By Yorozu S.

We provide a generalization of the end result received by way of C. Currais-Bosch. Weconsider the -operator linked to a transverse Killing box v on acomplete foliated Riemannian manifold . lower than a definite assumption,we turn out that, for every belongs to the Lie algebra of the linearholonomy crew . a unique case of our end result, the model of the foliationby issues, implies the implications given by means of B. Kostant (compact case) andC. Currfis-Bosch (non-compact case).

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1), we have a = −1, b = 2, and c = 0. Since a < 0, the graph of the function has a shape. The graph intersects the x-axis at the origin since y = 2x − x2 = when x = 0. The other intercept (intersection) with the x-axis is the other root of the equation 2x − x2 = 0, namely x = 2. This is evident since 2x − x2 = (2 − x)x. Therefore, one of the roots of the equation 2x − x2 = 0 is x = 0. The other root is x = 2. 3, and the graph of the function is shown in Fig. 3. Finally, we consider the function f (x) = x2 − 2x + 2.

The corresponding price, P0 , and quantity, Q0 , are called the equilibrium price and quantity. It is also of interest to observe the eﬀect of a shift of the market price away from its equilibrium price. 12 The demand and supply equations of a good are given by 4P = −Qd + 240, 5P = Qs + 30. 42 Elements of Mathematics for Economics and Finance Determine the equilibrium price and quantity. Solution. At market equilibrium, we have Qd = Qs = Q, say , where Q is the equilibrium quantity. In this case, the demand and supply equations become 4P = −Q + 240, 5P = Q + 30.

1 Introduction In this book, we will be concerned primarily with the analysis of the relationship between two or more variables. For example, we will be interested in the relationship between economic entities or variables such as – total cost and output, – price and quantity in an analysis of demand and supply, – production and factors of production such as labour and capital. If one variable, say y, changes in an entirely predictable way in terms of another variable, say x, then, under certain conditions (to be deﬁned precisely in Chapter 4), we say that y is a function of x.