On Some Functions Involving the lcm and gcd of Integer Tuples


 Nigel Jordan
 4 years ago
 Views:
Transcription
1 SCIENTIFIC PUBLICATIONS OF THE STATE UNIVERSITY OF NOVI PAZAR SER. A: APPL. MATH. INFORM. AND MECH. vol. 6, 2 (2014), On Some Functions Involving the lcm and gcd of Intege Tuples O. Bagdasa Abstact: In this pape an explicit fomula fo the numbe of tuples of positive integes having the same lowest common multiple n is deived, and some of the popeties of the esulting aithmetic function ae analyzed. The tuples having also the same geatest common diviso ae investigated, while some novel o existing intege sequences ae ecoveed as paticula cases. A fomula linking the gcd and lcm fo tuples of integes is also pesented. Keywods: divisibility, lowest common multiple, geatest common diviso, intege sequences 1 Intoduction The least common multiple of two natual numbes a and b, is usually denoted by lcm(a,b) o a,b, and is the smallest numbe divisible by both a and b 5, 5.1, p.48. The dual notion is the geatest common diviso, denoted by gcd(a,b) o (a,b) which is the lagest numbe that divides both a and b. The lcm and gcd can also be defined fo any ktuple of natual numbes a 1,...,a k, whee k 2. Fo n = p n 1, the numbe of odeed pais (a,b) having the same lcm n is {(a,b) : lcm(a,b) = n} = (2n 1 + 1)(2n 2 + 1) (2n + 1), (1) whee is the cadinality of a set 10. If n is squaefee we have 3 ω(n) 4, ex. 2.4, p.101), whee ω(n) denotes the numbe of pime divisos of n. The numbe of elatively pime odeed pais (a,b) having the same lcm n is (see 8) {(a,b) : gcd(a,b) = 1, lcm(a,b) = n} = 2 ω(n). (2) A popety linking the lcm and gcd of the intege pai (a,b) is (see 5) gcd(a,b)lcm(a,b) = ab. (3) The aim of this pape is to extend the above esults to geneal tuples of integes. Manuscipt eceived Mach 21 ; accepted June 28 O. Bagdasa is with the School of Computing and Mathematics, Univesity of Deby, Kedleston Road, Deby DE22 1GB, England, U.K., o.bagdasa deby.ac.uk 91
2 92 O. Bagdasa The numbe of ktuples of positive integes with the least common multiple n is LCM(n;k) = {(a 1,...,a k ) : lcm(a 1,...,a k ) = n}, (4) and some identities and inequalities involving the above aithmetic function ae pesented. A few sequences indexed in the OnLine Encyclopedia of Intege Sequences (OEIS) 7, ae obtained as paticula cases. The numbe of odeed ktuples with the same gcd d and lcm n is defined by GL(d,n;k) = {(a 1,...,a k ) : gcd(a 1,...,a k ) = d, lcm(a 1,...,a k ) = n}, (5) and a numbe of popeties of this function ae analyzed. In the pocess, some families of intege sequences not cuently indexed in the OEIS ae poduced. Finally, a popety linking the lcm and gcd fo ktuples of integes which genealizes fomula (3) is pesented, accompanied by a bief poof based on 9. 2 Main esults In this section the poofs fo the fomulae of LCM and GL ae povided, along with a esult linking the lcm and gcd computed fo ktuples of integes. 2.1 Tuples of integes with the same lcm This section pesents the fomula fo the numbe of odeed ktuples whose least common multiple is a natual numbe n. Some of the popeties of this function ae then investigated. Theoem 2.1 Let k and n be natuals numbes. If n has the factoization n = p n 1 the numbe of odeed ktuples whose lcm is n is given by the fomula LCM(n;k) =, (n i + 1) k n k i. (6) Poof. Let {a 1,...,a k } be a ktuple satisfying a 1,...,a k = n. The numbes a 1,...,a k need to be divisos of n, theefoe they can be factoized as a j = p a 1 j 1 p a 2 j 2... pa j, j = 1,...,k, (7) with 0 a i j n i (i = 1,...,). Because thei lcm is n, one needs to have max{a i1,...,a ik } = n i, (8) fo each diviso p i, i = 1,...,. Fo each i {1,...,}, the total numbe of pais 0 a i j n i ( j = 1,...,k) is (n i + 1) k. On the othe hand, the numbe of odeed pais 0 a i j n i 1 ( j = 1,...,k) not having n i as a maximum is n k i. The numbe of pais satisfying elation (8) is theefoe (n i + 1) k n k i. Multiplying this fomula fo i = 1,..., one obtains (6).
3 On some functions involving the lcm and gcd of intege ktuples 93 Remak 2.1 Fo paticula values of k, one ecoves the following OEIS indexed intege sequences: A fo LCM(n; 2) (linked to numeous algebaic intepetations), and A070919, A070920, A fo LCM(n;3), LCM(n;4), LCM(n;5), espectively. The following esult illustates the independence of LCM(n;k) on the pime factos. Coollay 2.1 Let k be a natual numbe and n = p n 1, m = q n 1 1 qn qn, such that all numbes p 1,..., p,q 1,...,q ae distinct. Then LCM(m;k) = LCM(n;k). Poof. Fom (6), LCM(n; k) only depends on the multiplicities of the pime factos, and not on the pime factos themselves. In this case LCM(m;k) = (n i + 1) k n k i = LCM(n;k) Consideing elatively pime numbes m and n, one obtains the consequence below, which hee is poved using a diect method. Coollay 2.2 Let m,n be integes satisfying (m,n) = 1. The following popety holds LCM(m n;k) = LCM(m;k) LCM(n;k). (9) Poof. Conside the following factoizations of m = q m 1 1 qm qm s s and n = p n 1. As (m,n) = 1, it follows that p i q j, fo any combination of i = 1,..., and j = 1,...,s. This implies that the pime factoization of the poduct mn is m n = q m 1 1 qm 2 2 qm s s p n 1 2 pn, with the pime factos not necessaily in an inceasing ode. By fomula (6) one obtains LCM(m n;k) = (m i + 1) k m k s i (n j + 1) k n k j = LCM(m;k) LCM(n;k), j=1 which ends the poof. Hee ae ecalled some well known popeties of aithmetic functions 1, Chapte 2. An aithmetic function f (n) of the positive intege n is multiplicative if f (1) = 1 and fo any a and b copime, then f (ab) = f (a) f (b). An aithmetic function is said completely multiplicative if f (1) = 1 and f (ab) = f (a) f (b), even when a and b ae not copime. Remak 2.2 Fom Coollay 2.2, the function LCM(n;k) is multiplicative. Remak 2.3 Let a,b be positive integes and f : N N be a multiplicative aithmetic function. The following popety holds f (gcd(a,b)) f (lcm(a,b)) = f (a) f (b). (10) Poof. Let d = gcd(a,b). The numbes d, a/d and b/d ae copime. Using (3) one obtains f (d) f (lcm(a,b)) = f (d) f (d) f (a/d) f (b/d) = f (d) f (a/d) f (d) f (b/d) = f (a) f (b). This ends the poof.
4 94 O. Bagdasa Fo LCM(n;k) we can check diectly the popety elative to lcm and gcd. Coollay 2.3 Let a,b be natual numbes. The following popety holds: LCM(gcd(a,b);k) LCM(lcm(a,b);k) = LCM(a;k) LCM(b;k). (11) Poof. Assume that the factoizations of n,d,a,b ae n = p n 1, d = p d 1 1 pd pd s and a = p α 1 1 pα pα, b = p β 1 1 pβ pβ, whee some of the powes may be zeo. Using the elation (6), it is sufficient to pove the fomula fo just one fixed value i fom 1,...,. The contibution of the pime facto p i to fomula (11) is (n i + 1) k n k i (d i + 1) k di k = (α i + 1) k α k i This elation is tue, as d i = min{α i,β i } and n i = max{α i,β i }. (β i + 1) k βi k Some inequalities fo LCM(n;k) can also be poved, fo geneal values of m and n. Theoem 2.2 Let k, m and n be natual numbes. The following inequality holds LCM(m n;k) LCM(m;k) LCM(n;k) (12) To pove this theoem, we shall fist pove the following esult. Lemma 2.3 Let k 2, p be a pime numbe and α,β 1 natual numbes. Then LCM(p α+β ;k) LCM(p α ;k) LCM(p β ;k) (13) Poof. The inequality (13) is then equivalent to (α + β + 1) k (α + β) k (α + 1) k α k (β + 1) k β k. (14) Fom (6) we have LCM(p α+β ;k) = (α + β + 1) k (α + β) k = k 1 ) (α + β) l, then ( k l=0 l LCM(p α ;k) = (α +1) k α k = i=0 k 1 ( k ) i α i and LCM(p β ;k) = (β +1) k β k = k 1 ( k j=0 j) β j. Fo a fixed pai (i, j) with i, j chosen fom the set {0,...,k 1}, the tem α i β j appeas in the ighthand side, and its coefficient is ( k k i)( j). On the othe hand, the tem α i β j only appeas on the lefthand side if l = i + j k 1, and its coefficient is ( k l)( l i). To finalize the poof it emains to be shown that the latte coefficient is smalle than the fome. One can check that ( k )( k i j) k(k 1) (k i + 1) ( k )( l = l i) (k j)(k j 1) (k j i + 1) 1. The computational details of the poof ae left to the eade as an execise. Conside the factoizations of m = p m 1 1 pm pm and n = p n 1, whee m i,n i 0. Using Coollay 2.2 fo the pime factos p 1,..., p and thei powes, and Lemma 2.3 fo each of the indices i = 1,...,, one obtains (12). This ends the poof of the theoem. Remak 2.4 It is not difficult to pove that using paticula values that the inequality in Theoem 2.2 is stict fo k 2. Fo example, fo p = 2, α = β = 1 one obtains the elation 3 k 2 k < (2 k 1 k )(2 k 1 k ), which is tue fo all values k 2. This also poves that LCM(n;k) is not completely multiplicative.
5 On some functions involving the lcm and gcd of intege ktuples Tuples of integes with the same lcm and gcd This section evaluates the function GL(d, n; k) given by (5), enumeating the ktuples with fixed geatest common diviso d and least common multiple n. Some of its popeties and paticula instances ae also pesented, along with some novel numbe sequences. The following lemma epesents the motivation fo the esults in this section. Lemma 2.4 Let d < n be positive integes, such that d n. The numbe of odeed pais (a,b) with the same geatest common diviso d and least common multiple n is {(a,b) : gcd(a,b) = d, lcm(a,b) = n} = 2 ω(n/d). (15) whee ω(x) epesents the numbe of distinct pime divisos fo the intege x. Poof. Let the numbes d,n,a,b have the pime decompositions d = p d 1 1 pd 2 2 pd, n = p n 1 2 pn, a = p α 1 1 pα 2 2 pα, b = p β 1 1 pβ 2 2 pβ, whee 1 d i n i fo i = 1,...,. As d = (a,b) and n = a,b, d i = min{α i,β i } and n i = max{α i,β i } fo each i {1,...,}. Also, the numbes a and b ae distinct, o othewise d = n. Thee ae two possibilities. When d i = n i, one has α i = β i = d i = n i. Each choice of i I = {i {1,...,n} : d i < n i } geneates two possible pais (α i,β i ) {(d i,n i ),(n i,d i )}, hence in total thee ae 2 I distinct pais of powes. As the pime decomposition of n/d is n/d = p n 1 d 1 1 p n 2 d 2 2 p n d = p n i d i i, i I one obtains that I = ω(n/d). This ends the poof. Fo d = 1 one ecoves the esult (2), epesenting the numbe of unitay divisos, indexed as A in the OEIS. The following esult elates GL(d,n;k) to the set of elatively pime tuples, and shows that finding GL(d,n;k) can be educed to evaluating GL(1,n/d;k). Lemma 2.5 Let k and d n be natual numbes. If d = p d 1 1 pd pd and n = p n 1 the numbe of odeed ktuples whose gcd is d, and lcm is n satisfies the popety GL(d,n;k) = GL(1,n/d;k). (16) Poof. Let {a 1,...,a k } be a ktuple satisfying (a 1,...,a k ) = d and a 1,...,a k = n. Numbes a 1,...,a k ae multiples of d and divisos of n, theefoe they can be factoized as a j = p a 1 j 1 p a 2 j 2... pa j, j = 1,...,k, with d i a i j n i (i = 1,...,). Because thei gcd is d and lcm is n, one needs to have min{a i1,...,a ik } = d i, max{a i1,...,a ik } = n i, (17),
6 96 O. Bagdasa fo each p i, i = 1,...,. The numbe of tuples in (17) is the same as of the tuples satisfying min{a i1,...,a ik} = 0, max{a i1,...,a ik} = n i d i, which elate to the ktuples having the popeties (a 1,...,a k ) = 1 and a 1,...,a k = n/d. This ends the poof. Lemma 2.6 Let k and α be positive integes. The numbe of tuples (α 1,...,α k ) satisfying T(α;k) = {(α 1,...,α k ) : min(α 1,...,α k ) = 0, max(α 1,...,α k ) = α} (18) is given by the fomula T(α;k) = (α + 1) k 2α k + (α 1) k. (19) Poof. Conside the tem T(α;k+1) epesenting the numbe of (k+1)tuples (α 1,...,α k+1 ) whose min is 0 and max is α. Thee ae thee distinct cases, depending on the value of α k α k+1 = 0. In this case the ktuple (α 1,...,α k ) has to satisfy max(α 1,...,α k ) = α. Following the agument used in Theoem 2.1, the numbe of ktuples having this popety is (α + 1) k α k. 2. α k+1 = n. Hee the estiction on the ktuple (α 1,...,α k ) is min(α 1,...,α k ) = 0. Similaly as above, the numbe of ktuples with this popety is (α + 1) k α k < α k+1 < n. Fo each of the (α 1) possible values of α k+1, the ktuples (α 1,...,α k ) satisfy min(α 1,...,α k ) = 0 and max(α 1,...,α k ) = α, and thei numbe is T(α;k). By counting all the (k + 1)tuples fo a fixed α, the tems T(α;k) satisfy the ecuence T(α;k + 1) = 2 (α + 1) k α k + (α 1)T(α;k) (20) Fo α 1 one has T(α;0) = T(α;1) = 0, while fom the poof of Lemma 2.5, T(α;2) = 2. Witing the ecusion until T(α;1), one obtains T(α;k + 1) = 2 = 2 k i=0 k i=0 (α 1) i (α + 1) k i α k i + (α 1) k+1 T (α;0) (α 1) i (α + 1) k i (α 1) i α k i (α + 1) k+1 α k+1 = 2 2 αk+1 (α 1) k+1 1 = (α + 1) k+1 2α k+1 + (α 1) k+1, whee the elation a k+1 b k+1 = (a b) ( a k + a k 1 b + + ab k 1 + b k) valid fo any a,b was used. The esult poves (19), which could also be checked by mathematical induction. Hee the diect poof was pefeed, as it shows how the fomula can be deived.
7 On some functions involving the lcm and gcd of intege ktuples 97 Fo a simplified notation we may define L(n;k) := GL(1,n;k), fo positive integes k,n. The fomula fo L(n;k) now follows as a diect consequence of Lemma 2.6. Theoem 2.7 Let k and n be natuals numbes. If n has the factoization n = p n 1 the numbe of odeed ktuples whose gcd is 1 and lcm is n, is given by the fomula L(n;k) = (n i + 1) k 2n k i + (n i 1) k. (21) Poof. Let {a 1,...,a k } be a ktuple satisfying gcd(a 1,...,a k ) = 1 and lcma 1,...,a k = n. The numbes a 1,...,a k can be factoized as a j = p a 1 j 1 p a 2 j 2... pa j, j = 1,...,k, with 0 a i j n i (i = 1,...,). Because thei gcd is 1 and lcm is n, one needs to have min{a i1,...,a ik } = 0, max{a i1,...,a ik } = n i, fo each p i, i = 1,...,, which fom Lemma 2.6 poduces (n i + 1) k 2n k i +(n i 1) k tuples. By taking the poduct ove all factos p i, i = 1,..., one obtains (21). The following popeties of L(n; k) ae diect consequences of fomula (21), simila to the esults obtained fo LCM(n;k). Fo this eason these ae pesented hee without poofs. The fist popety suggests the exclusive dependence of L(n;k) on the pime factoization. Coollay 2.4 Let k be a natual numbe and n = p n 1, m = q n 1 1 qn qn, such that all numbes p 1,..., p,q 1,...,q ae distinct. Then L(m;k) = L(n;k). The following esults states the multiplicity of the aithmetic function L(n;k) fo k 2. Coollay 2.5 Let m,n be compime integes and k 2. The following popety holds The last esult is a consequence of the multiplicity. L(m n;k) = L(m;k) L(n;k). (22) Coollay 2.6 Let a,b be natual numbes. The following popety holds: L(gcd(a,b);k) L(lcm(a,b);k) = L(a;k) L(b;k). (23) Remak 2.5 Choosing m = n = 2 and k 2 one may check that L(m n;k) = 3 k 2 2 k + 1 < (2 k 2) (2 k 2) = L(m;k) L(n;k). This poves that L(n;k) is not completely multiplicative. Peliminay investigations suggest that the inequality L(m n; k) L(m; k) L(n; k) is tue. Howeve, a complete solution is not known to the autho.,
8 98 O. Bagdasa 2.3 Intege sequences elated to T(n;k) and L(n;k) A numbe of intege sequences can be obtained fom T(n;k), fo paticula values of k and n. These novel, o existing OEIS numbe sequences have a unified intepetation, epesenting the numbe of odeed ktuples whose minimum is zeo and maximum is n. Fo n = 1, one obtains T(1;k) = 2 k 2, which is the numbe of nonempty pope subsets fo a set with k elements, indexed as A in the OEIS. Fo n = 2, the sequence defined by T(2;k 1) = 3 k 1 2 k + 1, epesents the Stiling numbes of the second kind, indexed as A in the OEIS. The fist few tems of the sequences T(n;k) fo n = 3,4,5 give T(3;k) : 0,2,18,110,570,2702,12138,52670,223290,931502,... (24) T(4;k) : 0,2,24,194,1320,8162,47544,266114, , ,... (25) T(5;k) : 0,2,30,302,2550,19502,140070,963902, , ,... (26) not cuently indexed in the OEIS, suggesting that they ae pobably new numbe sequences. Fo k = 1,2,3, one obtains T(n;1) = 0, T(n;2) = 2 and T(n;1) = 3n, espectively. Fo k = 4, the sequence defined by T(n;4) = 12n 2 + 2, epesents the numbe of points on suface of hexagonal pism, indexed as A in the OEIS. Finally, fo k = 5 one obtains T(n;5) = 20n n, which is A (also, T(n + 1;5) poduces A101098). Seveal intege sequences ae obtained fom L(n; k), fo paticula values of k and n. Some special cases woth mentioning. Fo a pime numbe p, one has L(p;k) = 2 k 2. Also, the values L(n;1) = 0 and L(1;k) = 1 ae poduced, fo a positive intege n and k 2. If n = p n 1, the following esults ae obtained as paticula cases: L(n; 2) = (n i + 1) 2 2n 2 i + (n i 1) 2 = 2 ω(n), (27) L(n; 3) = L(n; 4) = L(n; 5) = (n i + 1) 3 2n 3 i + (n i 1) 3 = 6 ω(n) (n i + 1) 4 2n 4 i + (n i 1) 4 = 2 ω(n) (n i + 1) 5 2n 5 i + (n i 1) 5 = 10 ω(n) n i, (28) 6n 2 i + 1, (29) 2n 3 i + n i. (30) Fo some paticula values of k, a numbe of novel intege sequences can be ecoveed. Fo k = 2, one obtains L(n;2) = 2 ω(n), which is as seen befoe, is A in the OEIS. The fist few tems of the numbe sequences L(n;k) fo k = 3,4,5 give L(n;3) : 1,6,6,12,6,36,6,18,12,36,6,72,6,36,36,... (31) L(n;4) : 1,14,14,50,14,196,14,110,50,196,14,700,14,196,196,... (32) L(n;5) : 1,30,30,180,30,900,30,570,180,900,30,5400,30,900,900,... (33) not cuently indexed in the OEIS. The enumeation of esults in this section suggests that some of the sequences pesented ae new. Also, some existing sequences wee given a diffeent intepetation.
9 On some functions involving the lcm and gcd of intege ktuples A link between the lcm and gcd of ktuples of integes One can use the pime numbe factoization to pove the following elations, which link the lcm and gcd of ktuples of integes. Detailed poofs can be found in 9. Theoem 2.8 Let k 2 and a 1,...,a k be natual numbes. The following popety holds gcd(a i1,...,a iu ) 1 i lcm(a 1,a 2,...,a k ) = 1 <...<i u k gcd(a i1,...,a iv ), (34) 1 i 1 <...<i v k whee u is odd and v is even. The dual of this theoem can be witten as Theoem 2.9 Let k 2 and a 1,...,a k be natual numbes. The following popety holds lcm(a i1,...,a iu ) 1 i gcd(a 1,a 2,...,a k ) = 1 <...<i u k lcm(a i1,...,a iv ), (35) 1 i 1 <...<i v k whee u is odd and v is even. Sketch of poof: Assuming that the multiplicities of p in a 1,...,a k ae m 1,...,m k, the poblem of poving (34) can be educed to poving the identity below. max(m 1,...,m k ) = 1 i 1 <...<i u n whee u is odd and v is even. 3 Discussion min(m i1,...,m iu ) 1 i 1 <...<i v n min(m i1,...,m iv ), (36) The eseach oiginated in the counting poblem solved by the authos in 2. Thee, the Hoadam sequences with a fixed peiod wee enumeated using a fomula involving all the pais of numbes having the same lcm, given by fomula (1). In this pape an explicit fomula fo the numbe of intege ktuples with the same lcm n denoted by LCM(n;k), was established. Some popeties of this aithmetic function wee exploed, being shown that LCM(n; k) is multiplicative, but not completely multiplicative. The geneal fomula plays a key ole in the enumeation of genealized peiodic Hoadam sequences, chaacteized in 3. The intege sequences LCM(n; 2), LCM(n; 3), LCM(n; 4) and LCM(n;5) wee found to be indexed in the OEIS. The function GL(d,n;k) counting the ktuples with same gcd d and lcm n was defined. Most of the popeties wee linked to the aithmetic function L(n;k) := GL(1,n;k), whose fomula was given. Function L(n; k) was multiplicative, but not completely multiplicative. Function T(n;k), enumeating ktuples whose min is zeo and max is n was also analyzed. Novel, o existing OEIS intege sequences wee poduced fom T(n;k) and L(n;k). Finally, two dual popeties linking the gcd and lcm of ktuples wee pesented.
10 100 O. Bagdasa Refeences 1 T. M. APOSTOL, Intoduction to Analytic Numbe Theoy, Spinge, New Yok, O. BAGDASAR and P. J. LARCOMBE, On the numbe of complex Hoadam sequences with a fixed peiod, Fibonacci Quat., Vol. 51, 4 (2013), O. Bagdasa and P. J. Lacombe, On the chaacteization of peiodic genealized Hoadam sequences, J. Diffe. Equ. Appl. (accepted). 4 R. CRANDALL and C. POMERANCE, Pime Numbes: A Computational Pespective, Spinge, New Yok, G. H. HARDY and E. M. WRIGHT An Intoduction to the Theoy of Numbes, Oxfod Univesity Pess, Oxfod, Fifth edition, Octogon Mathematical Magazine, Collection, The OnLine Encyclopedia of Intege Sequences, OEIS Foundation Inc R. R. SITARAMACHANDRA and D. SURYANABAYANA, The numbe of pais of integes with L.C.M. x, ARCH. MATH., Vol XXI (1970), D. VĂLCAN and O. D. BAGDASAR, Genealizations of some divisibility elations in N, Ceative Math. & Inf., Vol. 18, 1 (2009), F. ZUMHING and T. ANDREESCU, A Path To Combinatoics fo Undegaduates, Bikhause, 2004.
Nontrivial lower bounds for the least common multiple of some finite sequences of integers
J. Numbe Theoy, 15 (007), p. 393411. Nontivial lowe bounds fo the least common multiple of some finite sequences of integes Bai FARHI bai.fahi@gmail.com Abstact We pesent hee a method which allows to
More informationSymmetric polynomials and partitions Eugene Mukhin
Symmetic polynomials and patitions Eugene Mukhin. Symmetic polynomials.. Definition. We will conside polynomials in n vaiables x,..., x n and use the shotcut p(x) instead of p(x,..., x n ). A pemutation
More informationContinuous Compounding and Annualization
Continuous Compounding and Annualization Philip A. Viton Januay 11, 2006 Contents 1 Intoduction 1 2 Continuous Compounding 2 3 Pesent Value with Continuous Compounding 4 4 Annualization 5 5 A Special Poblem
More informationSeshadri constants and surfaces of minimal degree
Seshadi constants and sufaces of minimal degee Wioletta Syzdek and Tomasz Szembeg Septembe 29, 2007 Abstact In [] we showed that if the multiple point Seshadi constants of an ample line bundle on a smooth
More informationWeek 34: Permutations and Combinations
Week 34: Pemutations and Combinations Febuay 24, 2016 1 Two Counting Pinciples Addition Pinciple Let S 1, S 2,, S m be disjoint subsets of a finite set S If S S 1 S 2 S m, then S S 1 + S 2 + + S m Multiplication
More informationSaturated and weakly saturated hypergraphs
Satuated and weakly satuated hypegaphs Algebaic Methods in Combinatoics, Lectues 67 Satuated hypegaphs Recall the following Definition. A family A P([n]) is said to be an antichain if we neve have A B
More informationSTUDENT RESPONSE TO ANNUITY FORMULA DERIVATION
Page 1 STUDENT RESPONSE TO ANNUITY FORMULA DERIVATION C. Alan Blaylock, Hendeson State Univesity ABSTRACT This pape pesents an intuitive appoach to deiving annuity fomulas fo classoom use and attempts
More informationSpirotechnics! September 7, 2011. Amanda Zeringue, Michael Spannuth and Amanda Zeringue Dierential Geometry Project
Spiotechnics! Septembe 7, 2011 Amanda Zeingue, Michael Spannuth and Amanda Zeingue Dieential Geomety Poject 1 The Beginning The geneal consensus of ou goup began with one thought: Spiogaphs ae awesome.
More information2. TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES
. TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES In ode to etend the definitions of the si tigonometic functions to geneal angles, we shall make use of the following ideas: In a Catesian coodinate sstem, an
More informationThe Binomial Distribution
The Binomial Distibution A. It would be vey tedious if, evey time we had a slightly diffeent poblem, we had to detemine the pobability distibutions fom scatch. Luckily, thee ae enough similaities between
More informationInteger sequences from walks in graphs
otes on umbe Theoy and Discete Mathematics Vol. 9, 3, o. 3, 78 84 Intege seuences fom walks in gahs Enesto Estada, and José A. de la Peña Deatment of Mathematics and Statistics, Univesity of Stathclyde
More informationMULTIPLE SOLUTIONS OF THE PRESCRIBED MEAN CURVATURE EQUATION
MULTIPLE SOLUTIONS OF THE PRESCRIBED MEAN CURVATURE EQUATION K.C. CHANG AND TAN ZHANG In memoy of Pofesso S.S. Chen Abstact. We combine heat flow method with Mose theoy, supe and subsolution method with
More informationUNIT CIRCLE TRIGONOMETRY
UNIT CIRCLE TRIGONOMETRY The Unit Cicle is the cicle centeed at the oigin with adius unit (hence, the unit cicle. The equation of this cicle is + =. A diagam of the unit cicle is shown below: + =   
More informationChapter 3 Savings, Present Value and Ricardian Equivalence
Chapte 3 Savings, Pesent Value and Ricadian Equivalence Chapte Oveview In the pevious chapte we studied the decision of households to supply hous to the labo maket. This decision was a static decision,
More informationRisk Sensitive Portfolio Management With CoxIngersollRoss Interest Rates: the HJB Equation
Risk Sensitive Potfolio Management With CoxIngesollRoss Inteest Rates: the HJB Equation Tomasz R. Bielecki Depatment of Mathematics, The Notheasten Illinois Univesity 55 Noth St. Louis Avenue, Chicago,
More informationSeparation probabilities for products of permutations
Sepaation pobabilities fo poducts of pemutations Olivie Benadi, Rosena R. X. Du, Alejando H. Moales and Richad P. Stanley Mach 1, 2012 Abstact We study the mixing popeties of pemutations obtained as a
More informationFigure 2. So it is very likely that the Babylonians attributed 60 units to each side of the hexagon. Its resulting perimeter would then be 360!
1. What ae angles? Last time, we looked at how the Geeks intepeted measument of lengths. Howeve, as fascinated as they wee with geomety, thee was a shape that was much moe enticing than any othe : the
More informationAn Introduction to Omega
An Intoduction to Omega Con Keating and William F. Shadwick These distibutions have the same mean and vaiance. Ae you indiffeent to thei iskewad chaacteistics? The Finance Development Cente 2002 1 Fom
More informationINITIAL MARGIN CALCULATION ON DERIVATIVE MARKETS OPTION VALUATION FORMULAS
INITIAL MARGIN CALCULATION ON DERIVATIVE MARKETS OPTION VALUATION FORMULAS Vesion:.0 Date: June 0 Disclaime This document is solely intended as infomation fo cleaing membes and othes who ae inteested in
More informationFXA 2008. Candidates should be able to : Describe how a mass creates a gravitational field in the space around it.
Candidates should be able to : Descibe how a mass ceates a gavitational field in the space aound it. Define gavitational field stength as foce pe unit mass. Define and use the peiod of an object descibing
More informationCHAPTER 10 Aggregate Demand I
CHAPTR 10 Aggegate Demand I Questions fo Review 1. The Keynesian coss tells us that fiscal policy has a multiplied effect on income. The eason is that accoding to the consumption function, highe income
More informationChannel selection in ecommerce age: A strategic analysis of coop advertising models
Jounal of Industial Engineeing and Management JIEM, 013 6(1):89103 Online ISSN: 0130953 Pint ISSN: 013843 http://dx.doi.og/10.396/jiem.664 Channel selection in ecommece age: A stategic analysis of
More informationExplicit, analytical solution of scaling quantum graphs. Abstract
Explicit, analytical solution of scaling quantum gaphs Yu. Dabaghian and R. Blümel Depatment of Physics, Wesleyan Univesity, Middletown, CT 064590155, USA Email: ydabaghian@wesleyan.edu (Januay 6, 2003)
More informationValuation of Floating Rate Bonds 1
Valuation of Floating Rate onds 1 Joge uz Lopez us 316: Deivative Secuities his note explains how to value plain vanilla floating ate bonds. he pupose of this note is to link the concepts that you leaned
More informationCONCEPTUAL FRAMEWORK FOR DEVELOPING AND VERIFICATION OF ATTRIBUTION MODELS. ARITHMETIC ATTRIBUTION MODELS
CONCEPUAL FAMEOK FO DEVELOPING AND VEIFICAION OF AIBUION MODELS. AIHMEIC AIBUION MODELS Yui K. Shestopaloff, is Diecto of eseach & Deelopment at SegmentSoft Inc. He is a Docto of Sciences and has a Ph.D.
More informationCoordinate Systems L. M. Kalnins, March 2009
Coodinate Sstems L. M. Kalnins, Mach 2009 Pupose of a Coodinate Sstem The pupose of a coodinate sstem is to uniquel detemine the position of an object o data point in space. B space we ma liteall mean
More informationClosedform expressions for integrals of traveling wave reductions of integrable lattice equations
Closedfom expessions fo integals of taveling wave eductions of integable lattice equations Dinh T Tan, Pete H van de Kamp, G R W Quispel Depatment of Mathematics, La Tobe Univesity, Victoia 386, Austalia
More informationFinancing Terms in the EOQ Model
Financing Tems in the EOQ Model Habone W. Stuat, J. Columbia Business School New Yok, NY 1007 hws7@columbia.edu August 6, 004 1 Intoduction This note discusses two tems that ae often omitted fom the standad
More informationON THE (Q, R) POLICY IN PRODUCTIONINVENTORY SYSTEMS
ON THE R POLICY IN PRODUCTIONINVENTORY SYSTEMS Saifallah Benjaafa and JoonSeok Kim Depatment of Mechanical Engineeing Univesity of Minnesota Minneapolis MN 55455 Abstact We conside a poductioninventoy
More informationSkills Needed for Success in Calculus 1
Skills Needed fo Success in Calculus Thee is much appehension fom students taking Calculus. It seems that fo man people, "Calculus" is snonmous with "difficult." Howeve, an teache of Calculus will tell
More informationHow To Find The Optimal Stategy For Buying Life Insuance
Life Insuance Puchasing to Reach a Bequest Ehan Bayakta Depatment of Mathematics, Univesity of Michigan Ann Abo, Michigan, USA, 48109 S. David Pomislow Depatment of Mathematics, Yok Univesity Toonto, Ontaio,
More informationAn Efficient Group Key Agreement Protocol for Ad hoc Networks
An Efficient Goup Key Ageement Potocol fo Ad hoc Netwoks Daniel Augot, Raghav haska, Valéie Issany and Daniele Sacchetti INRIA Rocquencout 78153 Le Chesnay Fance {Daniel.Augot, Raghav.haska, Valéie.Issany,
More information1240 ev nm 2.5 ev. (4) r 2 or mv 2 = ke2
Chapte 5 Example The helium atom has 2 electonic enegy levels: E 3p = 23.1 ev and E 2s = 20.6 ev whee the gound state is E = 0. If an electon makes a tansition fom 3p to 2s, what is the wavelength of the
More information2 r2 θ = r2 t. (3.59) The equal area law is the statement that the term in parentheses,
3.4. KEPLER S LAWS 145 3.4 Keple s laws You ae familia with the idea that one can solve some mechanics poblems using only consevation of enegy and (linea) momentum. Thus, some of what we see as objects
More informationOverencryption: Management of Access Control Evolution on Outsourced Data
Oveencyption: Management of Access Contol Evolution on Outsouced Data Sabina De Capitani di Vimecati DTI  Univesità di Milano 26013 Cema  Italy decapita@dti.unimi.it Stefano Paaboschi DIIMM  Univesità
More informationChapter 4: Matrix Norms
EE448/58 Vesion.0 John Stensby Chate 4: Matix Noms The analysis of matixbased algoithms often equies use of matix noms. These algoithms need a way to quantify the "size" of a matix o the "distance" between
More informationLecture 16: Color and Intensity. and he made him a coat of many colours. Genesis 37:3
Lectue 16: Colo and Intensity and he made him a coat of many colous. Genesis 37:3 1. Intoduction To display a pictue using Compute Gaphics, we need to compute the colo and intensity of the light at each
More informationest using the formula I = Prt, where I is the interest earned, P is the principal, r is the interest rate, and t is the time in years.
9.2 Inteest Objectives 1. Undestand the simple inteest fomula. 2. Use the compound inteest fomula to find futue value. 3. Solve the compound inteest fomula fo diffeent unknowns, such as the pesent value,
More informationAMB111F Financial Maths Notes
AMB111F Financial Maths Notes Compound Inteest and Depeciation Compound Inteest: Inteest computed on the cuent amount that inceases at egula intevals. Simple inteest: Inteest computed on the oiginal fixed
More informationAN IMPLEMENTATION OF BINARY AND FLOATING POINT CHROMOSOME REPRESENTATION IN GENETIC ALGORITHM
AN IMPLEMENTATION OF BINARY AND FLOATING POINT CHROMOSOME REPRESENTATION IN GENETIC ALGORITHM Main Golub Faculty of Electical Engineeing and Computing, Univesity of Zageb Depatment of Electonics, Micoelectonics,
More informationA Capacitated Commodity Trading Model with Market Power
A Capacitated Commodity Tading Model with Maket Powe Victo MatínezdeAlbéniz Josep Maia Vendell Simón IESE Business School, Univesity of Navaa, Av. Peason 1, 08034 Bacelona, Spain VAlbeniz@iese.edu JMVendell@iese.edu
More informationDatabase Management Systems
Contents Database Management Systems (COP 5725) D. Makus Schneide Depatment of Compute & Infomation Science & Engineeing (CISE) Database Systems Reseach & Development Cente Couse Syllabus 1 Sping 2012
More informationThe Supply of Loanable Funds: A Comment on the Misconception and Its Implications
JOURNL OF ECONOMICS ND FINNCE EDUCTION Volume 7 Numbe 2 Winte 2008 39 The Supply of Loanable Funds: Comment on the Misconception and Its Implications. Wahhab Khandke and mena Khandke* STRCT Recently FieldsHat
More informationQuantity Formula Meaning of variables. 5 C 1 32 F 5 degrees Fahrenheit, 1 bh A 5 area, b 5 base, h 5 height. P 5 2l 1 2w
1.4 Rewite Fomulas and Equations Befoe You solved equations. Now You will ewite and evaluate fomulas and equations. Why? So you can apply geometic fomulas, as in Ex. 36. Key Vocabulay fomula solve fo a
More informationQuestions & Answers Chapter 10 Software Reliability Prediction, Allocation and Demonstration Testing
M13914 Questions & Answes Chapte 10 Softwae Reliability Pediction, Allocation and Demonstation Testing 1. Homewok: How to deive the fomula of failue ate estimate. λ = χ α,+ t When the failue times follow
More informationThe LCOE is defined as the energy price ($ per unit of energy output) for which the Net Present Value of the investment is zero.
Poject Decision Metics: Levelized Cost of Enegy (LCOE) Let s etun to ou wind powe and natual gas powe plant example fom ealie in this lesson. Suppose that both powe plants wee selling electicity into the
More informationArticle Two pricing mechanisms in sponsored search advertising
econsto www.econsto.eu De OpenAccessPublikationsseve de ZBW LeibnizInfomationszentum Witschaft The Open Access Publication Seve of the ZBW Leibniz Infomation Cente fo Economics Yang, Wei; Feng, Youyi;
More informationA framework for the selection of enterprise resource planning (ERP) system based on fuzzy decision making methods
A famewok fo the selection of entepise esouce planning (ERP) system based on fuzzy decision making methods Omid Golshan Tafti M.s student in Industial Management, Univesity of Yazd Omidgolshan87@yahoo.com
More informationLiquidity and Insurance for the Unemployed*
Fedeal Reseve Bank of Minneapolis Reseach Depatment Staff Repot 366 Decembe 2005 Liquidity and Insuance fo the Unemployed* Robet Shime Univesity of Chicago and National Bueau of Economic Reseach Iván Wening
More informationChris J. Skinner The probability of identification: applying ideas from forensic statistics to disclosure risk assessment
Chis J. Skinne The pobability of identification: applying ideas fom foensic statistics to disclosue isk assessment Aticle (Accepted vesion) (Refeeed) Oiginal citation: Skinne, Chis J. (2007) The pobability
More informationLiquidity and Insurance for the Unemployed
Liquidity and Insuance fo the Unemployed Robet Shime Univesity of Chicago and NBER shime@uchicago.edu Iván Wening MIT, NBER and UTDT iwening@mit.edu Fist Daft: July 15, 2003 This Vesion: Septembe 22, 2005
More informationConcept and Experiences on using a Wikibased System for Softwarerelated Seminar Papers
Concept and Expeiences on using a Wikibased System fo Softwaeelated Semina Papes Dominik Fanke and Stefan Kowalewski RWTH Aachen Univesity, 52074 Aachen, Gemany, {fanke, kowalewski}@embedded.wthaachen.de,
More informationModel Question Paper Mathematics Class XII
Model Question Pape Mathematics Class XII Time Allowed : 3 hous Maks: 100 Ma: Geneal Instuctions (i) The question pape consists of thee pats A, B and C. Each question of each pat is compulsoy. (ii) Pat
More informationThe transport performance evaluation system building of logistics enterprises
Jounal of Industial Engineeing and Management JIEM, 213 6(4): 194114 Online ISSN: 213953 Pint ISSN: 2138423 http://dx.doi.og/1.3926/jiem.784 The tanspot pefomance evaluation system building of logistics
More informationVector Calculus: Are you ready? Vectors in 2D and 3D Space: Review
Vecto Calculus: Ae you eady? Vectos in D and 3D Space: Review Pupose: Make cetain that you can define, and use in context, vecto tems, concepts and fomulas listed below: Section 7.7. find the vecto defined
More informationAn Analysis of Manufacturer Benefits under Vendor Managed Systems
An Analysis of Manufactue Benefits unde Vendo Managed Systems Seçil Savaşaneil Depatment of Industial Engineeing, Middle East Technical Univesity, 06531, Ankaa, TURKEY secil@ie.metu.edu.t Nesim Ekip 1
More informationApproximation Algorithms for Data Management in Networks
Appoximation Algoithms fo Data Management in Netwoks Chistof Kick Heinz Nixdof Institute and Depatment of Mathematics & Compute Science adebon Univesity Gemany kueke@upb.de Haald Räcke Heinz Nixdof Institute
More informationMechanics 1: Work, Power and Kinetic Energy
Mechanics 1: Wok, Powe and Kinetic Eneg We fist intoduce the ideas of wok and powe. The notion of wok can be viewed as the bidge between Newton s second law, and eneg (which we have et to define and discuss).
More informationResearch on Risk Assessment of the Transformer Based on Life Cycle Cost
ntenational Jounal of Smat Gid and lean Enegy eseach on isk Assessment of the Tansfome Based on Life ycle ost Hui Zhou a, Guowei Wu a, Weiwei Pan a, Yunhe Hou b, hong Wang b * a Zhejiang Electic Powe opoation,
More informationSoftware Engineering and Development
I T H E A 67 Softwae Engineeing and Development SOFTWARE DEVELOPMENT PROCESS DYNAMICS MODELING AS STATE MACHINE Leonid Lyubchyk, Vasyl Soloshchuk Abstact: Softwae development pocess modeling is gaining
More informationIlona V. Tregub, ScD., Professor
Investment Potfolio Fomation fo the Pension Fund of Russia Ilona V. egub, ScD., Pofesso Mathematical Modeling of Economic Pocesses Depatment he Financial Univesity unde the Govenment of the Russian Fedeation
More informationCapital Investment and Liquidity Management with collateralized debt.
TSE 54 Novembe 14 Capital Investment and Liquidity Management with collatealized debt. Ewan Piee, Stéphane Villeneuve and Xavie Wain 7 Capital Investment and Liquidity Management with collatealized debt.
More informationSemipartial (Part) and Partial Correlation
Semipatial (Pat) and Patial Coelation his discussion boows heavily fom Applied Multiple egession/coelation Analysis fo the Behavioal Sciences, by Jacob and Paticia Cohen (975 edition; thee is also an updated
More informationSecure SmartcardBased Fingerprint Authentication
Secue SmatcadBased Fingepint Authentication [full vesion] T. Chales Clancy Compute Science Univesity of Mayland, College Pak tcc@umd.edu Nega Kiyavash, Dennis J. Lin Electical and Compute Engineeing Univesity
More informationPRICING MODEL FOR COMPETING ONLINE AND RETAIL CHANNEL WITH ONLINE BUYING RISK
PRICING MODEL FOR COMPETING ONLINE AND RETAIL CHANNEL WITH ONLINE BUYING RISK Vaanya Vaanyuwatana Chutikan Anunyavanit Manoat Pinthong Puthapon Jaupash Aussaavut Dumongsii Siinhon Intenational Institute
More informationLoad Balancing in Processor Sharing Systems
Load Balancing in ocesso Shaing Systems Eitan Altman INRIA Sophia Antipolis 2004, oute des Lucioles 06902 Sophia Antipolis, Fance altman@sophia.inia.f Utzi Ayesta LAASCNRS Univesité de Toulouse 7, Avenue
More informationLoad Balancing in Processor Sharing Systems
Load Balancing in ocesso Shaing Systems Eitan Altman INRIA Sophia Antipolis 2004, oute des Lucioles 06902 Sophia Antipolis, Fance altman@sophia.inia.f Utzi Ayesta LAASCNRS Univesité de Toulouse 7, Avenue
More informationChapte 3 Is Gavitation A Results Of Asymmetic Coulomb Chage Inteactions? Jounal of Undegaduate Reseach èjurè Univesity of Utah è1992è, Vol. 3, No. 1, pp. 56í61. Jeæey F. Gold Depatment of Physics, Depatment
More informationPromised LeadTime Contracts Under Asymmetric Information
OPERATIONS RESEARCH Vol. 56, No. 4, July August 28, pp. 898 915 issn 3364X eissn 15265463 8 564 898 infoms doi 1.1287/ope.18.514 28 INFORMS Pomised LeadTime Contacts Unde Asymmetic Infomation Holly
More informationYARN PROPERTIES MEASUREMENT: AN OPTICAL APPROACH
nd INTERNATIONAL TEXTILE, CLOTHING & ESIGN CONFERENCE Magic Wold of Textiles Octobe 03 d to 06 th 004, UBROVNIK, CROATIA YARN PROPERTIES MEASUREMENT: AN OPTICAL APPROACH Jana VOBOROVA; Ashish GARG; Bohuslav
More informationMechanics 1: Motion in a Central Force Field
Mechanics : Motion in a Cental Foce Field We now stud the popeties of a paticle of (constant) ass oving in a paticula tpe of foce field, a cental foce field. Cental foces ae ve ipotant in phsics and engineeing.
More informationReduced Pattern Training Based on Task Decomposition Using Pattern Distributor
> PNN05P762 < Reduced Patten Taining Based on Task Decomposition Using Patten Distibuto ShengUei Guan, Chunyu Bao, and TseNgee Neo Abstact Task Decomposition with Patten Distibuto (PD) is a new task
More informationThe Role of Gravity in Orbital Motion
! The Role of Gavity in Obital Motion Pat of: Inquiy Science with Datmouth Developed by: Chistophe Caoll, Depatment of Physics & Astonomy, Datmouth College Adapted fom: How Gavity Affects Obits (Ohio State
More informationPAN STABILITY TESTING OF DC CIRCUITS USING VARIATIONAL METHODS XVIII  SPETO  1995. pod patronatem. Summary
PCE SEMINIUM Z PODSTW ELEKTOTECHNIKI I TEOII OBWODÓW 8  TH SEMIN ON FUNDMENTLS OF ELECTOTECHNICS ND CICUIT THEOY ZDENĚK BIOLEK SPŠE OŽNO P.., CZECH EPUBLIC DLIBO BIOLEK MILITY CDEMY, BNO, CZECH EPUBLIC
More informationQuestions for Review. By buying bonds This period you save s, next period you get s(1+r)
MACROECONOMICS 2006 Week 5 Semina Questions Questions fo Review 1. How do consumes save in the twopeiod model? By buying bonds This peiod you save s, next peiod you get s() 2. What is the slope of a consume
More informationThings to Remember. r Complete all of the sections on the Retirement Benefit Options form that apply to your request.
Retiement Benefit 1 Things to Remembe Complete all of the sections on the Retiement Benefit fom that apply to you equest. If this is an initial equest, and not a change in a cuent distibution, emembe to
More informationSupplementary Material for EpiDiff
Supplementay Mateial fo EpiDiff Supplementay Text S1. Pocessing of aw chomatin modification data In ode to obtain the chomatin modification levels in each of the egions submitted by the use QDCMR module
More informationEfficient Redundancy Techniques for Latency Reduction in Cloud Systems
Efficient Redundancy Techniques fo Latency Reduction in Cloud Systems 1 Gaui Joshi, Emina Soljanin, and Gegoy Wonell Abstact In cloud computing systems, assigning a task to multiple seves and waiting fo
More informationUncertain Version Control in Open Collaborative Editing of TreeStructured Documents
Uncetain Vesion Contol in Open Collaboative Editing of TeeStuctued Documents M. Lamine Ba Institut Mines Télécom; Télécom PaisTech; LTCI Pais, Fance mouhamadou.ba@ telecompaistech.f Talel Abdessalem
More informationBasic Financial Mathematics
Financial Engineeing and Computations Basic Financial Mathematics Dai, TianShy Outline Time Value of Money Annuities Amotization Yields Bonds Time Value of Money PV + n = FV (1 + FV: futue value = PV
More informationCONCEPT OF TIME AND VALUE OFMONEY. Simple and Compound interest
CONCEPT OF TIME AND VALUE OFMONEY Simple and Compound inteest What is the futue value of shs 10,000 invested today to ean an inteest of 12% pe annum inteest payable fo 10 yeas and is compounded; a. Annually
More informationPatent renewals and R&D incentives
RAND Jounal of Economics Vol. 30, No., Summe 999 pp. 97 3 Patent enewals and R&D incentives Fancesca Conelli* and Mak Schankeman** In a model with moal hazad and asymmetic infomation, we show that it can
More informationCarterPenrose diagrams and black holes
CatePenose diagams and black holes Ewa Felinska The basic intoduction to the method of building Penose diagams has been pesented, stating with obtaining a Penose diagam fom Minkowski space. An example
More informationThe impact of migration on the provision. of UK public services (SRG.10.039.4) Final Report. December 2011
The impact of migation on the povision of UK public sevices (SRG.10.039.4) Final Repot Decembe 2011 The obustness The obustness of the analysis of the is analysis the esponsibility is the esponsibility
More informationStrategic Betting for Competitive Agents
Stategic Betting fo Competitive Agents Liad Wagman Depatment of Economics Duke Univesity Duham, NC, USA liad.wagman@duke.edu Vincent Conitze Depts. of Compute Science and Economics Duke Univesity Duham,
More informationMETHODOLOGICAL APPROACH TO STRATEGIC PERFORMANCE OPTIMIZATION
ETHODOOGICA APPOACH TO STATEGIC PEFOANCE OPTIIZATION ao Hell * Stjepan Vidačić ** Željo Gaača *** eceived: 4. 07. 2009 Peliminay communication Accepted: 5. 0. 2009 UDC 65.02.4 This pape pesents a matix
More informationMATHEMATICAL SIMULATION OF MASS SPECTRUM
MATHEMATICA SIMUATION OF MASS SPECTUM.Beánek, J.Knížek, Z. Pulpán 3, M. Hubálek 4, V. Novák Univesity of South Bohemia, Ceske Budejovice, Chales Univesity, Hadec Kalove, 3 Univesity of Hadec Kalove, Hadec
More informationIt is required to solve the heatcondition equation for the excesstemperature function:
Jounal of Engineeing Physics and Themophysics. Vol. 73. No. 5. 2 METHOD OF PAIED INTEGAL EQUATIONS WITH LPAAMETE IN POBLEMS OF NONSTATIONAY HEAT CONDUCTION WITH MIXED BOUNDAY CONDITIONS FO AN INFINITE
More informationLab #7: Energy Conservation
Lab #7: Enegy Consevation Photo by Kallin http://www.bungeezone.com/pics/kallin.shtml Reading Assignment: Chapte 7 Sections 1,, 3, 5, 6 Chapte 8 Sections 14 Intoduction: Pehaps one of the most unusual
More informationPhysics 235 Chapter 5. Chapter 5 Gravitation
Chapte 5 Gavitation In this Chapte we will eview the popeties of the gavitational foce. The gavitational foce has been discussed in geat detail in you intoductoy physics couses, and we will pimaily focus
More informationYIELD TO MATURITY ACCRUED INTEREST QUOTED PRICE INVOICE PRICE
YIELD TO MATURITY ACCRUED INTEREST QUOTED PRICE INVOICE PRICE Septembe 1999 Quoted Rate Teasuy Bills [Called Banke's Discount Rate] d = [ P 1  P 1 P 0 ] * 360 [ N ] d = Bankes discount yield P 1 = face
More informationGravitational Mechanics of the MarsPhobos System: Comparing Methods of Orbital Dynamics Modeling for Exploratory Mission Planning
Gavitational Mechanics of the MasPhobos System: Compaing Methods of Obital Dynamics Modeling fo Exploatoy Mission Planning Alfedo C. Itualde The Pennsylvania State Univesity, Univesity Pak, PA, 6802 This
More informationCHAPTER 5 GRAVITATIONAL FIELD AND POTENTIAL
CHATER 5 GRAVITATIONAL FIELD AND OTENTIAL 5. Intoduction. This chapte deals with the calculation of gavitational fields and potentials in the vicinity of vaious shapes and sizes of massive bodies. The
More informationTORQUE AND ANGULAR MOMENTUM IN CIRCULAR MOTION
MISN034 TORQUE AND ANGULAR MOMENTUM IN CIRCULAR MOTION shaft TORQUE AND ANGULAR MOMENTUM IN CIRCULAR MOTION by Kiby Mogan, Chalotte, Michigan 1. Intoduction..............................................
More informationContingent capital with repeated interconversion between debt and equity
Contingent capital with epeated inteconvesion between debt and equity Zhaojun Yang 1, Zhiming Zhao School of Finance and Statistics, Hunan Univesity, Changsha 410079, China Abstact We develop a new type
More informationHow To Write A Theory Of The Concept Of The Mind In A Quey
Jounal of Atificial Intelligence Reseach 31 (2008) 157204 Submitted 06/07; published 01/08 Conjunctive Quey Answeing fo the Desciption Logic SHIQ Bite Glimm Ian Hoocks Oxfod Univesity Computing Laboatoy,
More informationPeertoPeer File Sharing Game using Correlated Equilibrium
PeetoPee File Shaing Game using Coelated Equilibium Beibei Wang, Zhu Han, and K. J. Ray Liu Depatment of Electical and Compute Engineeing and Institute fo Systems Reseach, Univesity of Mayland, College
More informationFI3300 Corporate Finance
Leaning Objectives FI00 Copoate Finance Sping Semeste 2010 D. Isabel Tkatch Assistant Pofesso of Finance Calculate the PV and FV in multipeiod multicf timevalueofmoney poblems: Geneal case Pepetuity
More informationInstituto Superior Técnico Av. Rovisco Pais, 1 1049001 Lisboa Email: virginia.infante@ist.utl.pt
FATIGUE LIFE TIME PREDICTIO OF POAF EPSILO TB30 AIRCRAFT  PART I: IMPLEMETATIO OF DIFERET CYCLE COUTIG METHODS TO PREDICT THE ACCUMULATED DAMAGE B. A. S. Seano 1, V. I. M.. Infante 2, B. S. D. Maado
More informationData Center Demand Response: Avoiding the Coincident Peak via Workload Shifting and Local Generation
(213) 1 28 Data Cente Demand Response: Avoiding the Coincident Peak via Wokload Shifting and Local Geneation Zhenhua Liu 1, Adam Wieman 1, Yuan Chen 2, Benjamin Razon 1, Niangjun Chen 1 1 Califonia Institute
More information