On the degrees of irreducible factors of higher order Bernoulli polynomials

Save this PDF as:

Size: px
Start display at page:

Download "On the degrees of irreducible factors of higher order Bernoulli polynomials"

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 p-eisensein 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 p-adic poles of he coefficiens in erms of he base p expansion of n. The p-eisensein 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 p-eisensein. 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 p-adic 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 p-eisensein 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 p-eisensein behavior of he firs and higher order Bernoulli polynomials. Our main ool is an analysis of he naure of he p-adic 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 non-rivial facor for n odd. The resuls in his paper enable us o deermine he insances of p-eisensein 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 = p-adic 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 p-eisensein 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 p-eisensein behavior; f(x is said o be p-eisensein if i is p-eisensein o degree 0. The following are sandard and/or easy o demonsrae (cf. [10, Ch. 4]. (1 If f(x is p-eisensein down o degree d and c Z, hen so is f(x+c. (2 If f(x is p-eisensein down o degree d, hen f(x has an irreducible facor of degree n d. (In paricular, p-eisensein 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 p-adic expansions m (Lucas s Theorem. I is convenien o inroduce some non-sandard 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 non-zero 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 non-zero digis in all places below he op digi of n, down o he boom non-zero digi of n. (I has a non-zero 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 non-zero digis in differen places, his lemma follows from he previous wo. 3. p-adic 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 non-negaive 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 non-zero 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 2-power, 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 p-eisensein behavior of hese polynomials. When we say ha A n (x, is p-eisensein 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 p-eisensein behavior. The nex heorem, which esablishes all insances of p-eisensein behavior follows immediaely from he preceding discussion. Theorem 3.5. If A n (x, is p-eisensein down o degree d, hen d = n N(n l for some boom segmen l of he p-adic expansion of n wih S(n l. Conversely, if d = n N(n l, hen A n (x, (n l is p-eisensein down o degree d; furhermore, A n (x, is p-eisensein 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 2-Eisensein 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 2-Eisensein 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 p-eisensein 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 p-eisensein 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 p-eisensein 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 p-eisensein. (iii A n (x, n is p-eisensein. (iv A n (x, 1 is p-eisensein. ( Furhermore, if hese condiions hold, A n (x, is p-eisensein iff n. R e m a r k 4. I follows ha if n = (p r and k n, hen A n (x, is p-eisensein. 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 p-eisensein for all k {1,..., n} iff n = (p r, r 0. (If n (p r and n f is he lowes non-zero 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 p-eisensein, 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 p-eisensein down o degree l. (iii A n (x, m is p-eisensein down o degree l. Furhermore, ( if hese condiions ( hold, A n (x, is p-eisensein 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 p-eisensein down o degree 1 (so here is an irreducible facor of degree n 1. Also A n (x, is p-eisensein down o degree 1 iff p (so N(n = n i. n 1 and p n i, where i is he larges p-power < 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 p-eisensein ( 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 p-eisensein 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 3-Eisensein 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 p-eisensein 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 p-eisensein 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 p-eisensein 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 p-adic 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

17 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

Duration 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 UVA-F-38 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

Random Walk in 1-D. 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

Chapter 7. Response of First-Order RL and RC Circuits

Chaper 7. esponse of Firs-Order 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

ANALYSIS 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,

Forecasting and Information Sharing in Supply Chains Under Quasi-ARMA Demand

Forecasing and Informaion Sharing in Supply Chains Under Quasi-ARMA Demand Avi Giloni, Clifford Hurvich, Sridhar Seshadri July 9, 2009 Absrac In his paper, we revisi he problem of demand propagaion in

Niche 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.

Optimal 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

A 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.

Option Put-Call Parity Relations When the Underlying Security Pays Dividends

Inernaional Journal of Business and conomics, 26, Vol. 5, No. 3, 225-23 Opion Pu-all Pariy Relaions When he Underlying Securiy Pays Dividends Weiyu Guo Deparmen of Finance, Universiy of Nebraska Omaha,

MTH6121 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

Lectures # 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 ζ-funcion o skech an argumen which would give an acual formula for π( and sugges

Second 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

Usefulness 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,

4 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 discree-ime linear sysem has oupus y[n] for he given inpus x[n] as shown in Figure P4.1-1.

Appendix A: Area. 1 Find the radius of a circle that has circumference 12 inches.

Appendi A: Area worked-ou s o Odd-Numbered Eercises Do no read hese worked-ou s before aemping o do he eercises ourself. Oherwise ou ma mimic he echniques shown here wihou undersanding he ideas. Bes wa

2.5 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

cooking 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

nonlocal conditions.

ISSN 1749-3889 prin, 1749-3897 online Inernaional Journal of Nonlinear Science Vol.11211 No.1,pp.3-9 Boundary Value Problem for Some Fracional Inegrodifferenial Equaions wih Nonlocal Condiions Mohammed

Circuit 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

Analogue 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: 0-7-380-7 Ifeachor

Economics 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

Single-machine Scheduling with Periodic Maintenance and both Preemptive and. Non-preemptive jobs in Remanufacturing System 1

Absrac number: 05-0407 Single-machine Scheduling wih Periodic Mainenance and boh Preempive and Non-preempive jobs in Remanufacuring Sysem Liu Biyu hen Weida (School of Economics and Managemen Souheas Universiy

Term 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 1-1-1 Nojihigashi, Kusasu, Shiga 525-8577, Japan E-mail:

CHARGE 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:

PATHWISE 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

arxiv: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, A-090 Wien, Ausria e-mail:

Present 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

Permutations 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 k-value for he middle erm, divide

Working 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 Open-Access-Publikaionsserver der ZBW Leibniz-Informaionszenrum Wirschaf The Open Access Publicaion Server of he ZBW Leibniz Informaion Cenre for Economics Biagini, Francesca;

A Re-examination of the Joint Mortality Functions

Norh merican cuarial Journal Volume 6, Number 1, p.166-170 (2002) Re-eaminaion of he Join Morali Funcions bsrac. Heekung Youn, rkad Shemakin, Edwin Herman Universi of S. Thomas, Sain Paul, MN, US Morali

Inductance 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

PROFIT 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

The 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

ARCH 2013.1 Proceedings

Aricle from: ARCH 213.1 Proceedings Augus 1-4, 212 Ghislain Leveille, Emmanuel Hamel A renewal model for medical malpracice Ghislain Léveillé École d acuaria Universié Laval, Québec, Canada 47h ARC Conference

Differential 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

INDEPENDENT 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

Morningstar 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

Cointegration: The Engle and Granger approach

Coinegraion: The Engle and Granger approach Inroducion Generally one would find mos of he economic variables o be non-saionary I(1) variables. Hence, any equilibrium heories ha involve hese variables require

Improper 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 [,

Appendix 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\223489-00\4

1. 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..

A 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

11/6/2013. Chapter 14: Dynamic AD-AS. Introduction. Introduction. Keeping track of time. The model s elements

Inroducion Chaper 14: Dynamic D-S 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 cuing-edge

ANALYTIC 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

Entropy: 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

The 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

4.2 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.

A 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

Journal 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 Yi-Kang Liu, (yikang@gwu.edu), George Washingon Universiy ABSTRACT The advanage of Mone Carlo

3 Runge-Kutta Methods

3 Runge-Kua Mehods In conras o he mulisep mehods of he previous secion, Runge-Kua mehods are single-sep mehods however, muliple sages per sep. They are moivaed by he dependence of he Taylor mehods on he

AP Calculus AB 2013 Scoring Guidelines

AP Calculus AB 1 Scoring Guidelines The College Board The College Board is a mission-driven no-for-profi organizaion ha connecs sudens o college success and opporuniy. Founded in 19, he College Board was

Table 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

9. 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

SEMIMARTINGALE STOCHASTIC APPROXIMATION PROCEDURE AND RECURSIVE ESTIMATION. Chavchavadze Ave. 17 a, Tbilisi, Georgia, E-mail: 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.

The 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 ime-series smoohing forecasing mehods. Various models are discussed,

Stochastic 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

A 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

BALANCE OF PAYMENTS. First quarter 2008. Balance of payments

BALANCE OF PAYMENTS DATE: 2008-05-30 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

UNDERSTANDING 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 Erlangen-Nuremberg Lange Gasse

Graduate 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.

Density Dependence. births are a decreasing function of density b(n) and deaths are an increasing function of density d(n).

FW 662 Densiy-dependen 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. Long-erm

Chapter 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

Life 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, DK-2100 Copenhagen Ø, Denmark PFA Pension,

Modeling Stock Price Dynamics with Fuzzy Opinion Networks

Modeling Sock Price Dynamics wih Fuzzy Opinion Neworks Li-Xin Wang Deparmen of Auomaion Science and Technology Xian Jiaoong Universiy, Xian, P.R. China Email: lxwang@mail.xju.edu.cn Key words: Sock price

Measuring macroeconomic volatility Applications to export revenue data, 1970-2005

FONDATION POUR LES ETUDES ET RERS LE DEVELOPPEMENT INTERNATIONAL Measuring macroeconomic volailiy Applicaions o expor revenue daa, 1970-005 by Joël Cariolle Policy brief no. 47 March 01 The FERDI is a

= r t dt + σ S,t db S t (19.1) with interest rates given by a mean reverting Ornstein-Uhlenbeck or Vasicek process,

Chaper 19 The Black-Scholes-Vasicek Model The Black-Scholes-Vasicek model is given by a sandard ime-dependen Black-Scholes model for he sock price process S, wih ime-dependen bu deerminisic volailiy σ

Part 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

Steps 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.

Acceleration 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

A 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

Return 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

A Generalized Bivariate Ornstein-Uhlenbeck Model for Financial Assets

A Generalized Bivariae Ornsein-Uhlenbeck Model for Financial Asses Romy Krämer, Mahias Richer Technische Universiä Chemniz, Fakulä für Mahemaik, 917 Chemniz, Germany Absrac In his paper, we sudy mahemaical

WHAT 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

TEMPORAL 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.

4. 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

Optimal 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

RC (Resistor-Capacitor) Circuits. AP Physics C

(Resisor-Capacior 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

Communication Networks II Contents

3 / 1 -- Communicaion Neworks II (Görg) -- www.comnes.uni-bremen.de Communicaion Neworks II Conens 1 Fundamenals of probabiliy heory 2 Traffic in communicaion neworks 3 Sochasic & Markovian Processes (SP

The 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

The 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.

Mathematics 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

INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES

INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES OPENGAMMA QUANTITATIVE RESEARCH Absrac. Exchange-raded ineres rae fuures and heir opions are described. The fuure opions include hose paying

Small and Large Trades Around Earnings Announcements: Does Trading Behavior Explain Post-Earnings-Announcement Drift?

Small and Large Trades Around Earnings Announcemens: Does Trading Behavior Explain Pos-Earnings-Announcemen Drif? Devin Shanhikumar * Firs Draf: Ocober, 2002 This Version: Augus 19, 2004 Absrac This paper

DYNAMIC 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

A 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:

TWO OPTIMAL CONTROL PROBLEMS IN CANCER CHEMOTHERAPY WITH DRUG RESISTANCE

Annals of he Academy of Romanian Scieniss Series on Mahemaics and is Applicaions ISSN 266-6594 Volume 3, Number 2 / 211 TWO OPTIMAL CONTROL PROBLEMS IN CANCER CHEMOTHERAPY WITH DRUG RESISTANCE Werner Krabs

5.8 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) ()

Hedging 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 buy-side of a forward/fuures

Answer, Key Homework 2 David McIntyre 45123 Mar 25, 2004 1

Answer, Key Homework 2 Daid McInyre 4123 Mar 2, 2004 1 This prin-ou should hae 1 quesions. Muliple-choice quesions may coninue on he ne column or page find all choices before making your selecion. The

Continuous Families of Embedded Solitons in the Third-Order Nonlinear Schrödinger Equation

Coninuous Families of Embedded Solions in he Third-Order Nonlinear Schrödinger Equaion By J. Yang and T. R. Akylas The nonlinear Schrödinger equaion wih a hird-order dispersive erm is considered. Infinie

Technical 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,

Dependent 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

Pricing Fixed-Income Derivaives wih he Forward-Risk Adjused Measure Jesper Lund Deparmen of Finance he Aarhus School of Business DK-8 Aarhus V, Denmark E-mail: jel@hha.dk Homepage: www.hha.dk/~jel/ Firs

Chapter 2: Principles of steady-state converter analysis

Chaper 2 Principles of Seady-Sae Converer Analysis 2.1. Inroducion 2.2. Inducor vol-second balance, capacior charge balance, and he small ripple approximaion 2.3. Boos converer example 2.4. Cuk converer

Signal Processing and Linear Systems I

Sanford Universiy Summer 214-215 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 14-15, Gibbons

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