By K. FARAHMAND
ISBN-10: 0582356229
ISBN-13: 9780582356221
Subject matters in Random Polynomials offers a rigorous and accomplished therapy of the mathematical habit of other kinds of random polynomials. those polynomials-the topic of intensive contemporary research-have many functions in physics, economics, and facts. the most effects are awarded in any such model that they are often understood and utilized by readers whose wisdom of chance accommodates little greater than easy chance conception and stochastic procedures.
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Extra info for Topics in Random Polynomials
Example text
1 are conveniently simplifications of those used in Kac [71]. 1 If the coefficients of P{x) are independent standard normal random variables, then the mathematical expectation of the number of real roots of P{x) = 0 satisfies 2 E N {—oo,oo) ~ —logn. 12) where A2 - C^. Since the distribution of the a, and the —aj,{j = 0, 1, 2, . . , n — 1) is the same, E N {—cx>,0) = E N { 0 ,oo). j{l/xY~^~^ is in the form of P ( l/ x ) , we conclude E N { 0 , 1) = E N { 1 , oo). Similarly E N {—oo, —1) = E N {—1 , 0 ).
Ak < ak+i = b be abscissas of the turning points. We therefore have / co s{ /^ (i)} |e '(i)| dt = Y . Ja co s{ /^ (i)} |^ '(i)| dt Joij = E - / j =0 •'“j ^ A + {sin cos{/^(i)}^'(i) dt (a^+i) - sin f^ { a j )} ~ 7 ’ where we allocate the positive sign if ^(i) is increasing between aj and aj+i and the negative sign if it is decreasing in th a t interval. j)}] ^ i =0 N{a,b). Exam ple The following example, due to Kac [73], shows an easy application of the above theorem to random polynomials. We apply the theorem to a wide class of polynomials defined in ( 1.
20), we have Now we evaluate /i( l,o o ) . 29) we obtain / . ( 1, 00) < / “ 4 ^ ^ Jl 7tA2 * := { 1 - '> % )} ■ '" Jt3 7r(l - y '■) dy, and therefore the result for the case of A = 0 in the previous section can be used. 16) we have /i(l,o o ) < < ,r(l - j/2) A -. 7t(1 - y2) dy ; ^ l o g n + 0(log4/^n). 31) In the following we obtain a lower lim it for 7 i(l,o o ). Qxf{y) = f { J { n - A ) /n ) < O/n^e^, where the maximum is taken over 38 0 < y < 1 — 1/n . 29) we have B2 A2 f y 2 n -4 (i_ ^ 2 ) I l { l - h ^ ( y ) } ( l - y 2n)2 (i + y ^ ) ( i - y ^ " ) W ( 1 - y^)^ - 2n(l - y^)| n2y2"-4(l _ y2^3 ^ ^ { l - / i 2 ( y ) } ( l - y 2 ’^)2 n^y2n-4(l _ y2)3 (1 —j/2n p _ |j 2y 2n - 2^J _ y 2^2 „2y2n-4(i _ y2^3 (1 - e-2)2 - 4e-2 4 5 y \, < .