By R. Beals

As soon as upon a time scholars of arithmetic and scholars of technological know-how or engineering took a similar classes in mathematical research past calculus. Now it's common to split" complicated arithmetic for technological know-how and engi neering" from what could be known as "advanced mathematical research for mathematicians." it kind of feels to me either priceless and well timed to aim a reconciliation. The separation among varieties of classes has bad results. Mathe matics scholars opposite the old improvement of research, studying the unifying abstractions first and the examples later (if ever). technology scholars examine the examples as taught generations in the past, lacking glossy insights. a decision among encountering Fourier sequence as a minor example of the repre sentation thought of Banach algebras, and encountering Fourier sequence in isolation and constructed in an advert hoc demeanour, is not any selection in any respect. you will realize those difficulties, yet much less effortless to counter the legiti mate pressures that have resulted in a separation. sleek arithmetic has broadened our views via abstraction and impressive generalization, whereas constructing options that can deal with classical theories in a definitive means. nevertheless, the applier of arithmetic has endured to wish various yes instruments and has now not had the time to procure the broadest and so much definitive grasp-to study invaluable and enough stipulations while uncomplicated enough stipulations will serve, or to benefit the final framework surround ing varied examples.

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**Sample text**

Let P(z) be a polynomial of degree n, whose zeros (assumed all real) are denoted by a, , a2 , - - , an , with a; 5 a;+1 , and let the zeros of the derivawith b; _< b;+1 . If the respective loci tive P'(z) be denoted by b, , b2 , , ak + h, then the reof the points a1 , a2 , - - , a are the intervals ak - h ;5 z spective loci of the critical points of the polynomial [i (z - ak) are the intervals bk-h<=z5bk+h. COROLLARY 3. Let P(z) be a polynomial of degree n, whose zeros a, , a2 , , an are all real, with a; ;9 a7}1 , and let the zeros of P(z) be b1, b2 , with , b; 5 b;+1 .

However, we first treat the special case of polynomials all of whose zeros are real. 3, that the critical points are also all real. A general result [Walsh, 1922] on the location of those critical points is THEOREM 1. Denote by Ik the interval ak 5 z 5 bk , k = 1, 2, , n. Let the critical points of III (z - ak) be c; , with c; _< c;+i and the critical points of Hi (z - bk) be d; , with d; 5 d;+i. Then if the intervals I, are the respective loci of the points a; , the locus of the k-th critical point (in algebraic magnitude) of the variable polynomial p(z) = III (z - a,) consists of the interval Ik: ck _<- z 5 dk , n - 1.

Let p(z) be a real polynomial with precisely two pairs of non-real zeros: a1i a1 , a2 , a2 , and with no real zeros. Let the Jensen circle Ci (j = 1, 2) 40 CHAPTER II. REAL POLYNOMIALS corresponding to a; and a; intersect the axis of reals in points z and z' , with z' < z;' , and suppose we have Z`1' < z2 . Let the tangent to the circle zla2a2 at a2 cut the axis of reals in z1 and the tangent to the circle z' a2«2 at a2 cut the axis of reals in z2 , and let A1; (P = 1, 2) be the circular arc bounded by at and a1 of angular measure less than it which is tangent to the lines alzk and atz1..