On the degrees of irreducible factors of higher order Bernoulli polynomials


 Lester Melton
 3 years ago
 Views:
Transcription
1 ACTA ARITHMETICA LXII.4 (1992 On he degrees of irreducible facors of higher order Bernoulli polynomials by Arnold Adelberg (Grinnell, Ia. 1. Inroducion. In his paper, we generalize he curren resuls on he peisensein behavior of firs and higher order Bernoulli polynomials [4], [6 9], using he machinery of [1]. In so doing, we provide a broader framework for he known resuls, all of which are eiher immediae consequences or special cases of our more general resuls. Because of an explici formula for he coefficiens in erms of falling facorials esablished in [1], he polynomials A n (x, which we consider here are acually ranslaes of he sandard higher order Bernoulli polynomials B n (ω (x, bu all of our significan resuls apply equally well o he sandard polynomials and heir ordinary coefficiens, wih ω = n k + 1. The main resuls are summarized using sandard noaions in he research announcemen [2]. Our approach differs from he usual one in ha we make no use of congruence properies of he Bernoulli numbers, and in paricular do no use he von Saud Clausen Theorem, which is an essenial ingredien of he usual approach. Insead we characerize he behavior of he padic poles of he coefficiens in erms of he base p expansion of n. The peisensein siuaion occurs when he highes order pole is simple. The Bernoulli polynomials B n (x are defined by (cf. [11, 12] e x e 1 = n=0 B n (x n n!. The arbirary order Bernoulli polynomials B n (ω ω e x = ( e 1 n=0 B n (ω (x n n!, (x are defined by which are called higher order, or more precisely firs or higher order, if ω {1, 2,..., n}. They are monic degree n raional polynomials. In [4], Carliz showed ha if n = k(p r where 0 < k < p, hen B n (x
2 330 A. Adelberg is irreducible (always wih respec o Q unless oherwise indicaed, namely ha pb n (x is peisensein. He also showed ha if 2m + 1 = k( + 1 where p is an odd prime and 1 k p, hen B 2m+1 /x(x 1 2 (x 1 has an irreducible facor of degree 2m + 1 p. Le S(n = S(n, p be he padic digi sum of n, i.e., if n = n i p i is he base p expansion, where all digis saisfy 0 n i, hen S(n = n i. McCarhy showed in [7] ha pb 2m (x is peisensein iff S(2m =. In [6], Kimura defined a funcion N(n, p, deermined by he base p expansion of n, which he used o srenghen he resuls of Carliz and McCarhy; namely, he showed ha if S(n, p, hen B n (x has an irreducible facor of degree N(n, p. We will generalize Kimura s and McCarhy s resuls and deermine all insances of even parial peisensein behavior of he firs and higher order Bernoulli polynomials. Our main ool is an analysis of he naure of he padic poles of he coefficiens, which urns ou o be remarkably regular and uniform. In he noaion of [1], he polynomials we acually consider are A n (x, s = n!a n (x, 0, s, wih exponenial generaing funcion ( s ln(1 + (1 + x = n=0 A n (x, s n n!. These are he Narumi polynomials [12, p. 127]. I has been shown ha A n (x, s = B n (n+s+1 (x + 1, so in paricular, A n (x, n = B n (x + 1 for firs order, wih s { 1,..., n} for ω {1,..., n}. To esablish he boundaries of our sudy, i should be noed ha A n (x, 0 = (x n and A n (x, (n + 1 = (1 + x n. We will show ha he Kimura bound [6, Theorems 1, 2], alhough sharp for s = n, is far from bes for he full range s { 1,..., n}. Finally, i has been shown ha if n is odd hen 2x (n + s 1 A n (x, s and i was proved in [1] ha A n (x, s is absoluely irreducible for n even > 0, and ha he above linear facor is he only nonrivial facor for n odd. The resuls in his paper enable us o deermine he insances of peisensein behavior leading o raional irreducibiliy of his ype. 2. Preliminaries. Throughou his paper, p will be a fixed bu arbirary prime, and all numbers are assumed o be posiive inegers, unless oherwise indicaed. ν p = padic valuaion of Q, i.e. ν p (m = highes power of p m if m Z and ν p (m/n = ν p (m ν p (n, wih ν p (0 =. If b > 0, we say r has a pole of order b if ν p (r = b.
3 Higher order Bernoulli polynomials 331 We say ha f(x = n i=0 a ix n i Q[x] is peisensein down o degree d if ( ν p (a 0 = 1, ν p (a i > 0 if 0 < i < n d, ν p (a n d = 0, and ν p (a i 0 for i > n d. We refer o his as (parial peisensein behavior; f(x is said o be peisensein if i is peisensein o degree 0. The following are sandard and/or easy o demonsrae (cf. [10, Ch. 4]. (1 If f(x is peisensein down o degree d and c Z, hen so is f(x+c. (2 If f(x is peisensein down o degree d, hen f(x has an irreducible facor of degree n d. (In paricular, peisensein implies irreducible. (3 If f(x = n i=0 b i(x n i in erms of he falling facorials, he coefficiens (a i of f(x saisfy ( iff he coefficiens (b i saisfy (. We also need some sandard resuls abou ν p of facorials and binomial coefficiens, where [x] = ineger par of x and x = leas ineger x. [ ] n (a ν p (n! = p e = n S(n n, whence ν p (n! 1 (b e=1 ( n + m S(n + S(m S(n + m ν p =. n if n > 0. (In paricular, S(n + m S(n + S(m. ( n + m (c ν p = # of carries in base p addiion of n and m. n ( n (d p iff n i m i for all digis of he padic expansions m (Lucas s Theorem. I is convenien o inroduce some nonsandard noaions, namely ( ( n m + δ m n if p and m δ if m m + δ, i.e. p. m m By (d, is a parial order, and clearly m n implies m n. Obviously if m and m have nonzero digis in differen places, hen (m + m n iff m n and m n. (Observe ha n > m iff n i > m i for he larges i such ha n i m i. Also by (c, m δ iff S(m + δ = S(m + S(δ. Finally, since ( ( k + m 1 = ( 1 m, m m
4 332 A. Adelberg we have ( p iff k 1 m. m The following lemma can be proved wihou oo much difficuly. Lemma 2.1. Le n > 0 and suppose ha n. Le q = n/( = s i=r q ip i, wih q r 0. The following are equivalen: ( n (i p. q (ii q r > q r+1 q r+2... q s. (iii S(n =. P r o o f. n 1 + q = pq 1 = s i=r+1 q i p i+1 + (q r 1p r+1 + ( r p i. The equivalence of (i and (ii follows immediaely by Lucas s Theorem. n = (q = s+1 i=r+2 i=0 (q i 1 q i p i + (q r q r+1 1p r+1 + (p q r p r. I follows ha if (ii holds, hen S(n =, by elescoping he digi sum. Suppose on he oher hand ha (ii fails. If q r q r+1, hen S(n p + q r q r p q r = p q r+1 + >, while if q r > q r+1 and f is he lowes place where q f 1 < q f, he same argumen shows S(n p q f + >. R e m a r k 1. The preceding lemma shows ha if S(n = hen n/( has nonzero digis in all places below he op digi of n, down o he boom nonzero digi of n. (I has a nonzero digi in he place of he op digi iff n = (p r. We now urn o Kimura s funcion N(n, p = N(n which we will use only when S(n. In his case, N(n = he smalles such ha n and S(. This is equivalen o N(n = smalles > 0 such ha n and, which is essenially Kimura s definiion [6, Lemma 4]. Thus if n = m i=0 n ip i is he base p expansion and r is such ha r i=0 n i while r+1 i=0 n i >, hen r N(n = i=0 ( n i p i + so ha S(N(n =. The nex lemma follows easily from (c. r n i p r+1, Lemma 2.2. Suppose ha δ and δ δ/(. Then δ = 0. i=0
5 Higher order Bernoulli polynomials 333 P r o o f. Observe ha δ + δ/( = δp/(. Hence if δ 0 and f is he lowes place where he digi q f 0, hen here is a carry in place f for he base p sum. We can deduce he following lemma, characerizing N(n. Lemma 2.3. Suppose N(n n and. Then p = N(n. P r o o f. If δ = N(n, hen n 1 + = ( N(n 1 + N(n ( + δ + δ. ( n Since here is no carry in he sum of he op digi of N(n and he boom digi of δ, and oherwise he wo summands have nonzero digis in differen places, his lemma follows from he previous wo. 3. padic analysis of he higher order Bernoulli polynomials. Recall ha from [1, 3.1ii and 3.2ii], n A n (x, s = b i (s(x n i, where b i (s = w(u=i i=0 ( ( s d(u 1 (n i d(u u 1 u u 1 3 u 2,... where he sum is over all sequences (u = (u 1, u 2,... = (r 1 r 2, r 2 r 3,... of nonnegaive inegers, evenually zero, wih weigh w(u = ju j = i and arbirary degree d(u = u j. I is convenien o abbreviae he erm ( ( s d(u 1 (s d(u τ u = (n w(u d(u u 1 u u 1 3 u 2 = (n w(u... u! u, and o ake w(u as he weigh of τ u. We say ha (u or τ u is concenraed in place if u j = 0 for all j. Fix k {1,..., n} and le s =. Clearly if w(u <, hen ν p (τ u 0 since p firs occurs in he denominaor when j =, whereas if w(u = i and (u is concenraed in place, hen i and ( ( ν p (τ u = ν p (n i i i ( n! = ν p i ( (n i! + ν p i = S(n i S(n + ν p i. iff
6 334 A. Adelberg The following lemma shows ha he erms concenraed in place deermine he pole srucure. Lemma 3.1. If (u is no concenraed in place, hen τ u dominaes a erm of lower weigh, i.e. here exiss (u wih w(u < w(u and ν p (τ u ν p (τ u, whence τ u dominaes a erm of lower weigh concenraed in place. P r o o f. C a s e 1. Suppose here exiss i < p 1 wih u i > 0. Consruc (u ui by replacing u i by 0 and increasing u by 1. This clearly ui decreases degree and weigh, and since ν p (u i! 1, we ge he desired dominaion. C a s e 2. Suppose here exiss i > wih u i > 0. Consruc (u by replacing u i by 0 and increasing u by u i. This clearly preserves degree and decreases weigh. Since ν p ((u + u i! ν p (u! + ν p (u i!, he dominaion is obvious unless i = kp α 1, α > 1, where (k, p = 1. In his case, Hence and w(u w(u = u i (kp α p u i (α 1p. ν p ( u = ν p ( u + u i (α 1 ν p ((n w(u ν p ((n w(u + u i (α 1, since for any m, ν p ((m rp r. This again esablishes he dominaion. Now, if (u is no concenraed in place, repea he argumen wih (u replaced by (u. Since he weighs are decreasing, evenually we ge a erm concenraed in place dominaed by τ u. Le a {0, 1, 2,...}. We hen deduce immediaely from he preceding lemma and remarks, using sandard properies of valuaions of sums, Corollary 3.2. If is he smalles i such ha ν p (b i < a, hen (i, and if τ u is concenraed in place and has weigh, hen ν p (τ u < a, (ii ν p (τ u a for all (u wih w(u excep for he erm τ u in (i, S(n S(n (iii a if i < and i, + ν p < a and S(n i S(n + ν p i
7 (iv if i <, i, p Higher order Bernoulli polynomials 335 ( n and S(i ( > a, hen p i. i The las condiion follows from he second par of he previous one. I is no hard o furher describe and indeed characerize by a heorem which implies ha he firs pole is simple, he second higher pole has order 2, ec., and which enables us o deermine where hey occur from he base p expansion of n. All subsequen resuls of his paper follow fairly easily from his heorem. Le a {0, 1, 2,...} as above. We say ha l is a segmen of n, saring in place r, if n = m i=0 n ip i and l = s i=r n ip i. (This includes l = 0 if s < r. If r = 0, we call l a boom segmen. Theorem 3.3. If is he smalles i such ha ν p (b i < a, hen ( (i p, ( n (ii p, S(n S(n (iii = S( = 1 a, (iv ν p (b = a 1, (v he only digis of n ha and n can share are op digis of segmens of divisible by, (vi is he smalles i such ha ( n i, p, i S(i > a and p ( i. P r o o f. Assume ha S( = and ha /( /(, e.g. = N(. Le 1 =, so 0 1 <, 1, and 1 /( /(, whence by (iii of he preceding corollary, a S(n 1 S(n = S(n 1 S(n + ν p 1 S( S(n S(n S(n S(n S(n S(n + ( + ν p 1 + ν p a. ( + ν p 1
8 336 A. Adelberg Thus all he inequaliies are acually equaliies, so ( S(n S(n ν p (b = + ν p = a 1, ( ( ν p 1 = ν p, and S(n 1 = S( + S(n, i.e. n. ( I follows ha ν p = 0, since if no, and r is he lowes place where ( here is a carry base p for k 1 + /(, i.e. (k 1 r + p, ( r and if = p r (, hen 1 /( and /( agree in all r ( 1 places excep place r and = 0, so ν p 1 conradicing he( above equaion. Nex assume r ( < ν p, 0, and le = (p r. Then n, r whence (n r = 0. Therefore n has nonzero digis only in he places where /( is zero. Now, le 0 = N(. Then n 0 and n has zero digis in all he places of 0 excep possibly he op place. Nex consider 1 = N( 0. The same argumen shows n 1, and if 1 and 0 share a digi of n in place r, hen (n r = 0. Coninue he process. Since, is he sum of hese i, so n, and he heorem follows. We can deduce easily from he preceding heorem and Lemmas 2.2 and 2.3 Corollary 3.4. Le l be a boom segmen of n wih S(n l. Le k = n l. Then = N(n l is he smalles i such ha ν p (b i < 0, ν p (b = 1, and ν p (b i 1 for all i. P r o o f. Observe ha = = N(n l, following he noaion of he previous proof. If l < l is any smaller segmen of n, hen k 1 has all digis in he places of l, so p N(n l. Since p N(n l ( and p N(n l l 1 N(n l
9 Higher order Bernoulli polynomials 337 if l 1 is a boom segmen of n l N(n l by he lemmas, he resul follows immediaely. R e m a r k 2. Le j be he smalles i such ha ν p (b i < j for j = 0, 1,..., a. Then ν p (b i = j 1 if i = j by (iv, so here are no gaps in he orders of he poles. We can use (v o consruc as follows: here is a (unique sequence l 0, l 1,..., l a of segmens of n such ha l j+1 is a boom segmen of n j j λ=0 l λ and ( j+1 j+1 j = N n j l λ = N j+1. Thus he higher order poles come in he naural order. In paricular, 0 = N(n l 0 = N 0 gives he firs pole (simple, where l 0 is a boom segmen of n, and 1 = 0 + N(n 0 l 0 l 1 gives he firs higher pole (double, where l 1 is a boom segmen of n 0 l 0. Hence j = j λ=0 N λ, where S(N λ =, and he N λ give he degree differences for he successively higher order poles. The sequences ( j and (l j can be deermined recursively ogeher from n and k as follows: l j+1 is he smalles segmen l of n j saring in he place of he highes digi of j such ha p ( λ=0 N(n j j λ=0 l λ l and j+1 j = N(n j j+1 λ=0 l λ, wih iniial condiion 1 = 0. [ I follows ] ha k = 1 gives he bigges possible pole, which has order S(n, wih all l j = 0. In paricular, here are no poles if S(n <. I is no hard o show ha we can urn around he preceding consrucion, namely if is consruced as in he preceding paragraph, hen here exiss k such ha j = j λ=0 N λ is he firs i such ha ν p (b i < j for j = 0, 1,..., a, and ν p (b i a 1 for all i. Thus exhibis he complee singulariy paern for such k. I can be seen ha one way o choose k is o sar wih n l 0, subrac all he N λ for λ > 0 and also he highes place conribuion of N 0 if N 0 and N 1 share a digi of n. The case p = 2 is simples o describe since S(N = iff N is a 2power, and here is no sharing of digis. In his case, N 0, N 1,..., N a are jus successively higher powers of 2 ha occur in he base 2 expansion of n, and k = n l 0 a λ=1 N λ = n l 0 ( a 0. Now we urn o he peisensein behavior of hese polynomials. When we say ha A n (x, is peisensein down o degree d, we acually refer o pa n (x,. The condiion ha we need is a simple highes order pole, which by he above analysis is equivalen o having a (simple pole, bu no,
10 338 A. Adelberg pole of order wo, i.e. ν p (b 0 = 0, ν p (b i 0 for i < n d, ν p (b n d = 1, ν p (b i 1 all i. If S(n <, hen (S(n i S(n/( > 1 for all i, so as previously noed here is no peisensein behavior. The nex heorem, which esablishes all insances of peisensein behavior follows immediaely from he preceding discussion. Theorem 3.5. If A n (x, is peisensein down o degree d, hen d = n N(n l for some boom segmen l of he padic expansion of n wih S(n l. Conversely, if d = n N(n l, hen A n (x, (n l is peisensein down o degree d; furhermore, A n (x, is peisensein down o degree d iff he following hree condiions hold: N(n l (i p (ii p ( (iii p N(n l for all boom segmens l of n such ha l < l,, and N(n l l 1 N(n l if l 1 is any boom segmen of n l N(n l. (This condiion assumes S(n l S(l 1 (, so i is vacuous if S(n l < 2(. The case p = 2 of his heorem is paricularly easy o sae: If A n (x, is 2Eisensein down o degree d, hen d = n 2 f for some power 2 f in he base 2 expansion ( of n. Furhermore, ( A n (x, is 2Eisensein down o degree n 2 f iff 2 and 2 for all oher powers 2 g in he base 2 2 f 2 g expansion of n iff he base 2 expansion of k 1 conains all he 2 g bu does no conain 2 f iff n 2 f k 1 iff k 1 = n 2 f + m, where m is any sum of lower powers of 2 no in he base 2 expansion of n. (Observe ha k = n l if m is all hese lower powers, where l is he boom segmen of n below 2 f. The following corollaries are special cases. Take l = 0 o ge he firs hree corollaries, and l = n 0 o ge he las hree. Corollary 3.6 (Kimura. A n (x, n is peisensein down o degree n N(n. R e m a r k 3. I follows immediaely ha if S(n ( < 2(p 1, hen A n (x, is peisensein down o degree n N(n iff p N(n. Since N(n = n iff S(n =, for our resriced use of N(n, we ge Corollary 3.7 (McCarhy. A n (x, n is peisensein iff S(n =.
11 p We can sharpen his as follows: Higher order Bernoulli polynomials 339 Corollary 3.8. The following are equivalen: (i S(n =. (ii There exiss k {1,..., n} such ha A n (x, is peisensein. (iii A n (x, n is peisensein. (iv A n (x, 1 is peisensein. ( Furhermore, if hese condiions hold, A n (x, is peisensein iff n. R e m a r k 4. I follows ha if n = (p r and k n, hen A n (x, is peisensein. In paricular, if n = 2 r and k n, hen A n (x, is 2 Eisensein. In fac, i is easy o deduce ha A n (x, is peisensein for all k {1,..., n} iff n = (p r, r 0. (If n (p r and n f is he lowes nonzero digi, hen le k 1 = (p f, so k n and k 1 n. More generally, if n = m(p r wih 1 m < p, hen 0 < k (p mp r and n k > (m 1p r+1 each imply ha A n (x, is peisensein, namely n = (m 1p r+1 + (p mp r, so S(n =, and each condiion implies ha k 1 n/(. Compare his wih [8, Theorems 1, 2, 4] and [9, Theorems 1, 2]. Corollary 3.9 (where l = n 0. Suppose ha n = m + l where l > 0 and p (n l. Then he following are equivalen: (i S(m = and p m. (ii There exiss k {1,..., n} such ha A n (x, is peisensein down o degree l. (iii A n (x, m is peisensein down o degree l. Furhermore, ( if hese condiions ( hold, A n (x, is peisensein down o degree l iff p m and p N(n. P r o o f. Observe ha he condiions p (n l and p n l are equivalen o l = n 0. The resul hus follows immediaely from he heorem. Corollary 3.10 (where l = n 0 = 1. Suppose ha p n 1 and S(n 1 =. Then A n (x, (n 1 is peisensein down o degree 1 (so here is an irreducible facor of degree n 1. Also A n (x, is peisensein down o degree 1 iff p (so N(n = n i. n 1 and p n i, where i is he larges ppower < n
12 340 A. Adelberg We have no seen anyhing like he following simple corollary in he lieraure. Corollary 3.11 (where l = n 0 = 1, p = 2. If n = 2 r + 1, wih r > 0, hen A n (x, ( is peisensein ( down o degree 1 iff k is even. (This is equivalen o 2 2 r and 2. 1 These corollaries give he irreducibiliy of cerain insances of A n (x, /(2x (n k 1 for n odd. We give some addiional illusraive compuaions beyond he case n = 2 r + 1, where he Kimura bound only gives he rivial resul ha here is an irreducible facor of degree 1, since N(n = 1. Examples. (1 Le n = 13, p = 3, l = 1. Then S(12 = 2 = and p n 1, from which i follows ha A n (x, is peisensein down o degree 1 for k = 2, 3, 11, 12, whence A n (x, /(2x (n k 1 is irreducible for hese values. Since N(13 = 4, Kimura s analysis only gives an irreducible facor of degree 4, and McCarhy s higher order heorem [8, Theorem 3], wih p = 5, only gives an irreducible facor of degree 8. The preceding analysis can also be useful when n is even, e.g. (2 n = 14, p = 3, l = 2. Then A 14 (x, is 3Eisensein down o degree 2 if k = 3 or 12, so here is an irreducible facor of degree 12. On he oher hand, N(14 = 2, so Kimura s bound only gives an irreducible facor of degree 2. I is of course known ha B 14 (x is irreducible [5]. (3 Finally, we consider he excepional case n = 11, where here is an addiional irreducible quadraic facor for k = 11 (cf. [3]. Wih p = 2, N(11 = 1 and N(10 = 2, which do no help much, bu N(8 = 8, so A 11 (x, 8 has an irreducible facor of degree 8. All of he peisensein resuls for higher order Bernoulli polynomials which we have seen in he lieraure can be readily derived from Theorem 3.5 and is corollaries, usually considerably srenghened. As an example, consider he following heorem of McCarhy, wih noaions slighly changed o conform wih our usage. Theorem M [8, Theorem 3]. Le p be an odd prime, and assume ha n = r( + 1 where 0 r 1 p α. Then B n (α (x has an irreducible facor of degree n p. We acually prove a sronger if and only if heorem, which conains he preceding resul as one direcion of he special case where α < p.
13 Higher order Bernoulli polynomials 341 Theorem M. Le p be an odd prime and 1 α n. Assume ha n 1 and 2 (n 1/( p. Then B n (α (x is peisensein down o degree p iff n p α p α. p P r o o f. Le r = (n 1/(, so n = r( + 1 and (n p/( = r 1. (Thus if α < p, Theorem M follows from our Theorem 3.5; he case r = 1 is rivial since hen n = p and Theorem M is vacuous. α Le q =, so 0 qp α < p. Since rp n α, we have r q. p Bu n = (r 1p + (p r + 1 so S(n = p by our hypoheses, whence S(n < 2( since p is odd. Thus if k = n + 1 α, hen B (α N(n. n (x is peisensein down o degree n N(n iff p ( Since N(n = n p = (r 1(, we mus deermine when p, r 1 i.e. when k 1 r 1. Bu k 1 + r 1 = rp α = (r qp + (qp α and 0 r q, r 1, qp α, so he resul follows immediaely from Lucas s Theorem. References [1] A. A d e l b e r g, A finie difference approach o degenerae Bernoulli and Sirling polynomials, Discree Mah., submied. [2], Irreducible facors and padic poles of higher order Bernoulli polynomials, C. R. Mah. Rep. Acad. Sci. Canada 14 (1992, [3] J. Brillhar, On he Euler and Bernoulli polynomials, J. Reine Angew. Mah. 234 (1969, [4] L. Carliz, Noe on irreducibiliy of he Bernoulli and Euler polynomials, Duke Mah. J. 19 (1952, [5], The irreducibiliy of he Bernoulli polynomial B 14 (x, Mah. Comp. 19 (1965, [6] N. Kimura, On he degree of an irreducible facor of he Bernoulli polynomials, Aca Arih. 50 (1988, [7] P. J. M c C a r h y, Irreducibiliy of cerain Bernoulli polynomials, Amer. Mah. Monhly 68 (1961, [8], Some irreducibiliy heorems for Bernoulli polynomials of higher order, Duke Mah. J. 27 (1960, [9], Irreducibiliy of Bernoulli polynomials of higher order, Canad. J. Mah. 14 (1962, [10] I. Niven, H. Zuckerman and H. Mongomery, An Inroducion o he Theory of Numbers, 5h ed., Wiley, 1991.
14 342 A. Adelberg [11] N. E. Nörlund, Differenzenrechnung, Springer, [12] S. Roman, The Umbral Calculus, Academic Press, DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE GRINNELL COLLEGE GRINNELL, IOWA 50112, U.S.A. Received on and in revised form on (2167
The Transport Equation
The Transpor Equaion Consider a fluid, flowing wih velociy, V, in a hin sraigh ube whose cross secion will be denoed by A. Suppose he fluid conains a conaminan whose concenraion a posiion a ime will be
More information17 Laplace transform. Solving linear ODE with piecewise continuous right hand sides
7 Laplace ransform. Solving linear ODE wih piecewise coninuous righ hand sides In his lecure I will show how o apply he Laplace ransform o he ODE Ly = f wih piecewise coninuous f. Definiion. A funcion
More informationDuration and Convexity ( ) 20 = Bond B has a maturity of 5 years and also has a required rate of return of 10%. Its price is $613.
Graduae School of Business Adminisraion Universiy of Virginia UVAF38 Duraion and Convexiy he price of a bond is a funcion of he promised paymens and he marke required rae of reurn. Since he promised
More informationRandom Walk in 1D. 3 possible paths x vs n. 5 For our random walk, we assume the probabilities p,q do not depend on time (n)  stationary
Random Walk in D Random walks appear in many cones: diffusion is a random walk process undersanding buffering, waiing imes, queuing more generally he heory of sochasic processes gambling choosing he bes
More informationChapter 7. Response of FirstOrder RL and RC Circuits
Chaper 7. esponse of FirsOrder L and C Circuis 7.1. The Naural esponse of an L Circui 7.2. The Naural esponse of an C Circui 7.3. The ep esponse of L and C Circuis 7.4. A General oluion for ep and Naural
More informationANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS
ANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS R. Caballero, E. Cerdá, M. M. Muñoz and L. Rey () Deparmen of Applied Economics (Mahemaics), Universiy of Málaga,
More informationForecasting and Information Sharing in Supply Chains Under QuasiARMA Demand
Forecasing and Informaion Sharing in Supply Chains Under QuasiARMA Demand Avi Giloni, Clifford Hurvich, Sridhar Seshadri July 9, 2009 Absrac In his paper, we revisi he problem of demand propagaion in
More informationNiche Market or Mass Market?
Niche Marke or Mass Marke? Maxim Ivanov y McMaser Universiy July 2009 Absrac The de niion of a niche or a mass marke is based on he ranking of wo variables: he monopoly price and he produc mean value.
More informationOptimal Investment and Consumption Decision of Family with Life Insurance
Opimal Invesmen and Consumpion Decision of Family wih Life Insurance Minsuk Kwak 1 2 Yong Hyun Shin 3 U Jin Choi 4 6h World Congress of he Bachelier Finance Sociey Torono, Canada June 25, 2010 1 Speaker
More informationA NOTE ON THE ALMOST EVERYWHERE CONVERGENCE OF ALTERNATING SEQUENCES WITH DUNFORD SCHWARTZ OPERATORS
C O L L O Q U I U M M A T H E M A T I C U M VOL. LXII 1991 FASC. I A OTE O THE ALMOST EVERYWHERE COVERGECE OF ALTERATIG SEQUECES WITH DUFORD SCHWARTZ OPERATORS BY RYOTARO S A T O (OKAYAMA) 1. Inroducion.
More informationOption PutCall Parity Relations When the Underlying Security Pays Dividends
Inernaional Journal of Business and conomics, 26, Vol. 5, No. 3, 22523 Opion Puall Pariy Relaions When he Underlying Securiy Pays Dividends Weiyu Guo Deparmen of Finance, Universiy of Nebraska Omaha,
More informationMTH6121 Introduction to Mathematical Finance Lesson 5
26 MTH6121 Inroducion o Mahemaical Finance Lesson 5 Conens 2.3 Brownian moion wih drif........................... 27 2.4 Geomeric Brownian moion........................... 28 2.5 Convergence of random
More informationLectures # 5 and 6: The Prime Number Theorem.
Lecures # 5 and 6: The Prime Number Theorem Noah Snyder July 8, 22 Riemann s Argumen Riemann used his analyically coninued ζfuncion o skech an argumen which would give an acual formula for π( and sugges
More informationSecond Order Linear Differential Equations
Second Order Linear Differenial Equaions Second order linear equaions wih consan coefficiens; Fundamenal soluions; Wronskian; Exisence and Uniqueness of soluions; he characerisic equaion; soluions of homogeneous
More informationUsefulness of the Forward Curve in Forecasting Oil Prices
Usefulness of he Forward Curve in Forecasing Oil Prices Akira Yanagisawa Leader Energy Demand, Supply and Forecas Analysis Group The Energy Daa and Modelling Cener Summary When people analyse oil prices,
More information4 Convolution. Recommended Problems. x2[n] 1 2[n]
4 Convoluion Recommended Problems P4.1 This problem is a simple example of he use of superposiion. Suppose ha a discreeime linear sysem has oupus y[n] for he given inpus x[n] as shown in Figure P4.11.
More informationAppendix A: Area. 1 Find the radius of a circle that has circumference 12 inches.
Appendi A: Area workedou s o OddNumbered Eercises Do no read hese workedou s before aemping o do he eercises ourself. Oherwise ou ma mimic he echniques shown here wihou undersanding he ideas. Bes wa
More information2.5 Life tables, force of mortality and standard life insurance products
Soluions 5 BS4a Acuarial Science Oford MT 212 33 2.5 Life ables, force of moraliy and sandard life insurance producs 1. (i) n m q represens he probabiliy of deah of a life currenly aged beween ages + n
More informationcooking trajectory boiling water B (t) microwave 0 2 4 6 8 101214161820 time t (mins)
Alligaor egg wih calculus We have a large alligaor egg jus ou of he fridge (1 ) which we need o hea o 9. Now here are wo accepable mehods for heaing alligaor eggs, one is o immerse hem in boiling waer
More informationnonlocal conditions.
ISSN 17493889 prin, 17493897 online Inernaional Journal of Nonlinear Science Vol.11211 No.1,pp.39 Boundary Value Problem for Some Fracional Inegrodifferenial Equaions wih Nonlocal Condiions Mohammed
More informationCircuit Types. () i( t) ( )
Circui Types DC Circuis Idenifying feaures: o Consan inpus: he volages of independen volage sources and currens of independen curren sources are all consan. o The circui does no conain any swiches. All
More informationAnalogue and Digital Signal Processing. First Term Third Year CS Engineering By Dr Mukhtiar Ali Unar
Analogue and Digial Signal Processing Firs Term Third Year CS Engineering By Dr Mukhiar Ali Unar Recommended Books Haykin S. and Van Veen B.; Signals and Sysems, John Wiley& Sons Inc. ISBN: 073807 Ifeachor
More informationEconomics Honors Exam 2008 Solutions Question 5
Economics Honors Exam 2008 Soluions Quesion 5 (a) (2 poins) Oupu can be decomposed as Y = C + I + G. And we can solve for i by subsiuing in equaions given in he quesion, Y = C + I + G = c 0 + c Y D + I
More informationSinglemachine Scheduling with Periodic Maintenance and both Preemptive and. Nonpreemptive jobs in Remanufacturing System 1
Absrac number: 050407 Singlemachine Scheduling wih Periodic Mainenance and boh Preempive and Nonpreempive jobs in Remanufacuring Sysem Liu Biyu hen Weida (School of Economics and Managemen Souheas Universiy
More informationTerm Structure of Prices of Asian Options
Term Srucure of Prices of Asian Opions Jirô Akahori, Tsuomu Mikami, Kenji Yasuomi and Teruo Yokoa Dep. of Mahemaical Sciences, Risumeikan Universiy 111 Nojihigashi, Kusasu, Shiga 5258577, Japan Email:
More informationCHARGE AND DISCHARGE OF A CAPACITOR
REFERENCES RC Circuis: Elecrical Insrumens: Mos Inroducory Physics exs (e.g. A. Halliday and Resnick, Physics ; M. Sernheim and J. Kane, General Physics.) This Laboraory Manual: Commonly Used Insrumens:
More informationPATHWISE PROPERTIES AND PERFORMANCE BOUNDS FOR A PERISHABLE INVENTORY SYSTEM
PATHWISE PROPERTIES AND PERFORMANCE BOUNDS FOR A PERISHABLE INVENTORY SYSTEM WILLIAM L. COOPER Deparmen of Mechanical Engineering, Universiy of Minnesoa, 111 Church Sree S.E., Minneapolis, MN 55455 billcoop@me.umn.edu
More informationarxiv:math/0111328v1 [math.co] 30 Nov 2001
arxiv:mah/038v [mahco 30 Nov 00 EVALUATIONS OF SOME DETERMINANTS OF MATRICES RELATED TO THE PASCAL TRIANGLE C Kraenhaler Insiu für Mahemaik der Universiä Wien, Srudlhofgasse 4, A090 Wien, Ausria email:
More informationPresent Value Methodology
Presen Value Mehodology Econ 422 Invesmen, Capial & Finance Universiy of Washingon Eric Zivo Las updaed: April 11, 2010 Presen Value Concep Wealh in Fisher Model: W = Y 0 + Y 1 /(1+r) The consumer/producer
More informationPermutations and Combinations
Permuaions and Combinaions Combinaorics Copyrigh Sandards 006, Tes  ANSWERS Barry Mabillard. 0 www.mah0s.com 1. Deermine he middle erm in he expansion of ( a b) To ge he kvalue for he middle erm, divide
More informationWorking Paper On the timing option in a futures contract. SSE/EFI Working Paper Series in Economics and Finance, No. 619
econsor www.econsor.eu Der OpenAccessPublikaionsserver der ZBW LeibnizInformaionszenrum Wirschaf The Open Access Publicaion Server of he ZBW Leibniz Informaion Cenre for Economics Biagini, Francesca;
More informationA Reexamination of the Joint Mortality Functions
Norh merican cuarial Journal Volume 6, Number 1, p.166170 (2002) Reeaminaion of he Join Morali Funcions bsrac. Heekung Youn, rkad Shemakin, Edwin Herman Universi of S. Thomas, Sain Paul, MN, US Morali
More informationInductance and Transient Circuits
Chaper H Inducance and Transien Circuis Blinn College  Physics 2426  Terry Honan As a consequence of Faraday's law a changing curren hrough one coil induces an EMF in anoher coil; his is known as muual
More informationPROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE
Profi Tes Modelling in Life Assurance Using Spreadshees PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE Erik Alm Peer Millingon 2004 Profi Tes Modelling in Life Assurance Using Spreadshees
More informationThe Greek financial crisis: growing imbalances and sovereign spreads. Heather D. Gibson, Stephan G. Hall and George S. Tavlas
The Greek financial crisis: growing imbalances and sovereign spreads Heaher D. Gibson, Sephan G. Hall and George S. Tavlas The enry The enry of Greece ino he Eurozone in 2001 produced a dividend in he
More informationARCH 2013.1 Proceedings
Aricle from: ARCH 213.1 Proceedings Augus 14, 212 Ghislain Leveille, Emmanuel Hamel A renewal model for medical malpracice Ghislain Léveillé École d acuaria Universié Laval, Québec, Canada 47h ARC Conference
More informationDifferential Equations. Solving for Impulse Response. Linear systems are often described using differential equations.
Differenial Equaions Linear sysems are ofen described using differenial equaions. For example: d 2 y d 2 + 5dy + 6y f() d where f() is he inpu o he sysem and y() is he oupu. We know how o solve for y given
More informationINDEPENDENT MARGINALS OF OPERATOR LÉVY S PROBABILITY MEASURES ON FINITE DIMENSIONAL VECTOR SPACES
Journal of Applied Analysis 1, 1 (1995), pp. 39 45 INDEPENDENT MARGINALS OF OPERATOR LÉVY S PROBABILITY MEASURES ON FINITE DIMENSIONAL VECTOR SPACES A. LUCZAK Absrac. We find exponens of independen marginals
More informationMorningstar Investor Return
Morningsar Invesor Reurn Morningsar Mehodology Paper Augus 31, 2010 2010 Morningsar, Inc. All righs reserved. The informaion in his documen is he propery of Morningsar, Inc. Reproducion or ranscripion
More informationCointegration: The Engle and Granger approach
Coinegraion: The Engle and Granger approach Inroducion Generally one would find mos of he economic variables o be nonsaionary I(1) variables. Hence, any equilibrium heories ha involve hese variables require
More informationImproper Integrals. Dr. Philippe B. laval Kennesaw State University. September 19, 2005. f (x) dx over a finite interval [a, b].
Improper Inegrls Dr. Philippe B. lvl Kennesw Se Universiy Sepember 9, 25 Absrc Noes on improper inegrls. Improper Inegrls. Inroducion In Clculus II, sudens defined he inegrl f (x) over finie inervl [,
More informationAppendix D Flexibility Factor/Margin of Choice Desktop Research
Appendix D Flexibiliy Facor/Margin of Choice Deskop Research Cheshire Eas Council Cheshire Eas Employmen Land Review Conens D1 Flexibiliy Facor/Margin of Choice Deskop Research 2 Final Ocober 2012 \\GLOBAL.ARUP.COM\EUROPE\MANCHESTER\JOBS\200000\22348900\4
More information1. y 5y + 6y = 2e t Solution: Characteristic equation is r 2 5r +6 = 0, therefore r 1 = 2, r 2 = 3, and y 1 (t) = e 2t,
Homework6 Soluions.7 In Problem hrough 4 use he mehod of variaion of parameers o find a paricular soluion of he given differenial equaion. Then check your answer by using he mehod of undeermined coeffiens..
More informationA UNIFIED APPROACH TO MATHEMATICAL OPTIMIZATION AND LAGRANGE MULTIPLIER THEORY FOR SCIENTISTS AND ENGINEERS
A UNIFIED APPROACH TO MATHEMATICAL OPTIMIZATION AND LAGRANGE MULTIPLIER THEORY FOR SCIENTISTS AND ENGINEERS RICHARD A. TAPIA Appendix E: Differeniaion in Absrac Spaces I should be no surprise ha he differeniaion
More information11/6/2013. Chapter 14: Dynamic ADAS. Introduction. Introduction. Keeping track of time. The model s elements
Inroducion Chaper 14: Dynamic DS dynamic model of aggregae and aggregae supply gives us more insigh ino how he economy works in he shor run. I is a simplified version of a DSGE model, used in cuingedge
More informationANALYTIC PROOF OF THE PRIME NUMBER THEOREM
ANALYTIC PROOF OF THE PRIME NUMBER THEOREM RYAN SMITH, YUAN TIAN Conens Arihmeical Funcions Equivalen Forms of he Prime Number Theorem 3 3 The Relaionshi Beween Two Asymoic Relaions 6 4 Dirichle Series
More informationEntropy: From the Boltzmann equation to the Maxwell Boltzmann distribution
Enropy: From he Bolzmann equaion o he Maxwell Bolzmann disribuion A formula o relae enropy o probabiliy Ofen i is a lo more useful o hink abou enropy in erms of he probabiliy wih which differen saes are
More informationThe Heisenberg group and Pansu s Theorem
The Heisenberg group and Pansu s Theorem July 31, 2009 Absrac The goal of hese noes is o inroduce he reader o he Heisenberg group wih is Carno Carahéodory meric and o Pansu s differeniaion heorem. As
More information4.2 Trigonometric Functions; The Unit Circle
4. Trigonomeric Funcions; The Uni Circle Secion 4. Noes Page A uni circle is a circle cenered a he origin wih a radius of. Is equaion is as shown in he drawing below. Here he leer represens an angle measure.
More informationA Mathematical Description of MOSFET Behavior
10/19/004 A Mahemaical Descripion of MOSFET Behavior.doc 1/8 A Mahemaical Descripion of MOSFET Behavior Q: We ve learned an awful lo abou enhancemen MOSFETs, bu we sill have ye o esablished a mahemaical
More informationJournal Of Business & Economics Research September 2005 Volume 3, Number 9
Opion Pricing And Mone Carlo Simulaions George M. Jabbour, (Email: jabbour@gwu.edu), George Washingon Universiy YiKang Liu, (yikang@gwu.edu), George Washingon Universiy ABSTRACT The advanage of Mone Carlo
More information3 RungeKutta Methods
3 RungeKua Mehods In conras o he mulisep mehods of he previous secion, RungeKua mehods are singlesep mehods however, muliple sages per sep. They are moivaed by he dependence of he Taylor mehods on he
More informationAP Calculus AB 2013 Scoring Guidelines
AP Calculus AB 1 Scoring Guidelines The College Board The College Board is a missiondriven noforprofi organizaion ha connecs sudens o college success and opporuniy. Founded in 19, he College Board was
More informationTable of contents Chapter 1 Interest rates and factors Chapter 2 Level annuities Chapter 3 Varying annuities
Table of conens Chaper 1 Ineres raes and facors 1 1.1 Ineres 2 1.2 Simple ineres 4 1.3 Compound ineres 6 1.4 Accumulaed value 10 1.5 Presen value 11 1.6 Rae of discoun 13 1.7 Consan force of ineres 17
More information9. Capacitor and Resistor Circuits
ElecronicsLab9.nb 1 9. Capacior and Resisor Circuis Inroducion hus far we have consider resisors in various combinaions wih a power supply or baery which provide a consan volage source or direc curren
More informationSEMIMARTINGALE STOCHASTIC APPROXIMATION PROCEDURE AND RECURSIVE ESTIMATION. Chavchavadze Ave. 17 a, Tbilisi, Georgia, Email: toronj333@yahoo.
SEMIMARTINGALE STOCHASTIC APPROXIMATION PROCEDURE AND RECURSIVE ESTIMATION N. LAZRIEVA, 2, T. SHARIA 3, 2 AND T. TORONJADZE Georgian American Universiy, Business School, 3, Alleyway II, Chavchavadze Ave.
More informationThe naive method discussed in Lecture 1 uses the most recent observations to forecast future values. That is, Y ˆ t + 1
Business Condiions & Forecasing Exponenial Smoohing LECTURE 2 MOVING AVERAGES AND EXPONENTIAL SMOOTHING OVERVIEW This lecure inroduces imeseries smoohing forecasing mehods. Various models are discussed,
More informationStochastic Optimal Control Problem for Life Insurance
Sochasic Opimal Conrol Problem for Life Insurance s. Basukh 1, D. Nyamsuren 2 1 Deparmen of Economics and Economerics, Insiue of Finance and Economics, Ulaanbaaar, Mongolia 2 School of Mahemaics, Mongolian
More informationA Note on Using the Svensson procedure to estimate the risk free rate in corporate valuation
A Noe on Using he Svensson procedure o esimae he risk free rae in corporae valuaion By Sven Arnold, Alexander Lahmann and Bernhard Schwezler Ocober 2011 1. The risk free ineres rae in corporae valuaion
More informationBALANCE OF PAYMENTS. First quarter 2008. Balance of payments
BALANCE OF PAYMENTS DATE: 20080530 PUBLISHER: Balance of Paymens and Financial Markes (BFM) Lena Finn + 46 8 506 944 09, lena.finn@scb.se Camilla Bergeling +46 8 506 942 06, camilla.bergeling@scb.se
More informationUNDERSTANDING THE DEATH BENEFIT SWITCH OPTION IN UNIVERSAL LIFE POLICIES. Nadine Gatzert
UNDERSTANDING THE DEATH BENEFIT SWITCH OPTION IN UNIVERSAL LIFE POLICIES Nadine Gazer Conac (has changed since iniial submission): Chair for Insurance Managemen Universiy of ErlangenNuremberg Lange Gasse
More informationGraduate Macro Theory II: Notes on Neoclassical Growth Model
Graduae Macro Theory II: Noes on Neoclassical Growh Model Eric Sims Universiy of Nore Dame Spring 2011 1 Basic Neoclassical Growh Model The economy is populaed by a large number of infiniely lived agens.
More informationDensity Dependence. births are a decreasing function of density b(n) and deaths are an increasing function of density d(n).
FW 662 Densiydependen populaion models In he previous lecure we considered densiy independen populaion models ha assumed ha birh and deah raes were consan and no a funcion of populaion size. Longerm
More informationChapter 2 Problems. 3600s = 25m / s d = s t = 25m / s 0.5s = 12.5m. Δx = x(4) x(0) =12m 0m =12m
Chaper 2 Problems 2.1 During a hard sneeze, your eyes migh shu for 0.5s. If you are driving a car a 90km/h during such a sneeze, how far does he car move during ha ime s = 90km 1000m h 1km 1h 3600s = 25m
More informationLife insurance cash flows with policyholder behaviour
Life insurance cash flows wih policyholder behaviour Krisian Buchard,,1 & Thomas Møller, Deparmen of Mahemaical Sciences, Universiy of Copenhagen Universiesparken 5, DK2100 Copenhagen Ø, Denmark PFA Pension,
More informationModeling Stock Price Dynamics with Fuzzy Opinion Networks
Modeling Sock Price Dynamics wih Fuzzy Opinion Neworks LiXin Wang Deparmen of Auomaion Science and Technology Xian Jiaoong Universiy, Xian, P.R. China Email: lxwang@mail.xju.edu.cn Key words: Sock price
More informationMeasuring macroeconomic volatility Applications to export revenue data, 19702005
FONDATION POUR LES ETUDES ET RERS LE DEVELOPPEMENT INTERNATIONAL Measuring macroeconomic volailiy Applicaions o expor revenue daa, 1970005 by Joël Cariolle Policy brief no. 47 March 01 The FERDI is a
More information= r t dt + σ S,t db S t (19.1) with interest rates given by a mean reverting OrnsteinUhlenbeck or Vasicek process,
Chaper 19 The BlackScholesVasicek Model The BlackScholesVasicek model is given by a sandard imedependen BlackScholes model for he sock price process S, wih imedependen bu deerminisic volailiy σ
More informationPart 1: White Noise and Moving Average Models
Chaper 3: Forecasing From Time Series Models Par 1: Whie Noise and Moving Average Models Saionariy In his chaper, we sudy models for saionary ime series. A ime series is saionary if is underlying saisical
More informationSteps for D.C Analysis of MOSFET Circuits
10/22/2004 Seps for DC Analysis of MOSFET Circuis.doc 1/7 Seps for D.C Analysis of MOSFET Circuis To analyze MOSFET circui wih D.C. sources, we mus follow hese five seps: 1. ASSUME an operaing mode 2.
More informationAcceleration Lab Teacher s Guide
Acceleraion Lab Teacher s Guide Objecives:. Use graphs of disance vs. ime and velociy vs. ime o find acceleraion of a oy car.. Observe he relaionship beween he angle of an inclined plane and he acceleraion
More informationA Probability Density Function for Google s stocks
A Probabiliy Densiy Funcion for Google s socks V.Dorobanu Physics Deparmen, Poliehnica Universiy of Timisoara, Romania Absrac. I is an approach o inroduce he Fokker Planck equaion as an ineresing naural
More informationReturn Calculation of U.S. Treasury Constant Maturity Indices
Reurn Calculaion of US Treasur Consan Mauri Indices Morningsar Mehodolog Paper Sepeber 30 008 008 Morningsar Inc All righs reserved The inforaion in his docuen is he proper of Morningsar Inc Reproducion
More informationA Generalized Bivariate OrnsteinUhlenbeck Model for Financial Assets
A Generalized Bivariae OrnseinUhlenbeck Model for Financial Asses Romy Krämer, Mahias Richer Technische Universiä Chemniz, Fakulä für Mahemaik, 917 Chemniz, Germany Absrac In his paper, we sudy mahemaical
More informationWHAT ARE OPTION CONTRACTS?
WHAT ARE OTION CONTRACTS? By rof. Ashok anekar An oion conrac is a derivaive which gives he righ o he holder of he conrac o do 'Somehing' bu wihou he obligaion o do ha 'Somehing'. The 'Somehing' can be
More informationTEMPORAL PATTERN IDENTIFICATION OF TIME SERIES DATA USING PATTERN WAVELETS AND GENETIC ALGORITHMS
TEMPORAL PATTERN IDENTIFICATION OF TIME SERIES DATA USING PATTERN WAVELETS AND GENETIC ALGORITHMS RICHARD J. POVINELLI AND XIN FENG Deparmen of Elecrical and Compuer Engineering Marquee Universiy, P.O.
More information4. International Parity Conditions
4. Inernaional ariy ondiions 4.1 urchasing ower ariy he urchasing ower ariy ( heory is one of he early heories of exchange rae deerminaion. his heory is based on he concep ha he demand for a counry's currency
More informationOptimal Stock Selling/Buying Strategy with reference to the Ultimate Average
Opimal Sock Selling/Buying Sraegy wih reference o he Ulimae Average Min Dai Dep of Mah, Naional Universiy of Singapore, Singapore Yifei Zhong Dep of Mah, Naional Universiy of Singapore, Singapore July
More informationRC (ResistorCapacitor) Circuits. AP Physics C
(ResisorCapacior Circuis AP Physics C Circui Iniial Condiions An circui is one where you have a capacior and resisor in he same circui. Suppose we have he following circui: Iniially, he capacior is UNCHARGED
More informationCommunication Networks II Contents
3 / 1  Communicaion Neworks II (Görg)  www.comnes.unibremen.de Communicaion Neworks II Conens 1 Fundamenals of probabiliy heory 2 Traffic in communicaion neworks 3 Sochasic & Markovian Processes (SP
More informationThe Roos of Lisp paul graham Draf, January 18, 2002. In 1960, John McCarhy published a remarkable paper in which he did for programming somehing like wha Euclid did for geomery. 1 He showed how, given
More informationThe option pricing framework
Chaper 2 The opion pricing framework The opion markes based on swap raes or he LIBOR have become he larges fixed income markes, and caps (floors) and swapions are he mos imporan derivaives wihin hese markes.
More informationMathematics in Pharmacokinetics What and Why (A second attempt to make it clearer)
Mahemaics in Pharmacokineics Wha and Why (A second aemp o make i clearer) We have used equaions for concenraion () as a funcion of ime (). We will coninue o use hese equaions since he plasma concenraions
More informationINTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES
INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES OPENGAMMA QUANTITATIVE RESEARCH Absrac. Exchangeraded ineres rae fuures and heir opions are described. The fuure opions include hose paying
More informationSmall and Large Trades Around Earnings Announcements: Does Trading Behavior Explain PostEarningsAnnouncement Drift?
Small and Large Trades Around Earnings Announcemens: Does Trading Behavior Explain PosEarningsAnnouncemen Drif? Devin Shanhikumar * Firs Draf: Ocober, 2002 This Version: Augus 19, 2004 Absrac This paper
More informationDYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS
DYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS Hong Mao, Shanghai Second Polyechnic Universiy Krzyszof M. Osaszewski, Illinois Sae Universiy Youyu Zhang, Fudan Universiy ABSTRACT Liigaion, exper
More informationA Universal Pricing Framework for Guaranteed Minimum Benefits in Variable Annuities *
A Universal Pricing Framework for Guaraneed Minimum Benefis in Variable Annuiies * Daniel Bauer Deparmen of Risk Managemen and Insurance, Georgia Sae Universiy 35 Broad Sree, Alana, GA 333, USA Phone:
More informationTWO OPTIMAL CONTROL PROBLEMS IN CANCER CHEMOTHERAPY WITH DRUG RESISTANCE
Annals of he Academy of Romanian Scieniss Series on Mahemaics and is Applicaions ISSN 2666594 Volume 3, Number 2 / 211 TWO OPTIMAL CONTROL PROBLEMS IN CANCER CHEMOTHERAPY WITH DRUG RESISTANCE Werner Krabs
More information5.8 Resonance 231. The study of vibrating mechanical systems ends here with the theory of pure and practical resonance.
5.8 Resonance 231 5.8 Resonance The sudy of vibraing mechanical sysems ends here wih he heory of pure and pracical resonance. Pure Resonance The noion of pure resonance in he differenial equaion (1) ()
More informationHedging with Forwards and Futures
Hedging wih orwards and uures Hedging in mos cases is sraighforward. You plan o buy 10,000 barrels of oil in six monhs and you wish o eliminae he price risk. If you ake he buyside of a forward/fuures
More informationAnswer, Key Homework 2 David McIntyre 45123 Mar 25, 2004 1
Answer, Key Homework 2 Daid McInyre 4123 Mar 2, 2004 1 This prinou should hae 1 quesions. Muliplechoice quesions may coninue on he ne column or page find all choices before making your selecion. The
More informationContinuous Families of Embedded Solitons in the ThirdOrder Nonlinear Schrödinger Equation
Coninuous Families of Embedded Solions in he ThirdOrder Nonlinear Schrödinger Equaion By J. Yang and T. R. Akylas The nonlinear Schrödinger equaion wih a hirdorder dispersive erm is considered. Infinie
More informationTechnical Appendix to Risk, Return, and Dividends
Technical Appendix o Risk, Reurn, and Dividends Andrew Ang Columbia Universiy and NBER Jun Liu UC San Diego This Version: 28 Augus, 2006 Columbia Business School, 3022 Broadway 805 Uris, New York NY 10027,
More informationDependent Interest and Transition Rates in Life Insurance
Dependen Ineres and ransiion Raes in Life Insurance Krisian Buchard Universiy of Copenhagen and PFA Pension January 28, 2013 Absrac In order o find marke consisen bes esimaes of life insurance liabiliies
More informationPricing FixedIncome Derivaives wih he ForwardRisk Adjused Measure Jesper Lund Deparmen of Finance he Aarhus School of Business DK8 Aarhus V, Denmark Email: jel@hha.dk Homepage: www.hha.dk/~jel/ Firs
More informationChapter 2: Principles of steadystate converter analysis
Chaper 2 Principles of SeadySae Converer Analysis 2.1. Inroducion 2.2. Inducor volsecond balance, capacior charge balance, and he small ripple approximaion 2.3. Boos converer example 2.4. Cuk converer
More informationSignal Processing and Linear Systems I
Sanford Universiy Summer 214215 Signal Processing and Linear Sysems I Lecure 5: Time Domain Analysis of Coninuous Time Sysems June 3, 215 EE12A:Signal Processing and Linear Sysems I; Summer 1415, Gibbons
More informationSTABILITY OF LOAD BALANCING ALGORITHMS IN DYNAMIC ADVERSARIAL SYSTEMS
STABILITY OF LOAD BALANCING ALGORITHMS IN DYNAMIC ADVERSARIAL SYSTEMS ELLIOT ANSHELEVICH, DAVID KEMPE, AND JON KLEINBERG Absrac. In he dynamic load balancing problem, we seek o keep he job load roughly
More informationChabot College Physics Lab RC Circuits Scott Hildreth
Chabo College Physics Lab Circuis Sco Hildreh Goals: Coninue o advance your undersanding of circuis, measuring resisances, currens, and volages across muliple componens. Exend your skills in making breadboard
More informationThe Torsion of Thin, Open Sections
EM 424: Torsion of hin secions 26 The Torsion of Thin, Open Secions The resuls we obained for he orsion of a hin recangle can also be used be used, wih some qualificaions, for oher hin open secions such
More information