
�PAGE �82�
Matemática 1 – Professor Paulo Cesar Pfaltzgraff Ferreira
Passando inicialmente os números para a forma retangular,
Temos também que:
 
Fig. 1.23
Exemplo 1.18
Resolva a equação 
 para 
 e verifique a solução geometricamente3
Solução:
Temos que:
 (*)
onde 
 e 
donde,
 	=1
Substituindo na equação (*), verificamos que somente o valor 
 é compatível.
A verificação gráfica é imediata, visto que 
 é a distância entre os pontos definidos pelos complexos 
 e 
.
Sendo 
, temos que 
, e o lugar geométrico representado por 
, quando ( varia ao longo do intervalo 
, é uma circunferência de raio unitário centrada na origem.
Sendo 
, a situação é a representada na figura a seguir: 
Fig. 1.24
É fácil verificar que teremos 
 quando ( assumir o valor 
.
Multiplicação
A multiplicação de grandezas na forma retangular é dada por:
Lembramos que 
 segue-se que:
 (62)
Já na forma exponencial,
o que nos permite então escrever:
 (63)
Conclusões:
1.ª) Da equação (63) temos que:
 (64)
e
 (65)
2.ª) Para 
 e 
 vale então estabelecer a seguinte equação:
ou seja,
 (66)
3.ª) Também não é difícil mostrar que
 (67)
Exemplo 1.19
Multiplicar os seguintes números complexos:
 e 
 e 
 e 
Solução:
�� EMBED Equation.3 
Exemplo 1.20
Passar o número complexo 
 para as formas polar e cartesiana.
Solução:
Este é uma excelente exemplo, pois, lembrando a forma exponencial de um complexo, 
, parece que estamos diante de um absurdo, qual seja um numero com módulo negativo. Acontece que aí não existe módulo negativo, mas sim uma “multiplicação implícita”, conforme veremos a seguir:
 
que é a forma polar.
A forma cartesiana é facilmente obtida à partir da forma polar, ou seja:
Observação: As calculadoras eletrônicas estão em um estágio de desenvolvimento tão elevado que, aquelas que tem as rotinas RET ( POL e POL ( RET, assimilariam a transformação 
 diretamente para a forma cartesiana, pois, quando se entra com 
, o software da calculadora entende que isto não é simplesmente módulo, e que existe uma multiplicação implícita. Está duvidando? Pois então pegue uma e execute a operação!
Divisão
A divisão de duas grandezas complexas, 
, é definida como 
 se 
.
Em coordenadas retangulares temos:
onde o processo de racionalização foi efetuado utilizando-se o complexo conjugado do denomi​nador.
Finalmente,
 (68)
e na forma exponencial,
o que nos conduz a
 (69)
Conclusões:
1ª) Da equação (69) concluímos que:
 (70)
e
. (71)
2ª) Não é difícil mostrar que
, sendo 
 (72)
3ª) Fica então evidente que a multiplicação e a divisão de grandezas complexas são mais facilmente efetuadas na forma polar, a menos que, conforme já dito anteriormente, se tenha uma calculadora eletrônica mais sofisticada.
4ª) É importante notar que multiplicar uma grandeza complexa por 
 não altera o seu módulo, mas soma 90º ao seu ângulo de fase. Raciocinando em termos da representação por meio de segmento orientado no plano complexo, a multiplicação por j gira o segmento orientado de 90º no sentido anti-horário. De modo análogo, a multiplicação por 
 também não altera o módulo da grandeza mas, neste caso, há uma subtração de 90º na fase, ou seja, o segmento orientado é agora girado de 90º no sentido horário.
5ª) Similarmente, se multiplicarmos um número complexo por 
, não alteramos o seu módulo; apenas acrescentamos ( ao seu ângulo de fase ou, em outras palavras: giramos o segmento orientado que representa o complexo de um ângulo ( no sentido anti-horário. Se a multiplicação for por 
 o giro será no sentido horário.
6ª) Das propriedades e definições vistas até então resultam as leis comutativa, associativa e distributiva usuais:
 (73)
 (74)
 (75)
 (76)
 (77)
 (78)
Exemplo 1.21
Dividir os seguintes números complexos:
 e 
 e 
 e 
Solução:
Exemplo 1.22
Determinar o resultado da expressão
Solução:
Temos então:
Potenciação
Consideremos, inicialmente, um número complexo genérico
Procedamos agora a potenciação deste número, ou seja,
.
Temos então:
Assim sendo vem que:
porém, da identidade de Euler,
o que nos permite escrever
 (79)
Daí concluímos que se
podemos exprimir a potência nas seguintes formas:
 (80)
 (81)
 (82), também conhecida como 1.ª fórmula de De Moivre.
Considerando a parte assinalada com asterisco na equação (79), concluímos também que: 
 (83), que é reconhecida como sendo a identidade de De Moivre.
Exemplo 1.23
Calcular 
 utilizando (a) a forma exponencial e (b) a 1.ª fórmula de De Moivre.
Solução:
Temos que:
Logo,
.
Assim sendo,
Exemplo 1.24
Calcular 
 utilizando (a) a forma exponencial e (b) a 1.ª fórmula de De Moivre.
Solução:
Temos que:
Logo, 
Assim sendo,
Exemplo 1.25
Determinar o resultado da expressão 
 tanto na forma polar quanto na retangular.
Solução:
Inicialmente vamos passar cada um dos fatores para a forma polar:
Solução Alternativa:
Vamos manter os fatores na forma retangular e racionalizar a fração resultante:
Radiciação:
Diz-se que um número w é a raiz n-ésima de um número complexo z se
que é equivalente a
.
Para determinar as n raízes distintas do número z vamos considerá-lo em sua forma trigonométrica
e representemos, também em forma trigonométrica, a raiz que desejamos encontrar:
.
Utilizando a 1.ª fórmula de De Moivre, a equação 
 assume a seguinte forma:
.
Uma vez que a igualdade dos números complexos requer a igualdade das partes reais e das partes imaginárias, separadamente, devemos ter:
e
Tais equações, por sua vez, são equivalentes a
e
		
ou seja,
 e 
Seque-se então a expressão conhecida como 2ª fórmula de De Moivre: 
�� EMBED Word.Picture.8 
sendo k = 0, 1, 2, ..., n – 1.
 (84a)
Que também pode ser expressa para o argumento em graus,
�� EMBED Word.Picture.8 
sendo k=0, 1, 2, ..., n – 1.
 (84b)
Esta fórmula produz n raízes distintas 
todas com o mesmo módulo e com argumentos 
, k = 0, 1, 2, ..., n – 1,
que estão situadas sobre a circunferência centrada na origem e com raio 
, sendo os vértices de um polígono regular de n lados, conforme ilustrado a seguir:
Fig. 1.25
Casos particulares:
1º) Raízes da unidade:
 Quando z = 1, o ângulo 
 assume o valor zero e a fórmula (84) reduz-se a :
sendo k= 0, 1, 2... n-1
 (85)
Considerando
e utilizando a identidade de De Moivre, vemos que as n-ésimas raízes da unidade são dadas por: 
A figura 1.26 ilustra as raízes no caso n = 6, onde 
Fig. 1.26
2º) Raízes quadradas:
 (86)
3º) Raízes cúbicas:
 (87)
Exemplo 1.26
	Determine os valores das seguintes raízes: 
a) 
;
b) 
;
c) 
d) 
e represente-as no plano complexo.
Solução:
Temos que 
Pela expressão (86):
=
=
Fig. 1.27
Pela expressão (87),
Fig. 1.28
:
Temos que z = 1 e, pela expressão (85), com n = 8, 
 
 
 sendo k = 0, 1, 2, ... , 7 no presente caso.
Assim sendo,
 
Fig. 1.29
:
Temos que 
Pela expressão (86),Fig. 1.30
Exemplo 1.27
Determinar o conjunto-solução em C da equação 
.
Solução:
Temos:
isto significa que devemos calcular as raízes quartas de z = – 4. Temos então:
Utilizando a expressão (84), com n = 4, 
, sendo k = 0, 1, 2, 3 no presente caso. Assim sendo,
Logo o conjunto solução é:
A Desigualdade do Triângulo
Em alguns trabalhos sobre números complexos, este é um item que aparece logo no começo, visto que, no mais das vezes, é apresentada uma demonstração para ela baseada puramente em uma propriedade geométrica dos triângulos. Nesta oportunidade, vamos também apresentar uma demonstração analítica , pelo que optamos por aguardar um maior amadurecimento do estudante com relação aos vários conceitos básicos.
Vamos então considerar dois pontos do plano complexo associados aos números 
 e 
, conforme apresentado na figura 1.20.
	Temos então:
	
 (88)
A demonstração geométrica segue o fato de que os pontos
 são os vértices de um triângulo de lados 
, e um lado não pode exceder a soma dos outros dois.
	É também interessante notar que a desigualdade se torna uma igualdade quando os pontos 
 são colineares.
	Para demonstrar a desigualdade algebricamente vamos escrever, baseados nas expressões que envolvem complexos conjugados, que 
porém
3 A verificação geométrica da solução talvez seja melhor apreciada após o estudo da subseção 1.14.6.
_1063773723.unknown
_1063793138.unknown
_1066208120.doc
� EMBED Equation.3 ���
_1066107178.unknown
_1066107724.unknown
_1066110072.unknown
_1066110258.unknown
_1066110378.unknown
_1066110936.unknown
_1066208109.unknown
_1066110533.unknown
_1066110371.unknown
_1066110098.unknown
_1066110121.unknown
_1066110074.unknown
_1066107826.unknown
_1066108106.unknown
_1066109998.unknown
_1066107840.unknown
_1066107820.unknown
_1066107821.unknown
_1066107748.unknown
_1066107237.unknown
_1066107571.unknown
_1066107612.unknown
_1066107386.unknown
_1066107223.unknown
_1066107235.unknown
_1066107181.unknown
_1066106921.unknown
_1066106965.unknown
_1066107001.unknown
_1066106923.unknown
_1066105982.unknown
_1066106091.unknown
_1066105585.unknown
_1090137608.unknown
_1090149992.unknown
_1090190321.unknown
_1090190651.unknown
_1090190882.unknown
_1090190948.unknown
_1090191070.unknown
_1090191094.unknown
_1090191182.doc
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
_1061876463.unknown
_1061876626.unknown
_1061876698.unknown
_1090191124.unknown
_1090191144.unknown
_1090191163.unknown
_1090191116.unknown
_1061876682.unknown
_1061876540.unknown
_1061876586.unknown
_1061876536.unknown
_1061874616.unknown
_1061875700.unknown
_1061876397.unknown
_1061875580.unknown
_1061872892.unknown
_1061872904.unknown
_1061872877.unknown
_1090191020.doc
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
_1061873224.unknown
_1061874555.unknown
_1061874617.unknown
_1061875580.unknown
_1061875758.unknown
_1090190978.unknown
_1090190984.unknown
_1090190996.unknown
_1090190969.unknown
_1061875700.unknown
_1061875746.unknown
_1061875670.unknown
_1061874738.unknown
_1061875565.unknown
_1061874663.unknown
_1061874594.unknown
_1061874616.unknown
_1061874578.unknown
_1061874437.unknown
_1061874461.unknown
_1061874551.unknown
_1061874456.unknown
_1061873687.unknown
_1061873724.unknown
_1061873609.unknown
_1061872917.unknown
_1061872944.unknown
_1061872957.unknown
_1061872930.unknown
_1061872892.unknown
_1061872904.unknown
_1061872877.unknown
_1090190903.unknown
_1090190947.unknown
_1090190890.doc
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
_1061873224.unknown
_1061873599.unknown
_1061873687.unknown
_1090190801.unknown
_1090190832.unknown
_1061873724.unknown
_1061873609.unknown
_1061873536.unknown
_1061873554.unknown
_1061873474.unknown
_1061872917.unknown
_1061872944.unknown
_1061872957.unknown
_1061872930.unknown
_1061872892.unknown
_1061872904.unknown
_1061872877.unknown
_1090190835.unknown
_1090190861.unknown
_1090190728.unknown
_1090190483.unknown
_1090190635.doc
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
1
_1061872157.unknown
_1061872446.unknown
_1090190546.unknown
_1090190596.unknown
_1090190597.unknown
_1090190559.unknown
_1061872492.unknown
_1061872235.unknown
_1061872248.unknown
_1061872221.unknown
_1061871997.unknown
_1061872022.unknown
_1061872129.unknown
_1061872009.unknown
_1061871973.unknown
_1061871983.unknown
_1061871960.unknown
_1090190446.unknown
_1090153414.doc
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
1
_1061871997.unknown
_1061872157.unknown
_1061872235.unknown
_1061872446.unknown
_1061872492.unknown
_1061872248.unknown
_1061872221.unknown
_1061872022.unknown
_1061872129.unknown
_1061872009.unknown
_1061871973.unknown
_1061871983.unknown
_1061871960.unknown_1090189989.doc
�EMBED Equation.3���
�EMBED Equation.3���
�EMBED Equation.3���
6,7
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
�EMBED Equation.3���
�EMBED Equation.3���
�EMBED Equation.3���
59º
�EMBED Equation.3���
�EMBED Equation.3���
_1061907223.unknown
_1061907362.unknown
_1062245440.unknown
_1062245554.unknown
_1062245696.unknown
_1062245880.unknown
_1062245900.unknown
_1062245904.unknown
_1062245899.unknown
_1062245873.unknown
_1062245655.unknown
_1062245666.unknown
_1062245647.unknown
_1062245495.unknown
_1062245551.unknown
_1062245553.unknown
_1062245507.unknown
_1062245490.unknown
_1062245491.unknown
_1062245472.unknown
_1061907431.unknown
_1062245357.unknown
_1062245378.unknown
_1062245046.unknown
_1061907428.unknown
_1061907429.unknown
_1061907378.unknown
_1061907286.unknown
_1061907320.unknown
_1061907330.unknown
_1061907289.unknown
_1061907264.unknown
_1061907270.unknown
_1061907227.unknown
_1061812314.unknown
_1061818731.unknown
_1061818862.unknown
_1061819198.unknown
_1061818841.unknown
_1061818660.unknown
_1061818693.unknown
_1061818597.unknown
_1061811037.unknown
_1061811042.unknown
_1061812271.unknown
_1061811039.unknown
_1061811032.unknown
_1061811035.unknown
_1061811029.unknown
_1090190093.unknown
_1090153485.doc
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
_1061872917.unknown
_1061873224.unknown
_1061873536.unknown
_1061873599.unknown
_1061873687.unknown
_1061873724.unknown
_1061873609.unknown
_1061873554.unknown
_1061873474.unknown
_1061872944.unknown
_1061872957.unknown
_1061872930.unknown
_1061872892.unknown
_1061872904.unknown
_1061872877.unknown
_1090151619.doc
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
1
_1061871997.unknown
_1061872157.unknown
_1061872235.unknown
_1061872446.unknown
_1061872492.unknown
_1061872248.unknown
_1061872221.unknown
_1061872022.unknown
_1061872129.unknown
_1061872009.unknown
_1061871973.unknown
_1061871983.unknown
_1061871960.unknown
_1090152042.doc
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
_1061872917.unknown
_1061873224.unknown
_1061873536.unknown
_1061873599.unknown
_1061873687.unknown
_1061873724.unknown
_1061873609.unknown
_1061873554.unknown
_1061873474.unknown
_1061872944.unknown
_1061872957.unknown
_1061872930.unknown
_1061872892.unknown
_1061872904.unknown
_1061872877.unknown
_1090150005.unknown
_1090139695.unknown
_1090139832.unknown
_1090139972.unknown
_1090139700.unknown
_1090139493.unknown
_1090139676.unknown
_1090139259.unknown
_1067069083.doc
� EMBED Equation.3 ���
_1066107178.unknown
_1066107724.unknown
_1066110072.unknown
_1066110258.unknown
_1066110378.unknown
_1066110936.unknown
_1067069066.unknown
_1066110533.unknown
_1066110371.unknown
_1066110098.unknown
_1066110121.unknown
_1066110074.unknown
_1066107826.unknown
_1066108106.unknown
_1066109998.unknown
_1066107840.unknown
_1066107820.unknown
_1066107821.unknown
_1066107748.unknown
_1066107237.unknown
_1066107571.unknown
_1066107612.unknown
_1066107386.unknown
_1066107223.unknown
_1066107235.unknown
_1066107181.unknown
_1066106921.unknown
_1066106965.unknown
_1066107001.unknown
_1066106923.unknown
_1066105982.unknown
_1066106091.unknown
_1066105585.unknown
_1067069569.doc
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
_1061870464.unknown
_1061870615.unknown
_1061870637.unknown
_1061870653.unknown
_1061870809.unknown
_1061870828.unknown
_1061870658.unknown
_1061870652.unknown
_1061870625.unknown
_1061870510.unknown
_1061870521.unknown
_1061870498.unknown
_1061870440.unknown
_1061870453.unknown
_1061870414.unknown
_1067070443.unknown
_1067070582.unknown
_1067070741.doc
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
_1061874616.unknown
_1061876463.unknown
_1061876540.unknown
_1061876626.unknown
_1061876682.unknown
_1061876698.unknown
_1061876586.unknown
_1061876536.unknown
_1061875700.unknown
_1061876397.unknown
_1061875580.unknown
_1061872892.unknown
_1061872904.unknown
_1061872877.unknown
_1067147019.doc
� EMBED Equation.3 ���
_1066107724.unknown
_1066110936.unknown
_1066120878.unknown
_1066122289.unknown
_1066123143.unknown
_1066123194.unknown
_1066123553.unknown
_1066123929.unknown
_1067146984.unknown
_1066123425.unknown
_1066123159.unknown
_1066122488.unknown
_1066123021.unknown
_1066122316.unknown
_1066122024.unknown
_1066122037.unknown
_1066120905.unknown
_1066111713.unknown
_1066112017.unknown
_1066113696.unknown
_1066111859.unknown
_1066111566.unknown_1066111704.unknown
_1066111281.unknown
_1066110072.unknown
_1066110258.unknown
_1066110378.unknown
_1066110533.unknown
_1066110371.unknown
_1066110098.unknown
_1066110121.unknown
_1066110074.unknown
_1066107826.unknown
_1066108106.unknown
_1066109998.unknown
_1066107840.unknown
_1066107820.unknown
_1066107821.unknown
_1066107748.unknown
_1066107178.unknown
_1066107237.unknown
_1066107571.unknown
_1066107612.unknown
_1066107386.unknown
_1066107223.unknown
_1066107235.unknown
_1066107181.unknown
_1066106921.unknown
_1066106965.unknown
_1066107001.unknown
_1066106923.unknown
_1066105982.unknown
_1066106091.unknown
_1066105585.unknown
_1067070492.doc
� EMBED Equation.3 ���
_1066107724.unknown
_1066110072.unknown
_1066110936.unknown
_1066120878.unknown
_1066122289.unknown
_1066122488.unknown
_1066123367.unknown
_1067070472.unknown
_1066123268.unknown
_1066122316.unknown
_1066122024.unknown
_1066122037.unknown
_1066120905.unknown
_1066111713.unknown
_1066112017.unknown
_1066113696.unknown
_1066111859.unknown
_1066111566.unknown
_1066111704.unknown
_1066111281.unknown
_1066110258.unknown
_1066110378.unknown
_1066110533.unknown
_1066110371.unknown
_1066110098.unknown
_1066110121.unknown
_1066110074.unknown
_1066107826.unknown
_1066108106.unknown
_1066109998.unknown
_1066107840.unknown
_1066107820.unknown
_1066107821.unknown
_1066107748.unknown
_1066107178.unknown
_1066107237.unknown
_1066107571.unknown
_1066107612.unknown
_1066107386.unknown
_1066107223.unknown
_1066107235.unknown
_1066107181.unknown
_1066106921.unknown
_1066106965.unknown
_1066107001.unknown
_1066106923.unknown
_1066105982.unknown
_1066106091.unknown
_1066105585.unknown
_1067070277.unknown
_1067070313.unknown
_1067070080.doc
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
� EMBED Equation.3 ���
_1055748951.unknown
_1055749127.unknown
_1055749182.unknown
_1055749206.unknown
_1055749169.unknown
_1055748968.unknown
_1055748883.unknown
_1055748932.unknown
_1055748838.unknown
_1067069331.unknown
_1067069403.unknown
_1067069196.doc
� EMBED Equation.3 ���
_1066120684.unknown
_1066120844.unknown
_1066120903.unknown
_1067069150.unknown
_1066120784.unknown
_1063772996.unknown
_1066119682.unknown
_1060267389.unknown
_1067068588.doc
� EMBED Equation.3 ���
_1066107724.unknown
_1066110072.unknown
_1066110258.unknown
_1066110936.unknown
_1066111566.unknown
_1066111713.unknown
_1067068575.unknown
_1066111704.unknown
_1066111281.unknown
_1066110378.unknown
_1066110533.unknown
_1066110371.unknown
_1066110098.unknown
_1066110121.unknown
_1066110074.unknown
_1066107826.unknown
_1066108106.unknown
_1066109998.unknown
_1066107840.unknown
_1066107820.unknown
_1066107821.unknown
_1066107748.unknown
_1066107178.unknown
_1066107237.unknown
_1066107571.unknown
_1066107612.unknown
_1066107386.unknown
_1066107223.unknown
_1066107235.unknown
_1066107181.unknown
_1066106921.unknown
_1066106965.unknown
_1066107001.unknown
_1066106923.unknown
_1066105982.unknown
_1066106091.unknown
_1066105585.unknown
_1067068943.unknown
_1067068982.unknown
_1067068897.unknown
_1066208807.doc
� EMBED Equation.3 ���
_1066107724.unknown
_1066110072.unknown
_1066110258.unknown
_1066110936.unknown
_1066121576.unknown
_1066121729.unknown
_1066208756.unknown
_1066121327.unknown
_1066110378.unknown
_1066110533.unknown
_1066110371.unknown
_1066110098.unknown
_1066110121.unknown
_1066110074.unknown
_1066107826.unknown
_1066108106.unknown
_1066109998.unknown
_1066107840.unknown
_1066107820.unknown
_1066107821.unknown
_1066107748.unknown
_1066107178.unknown
_1066107237.unknown
_1066107571.unknown
_1066107612.unknown
_1066107386.unknown
_1066107223.unknown
_1066107235.unknown
_1066107181.unknown
_1066106921.unknown
_1066106965.unknown
_1066107001.unknown
_1066106923.unknown
_1066105982.unknown
_1066106091.unknown
_1066105585.unknown
_1066208911.unknown
_1066209410.doc
� EMBED Equation.3 ���
_1066107724.unknown
_1066110936.unknown
_1066120878.unknown
_1066122289.unknown
_1066122488.unknown
_1066123620.unknown
_1066123662.unknown
_1066209399.unknown
_1066123021.unknown
_1066122316.unknown
_1066122024.unknown
_1066122037.unknown
_1066120905.unknown
_1066111713.unknown
_1066112017.unknown
_1066113696.unknown
_1066111859.unknown
_1066111566.unknown
_1066111704.unknown
_1066111281.unknown
_1066110072.unknown
_1066110258.unknown
_1066110378.unknown
_1066110533.unknown
_1066110371.unknown
_1066110098.unknown
_1066110121.unknown
_1066110074.unknown
_1066107826.unknown
_1066108106.unknown
_1066109998.unknown
_1066107840.unknown
_1066107820.unknown
_1066107821.unknown
_1066107748.unknown
_1066107178.unknown
_1066107237.unknown
_1066107571.unknown
_1066107612.unknown
_1066107386.unknown
_1066107223.unknown
_1066107235.unknown
_1066107181.unknown
_1066106921.unknown
_1066106965.unknown
_1066107001.unknown
_1066106923.unknown
_1066105982.unknown
_1066106091.unknown
_1066105585.unknown
_1066209845.unknown
_1066209916.unknown
_1066209827.unknown
_1066209178.unknown
_1066208872.doc
� EMBED Equation.3 ���
_1066107724.unknown
_1066110072.unknown
_1066110258.unknown
_1066110936.unknown
_1066121576.unknown
_1066208756.unknown
_1066208830.unknown
_1066121729.unknown
_1066121327.unknown
_1066110378.unknown
_1066110533.unknown
_1066110371.unknown
_1066110098.unknown
_1066110121.unknown
_1066110074.unknown
_1066107826.unknown
_1066108106.unknown
_1066109998.unknown
_1066107840.unknown
_1066107820.unknown
_1066107821.unknown
_1066107748.unknown
_1066107178.unknown
_1066107237.unknown
_1066107571.unknown
_1066107612.unknown
_1066107386.unknown
_1066107223.unknown
_1066107235.unknown
_1066107181.unknown
_1066106921.unknown
_1066106965.unknown
_1066107001.unknown
_1066106923.unknown
_1066105982.unknown
_1066106091.unknown_1066105585.unknown
_1066208640.unknown
_1066208680.unknown
_1066208638.unknown
_1066113128.unknown
_1066121031.doc
� EMBED Equation.3 ���
_1066107724.unknown
_1066110072.unknown
_1066110936.unknown
_1066111713.unknown
_1066112017.unknown
_1066120878.unknown
_1066120905.unknown
_1066113696.unknown
_1066111859.unknown
_1066111566.unknown
_1066111704.unknown
_1066111281.unknown
_1066110258.unknown
_1066110378.unknown
_1066110533.unknown
_1066110371.unknown
_1066110098.unknown
_1066110121.unknown
_1066110074.unknown
_1066107826.unknown
_1066108106.unknown
_1066109998.unknown
_1066107840.unknown
_1066107820.unknown
_1066107821.unknown
_1066107748.unknown
_1066107178.unknown
_1066107237.unknown
_1066107571.unknown
_1066107612.unknown
_1066107386.unknown
_1066107223.unknown
_1066107235.unknown
_1066107181.unknown
_1066106921.unknown
_1066106965.unknown
_1066107001.unknown
_1066106923.unknown
_1066105982.unknown
_1066106091.unknown
_1066105585.unknown
_1066121929.doc
� EMBED Equation.3 ���
_1066107724.unknown
_1066110072.unknown
_1066110936.unknown
_1066111713.unknown
_1066120878.unknown
_1066121143.unknown
_1066121486.unknown
_1066121911.unknown
_1066121933.unknown
_1066121660.unknown
_1066121217.unknown
_1066120905.unknown
_1066112017.unknown
_1066113696.unknown
_1066111859.unknown
_1066111566.unknown
_1066111704.unknown
_1066111281.unknown
_1066110258.unknown
_1066110378.unknown
_1066110533.unknown
_1066110371.unknown
_1066110098.unknown
_1066110121.unknown
_1066110074.unknown
_1066107826.unknown
_1066108106.unknown
_1066109998.unknown
_1066107840.unknown
_1066107820.unknown
_1066107821.unknown
_1066107748.unknown
_1066107178.unknown
_1066107237.unknown
_1066107571.unknown
_1066107612.unknown
_1066107386.unknown
_1066107223.unknown
_1066107235.unknown
_1066107181.unknown
_1066106921.unknown
_1066106965.unknown
_1066107001.unknown
_1066106923.unknown
_1066105982.unknown
_1066106091.unknown
_1066105585.unknown
_1066122285.doc
� EMBED Equation.3 ���
_1066107724.unknown
_1066110072.unknown
_1066110936.unknown
_1066111713.unknown
_1066120878.unknown
_1066122024.unknown
_1066122289.unknown
_1066122316.unknown
_1066122037.unknown
_1066120905.unknown
_1066112017.unknown
_1066113696.unknown
_1066111859.unknown
_1066111566.unknown
_1066111704.unknown
_1066111281.unknown
_1066110258.unknown
_1066110378.unknown
_1066110533.unknown
_1066110371.unknown
_1066110098.unknown
_1066110121.unknown
_1066110074.unknown
_1066107826.unknown
_1066108106.unknown
_1066109998.unknown
_1066107840.unknown
_1066107820.unknown
_1066107821.unknown
_1066107748.unknown
_1066107178.unknown
_1066107237.unknown
_1066107571.unknown
_1066107612.unknown
_1066107386.unknown
_1066107223.unknown
_1066107235.unknown
_1066107181.unknown
_1066106921.unknown
_1066106965.unknown
_1066107001.unknown
_1066106923.unknown
_1066105982.unknown
_1066106091.unknown
_1066105585.unknown
_1066123697.doc
� EMBED Equation.3 ���
_1066107724.unknown
_1066110936.unknown
_1066120878.unknown
_1066122289.unknown
_1066122488.unknown
_1066123620.unknown
_1066123662.unknown
_1066123700.unknown
_1066123021.unknown
_1066122316.unknown
_1066122024.unknown
_1066122037.unknown
_1066120905.unknown
_1066111713.unknown
_1066112017.unknown
_1066113696.unknown
_1066111859.unknown
_1066111566.unknown
_1066111704.unknown
_1066111281.unknown
_1066110072.unknown
_1066110258.unknown
_1066110378.unknown
_1066110533.unknown
_1066110371.unknown
_1066110098.unknown
_1066110121.unknown
_1066110074.unknown
_1066107826.unknown
_1066108106.unknown
_1066109998.unknown
_1066107840.unknown
_1066107820.unknown
_1066107821.unknown
_1066107748.unknown
_1066107178.unknown
_1066107237.unknown
_1066107571.unknown
_1066107612.unknown
_1066107386.unknown
_1066107223.unknown
_1066107235.unknown
_1066107181.unknown
_1066106921.unknown
_1066106965.unknown
_1066107001.unknown
_1066106923.unknown
_1066105982.unknown
_1066106091.unknown
_1066105585.unknown
_1066123898.doc
� EMBED Equation.3 ���
_1066107724.unknown
_1066110936.unknown
_1066120878.unknown
_1066122289.unknown
_1066123620.unknown
_1066123845.unknown
_1066123874.unknown
_1066123901.unknown
_1066123662.unknown
_1066122488.unknown
_1066123021.unknown
_1066122316.unknown
_1066122024.unknown
_1066122037.unknown
_1066120905.unknown
_1066111713.unknown
_1066112017.unknown
_1066113696.unknown
_1066111859.unknown
_1066111566.unknown
_1066111704.unknown
_1066111281.unknown
_1066110072.unknown
_1066110258.unknown
_1066110378.unknown
_1066110533.unknown
_1066110371.unknown
_1066110098.unknown
_1066110121.unknown
_1066110074.unknown
_1066107826.unknown
_1066108106.unknown
_1066109998.unknown
_1066107840.unknown
_1066107820.unknown
_1066107821.unknown
_1066107748.unknown
_1066107178.unknown
_1066107237.unknown
_1066107571.unknown
_1066107612.unknown
_1066107386.unknown
_1066107223.unknown
_1066107235.unknown
_1066107181.unknown
_1066106921.unknown
_1066106965.unknown
_1066107001.unknown
_1066106923.unknown
_1066105982.unknown
_1066106091.unknown
_1066105585.unknown
_1066123956.doc
� EMBED Equation.3 ���
_1066107724.unknown
_1066110936.unknown
_1066120878.unknown
_1066122289.unknown
_1066123620.unknown
_1066123845.unknown
_1066123901.unknown
_1066123959.unknown
_1066123874.unknown
_1066123662.unknown
_1066122488.unknown
_1066123021.unknown
_1066122316.unknown
_1066122024.unknown
_1066122037.unknown
_1066120905.unknown
_1066111713.unknown
_1066112017.unknown
_1066113696.unknown
_1066111859.unknown
_1066111566.unknown
_1066111704.unknown
_1066111281.unknown
_1066110072.unknown
_1066110258.unknown
_1066110378.unknown
_1066110533.unknown
_1066110371.unknown
_1066110098.unknown
_1066110121.unknown
_1066110074.unknown
_1066107826.unknown
_1066108106.unknown
_1066109998.unknown
_1066107840.unknown
_1066107820.unknown
_1066107821.unknown
_1066107748.unknown
_1066107178.unknown
_1066107237.unknown_1066107571.unknown
_1066107612.unknown
_1066107386.unknown
_1066107223.unknown
_1066107235.unknown
_1066107181.unknown
_1066106921.unknown
_1066106965.unknown
_1066107001.unknown
_1066106923.unknown
_1066105982.unknown
_1066106091.unknown
_1066105585.unknown
_1066124022.doc
� EMBED Equation.3 ���
_1066107724.unknown
_1066110936.unknown
_1066120878.unknown
_1066122289.unknown
_1066123620.unknown
_1066123845.unknown
_1066123901.unknown
_1066123959.unknown
_1066124024.unknown
_1066123874.unknown
_1066123662.unknown
_1066122488.unknown
_1066123021.unknown
_1066122316.unknown
_1066122024.unknown
_1066122037.unknown
_1066120905.unknown
_1066111713.unknown
_1066112017.unknown
_1066113696.unknown
_1066111859.unknown
_1066111566.unknown
_1066111704.unknown
_1066111281.unknown
_1066110072.unknown
_1066110258.unknown
_1066110378.unknown
_1066110533.unknown
_1066110371.unknown
_1066110098.unknown
_1066110121.unknown
_1066110074.unknown
_1066107826.unknown
_1066108106.unknown
_1066109998.unknown
_1066107840.unknown
_1066107820.unknown
_1066107821.unknown
_1066107748.unknown
_1066107178.unknown
_1066107237.unknown
_1066107571.unknown
_1066107612.unknown
_1066107386.unknown
_1066107223.unknown
_1066107235.unknown
_1066107181.unknown
_1066106921.unknown
_1066106965.unknown
_1066107001.unknown
_1066106923.unknown
_1066105982.unknown
_1066106091.unknown
_1066105585.unknown
_1066124046.doc
� EMBED Equation.3 ���
_1066107724.unknown
_1066110072.unknown
_1066110936.unknown
_1066111713.unknown
_1066120878.unknown
_1066122289.unknown
_1066122488.unknown
_1066123999.unknown
_1066124050.unknown
_1066122316.unknown
_1066122024.unknown
_1066122037.unknown
_1066120905.unknown
_1066112017.unknown
_1066113696.unknown
_1066111859.unknown
_1066111566.unknown
_1066111704.unknown
_1066111281.unknown
_1066110258.unknown
_1066110378.unknown
_1066110533.unknown
_1066110371.unknown
_1066110098.unknown
_1066110121.unknown
_1066110074.unknown
_1066107826.unknown
_1066108106.unknown
_1066109998.unknown
_1066107840.unknown
_1066107820.unknown
_1066107821.unknown
_1066107748.unknown
_1066107178.unknown
_1066107237.unknown
_1066107571.unknown
_1066107612.unknown
_1066107386.unknown
_1066107223.unknown
_1066107235.unknown
_1066107181.unknown
_1066106921.unknown
_1066106965.unknown
_1066107001.unknown
_1066106923.unknown
_1066105982.unknown
_1066106091.unknown
_1066105585.unknown
_1066123995.doc
� EMBED Equation.3 ���
_1066107724.unknown
_1066110072.unknown
_1066110936.unknown
_1066111713.unknown
_1066120878.unknown
_1066122024.unknown
_1066122289.unknown
_1066122488.unknown
_1066123999.unknown
_1066122316.unknown
_1066122037.unknown
_1066120905.unknown
_1066112017.unknown
_1066113696.unknown
_1066111859.unknown
_1066111566.unknown
_1066111704.unknown
_1066111281.unknown
_1066110258.unknown
_1066110378.unknown
_1066110533.unknown
_1066110371.unknown
_1066110098.unknown
_1066110121.unknown
_1066110074.unknown
_1066107826.unknown
_1066108106.unknown
_1066109998.unknown
_1066107840.unknown
_1066107820.unknown
_1066107821.unknown
_1066107748.unknown
_1066107178.unknown
_1066107237.unknown
_1066107571.unknown
_1066107612.unknown
_1066107386.unknown
_1066107223.unknown
_1066107235.unknown
_1066107181.unknown
_1066106921.unknown
_1066106965.unknown
_1066107001.unknown
_1066106923.unknown
_1066105982.unknown
_1066106091.unknown
_1066105585.unknown
_1066123927.doc
� EMBED Equation.3 ���
_1066107724.unknown
_1066110936.unknown
_1066120878.unknown
_1066122289.unknown
_1066122488.unknown
_1066123143.unknown
_1066123194.unknown
_1066123929.unknown
_1066123159.unknown
_1066123021.unknown
_1066122316.unknown
_1066122024.unknown
_1066122037.unknown
_1066120905.unknown
_1066111713.unknown
_1066112017.unknown
_1066113696.unknown
_1066111859.unknown
_1066111566.unknown
_1066111704.unknown
_1066111281.unknown
_1066110072.unknown
_1066110258.unknown
_1066110378.unknown
_1066110533.unknown
_1066110371.unknown
_1066110098.unknown
_1066110121.unknown
_1066110074.unknown
_1066107826.unknown
_1066108106.unknown
_1066109998.unknown
_1066107840.unknown
_1066107820.unknown
_1066107821.unknown
_1066107748.unknown
_1066107178.unknown
_1066107237.unknown
_1066107571.unknown
_1066107612.unknown
_1066107386.unknown
_1066107223.unknown
_1066107235.unknown
_1066107181.unknown
_1066106921.unknown
_1066106965.unknown
_1066107001.unknown
_1066106923.unknown
_1066105982.unknown
_1066106091.unknown
_1066105585.unknown
_1066123842.doc
� EMBED Equation.3 ���
_1066107724.unknown
_1066110936.unknown
_1066120878.unknown
_1066122289.unknown
_1066122488.unknown
_1066123620.unknown
_1066123662.unknown
_1066123845.unknown
_1066123021.unknown
_1066122316.unknown
_1066122024.unknown
_1066122037.unknown
_1066120905.unknown
_1066111713.unknown
_1066112017.unknown
_1066113696.unknown
_1066111859.unknown
_1066111566.unknown
_1066111704.unknown
_1066111281.unknown
_1066110072.unknown
_1066110258.unknown
_1066110378.unknown
_1066110533.unknown
_1066110371.unknown
_1066110098.unknown
_1066110121.unknown
_1066110074.unknown
_1066107826.unknown
_1066108106.unknown
_1066109998.unknown
_1066107840.unknown
_1066107820.unknown
_1066107821.unknown
_1066107748.unknown
_1066107178.unknown
_1066107237.unknown
_1066107571.unknown
_1066107612.unknown
_1066107386.unknown
_1066107223.unknown
_1066107235.unknown
_1066107181.unknown
_1066106921.unknown
_1066106965.unknown
_1066107001.unknown
_1066106923.unknown
_1066105982.unknown
_1066106091.unknown
_1066105585.unknown
_1066123871.doc
� EMBED Equation.3 ���
_1066107724.unknown
_1066110936.unknown
_1066120878.unknown
_1066122289.unknown
_1066122488.unknown
_1066123620.unknown
_1066123845.unknown
_1066123874.unknown
_1066123662.unknown
_1066123021.unknown
_1066122316.unknown
_1066122024.unknown
_1066122037.unknown
_1066120905.unknown
_1066111713.unknown
_1066112017.unknown_1066113696.unknown
_1066111859.unknown
_1066111566.unknown
_1066111704.unknown
_1066111281.unknown
_1066110072.unknown
_1066110258.unknown
_1066110378.unknown
_1066110533.unknown
_1066110371.unknown
_1066110098.unknown
_1066110121.unknown
_1066110074.unknown
_1066107826.unknown
_1066108106.unknown
_1066109998.unknown
_1066107840.unknown
_1066107820.unknown
_1066107821.unknown
_1066107748.unknown
_1066107178.unknown
_1066107237.unknown
_1066107571.unknown
_1066107612.unknown
_1066107386.unknown
_1066107223.unknown
_1066107235.unknown
_1066107181.unknown
_1066106921.unknown
_1066106965.unknown
_1066107001.unknown
_1066106923.unknown
_1066105982.unknown
_1066106091.unknown
_1066105585.unknown
_1066123805.doc
� EMBED Equation.3 ���
_1066107724.unknown
_1066110072.unknown
_1066110936.unknown
_1066111713.unknown
_1066120878.unknown
_1066122289.unknown
_1066122488.unknown
_1066123021.unknown
_1066123812.unknown
_1066122316.unknown
_1066122024.unknown
_1066122037.unknown
_1066120905.unknown
_1066112017.unknown
_1066113696.unknown
_1066111859.unknown
_1066111566.unknown
_1066111704.unknown
_1066111281.unknown
_1066110258.unknown
_1066110378.unknown
_1066110533.unknown
_1066110371.unknown
_1066110098.unknown
_1066110121.unknown
_1066110074.unknown
_1066107826.unknown
_1066108106.unknown
_1066109998.unknown
_1066107840.unknown
_1066107820.unknown
_1066107821.unknown
_1066107748.unknown
_1066107178.unknown
_1066107237.unknown
_1066107571.unknown
_1066107612.unknown
_1066107386.unknown
_1066107223.unknown
_1066107235.unknown
_1066107181.unknown
_1066106921.unknown
_1066106965.unknown
_1066107001.unknown
_1066106923.unknown
_1066105982.unknown
_1066106091.unknown
_1066105585.unknown
_1066123076.doc
� EMBED Equation.3 ���
_1066107724.unknown
_1066110072.unknown
_1066110936.unknown
_1066111713.unknown
_1066120878.unknown
_1066122289.unknown
_1066122529.unknown
_1066122900.unknown
_1066123081.unknown
_1066122316.unknown
_1066122024.unknown
_1066122037.unknown
_1066120905.unknown
_1066112017.unknown
_1066113696.unknown
_1066111859.unknown
_1066111566.unknown
_1066111704.unknown
_1066111281.unknown
_1066110258.unknown
_1066110378.unknown
_1066110533.unknown
_1066110371.unknown
_1066110098.unknown
_1066110121.unknown
_1066110074.unknown
_1066107826.unknown
_1066108106.unknown
_1066109998.unknown
_1066107840.unknown
_1066107820.unknown
_1066107821.unknown
_1066107748.unknown
_1066107178.unknown
_1066107237.unknown
_1066107571.unknown
_1066107612.unknown
_1066107386.unknown
_1066107223.unknown
_1066107235.unknown
_1066107181.unknown
_1066106921.unknown
_1066106965.unknown
_1066107001.unknown
_1066106923.unknown
_1066105982.unknown
_1066106091.unknown
_1066105585.unknown
_1066123287.doc
� EMBED Equation.3 ���
_1066107724.unknown
_1066110072.unknown
_1066110936.unknown
_1066120878.unknown
_1066122289.unknown
_1066122488.unknown
_1066123290.unknown
_1066123812.unknown
_1066123021.unknown
_1066122316.unknown
_1066122024.unknown
_1066122037.unknown
_1066120905.unknown
_1066111713.unknown
_1066112017.unknown
_1066113696.unknown
_1066111859.unknown
_1066111566.unknown
_1066111704.unknown
_1066111281.unknown
_1066110258.unknown
_1066110378.unknown
_1066110533.unknown
_1066110371.unknown
_1066110098.unknown
_1066110121.unknown
_1066110074.unknown
_1066107826.unknown
_1066108106.unknown
_1066109998.unknown
_1066107840.unknown
_1066107820.unknown
_1066107821.unknown
_1066107748.unknown
_1066107178.unknown
_1066107237.unknown
_1066107571.unknown
_1066107612.unknown
_1066107386.unknown
_1066107223.unknown
_1066107235.unknown
_1066107181.unknown
_1066106921.unknown
_1066106965.unknown
_1066107001.unknown
_1066106923.unknown
_1066105982.unknown
_1066106091.unknown
_1066105585.unknown
_1066123392.doc
� EMBED Equation.3 ���
_1066107724.unknown
_1066110936.unknown
_1066120878.unknown
_1066122289.unknown
_1066123394.unknown
_1066123662.unknown
_1066123874.unknown
_1066123901.unknown
_1066123959.unknown
_1066123845.unknown
_1066123620.unknown
_1066122488.unknown
_1066123021.unknown
_1066122316.unknown
_1066122024.unknown
_1066122037.unknown
_1066120905.unknown
_1066111713.unknown
_1066112017.unknown
_1066113696.unknown
_1066111859.unknown
_1066111566.unknown
_1066111704.unknown
_1066111281.unknown
_1066110072.unknown
_1066110258.unknown
_1066110378.unknown
_1066110533.unknown
_1066110371.unknown
_1066110098.unknown
_1066110121.unknown
_1066110074.unknown
_1066107826.unknown
_1066108106.unknown
_1066109998.unknown
_1066107840.unknown
_1066107820.unknown
_1066107821.unknown
_1066107748.unknown
_1066107178.unknown
_1066107237.unknown
_1066107571.unknown
_1066107612.unknown
_1066107386.unknown
_1066107223.unknown
_1066107235.unknown
_1066107181.unknown
_1066106921.unknown
_1066106965.unknown
_1066107001.unknown
_1066106923.unknown
_1066105982.unknown
_1066106091.unknown
_1066105585.unknown
_1066123461.doc
� EMBED Equation.3 ���
_1066107724.unknown
_1066110936.unknown
_1066120878.unknown
_1066122289.unknown
_1066122488.unknown
_1066123290.unknown
_1066123464.unknown
_1066123812.unknown
_1066123021.unknown
_1066122316.unknown
_1066122024.unknown
_1066122037.unknown
_1066120905.unknown
_1066111713.unknown
_1066112017.unknown
_1066113696.unknown
_1066111859.unknown
_1066111566.unknown
_1066111704.unknown
_1066111281.unknown
_1066110072.unknown
_1066110258.unknown
_1066110378.unknown
_1066110533.unknown
_1066110371.unknown
_1066110098.unknown
_1066110121.unknown
_1066110074.unknown
_1066107826.unknown
_1066108106.unknown
_1066109998.unknown
_1066107840.unknown
_1066107820.unknown
_1066107821.unknown
_1066107748.unknown
_1066107178.unknown
_1066107237.unknown
_1066107571.unknown
_1066107612.unknown
_1066107386.unknown
_1066107223.unknown
_1066107235.unknown
_1066107181.unknown
_1066106921.unknown
_1066106965.unknown
_1066107001.unknown
_1066106923.unknown
_1066105982.unknown_1066106091.unknown
_1066105585.unknown
_1066123581.doc
� EMBED Equation.3 ���
_1066107724.unknown
_1066110936.unknown
_1066120878.unknown
_1066122289.unknown
_1066123394.unknown
_1066123662.unknown
_1066123874.unknown
_1066123901.unknown
_1066123959.unknown
_1066123845.unknown
_1066123584.unknown
_1066123620.unknown
_1066123504.unknown
_1066122488.unknown
_1066123021.unknown
_1066122316.unknown
_1066122024.unknown
_1066122037.unknown
_1066120905.unknown
_1066111713.unknown
_1066112017.unknown
_1066113696.unknown
_1066111859.unknown
_1066111566.unknown
_1066111704.unknown
_1066111281.unknown
_1066110072.unknown
_1066110258.unknown
_1066110378.unknown
_1066110533.unknown
_1066110371.unknown
_1066110098.unknown
_1066110121.unknown
_1066110074.unknown
_1066107826.unknown
_1066108106.unknown
_1066109998.unknown
_1066107840.unknown
_1066107820.unknown
_1066107821.unknown
_1066107748.unknown
_1066107178.unknown
_1066107237.unknown
_1066107571.unknown
_1066107612.unknown
_1066107386.unknown
_1066107223.unknown
_1066107235.unknown
_1066107181.unknown
_1066106921.unknown
_1066106965.unknown
_1066107001.unknown
_1066106923.unknown
_1066105982.unknown
_1066106091.unknown
_1066105585.unknown
_1066123611.doc
� EMBED Equation.3 ���
_1066107724.unknown
_1066110936.unknown
_1066120878.unknown
_1066123394.unknown
_1066123620.unknown
_1066123845.unknown
_1066123901.unknown
_1066123959.unknown
_1066123874.unknown
_1066123662.unknown
_1066123584.unknown
_1066123614.unknown
_1066123504.unknown
_1066122289.unknown
_1066122488.unknown
_1066123021.unknown
_1066122316.unknown
_1066122024.unknown
_1066122037.unknown
_1066120905.unknown
_1066111713.unknown
_1066112017.unknown
_1066113696.unknown
_1066111859.unknown
_1066111566.unknown
_1066111704.unknown
_1066111281.unknown
_1066110072.unknown
_1066110258.unknown
_1066110378.unknown
_1066110533.unknown
_1066110371.unknown
_1066110098.unknown
_1066110121.unknown
_1066110074.unknown
_1066107826.unknown
_1066108106.unknown
_1066109998.unknown
_1066107840.unknown
_1066107820.unknown
_1066107821.unknown
_1066107748.unknown
_1066107178.unknown
_1066107237.unknown
_1066107571.unknown
_1066107612.unknown
_1066107386.unknown
_1066107223.unknown
_1066107235.unknown
_1066107181.unknown
_1066106921.unknown
_1066106965.unknown
_1066107001.unknown
_1066106923.unknown
_1066105982.unknown
_1066106091.unknown
_1066105585.unknown
_1066123501.doc
� EMBED Equation.3 ���
_1066107724.unknown
_1066110936.unknown
_1066120878.unknown
_1066122289.unknown
_1066123394.unknown
_1066123620.unknown
_1066123845.unknown
_1066123901.unknown
_1066123959.unknown
_1066123874.unknown
_1066123662.unknown
_1066123504.unknown
_1066122488.unknown
_1066123021.unknown
_1066122316.unknown
_1066122024.unknown
_1066122037.unknown
_1066120905.unknown
_1066111713.unknown
_1066112017.unknown
_1066113696.unknown
_1066111859.unknown
_1066111566.unknown
_1066111704.unknown
_1066111281.unknown
_1066110072.unknown
_1066110258.unknown
_1066110378.unknown
_1066110533.unknown
_1066110371.unknown
_1066110098.unknown
_1066110121.unknown
_1066110074.unknown
_1066107826.unknown
_1066108106.unknown
_1066109998.unknown
_1066107840.unknown
_1066107820.unknown
_1066107821.unknown
_1066107748.unknown
_1066107178.unknown
_1066107237.unknown
_1066107571.unknown
_1066107612.unknown
_1066107386.unknown
_1066107223.unknown
_1066107235.unknown
_1066107181.unknown
_1066106921.unknown
_1066106965.unknown
_1066107001.unknown
_1066106923.unknown
_1066105982.unknown
_1066106091.unknown
_1066105585.unknown
_1066123422.doc
� EMBED Equation.3 ���
_1066107724.unknown
_1066110936.unknown
_1066120878.unknown
_1066122289.unknown
_1066123143.unknown
_1066123194.unknown
_1066123425.unknown
_1066123929.unknown
_1066123159.unknown
_1066122488.unknown
_1066123021.unknown
_1066122316.unknown
_1066122024.unknown
_1066122037.unknown
_1066120905.unknown
_1066111713.unknown
_1066112017.unknown
_1066113696.unknown
_1066111859.unknown
_1066111566.unknown
_1066111704.unknown
_1066111281.unknown
_1066110072.unknown
_1066110258.unknown
_1066110378.unknown
_1066110533.unknown
_1066110371.unknown
_1066110098.unknown
_1066110121.unknown
_1066110074.unknown
_1066107826.unknown
_1066108106.unknown
_1066109998.unknown
_1066107840.unknown
_1066107820.unknown
_1066107821.unknown
_1066107748.unknown
_1066107178.unknown
_1066107237.unknown
_1066107571.unknown
_1066107612.unknown
_1066107386.unknown
_1066107223.unknown
_1066107235.unknown
_1066107181.unknown
_1066106921.unknown
_1066106965.unknown
_1066107001.unknown
_1066106923.unknown
_1066105982.unknown
_1066106091.unknown
_1066105585.unknown
_1066123307.doc
� EMBED Equation.3 ���
_1066107724.unknown
_1066110936.unknown
_1066120878.unknown
_1066122289.unknown
_1066122488.unknown
_1066123310.unknown
_1066123336.unknown
_1066123812.unknown
_1066123021.unknown
_1066122316.unknown
_1066122024.unknown
_1066122037.unknown
_1066120905.unknown
_1066111713.unknown
_1066112017.unknown
_1066113696.unknown
_1066111859.unknown
_1066111566.unknown
_1066111704.unknown
_1066111281.unknown
_1066110072.unknown
_1066110258.unknown
_1066110378.unknown
_1066110533.unknown
_1066110371.unknown
_1066110098.unknown
_1066110121.unknown
_1066110074.unknown
_1066107826.unknown
_1066108106.unknown
_1066109998.unknown
_1066107840.unknown
_1066107820.unknown
_1066107821.unknown
_1066107748.unknown
_1066107178.unknown
_1066107237.unknown
_1066107571.unknown
_1066107612.unknown
_1066107386.unknown
_1066107223.unknown
_1066107235.unknown
_1066107181.unknown
_1066106921.unknown
_1066106965.unknown
_1066107001.unknown
_1066106923.unknown
_1066105982.unknown
_1066106091.unknown
_1066105585.unknown
_1066123125.doc
� EMBED Equation.3 ���
_1066107724.unknown
_1066110072.unknown
_1066110936.unknown
_1066120878.unknown
_1066122289.unknown
_1066122488.unknown
_1066123143.unknown_1066123159.unknown
_1066123021.unknown
_1066122316.unknown
_1066122024.unknown
_1066122037.unknown
_1066120905.unknown
_1066111713.unknown
_1066112017.unknown
_1066113696.unknown
_1066111859.unknown
_1066111566.unknown
_1066111704.unknown
_1066111281.unknown
_1066110258.unknown
_1066110378.unknown
_1066110533.unknown
_1066110371.unknown
_1066110098.unknown
_1066110121.unknown
_1066110074.unknown
_1066107826.unknown
_1066108106.unknown
_1066109998.unknown
_1066107840.unknown
_1066107820.unknown
_1066107821.unknown
_1066107748.unknown
_1066107178.unknown
_1066107237.unknown
_1066107571.unknown
_1066107612.unknown
_1066107386.unknown
_1066107223.unknown
_1066107235.unknown
_1066107181.unknown
_1066106921.unknown
_1066106965.unknown
_1066107001.unknown
_1066106923.unknown
_1066105982.unknown
_1066106091.unknown
_1066105585.unknown
_1066123191.doc
� EMBED Equation.3 ���
_1066107724.unknown
_1066110936.unknown
_1066120878.unknown
_1066122289.unknown
_1066122488.unknown
_1066123143.unknown
_1066123159.unknown
_1066123194.unknown
_1066123021.unknown
_1066122316.unknown
_1066122024.unknown
_1066122037.unknown
_1066120905.unknown
_1066111713.unknown
_1066112017.unknown
_1066113696.unknown
_1066111859.unknown
_1066111566.unknown
_1066111704.unknown
_1066111281.unknown
_1066110072.unknown
_1066110258.unknown
_1066110378.unknown
_1066110533.unknown
_1066110371.unknown
_1066110098.unknown
_1066110121.unknown
_1066110074.unknown
_1066107826.unknown
_1066108106.unknown
_1066109998.unknown
_1066107840.unknown
_1066107820.unknown
_1066107821.unknown
_1066107748.unknown
_1066107178.unknown
_1066107237.unknown
_1066107571.unknown
_1066107612.unknown
_1066107386.unknown
_1066107223.unknown
_1066107235.unknown
_1066107181.unknown
_1066106921.unknown
_1066106965.unknown
_1066107001.unknown
_1066106923.unknown
_1066105982.unknown
_1066106091.unknown
_1066105585.unknown
_1066122486.doc
� EMBED Equation.3 ���
_1066107724.unknown
_1066110072.unknown
_1066110936.unknown
_1066111713.unknown
_1066120878.unknown
_1066122024.unknown
_1066122289.unknown
_1066122316.unknown
_1066122488.unknown
_1066122037.unknown
_1066120905.unknown
_1066112017.unknown
_1066113696.unknown
_1066111859.unknown
_1066111566.unknown
_1066111704.unknown
_1066111281.unknown
_1066110258.unknown
_1066110378.unknown
_1066110533.unknown
_1066110371.unknown
_1066110098.unknown
_1066110121.unknown
_1066110074.unknown
_1066107826.unknown
_1066108106.unknown
_1066109998.unknown
_1066107840.unknown
_1066107820.unknown
_1066107821.unknown
_1066107748.unknown
_1066107178.unknown
_1066107237.unknown
_1066107571.unknown
_1066107612.unknown
_1066107386.unknown
_1066107223.unknown
_1066107235.unknown
_1066107181.unknown
_1066106921.unknown
_1066106965.unknown
_1066107001.unknown
_1066106923.unknown
_1066105982.unknown
_1066106091.unknown
_1066105585.unknown
_1066122527.doc
� EMBED Equation.3 ���
_1066107724.unknown
_1066110072.unknown
_1066110936.unknown
_1066111713.unknown
_1066120878.unknown
_1066122024.unknown
_1066122289.unknown
_1066122529.unknown
_1066122900.unknown
_1066122316.unknown
_1066122037.unknown
_1066120905.unknown
_1066112017.unknown
_1066113696.unknown
_1066111859.unknown
_1066111566.unknown
_1066111704.unknown
_1066111281.unknown
_1066110258.unknown
_1066110378.unknown
_1066110533.unknown
_1066110371.unknown
_1066110098.unknown
_1066110121.unknown
_1066110074.unknown
_1066107826.unknown
_1066108106.unknown
_1066109998.unknown
_1066107840.unknown
_1066107820.unknown
_1066107821.unknown
_1066107748.unknown
_1066107178.unknown
_1066107237.unknown
_1066107571.unknown
_1066107612.unknown
_1066107386.unknown
_1066107223.unknown
_1066107235.unknown
_1066107181.unknown
_1066106921.unknown
_1066106965.unknown
_1066107001.unknown
_1066106923.unknown
_1066105982.unknown
_1066106091.unknown
_1066105585.unknown
_1066123019.doc
� EMBED Equation.3 ���
_1066107724.unknown
_1066110072.unknown
_1066110936.unknown
_1066111713.unknown
_1066120878.unknown
_1066122024.unknown
_1066122289.unknown
_1066122488.unknown
_1066123021.unknown
_1066122316.unknown
_1066122037.unknown
_1066120905.unknown
_1066112017.unknown
_1066113696.unknown
_1066111859.unknown
_1066111566.unknown
_1066111704.unknown
_1066111281.unknown
_1066110258.unknown
_1066110378.unknown
_1066110533.unknown
_1066110371.unknown
_1066110098.unknown
_1066110121.unknown
_1066110074.unknown
_1066107826.unknown
_1066108106.unknown
_1066109998.unknown
_1066107840.unknown
_1066107820.unknown
_1066107821.unknown
_1066107748.unknown
_1066107178.unknown
_1066107237.unknown
_1066107571.unknown
_1066107612.unknown
_1066107386.unknown
_1066107223.unknown
_1066107235.unknown
_1066107181.unknown
_1066106921.unknown
_1066106965.unknown
_1066107001.unknown
_1066106923.unknown
_1066105982.unknown
_1066106091.unknown
_1066105585.unknown
_1066122343.doc
� EMBED Equation.3 ���
_1066107724.unknown
_1066110072.unknown
_1066110936.unknown
_1066111713.unknown
_1066120878.unknown
_1066122024.unknown
_1066122289.unknown
_1066122316.unknown
_1066122345.unknown
_1066122037.unknown
_1066120905.unknown
_1066112017.unknown
_1066113696.unknown
_1066111859.unknown
_1066111566.unknown
_1066111704.unknown
_1066111281.unknown
_1066110258.unknown
_1066110378.unknown
_1066110533.unknown
_1066110371.unknown
_1066110098.unknown
_1066110121.unknown
_1066110074.unknown
_1066107826.unknown
_1066108106.unknown
_1066109998.unknown
_1066107840.unknown
_1066107820.unknown
_1066107821.unknown
_1066107748.unknown
_1066107178.unknown
_1066107237.unknown
_1066107571.unknown
_1066107612.unknown
_1066107386.unknown
_1066107223.unknown
_1066107235.unknown
_1066107181.unknown
_1066106921.unknown
_1066106965.unknown
_1066107001.unknown
_1066106923.unknown
_1066105982.unknown
_1066106091.unknown
_1066105585.unknown
_1066122020.doc
� EMBED Equation.3 ���
_1066107724.unknown_1066110072.unknown
_1066110936.unknown
_1066111713.unknown
_1066112017.unknown
_1066120878.unknown
_1066122024.unknown
_1066122037.unknown
_1066120905.unknown
_1066113696.unknown
_1066111859.unknown
_1066111566.unknown
_1066111704.unknown
_1066111281.unknown
_1066110258.unknown
_1066110378.unknown
_1066110533.unknown
_1066110371.unknown
_1066110098.unknown
_1066110121.unknown
_1066110074.unknown
_1066107826.unknown
_1066108106.unknown
_1066109998.unknown
_1066107840.unknown
_1066107820.unknown
_1066107821.unknown
_1066107748.unknown
_1066107178.unknown
_1066107237.unknown
_1066107571.unknown
_1066107612.unknown
_1066107386.unknown
_1066107223.unknown
_1066107235.unknown
_1066107181.unknown
_1066106921.unknown
_1066106965.unknown
_1066107001.unknown
_1066106923.unknown
_1066105982.unknown
_1066106091.unknown
_1066105585.unknown
_1066122194.doc
� EMBED Equation.3 ���
_1066107724.unknown
_1066110072.unknown
_1066110936.unknown
_1066111713.unknown
_1066120878.unknown
_1066122024.unknown
_1066122037.unknown
_1066122199.unknown
_1066120905.unknown
_1066112017.unknown
_1066113696.unknown
_1066111859.unknown
_1066111566.unknown
_1066111704.unknown
_1066111281.unknown
_1066110258.unknown
_1066110378.unknown
_1066110533.unknown
_1066110371.unknown
_1066110098.unknown
_1066110121.unknown
_1066110074.unknown
_1066107826.unknown
_1066108106.unknown
_1066109998.unknown
_1066107840.unknown
_1066107820.unknown
_1066107821.unknown
_1066107748.unknown
_1066107178.unknown
_1066107237.unknown
_1066107571.unknown
_1066107612.unknown
_1066107386.unknown
_1066107223.unknown
_1066107235.unknown
_1066107181.unknown
_1066106921.unknown
_1066106965.unknown
_1066107001.unknown
_1066106923.unknown
_1066105982.unknown
_1066106091.unknown
_1066105585.unknown
_1066121950.doc
� EMBED Equation.3 ���
_1066107724.unknown
_1066110072.unknown
_1066110936.unknown
_1066111713.unknown
_1066120878.unknown
_1066121143.unknown
_1066121486.unknown
_1066121871.unknown
_1066121957.unknown
_1066121660.unknown
_1066121217.unknown
_1066120905.unknown
_1066112017.unknown
_1066113696.unknown
_1066111859.unknown
_1066111566.unknown
_1066111704.unknown
_1066111281.unknown
_1066110258.unknown
_1066110378.unknown
_1066110533.unknown
_1066110371.unknown
_1066110098.unknown
_1066110121.unknown
_1066110074.unknown
_1066107826.unknown
_1066108106.unknown
_1066109998.unknown
_1066107840.unknown
_1066107820.unknown
_1066107821.unknown
_1066107748.unknown
_1066107178.unknown
_1066107237.unknown
_1066107571.unknown
_1066107612.unknown
_1066107386.unknown
_1066107223.unknown
_1066107235.unknown
_1066107181.unknown
_1066106921.unknown
_1066106965.unknown
_1066107001.unknown
_1066106923.unknown
_1066105982.unknown
_1066106091.unknown
_1066105585.unknown
_1066121445.doc
� EMBED Equation.3 ���
_1066107724.unknown
_1066110072.unknown
_1066110936.unknown
_1066111713.unknown
_1066112017.unknown
_1066120878.unknown
_1066121143.unknown
_1066121448.unknown
_1066120905.unknown
_1066113696.unknown
_1066111859.unknown
_1066111566.unknown
_1066111704.unknown
_1066111281.unknown
_1066110258.unknown
_1066110378.unknown
_1066110533.unknown
_1066110371.unknown
_1066110098.unknown
_1066110121.unknown
_1066110074.unknown
_1066107826.unknown
_1066108106.unknown
_1066109998.unknown
_1066107840.unknown
_1066107820.unknown
_1066107821.unknown
_1066107748.unknown
_1066107178.unknown
_1066107237.unknown
_1066107571.unknown
_1066107612.unknown
_1066107386.unknown
_1066107223.unknown
_1066107235.unknown
_1066107181.unknown
_1066106921.unknown
_1066106965.unknown
_1066107001.unknown
_1066106923.unknown
_1066105982.unknown
_1066106091.unknown
_1066105585.unknown
_1066121772.doc
� EMBED Equation.3 ���
_1066107724.unknown
_1066110072.unknown
_1066110936.unknown
_1066111713.unknown
_1066120878.unknown
_1066121143.unknown
_1066121486.unknown
_1066121660.unknown
_1066121871.unknown
_1066121217.unknown
_1066120905.unknown
_1066112017.unknown
_1066113696.unknown
_1066111859.unknown
_1066111566.unknown
_1066111704.unknown
_1066111281.unknown
_1066110258.unknown
_1066110378.unknown
_1066110533.unknown
_1066110371.unknown
_1066110098.unknown
_1066110121.unknown
_1066110074.unknown
_1066107826.unknown
_1066108106.unknown
_1066109998.unknown
_1066107840.unknown
_1066107820.unknown
_1066107821.unknown
_1066107748.unknown
_1066107178.unknown
_1066107237.unknown
_1066107571.unknown
_1066107612.unknown
_1066107386.unknown
_1066107223.unknown
_1066107235.unknown
_1066107181.unknown
_1066106921.unknown
_1066106965.unknown
_1066107001.unknown
_1066106923.unknown
_1066105982.unknown
_1066106091.unknown
_1066105585.unknown
_1066121904.doc
� EMBED Equation.3 ���
_1066107724.unknown
_1066110072.unknown
_1066110936.unknown
_1066111713.unknown
_1066120878.unknown
_1066121143.unknown
_1066121486.unknown
_1066121660.unknown
_1066121911.unknown
_1066121217.unknown
_1066120905.unknown
_1066112017.unknown
_1066113696.unknown
_1066111859.unknown
_1066111566.unknown
_1066111704.unknown
_1066111281.unknown
_1066110258.unknown
_1066110378.unknown
_1066110533.unknown
_1066110371.unknown
_1066110098.unknown
_1066110121.unknown
_1066110074.unknown
_1066107826.unknown
_1066108106.unknown
_1066109998.unknown
_1066107840.unknown
_1066107820.unknown
_1066107821.unknown
_1066107748.unknown
_1066107178.unknown
_1066107237.unknown
_1066107571.unknown
_1066107612.unknown
_1066107386.unknown
_1066107223.unknown
_1066107235.unknown
_1066107181.unknown
_1066106921.unknown
_1066106965.unknown
_1066107001.unknown
_1066106923.unknown
_1066105982.unknown
_1066106091.unknown
_1066105585.unknown
_1066121569.doc
� EMBED Equation.3 ���
_1066107724.unknown
_1066110072.unknown
_1066110258.unknown
_1066110378.unknown
_1066110936.unknown
_1066121576.unknown
_1066121729.unknown
_1066121327.unknown
_1066110533.unknown
_1066110371.unknown
_1066110098.unknown