A NOTE ON PERIODIC SOLUTIONS IN THE VICINITY OF A CENTER
|
|
- Aubrey Nicholson
- 6 years ago
- Views:
Transcription
1 proceedings of the american mathematical society Volume 81, umber 4, April 1981 A OTE O PERIODIC SOLUTIOS I THE VICIITY OF A CETER BAU-SE DU Abstract. We show that a small polynomial perturbation of a center will result in periodic solutions provided the coefficients of the polynomial satisfy inequality (2) below. The proof is based on an application of degree theory. 1. Introduction. In her book [1] (see also [2]), Cronin stated the following. Let hx(x,y, p) and h2(x, y, p) be polynomials in x, v and p and let /^(sin 9, cos 9, p), &2(sin 9, cos 9, p) be polynomials in sin 9, cos 9 and p. Assume that n( p) is a C ' function and i)0 = t/(0) ^ 0. Let w = [ 1 + pn( p)]_1 and t = wt. Then the following two-dimensional nonlinear differential system x' = y + (i[hx(x,y, p) + kx(sin t, cos t, p)], y' = -x + n[h2(x,y, p) + k2(sin r, cos t, p)], I' = J, has, for all sufficiently small p, at least one solution of period 2w(l + pij(p)). For the proof of this result, Dr. Cronin used the results of degree theory, in particular, the homotopy theorem and the odd mapping theorem as applied to a function ^C) which depends on the terms hx, h2, kx and k2. The quantity C represents the initial data for the solution which is periodic precisely when ^C) = 0. But it seems that, in some cases, one cannot apply the odd mapping theorem. For example, if we let rj( p) = 1, to = (1 + p)-', t = on and consider x' = y + p(x + 3v + cos t), y' = -x + p(x v sin t), (*) then 0(C) = (2rJ). (From now on, we shall use the same notation as Cronin [1, pp ].) So the degree of 0(C) in any domain is 0. In this case, we cannot conclude from degree theory that the differential system (*) has, for all sufficiently small p, at least one solution of period 2tt(1 + p). The purpose of this paper is to compute directly 0(C). We will show that if the coefficients of hx and h2 satisfy a simple inequality, then the differential system (1) has a periodic solution for p sufficiently small. The precise statement of our result is given in the next section, but we note that the inequality required is a generic condition on hx and h2, and consequently most systems satisfy the conclusion of Cronin's Theorem. The proof of our result appears in 3 and 4. Received by the editors March 6, Mathematics Subject Classification. Primary 34C American Mathematical Society /81/ /S02.50
2 580 BAU-SE DU 2. Statement of the theorem. For nonnegative integers n and k, we define i(-o*-^k'%+-;1 "«**<- 2n+\-k 2~Vl)*+7~"( *.)(2" + 1_ k) iîn + l<k<2n + l. Then, it is not hard to show that 4" r2* akn = ^- ( cos2n+i-k9-(-l)k sin* 0- e m Jo Let (2n + 1) n m j.. 2 A 6 (2n + 2) ' A" i* A" r iik = 0, (2n *) (* - 1) 2m (2n + 2) if k is even and 0 < k < 2n + 1, (2/1- k) k 2m (2/i + 2) if k is odd and 0 < k < 2«+ 1, (_1)" + 1.4" i (2n n + 1) tt - ; :-h-rr, if Jfc = 2/i (2«+ 2) ' 2«+l páx'y) = 2 a*,n-x;2',+1 ~^*' *-0 Ôn(^7) = 2nf,6^2n+1^\ fc-0 and let An = S (-l)'(%/2t^ - *2*+l, «2it+l>»). fc-0 fin = 2 (-1)*(û2*+l,n«2*+l,n + ^/2t.J' fc-0 ow we can state the following theorem. Theorem. Assume that hx(x,y, p) and h2(x,y, p) are polynomials in x, y and p. When p = 0, let hx(x,y, 0) = P(x,y) + 2 Pn(x,y), n-0 h2(x,y, 0) = Q(x,y) + 2 Qn(x, y), n-0
3 PERIODIC SOLUTIOS I THE VICIITY OF A CETER 581 where P(x, y), Q(x, y) are polynomials in x, y in which each term is of even degree. Define An, Bn as above, and let kx($rn 9, cos 9, p) and A:2(sin 9, cos 9, p) be polynomials in sin 9, cos 9 and p. Assume that r (n) is a Cl function and t/0 = tj(0). Let 03 = [1 + ptj(p)]"1 and T = at. If Ko + Ko? + (2i)o- «w + M' + 2 K2 + B2) * 0, then the following two-dimensional x' = y + fi{hx(x,y, n-l nonlinear differential system p) + kx(sin r, cos t, p)}, y' = -x + n{h2(x, v, p) + /c2(sin t, cos t, p)}, has, for all sufficiently small p, ai /easi o/ie? periodic solution of period 2nr(\ + p7,(p)). Corollary. //, in the above theorem, we let hx(x,y, 0) = a0 + axx + a2y + H.O.T., h2(x,y, 0) = b0 + bxx + b2y + H.O.T., and assume (ax + b2)2 + (2tj0 a2 + bx)2 = 0, then the conclusion of the above theorem holds. Remark. We call attention to the fact that this theory applies in cases where 3. Computation of 0(C). Let and let z = c, i'c2. When p = 0, we let -CD (2) (x\(x(9,0,c)\ m ( cxcos9 + c2sin0 \ _ {?> \y(9, 0, C)) \-cx sin 9 + c2 cos 9) ~ Í1 (e'9z + e-iez) ^(eiez - e~*z) In particular, when z = 1, one has x = cos 9, y = -sin 9. In Cronin's notation, one has where (C) = f 2wV C2 + //,(C" Cl) + Kl \ \ -2mj]0cx + H2(cx, c2) + K2 ) / Hi(cu ci)\ m cm hx(x,y, 0)cos 9 - h2(x,y, 0)sin 9\ \ H2(cx, c2) ) h \ hx(x, y, 0)sin 9 + h2(x, y, 0)cos 9 j -Í Re{[A1(x,^,0) + ia2(jc,>-,0)]e*} lm{[n,(x,v,0) + in2(x,v,0)]e*}
4 582 BAU-SE DU and / Kx \ r2mi A:,(sin 9, cos 9, 0)cos 9 - k2(sin 9, cos 9, 0)sin 9 \ \K2J ~ J0 \ kx(sin 9, cos 9, 0)sin 0 + fc2(sin 9, cos 0, 0)cos 9 J is a constant vector. Since hx, h2 are polynomials in x and y, it suffices to compute l" h(x,y)e'e d.9, where h(x,y) is a polynomial in x andy. Given two nonnegative integers m and n, and let h(x,y) = x"y". If m + n is even, then it is trivial that J"2,* h(x, y)e'9 d9 = 0. So we can assume that m + n is odd. If h(x,y) = x2n + l~kyk, where 0 < k < n, then { h(x,y)emd9 = f x2"+1-*yv*<0 i i*7r m, _. _ = ^>r(zz \zz)"z )* «*, -akn. Similarly, if A(x,y) = x2" + 1"fcy*, where n + 1 < it < 2n + 1, then f2nh(x,y)eie d9 = f2v"+1-*y V* d9 ikm, _. _ 2n + *-k. Uk+J-J k \(2n+l-k\ im, _,. _ ow assume *,<*,.y, o) = 2 ^x2"*1-^* = jv(*a (V-=0 /t2(x, y, 0) = ^,Bx2" + 1-*y* = ß (x, v), («> 0). *=o
5 PERIODIC SOLUTIOS I THE VICIITY OF A CETER 583 Then -2ir f "[hx(x,y, 0) + i«2(x,y, 0)]e's d9 Jo = 2nî\ak<n + ibkjf2\2"^-ye">d9 /t-o *-0 2n+l i m = 2 («*. + *Jt») ' IS" " (ZZ)"Z "a*,n *-0 Jo = 4»(«) " / -\n- 2 2 (-l)*02*,«a2m - 2 (-0**2*+l,««2* + l.«/t = 0 t-0 + '4>t(zz) z 2 (-1)*a2*+l,n«2*+l, + 2 (-l)%k,na2k,n k-0 k-0 IT IT n = (zz)"zan + i (zz)"zb (where v4n, /? are as defined in 2) = ^(zz)"z(an + ibn) = y(c? + c2)n(c, + ic2)(^ +./? ) w =?(c? + c22)fl[(c^n - c2bn) + i(cxbn + C2A )]. Therefore, the real and imaginary parts of S\?[P (x,y) + iq (x,y)]e (m/a")(c2 + c2)"(cxan - c2bn) and (m/an)(c2 + c2)"(cxbn + C2A ) respectively. Finally, we let hx(x,y, 0) = P(x,y) + 2?_0 P (x,y), and h2(x,y, 0) = Q(x,y) + 2^_o Q (x,y) where P(x,y) and Q(x,y) are polynomials in x, y in which each term is of even degree, and Pn(x, y), Q (x, y) are defined as above. Then C"[hx(x,y, 0) + ih2(x,y, 0)]eie d9 = 2 C'[PÁx,y) + iq (x,y)]e* db. J0 n~0j0 (ote that (2"p(x,y)em d9 = 0= (2 Q(x,y)e* d9 >o Jo = 2 i(*? + c22)"[(c1v4n - c2/? ) + /(c,^ + cvl,)]. n=0 ^ So the real and imaginary parts of /o*[«,(x,y, 0) + i7j2(x, y, 0)]e'9 d9 are 2ir_o(^/4")(c2 + c2)"(cxan - c2bn) and 2^0(^/4")(c2 + c2)\cxbn + ClAn) respectively. Therefore, ">(c c2) = 2 3(c? + c2)"(c^ - c2/? ), n-0 ^ are #2(^1. c2) = 2 (c? + c2)"(cxb + C2An), n-0 H where An, Bn are as defined in 2. So we have got the explicit formulas Hx(cx, c and H2(cx, c^. for
6 584 BAU-SE DU 4. Proof of the theorem. ow we have obtained the formulas for Hx(cx, c^ and H2(cx, c2>- So (C) = I 2mi C2 + H^Cl' C2) + Kl\ \ -2Tr/0c, + H2(cx, c2) + K2 ) = m m 2irn0c2 + m(cxa0 - c2b0) + 2 7«"(ci + c2)"(cian ~ C2BJ + *i n-l H -2,rr,0c, + m(cxb0 + c^0) + 2 m «(*? + 4)n(cxBn + C2A ) + K2 n-\ * ci(ao,o + *i,o) + c2(2i7o + bo,o - ai,o) + 2 S (*ï + 4) 2\" n-l 4" (c^ - c2bn) ic2 + c2\" Ci(-2i)o - V + ai,o) + c2(a0,o + >i,o) + 2 n-l ' An 4" 2 (c\bn + <VU (*;) If for some n > 1, A2 + B2 = 0, then JV0(C) is dominated at least by (c2 + c2)". For all sufficiently large C, one has 0(C) ^ 0, and we can apply the odd mapping theorem. On the other hand, if An = Bn = 0 for all 1 < n <./V and if (ao,o + 6i,o)2 + (2^,0 + *o,o - ai,o)2 ^ 0. ti"511 tc\ m J ci(ao,o + 6i,o) + c2(2tjo + *o,o- «i.o) \ + / *1 \ = 0 I C!(-2t,0 - b00 + axo) + c2(a00 + bxo) ) \K2) for exactly one C. So, again, for all sufficiently large C, the function Q(C) can never be zero and the odd mapping theorem applies too. But if a00 + bxo = 0 = 2î/o + *o,o ai,o an(* -An = Bn = 0 for 1 < n <, then the function 0(C) is constantly equal to (* ). So its degree is either undefined or equal to zero. Hence no conclusion can be drawn in this case. This completes the proof. Acknowledgments. I want to thank Professor G. R. Sell for his many valuable suggestions. References 1. J. Cronin, Fixed points and topological degree in nonlinear analysis, Math. Surveys, no. 11, Amer. Math. Soc., Providence, R. I., 1964, pp G. Lloyd, Degree theory, Cambridge Univ. Press, Cambridge and ew York, 1978, pp School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
(i) tn = f î {nk)tk(i - ty-\dx{t) (»>o).
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 102, Number 1, January 1988 A TAUBERIAN THEOREM FOR HAUSDORFF METHODS BRIAN KUTTNER AND MANGALAM R. PARAMESWARAN (Communicated by R. Daniel Mauldin)
More informationCalculation of the Ramanujan t-dirichlet. By Robert Spira
MATHEMATICS OF COMPUTATION, VOLUME 27, NUMBER 122, APRIL, 1973 Calculation of the Ramanujan t-dirichlet Series By Robert Spira Abstract. A method is found for calculating the Ramanujan T-Dirichlet series
More informationRoots and Coefficients Polynomials Preliminary Maths Extension 1
Preliminary Maths Extension Question If, and are the roots of x 5x x 0, find the following. (d) (e) Question If p, q and r are the roots of x x x 4 0, evaluate the following. pq r pq qr rp p q q r r p
More information2 Series Solutions near a Regular Singular Point
McGill University Math 325A: Differential Equations LECTURE 17: SERIES SOLUTION OF LINEAR DIFFERENTIAL EQUATIONS II 1 Introduction Text: Chap. 8 In this lecture we investigate series solutions for the
More informationA CONVERGENCE CRITERION FOR FOURIER SERIES
A CONVERGENCE CRITERION FOR FOURIER SERIES M. TOMIC 1.1. It is known that the condition (1) ta(xo,h) sf(xo + h) -f(x0) = oi-r--- -V A-*0, \ log  / for a special x0 does not imply convergence of the Fourier
More informationThings to remember: x n a 1. x + a 0. x n + a n-1. P(x) = a n. Therefore, lim g(x) = 1. EXERCISE 3-2
lim f() = lim (0.8-0.08) = 0, " "!10!10 lim f() = lim 0 = 0.!10!10 Therefore, lim f() = 0.!10 lim g() = lim (0.8 - "!10!10 0.042-3) = 1, " lim g() = lim 1 = 1.!10!0 Therefore, lim g() = 1.!10 EXERCISE
More informationLOCAL AND GLOBAL EXTREMA FOR FUNCTIONS OF SEVERAL VARIABLES
/. Austral. Math. Soc. {Series A) 29 (1980) 362-368 LOCAL AND GLOBAL EXTREMA FOR FUNCTIONS OF SEVERAL VARIABLES BRUCE CALVERT and M. K. VAMANAMURTHY (Received 18 April; revised 18 September 1979) Communicated
More informationREPRESENTATIONS FOR A SPECIAL SEQUENCE
REPRESENTATIONS FOR A SPECIAL SEQUENCE L. CARLITZ* RICHARD SCOVILLE Dyke University, Durham,!\!orth Carolina VERNERE.HOGGATTJR. San Jose State University, San Jose, California Consider the sequence defined
More information2. Second-order Linear Ordinary Differential Equations
Advanced Engineering Mathematics 2. Second-order Linear ODEs 1 2. Second-order Linear Ordinary Differential Equations 2.1 Homogeneous linear ODEs 2.2 Homogeneous linear ODEs with constant coefficients
More informationON HIGHER ORDER NONSINGULAR IMMERSIONS OF DOLD MANIFOLDS
PROCEEDINGS of the AMERICAN MATHEMATICAL SOCIETY Volume 39, Number 1, June 1973 ON HIGHER ORDER NONSINGULAR IMMERSIONS OF DOLD MANIFOLDS WEI-LUNG TING1 Abstract. In this paper we employ y-operations and
More information2 K cos nkx + K si" nkx) (1.1)
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 71, Number 1, August 1978 ON THE ABSOLUTE CONVERGENCE OF LACUNARY FOURIER SERIES J. R. PATADIA Abstract. Let / G L[ it, -n\ be 27r-periodic. Noble
More informationSMOOTHNESS OF FUNCTIONS GENERATED BY RIESZ PRODUCTS
SMOOTHNESS OF FUNCTIONS GENERATED BY RIESZ PRODUCTS PETER L. DUREN Riesz products are a useful apparatus for constructing singular functions with special properties. They have been an important source
More informationON EQUIVALENCE OF ANALYTIC FUNCTIONS TO RATIONAL REGULAR FUNCTIONS
J. Austral. Math. Soc. (Series A) 43 (1987), 279-286 ON EQUIVALENCE OF ANALYTIC FUNCTIONS TO RATIONAL REGULAR FUNCTIONS WOJC3ECH KUCHARZ (Received 15 April 1986) Communicated by J. H. Rubinstein Abstract
More informationMath 141: Lecture 11
Math 141: Lecture 11 The Fundamental Theorem of Calculus and integration methods Bob Hough October 12, 2016 Bob Hough Math 141: Lecture 11 October 12, 2016 1 / 36 First Fundamental Theorem of Calculus
More informationSeries Solutions of Differential Equations
Chapter 6 Series Solutions of Differential Equations In this chapter we consider methods for solving differential equations using power series. Sequences and infinite series are also involved in this treatment.
More informationMath Circles: Number Theory III
Math Circles: Number Theory III Centre for Education in Mathematics and Computing University of Waterloo March 9, 2011 A prime-generating polynomial The polynomial f (n) = n 2 n + 41 generates a lot of
More informationEVENTUAL DISCONJUGACY OF A LINEAR DIFFERENTIAL EQUATION. II WILLIAM F. TRENCH
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 94. Number 4. August 1985 EVENTUAL DISCONJUGACY OF A LINEAR DIFFERENTIAL EQUATION. II WILLIAM F. TRENCH Abstract. A sufficient condition is given
More informationSection Properties of Rational Expressions
88 Section. - Properties of Rational Expressions Recall that a rational number is any number that can be written as the ratio of two integers where the integer in the denominator cannot be. Rational Numbers:
More informationGROUP OF TRANSFORMATIONS*
A FUNDAMENTAL SYSTEM OF INVARIANTS OF A MODULAR GROUP OF TRANSFORMATIONS* BY JOHN SIDNEY TURNER 1. Introduction. Let G be any given group of g homogeneous linear transformations on the indeterminates xi,,
More informationVAROL'S PERMUTATION AND ITS GENERALIZATION. Bolian Liu South China Normal University, Guangzhou, P. R. of China {Submitted June 1994)
VROL'S PERMUTTION ND ITS GENERLIZTION Bolian Liu South China Normal University, Guangzhou, P. R. of China {Submitted June 994) INTRODUCTION Let S be the set of n! permutations II = a^x...a n of Z = (,,...,«}.
More informationMarch Algebra 2 Question 1. March Algebra 2 Question 1
March Algebra 2 Question 1 If the statement is always true for the domain, assign that part a 3. If it is sometimes true, assign it a 2. If it is never true, assign it a 1. Your answer for this question
More informationON JAMES' QUASI-REFLEXIVE BANACH SPACE
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 67, Number 2, December 1977 ON JAMES' QUASI-REFLEXIVE BANACH SPACE P. G. CASAZZA, BOR-LUH LIN AND R. H. LOHMAN Abstract. In the James' space /, there
More informationTHE POISSON TRANSFORM^)
THE POISSON TRANSFORM^) BY HARRY POLLARD The Poisson transform is defined by the equation (1) /(*)=- /" / MO- T J _M 1 + (X t)2 It is assumed that a is of bounded variation in each finite interval, and
More informationJ2 e-*= (27T)- 1 / 2 f V* 1 '»' 1 *»
THE NORMAL APPROXIMATION TO THE POISSON DISTRIBUTION AND A PROOF OF A CONJECTURE OF RAMANUJAN 1 TSENG TUNG CHENG 1. Summary. The Poisson distribution with parameter X is given by (1.1) F(x) = 23 p r where
More informationEXPANSION OF ANALYTIC FUNCTIONS IN TERMS INVOLVING LUCAS NUMBERS OR SIMILAR NUMBER SEQUENCES
EXPANSION OF ANALYTIC FUNCTIONS IN TERMS INVOLVING LUCAS NUMBERS OR SIMILAR NUMBER SEQUENCES PAUL F. BYRD San Jose State College, San Jose, California 1. INTRODUCTION In a previous article [_ 1J, certain
More informationUsing the Rational Root Theorem to Find Real and Imaginary Roots Real roots can be one of two types: ra...-\; 0 or - l (- - ONLl --
Using the Rational Root Theorem to Find Real and Imaginary Roots Real roots can be one of two types: ra...-\; 0 or - l (- - ONLl -- Consider the function h(x) =IJ\ 4-8x 3-12x 2 + 24x {?\whose graph is
More informationDISTRIBUTIONS FUNCTIONS OF PROBABILITY SOME THEOREMS ON CHARACTERISTIC. (1.3) +(t) = eitx df(x),
SOME THEOREMS ON CHARACTERISTIC FUNCTIONS OF PROBABILITY DISTRIBUTIONS 1. Introduction E. J. G. PITMAN UNIVERSITY OF TASMANIA Let X be a real valued random variable with probability measure P and distribution
More informationx 9 or x > 10 Name: Class: Date: 1 How many natural numbers are between 1.5 and 4.5 on the number line?
1 How many natural numbers are between 1.5 and 4.5 on the number line? 2 How many composite numbers are between 7 and 13 on the number line? 3 How many prime numbers are between 7 and 20 on the number
More informationMAC College Algebra
MAC 05 - College Algebra Name Review for Test 2 - Chapter 2 Date MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the exact distance between the
More informationON NONNEGATIVE COSINE POLYNOMIALS WITH NONNEGATIVE INTEGRAL COEFFICIENTS
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 120, Number I, January 1994 ON NONNEGATIVE COSINE POLYNOMIALS WITH NONNEGATIVE INTEGRAL COEFFICIENTS MIHAIL N. KOLOUNTZAKIS (Communicated by J. Marshall
More informationSo, eqn. to the bisector containing (-1, 4) is = x + 27y = 0
Q.No. The bisector of the acute angle between the lines x - 4y + 7 = 0 and x + 5y - = 0, is: Option x + y - 9 = 0 Option x + 77y - 0 = 0 Option x - y + 9 = 0 Correct Answer L : x - 4y + 7 = 0 L :-x- 5y
More informationPower Series Solutions to the Legendre Equation
Department of Mathematics IIT Guwahati The Legendre equation The equation (1 x 2 )y 2xy + α(α + 1)y = 0, (1) where α is any real constant, is called Legendre s equation. When α Z +, the equation has polynomial
More informationNewton, Fermat, and Exactly Realizable Sequences
1 2 3 47 6 23 11 Journal of Integer Sequences, Vol. 8 (2005), Article 05.1.2 Newton, Fermat, and Exactly Realizable Sequences Bau-Sen Du Institute of Mathematics Academia Sinica Taipei 115 TAIWAN mabsdu@sinica.edu.tw
More informationEVANESCENT SOLUTIONS FOR LINEAR ORDINARY DIFFERENTIAL EQUATIONS
EVANESCENT SOLUTIONS FOR LINEAR ORDINARY DIFFERENTIAL EQUATIONS Cezar Avramescu Abstract The problem of existence of the solutions for ordinary differential equations vanishing at ± is considered. AMS
More informationAN INEQUALITY OF TURAN TYPE FOR JACOBI POLYNOMIALS
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 32, Number 2, April 1972 AN INEQUALITY OF TURAN TYPE FOR JACOBI POLYNOMIALS GEORGE GASPER Abstract. For Jacobi polynomials P^^Hx), a, ß> 1, let RAX)
More informationMath Euler Cauchy Equations
1 Math 371 - Euler Cauchy Equations Erik Kjær Pedersen November 29, 2005 We shall now consider the socalled Euler Cauchy equation x 2 y + axy + by = 0 where a and b are constants. To solve this we put
More informationand A", where (1) p- 8/+ 1 = X2 + Y2 = C2 + 2D2, C = X=\ (mod4).
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 87, Number 3. March 1983 THE Odie PERIOD POLYNOMIAL RONALD J. EVANS1 Abstract. The coefficients and the discriminant of the octic period polynomial
More informationPRE-CALCULUS By: Salah Abed, Sonia Farag, Stephen Lane, Tyler Wallace, and Barbara Whitney
PRE-CALCULUS By: Salah Abed, Sonia Farag, Stephen Lane, Tyler Wallace, and Barbara Whitney MATH 141/14 1 Pre-Calculus by Abed, Farag, Lane, Wallace, and Whitney is licensed under the creative commons attribution,
More informationElementary Differential Equations, Section 2 Prof. Loftin: Practice Test Problems for Test Find the radius of convergence of the power series
Elementary Differential Equations, Section 2 Prof. Loftin: Practice Test Problems for Test 2 SOLUTIONS 1. Find the radius of convergence of the power series Show your work. x + x2 2 + x3 3 + x4 4 + + xn
More informationON THE CONVERGENCE OF DOUBLE SERIES
ON THE CONVERGENCE OF DOUBLE SERIES ALBERT WILANSKY 1. Introduction. We consider three convergence definitions for double series, denoted by (p), (
More informationA LOWER BOUND FOR THE FUNDAMENTAL FREQUENCY OF A CONVEX REGION
proceedings of the american mathematical society Volume 81, Number 1, January 1981 A LOWER BOUND FOR THE FUNDAMENTAL FREQUENCY OF A CONVEX REGION M. H. PROTTER1 Abstract. A lower bound for the first eigenvalue
More informationProblems in Algebra. 2 4ac. 2a
Problems in Algebra Section Polynomial Equations For a polynomial equation P (x) = c 0 x n + c 1 x n 1 + + c n = 0, where c 0, c 1,, c n are complex numbers, we know from the Fundamental Theorem of Algebra
More informationLecture 13: Series Solutions near Singular Points
Lecture 13: Series Solutions near Singular Points March 28, 2007 Here we consider solutions to second-order ODE s using series when the coefficients are not necessarily analytic. A first-order analogy
More informationDYSON'S CRANK OF A PARTITION
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 8, Number, April 988 DYSON'S CRANK OF A PARTITION GEORGE E. ANDREWS AND F. G. GARVAN. Introduction. In [], F. J. Dyson defined the rank
More informationFOURTH-ORDER TWO-POINT BOUNDARY VALUE PROBLEMS
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 104, Number 1, September 1988 FOURTH-ORDER TWO-POINT BOUNDARY VALUE PROBLEMS YISONG YANG (Communicated by Kenneth R. Meyer) ABSTRACT. We establish
More informationKing Fahd University of Petroleum and Minerals Prep-Year Math Program Math Term 161 Recitation (R1, R2)
Math 001 - Term 161 Recitation (R1, R) Question 1: How many rational and irrational numbers are possible between 0 and 1? (a) 1 (b) Finite (c) 0 (d) Infinite (e) Question : A will contain how many elements
More informationGRONWALL'S INEQUALITY FOR SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS IN TWO INDEPENDENT VARIABLES
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 33, Number I, May 1972 GRONWALL'S INEQUALITY FOR SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS IN TWO INDEPENDENT VARIABLES DONALD R. SNOW Abstract.
More informationMath 473: Practice Problems for Test 1, Fall 2011, SOLUTIONS
Math 473: Practice Problems for Test 1, Fall 011, SOLUTIONS Show your work: 1. (a) Compute the Taylor polynomials P n (x) for f(x) = sin x and x 0 = 0. Solution: Compute f(x) = sin x, f (x) = cos x, f
More informationMath 221 Notes on Rolle s Theorem, The Mean Value Theorem, l Hôpital s rule, and the Taylor-Maclaurin formula. 1. Two theorems
Math 221 Notes on Rolle s Theorem, The Mean Value Theorem, l Hôpital s rule, and the Taylor-Maclaurin formula 1. Two theorems Rolle s Theorem. If a function y = f(x) is differentiable for a x b and if
More informationTHE PRESERVATION OF CONVERGENCE OF MEASURABLE FUNCTIONS UNDER COMPOSITION
THE PRESERVATION OF CONVERGENCE OF MEASURABLE FUNCTIONS UNDER COMPOSITION ROBERT G. BARTLE AND JAMES T. JOICHI1 Let / be a real-valued measurable function on a measure space (S,, p.) and let
More informationFUNCTIONS OVER THE RESIDUE FIELD MODULO A PRIME. Introduction
FUNCTIONS OVER THE RESIDUE FIELD MODULO A PRIME DAVID LONDON and ZVI ZIEGLER (Received 7 March 966) Introduction Let F p be the residue field modulo a prime number p. The mappings of F p into itself are
More informationA101 ASSESSMENT Quadratics, Discriminant, Inequalities 1
Do the questions as a test circle questions you cannot answer Red (1) Solve a) 7x = x 2-30 b) 4x 2-29x + 7 = 0 (2) Solve the equation x 2 6x 2 = 0, giving your answers in simplified surd form [3] (3) a)
More informationSTABILIZATION BY A DIAGONAL MATRIX
STABILIZATION BY A DIAGONAL MATRIX C. S. BALLANTINE Abstract. In this paper it is shown that, given a complex square matrix A all of whose leading principal minors are nonzero, there is a diagonal matrix
More informationSUBSPACE LATTICES OF FINITE VECTOR SPACES ARE 5-GENERATED
SUBSPACE LATTICES OF FINITE VECTOR SPACES ARE 5-GENERATED LÁSZLÓ ZÁDORI To the memory of András Huhn Abstract. Let n 3. From the description of subdirectly irreducible complemented Arguesian lattices with
More informationIRREGULARITIES OF DISTRIBUTION. V WOLFGANG M. SCHMIDT
IRREGULARITIES OF DISTRIBUTION. V WOLFGANG M. SCHMIDT Abstract. It is shown that for any finite or countable number of points with positive weights on the sphere, there is always a spherical cap such that
More information3.2 Subspace. Definition: If S is a non-empty subset of a vector space V, and S satisfies the following conditions: (i).
. ubspace Given a vector spacev, it is possible to form another vector space by taking a subset of V and using the same operations (addition and multiplication) of V. For a set to be a vector space, it
More informationTRANSLATION PROPERTIES OF SETS OF POSITIVE UPPER DENSITY VITALY BERGELSON AND BENJAMIN WEISS1
PROCEEDINGS OF THE AMERICAN MATHEMATICAL Volume 94. Number 3, July 1985 SOCIETY TRANSLATION PROPERTIES OF SETS OF POSITIVE UPPER DENSITY VITALY BERGELSON AND BENJAMIN WEISS1 Abstract. Generalizing a result
More informationf. D that is, F dr = c c = [2"' (-a sin t)( -a sin t) + (a cos t)(a cost) dt = f2"' dt = 2
SECTION 16.4 GREEN'S THEOREM 1089 X with center the origin and radius a, where a is chosen to be small enough that C' lies inside C. (See Figure 11.) Let be the region bounded by C and C'. Then its positively
More informationThe Method of Frobenius
The Method of Frobenius R. C. Trinity University Partial Differential Equations April 7, 2015 Motivating example Failure of the power series method Consider the ODE 2xy +y +y = 0. In standard form this
More informationTHE DIFFERENCE BETWEEN CONSECUTIVE PRIME NUMBERS. IV. / = lim inf ^11-tl.
THE DIFFERENCE BETWEEN CONSECUTIVE PRIME NUMBERS. IV r. a. ran kin 1. Introduction. Let pn denote the nth prime, and let 0 be the lower bound of all positive numbers
More informationMath 304 Answers to Selected Problems
Math Answers to Selected Problems Section 6.. Find the general solution to each of the following systems. a y y + y y y + y e y y y y y + y f y y + y y y + 6y y y + y Answer: a This is a system of the
More informationHt) = - {/(* + t)-f(x-t)}, e(t) = - I^-á«.
ON THE ABSOLUTE SUMMABILITY OF THE CONJUGATE SERIES OF A FOURIER SERIES T. PATI 1.1. Let/( ) be a periodic function with period 2w and integrable (L) over ( ir, it). Let 1 (1.1.1) a0 + Y (an cos nt + bn
More information12d. Regular Singular Points
October 22, 2012 12d-1 12d. Regular Singular Points We have studied solutions to the linear second order differential equations of the form P (x)y + Q(x)y + R(x)y = 0 (1) in the cases with P, Q, R real
More informationSCTT The pqr-method august 2016
SCTT The pqr-method august 2016 A. Doledenok, M. Fadin, À. Menshchikov, A. Semchankau Almost all inequalities considered in our project are symmetric. Hence if plugging (a 0, b 0, c 0 ) into our inequality
More informationSQUARE ROOTS OF 2x2 MATRICES 1. Sam Northshield SUNY-Plattsburgh
SQUARE ROOTS OF x MATRICES Sam Northshield SUNY-Plattsburgh INTRODUCTION A B What is the square root of a matrix such as? It is not, in general, A B C D C D This is easy to see since the upper left entry
More informationTHE NUMBER OF INDEPENDENT DOMINATING SETS OF LABELED TREES. Changwoo Lee. 1. Introduction
Commun. Korean Math. Soc. 18 (2003), No. 1, pp. 181 192 THE NUMBER OF INDEPENDENT DOMINATING SETS OF LABELED TREES Changwoo Lee Abstract. We count the numbers of independent dominating sets of rooted labeled
More informationSolving Differential Equations Using Power Series
LECTURE 8 Solving Differential Equations Using Power Series We are now going to employ power series to find solutions to differential equations of the form () y + p(x)y + q(x)y = 0 where the functions
More informationSolving Differential Equations Using Power Series
LECTURE 25 Solving Differential Equations Using Power Series We are now going to employ power series to find solutions to differential equations of the form (25.) y + p(x)y + q(x)y = 0 where the functions
More informationPUTNAM PROBLEMS SEQUENCES, SERIES AND RECURRENCES. Notes
PUTNAM PROBLEMS SEQUENCES, SERIES AND RECURRENCES Notes. x n+ = ax n has the general solution x n = x a n. 2. x n+ = x n + b has the general solution x n = x + (n )b. 3. x n+ = ax n + b (with a ) can be
More informationMOMENT SEQUENCES AND BACKWARD EXTENSIONS OF SUBNORMAL WEIGHTED SHIFTS
J. Austral. Math. Soc. 73 (2002), 27-36 MOMENT SEQUENCES AND BACKWARD EXTENSIONS OF SUBNORMAL WEIGHTED SHIFTS THOMAS HOOVER, IL BONG JUNG and ALAN LAMBERT (Received 15 February 2000; revised 10 July 2001)
More informationChapter 8B - Trigonometric Functions (the first part)
Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 8B-I! Page 79 Chapter 8B - Trigonometric Functions (the first part) Recall from geometry that if 2 corresponding triangles have 2 angles of
More informationON THE SET OF ALL CONTINUOUS FUNCTIONS WITH UNIFORMLY CONVERGENT FOURIER SERIES
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 24, Number, November 996 ON THE SET OF ALL CONTINUOUS FUNCTIONS WITH UNIFORMLY CONVERGENT FOURIER SERIES HASEO KI Communicated by Andreas R. Blass)
More informationJACOBIANS OF QUATERNION POLYNOMIALS. 1. Introduction. c 0 c 1 c 2 c 3 C = c 1 c 0 c 3 c 2 c 2 c 3 c 0 c 1
BULLETIN OF THE HELLENIC MATHEMATICAL SOCIETY Volume 61, 2017 (31 40) JACOBIANS OF QUATERNION POLYNOMIALS TAKIS SAKKALIS Abstract. A map f from the quaternion skew field H to itself, can also be thought
More informationHonours Algebra 2, Assignment 8
Honours Algebra, Assignment 8 Jamie Klassen and Michael Snarski April 10, 01 Question 1. Let V be the vector space over the reals consisting of polynomials of degree at most n 1, and let x 1,...,x n be
More informationCONTENTS COLLEGE ALGEBRA: DR.YOU
1 CONTENTS CONTENTS Textbook UNIT 1 LECTURE 1-1 REVIEW A. p. LECTURE 1- RADICALS A.10 p.9 LECTURE 1- COMPLEX NUMBERS A.7 p.17 LECTURE 1-4 BASIC FACTORS A. p.4 LECTURE 1-5. SOLVING THE EQUATIONS A.6 p.
More information7.3 Singular points and the method of Frobenius
284 CHAPTER 7. POWER SERIES METHODS 7.3 Singular points and the method of Frobenius Note: or.5 lectures, 8.4 and 8.5 in [EP], 5.4 5.7 in [BD] While behaviour of ODEs at singular points is more complicated,
More informationApproximations for the Bessel and Struve Functions
MATHEMATICS OF COMPUTATION VOLUME 43, NUMBER 168 OCTOBER 1984. PAGES 551-556 Approximations for the Bessel and Struve Functions By J. N. Newman Abstract. Polynomials and rational-fraction approximations
More informationFORMAL TAYLOR SERIES AND COMPLEMENTARY INVARIANT SUBSPACES
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCITEY Volume 45, Number 1, July 1974 FORMAL TAYLOR SERIES AND COMPLEMENTARY INVARIANT SUBSPACES DOMINGO A. HERRERO X ABSTRACT. A class of operators (which includes
More informationSection 0.2 & 0.3 Worksheet. Types of Functions
MATH 1142 NAME Section 0.2 & 0.3 Worksheet Types of Functions Now that we have discussed what functions are and some of their characteristics, we will explore different types of functions. Section 0.2
More informationChapter 8. Exploring Polynomial Functions. Jennifer Huss
Chapter 8 Exploring Polynomial Functions Jennifer Huss 8-1 Polynomial Functions The degree of a polynomial is determined by the greatest exponent when there is only one variable (x) in the polynomial Polynomial
More information4 = [A, : at(a0)l Finally an amalgam A is finite if A (i = 0, ±1) are finite groups.
PROCEEDINGS of the AMERICAN MATHEMATICAL SOCIETY Volume 80, Number 1, September 1980 A CLASS OF FINITE GROUP-AMALGAMS DRAGOMIR Z. DJOKOVIC1 Abstract. Let A_\ and A, be finite groups such that A_ n Ax =
More informationBMT 2016 Orthogonal Polynomials 12 March Welcome to the power round! This year s topic is the theory of orthogonal polynomials.
Power Round Welcome to the power round! This year s topic is the theory of orthogonal polynomials. I. You should order your papers with the answer sheet on top, and you should number papers addressing
More informationIMMERSE 2008: Assignment 4
IMMERSE 2008: Assignment 4 4.) Let A be a ring and set R = A[x,...,x n ]. For each let R a = A x a...xa. Prove that is an -graded ring. a = (a,...,a ) R = a R a Proof: It is necessary to show that (a)
More information6 Second Order Linear Differential Equations
6 Second Order Linear Differential Equations A differential equation for an unknown function y = f(x) that depends on a variable x is any equation that ties together functions of x with y and its derivatives.
More informationr f l*llv»n = [/ [g*ix)]qw(x)dx < OO,
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 320, Number 2, August 1990 MAXIMAL FUNCTIONS ON CLASSICAL LORENTZ SPACES AND HARDY'S INEQUALITY WITH WEIGHTS FOR NONINCREASING FUNCTIONS MIGUEL
More informationPUTNAM PROBLEMS DIFFERENTIAL EQUATIONS. First Order Equations. p(x)dx)) = q(x) exp(
PUTNAM PROBLEMS DIFFERENTIAL EQUATIONS First Order Equations 1. Linear y + p(x)y = q(x) Muliply through by the integrating factor exp( p(x)) to obtain (y exp( p(x))) = q(x) exp( p(x)). 2. Separation of
More information1 Solving Algebraic Equations
Arkansas Tech University MATH 1203: Trigonometry Dr. Marcel B. Finan 1 Solving Algebraic Equations This section illustrates the processes of solving linear and quadratic equations. The Geometry of Real
More informationPreCalculus Notes. MAT 129 Chapter 5: Polynomial and Rational Functions. David J. Gisch. Department of Mathematics Des Moines Area Community College
PreCalculus Notes MAT 129 Chapter 5: Polynomial and Rational Functions David J. Gisch Department of Mathematics Des Moines Area Community College September 2, 2011 1 Chapter 5 Section 5.1: Polynomial Functions
More information^-,({«}))=ic^if<iic/n^ir=iic/, where e(n) is the sequence defined by e(n\m) = 8^, (the Kronecker delta).
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 70, Number 1, June 1978 COMPOSITION OPERATOR ON F AND ITS ADJOINT R. K. SINGH AND B. S. KOMAL Abstract. A necessary and sufficient condition for
More informationHere, logx denotes the natural logarithm of x. Mrsky [7], and later Cheo and Yien [2], proved that iz;.(»)=^io 8 "0(i).
Curtis Cooper f Robert E* Kennedy, and Milo Renberg Dept. of Mathematics, Central Missouri State University, Warrensburg, MO 64093-5045 (Submitted January 1997-Final Revision June 1997) 0. INTRODUCTION
More informationFRAGMENTABILITY OF ROTUND BANACH SPACES. Scott Sciffer. lim <Jl(x+A.y)- $(x) A-tO A
222 FRAGMENTABILITY OF ROTUND BANACH SPACES Scott Sciffer Introduction A topological. space X is said to be fragmented by a metric p if for every e > 0 and every subset Y of X there exists a nonempty relatively
More informationSection 5.2 Series Solution Near Ordinary Point
DE Section 5.2 Series Solution Near Ordinary Point Page 1 of 5 Section 5.2 Series Solution Near Ordinary Point We are interested in second order homogeneous linear differential equations with variable
More informationTHE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS
THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linear
More informationMATH 409 Advanced Calculus I Lecture 11: More on continuous functions.
MATH 409 Advanced Calculus I Lecture 11: More on continuous functions. Continuity Definition. Given a set E R, a function f : E R, and a point c E, the function f is continuous at c if for any ε > 0 there
More informationOn the Equation =<#>(«+ k)
MATHEMATICS OF COMPUTATION, VOLUME 26, NUMBER 11, APRIL 1972 On the Equation =(«+ k) By M. Lai and P. Gillard Abstract. The number of solutions of the equation (n) = >(n + k), for k ä 30, at intervals
More informationNote : This document might take a little longer time to print. more exam papers at : more exam papers at : more exam papers at : more exam papers at : more exam papers at : more exam papers at : more
More informationEE 4TM4: Digital Communications II. Channel Capacity
EE 4TM4: Digital Communications II 1 Channel Capacity I. CHANNEL CODING THEOREM Definition 1: A rater is said to be achievable if there exists a sequence of(2 nr,n) codes such thatlim n P (n) e (C) = 0.
More informationON THE HARMONIC SUMMABILITY OF DOUBLE FOURIER SERIES P. L. SHARMA
ON THE HARMONIC SUMMABILITY OF DOUBLE FOURIER SERIES P. L. SHARMA 1. Suppose that the function f(u, v) is integrable in the sense of Lebesgue, over the square ( ir, ir; it, it) and is periodic with period
More informationUNIFORM BOUNDS FOR BESSEL FUNCTIONS
Journal of Applied Analysis Vol. 1, No. 1 (006), pp. 83 91 UNIFORM BOUNDS FOR BESSEL FUNCTIONS I. KRASIKOV Received October 8, 001 and, in revised form, July 6, 004 Abstract. For ν > 1/ and x real we shall
More informationInteger-Valued Polynomials
Integer-Valued Polynomials LA Math Circle High School II Dillon Zhi October 11, 2015 1 Introduction Some polynomials take integer values p(x) for all integers x. The obvious examples are the ones where
More information