Cohomology of Siegel varieties by Mokrane A., Polo P., Tilouine J.

By Mokrane A., Polo P., Tilouine J.

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During the next month, he deposited $80, wrote a check for $120, made another deposit of $25, wrote two checks: one for $60 and the other for $32. He was also assessed a monthly service charge of $5. What was his balance at the end of the month? In Problems 147 and 148, write an inequality using an absolute value to describe each statement. 147. x is at least 6 units from 4. 148. x is more than 5 units from 2. 149. S. Voltage In the United States, normal household voltage is 110 volts. It is acceptable for the actual voltage x to differ from normal by at most 5 volts.

A more formal definition of absolute value is given next. The absolute value of a real number a, denoted by the symbol ƒ a ƒ , is defined by the rules ƒ a ƒ = a if a Ú 0 and ƒ a ƒ = -a if a 6 0 For example, since -4 6 0, the second rule must be used to get ƒ -4 ƒ = -1-42 = 4. EXAMPLE 3 Computing Absolute Value (a) ƒ 8 ƒ = 8 (b) ƒ 0 ƒ = 0 (c) ƒ -15 ƒ = -1-152 = 15 ᭹ Look again at Figure 13. The distance from -4 to 3 is 7 units. This distance is the difference 3 - 1-42, obtained by subtracting the smaller coordinate from the larger.

Collectively, the symbols 6, 7, …, and Ú are called inequality symbols. Note that a 6 b and b 7 a mean the same thing. It does not matter whether we write 2 6 3 or 3 7 2. Furthermore, if a 6 b or if b 7 a, then the difference b - a is positive. Do you see why? Using Inequality Symbols (a) 3 6 7 (d) -8 6 -4 (b) -8 7 -16 (e) 4 7 -1 (c) -6 6 0 (f) 8 7 0 ᭹ In Example 1(a), we conclude that 3 6 7 either because 3 is to the left of 7 on the real number line or because the difference, 7 - 3 = 4, is a positive real number.

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