Definite Integration in Mathematica V3.0


 Pearl Flynn
 2 years ago
 Views:
Transcription
1 Definite Integration in Mathematica V3. Victor Adamchi Introduction The aim of this aer is to rovide a short descrition of definite integration algorithms imlemented in Mathematica Version 3.. $Version Linu 3. HAril 5, 997L Proer Integrals All roer integrals in Mathematica are evaluated by means of the NewtonLeibni theorem a b f HL Ç = FHbL  FHaL where FHL is an antiderivative. It is wellnown that the NewtonLeibni formula in the given form does not hold any longer if the antiderivative FHL has singularities on an interval of integration Ha, bl. Let us consider the following integral + + þþ Ç where the integrand is a smooth integrable function on an interval H, L.
2 mier.nb + + PlotA þþþ, 8,, <E ú Grahics ú 3 As it follows from the Risch structure theorem the corresondent indefinite integral is doable in elementary functions int = + + þþþ Ç tani j þ   y   { tani j þþþþ + y  { If we simly substitute limits of integration into the antiderivative we get an incorrect result. D  D tan i j y  tan  J þþ 5 { N This is because the antiderivative is not a continuous function on an interval H, L. It has a jum at =, which is easy to see in the following grahic.
3 mier.nb 3 8,, <D ú Grahics ú The right way of alying the NewtonLeibni theorem is to tae into account an influence of the jum >, Direction > D  >, Direction > D + >, Direction > D  >, Direction > D tan i j y  tan  J þþ 5 { N Mathematica evaluates definite integrals in recisely that way. + + þþþ Ç tan i j y  tan  J þþ 5 { N The origin of discontinuities of antiderivatives along the ath of integration is not in the method of indefinite integration but rather in the integrand. In the discussed eamle, the integrand has four singular oles that become branch oints for the antiderivative ==, D == É Ã == É Ã == É Ã == É Connected in airs these oints mae two branch cuts. And the ath of integration crosses one of them. The following ContourPlot clearly eoses the roblem.
4 mier.nb ƒ. > + I ydd, 8, 3, <, 8y, 3, <, ContourShading > False, Contours >, PlotPoints >, Eilog > <, 8, <<D<D ú ContourGrahics ú We see that in a comle lane of the variable the antiderivative has two branch cuts (bold blac vertical lines) and the ath of integration, the line (,), intersects the right branch cut. Obviously, by varying the constant of integration we can change the form of the antiderivative so that we would get various forms of branch cuts. Here df HL we understand the constant of integration as a function f() such that þþ is ero. As a simle eamle let us d consider the stewise constant function!!!!!! þþþ!!!!!! SimlifyA þ E Thining hard, we can built an antiderivative that does not have a branch cut crossing a given interval of integration. However, we can never get rid of branch cuts! Analysis of the singularities of antiderivatives is a time consuming and sometimes heuristic rocess, esecially if trigonometric or secial functions are involved in antiderivatives. In the latter case Integrate may not be able to detect all singular oints on the interval of integration, which will result in a warning message Integrate::gener : Unable to chec convergence
5 mier.nb 5 You should ay attention to the message since it warns you that the result of the integration might be wrong. Imroer Integrals It is quite clear that the above rocedure cannot cover the whole variety of definite integrals. There are two reasons behind that. First, the corresondent indefinite integral cannot be eressed in finite terms of functions reresented in Mathematica. For instance, Ç coshsinhll Ç However, the definite integral with the secific limits of integration is doable. Ç J HL Second, even if an indefinite integral can be done, it requires a great deal of effort to find limits at the end oints. Here is an eamle, þþþ ƒ Ç seci j y { This is an imroer integral since the to limit is a singular oint of the integrand. The result of indefinite integration is ƒ Ç F H þþþ + +,;+ þþþ ; tan HLL tan + HL þþ þþ + To find the limit at = þþþþ one has to, first, develo the asymtotic eansion of the hyergeometric Gauss function at, and second, construct the asymtotic scale for the Puiseu series. Currently, Mathematica s Series structure is based on ower series that allow only rational eonents. Following we resent a short descrition of the algorithm for evaluating imroer integrals. The overall idea has been given in [] and [3]. Some ractical details regarding "logarithmic" cases are described in [] and [].
6 6 mier.nb The General Idea By means of the Mellin integral transform and Parseval s equality, a given imroer integral is transformed to a contour integral over the straight line Hg É, g+él in a comle lane of the arameter : f HL f J N þþþ Ç = þþþ É where f * HL and f * HL are Mellin transforms of f HL and f HL g+é gé f * HsL f * HsL s Ç s () f * HsL = f HL s Ç The real arameter g in the formula () is defined by conditions of eistence of Mellin transforms f * HL and f * HL. Finally, the residue theorem is used to evaluate the contour integral in the right side of the formula () þþþ É G m f HL Ç = Ç res = a = f HL where G is a closed contour, and a are oles of f() that lie in a domain bounded by G. The success of this scheme deends on two factors: first, the Mellin image of an integrand f HL f H þþþþ L must eist, and, second, it should be reresented in terms of Gamma functions. If these conditions are satisfied then the contour integral in (), called the MellinBarnes integral or the Meijer Gfunction, can almost always be eressed in finite terms of hyergeometric functions. This fact is nown as Slater s theorem (see [] and [5]). We said "almost always", since there is a secial case of the Gfunction when the latter cannot be reduced to hyergeometric functions but to their derivatives with resect to arameters. By analogy with linear differential equations with olynomial coefficients, such a singular case of the Gfunction is named a logarithmic case. The modified Bessel function K HL is such a classical eamle, since its series reresentation K HL =I HL logj N + È = y H + L i þ! j þ y { cannot be eressed via hyergeometrics. MellinBarnes Integrals These integrals are defined by (see[6]) þþþ É L ghsl s Ç s where g HsL = À n j= G Ha j + sl À m j= G Hb j  sl þ l GHc j + sl À j= GHd j  sl þþ É j=
7 mier.nb 7 and the contour L is a line that searates oles of G Ha j + sl from G Hb j  sl. Let us investigate when the integral eists. On a straight line L, = Hg É, g+él the real art of s ³ L is bounded, and ImHsL. Using the Stirling asymtotic formula GH +ÉyL =!!!!!!! y  þþþþ È  þþþþ i y j + O i j y þþþ y y {{, y we can deduce that the integrand ghsl s vanishes eonentially as Im(s) if m + n   l >, and arghl < þþþþ Hm + n   ll. If m + n   l =, then must be real and ositive. Some additional conditions are required here (see details in []). It is clear that we can not simly imly the residue theorem to this contour integral. We need initially to transform the contour L to the closed one. There are two ossibilities: we can either transform L into the left loo L , the contour encircling all oles of G Ha j + sl, or to the right loo L +, the contour encircling the oles of G Hb j  sl. The criteria of which contour should be chosen aears from the convergence of the integral along that contour. On the lefthand loo L , the imaginary art of s ³ L  is bounded and ReHsL . Assuming that arghsl < þþþþ, and maing use of the Stirling formula GHL =!!!!!!!  þþþþ È  J + OJ NN,, arghl < and the reflection formula of the Gamma function, we find that the MellinBarnes integral over the loo L  eists, if n + l  m  >. If n + l  m  =, then must be within the unit dis <. If is on a unit circle =, the integral converges if i n Re j Ç j= m a j + Ç j= b j + Ç j= l c j + Ç j= y d j < + n  { On the righthand loo L +, the imaginary art of s ³ L + is bounded and ReHsL +. Proceeding similarly to the above, we find that the MellinBarnes integral over the loo L + eists, if n + l  m  <. If n + l  m  =, then must be outside of the unit dis. Additional conditions are required if is on a unit circle. After determining the correct contour, the net ste is to calculate residues of the integrand. Since the integrand contains only Gamma functions, this tas is more or less formal. We don t even need to calculate residues of the integrand, but go straightforwardly to the generalied hyergeometric functions. The only obstacle is the logarithmic case, which occurs when the integrand has multile oles. We have to searate two subclasses here: the integrand that has a finite number of multile oles, and the integrand that has infinitely many multile oles. The Mathematica integration routine has a full imlementation of the former case. In the latter, the integration artificially restricted by the second order oles, since for the higher order oles it would lead to infinite sums with higher order olygamma functions. This class of infinite sums are etremely hard to deal with, symbolically and numerically. If such a situation is detected Integrate returns the Meijer Gfunction. However, there are some very secial transformations of the Gfunction, which could avoid buly infinite sums with olygamma functions, and give a nice result in terms of nown functions. In [] I demonstrated a few transformations that reduce the order of the Gfunction and mae it ossible to handle secial class of Bessel integrals in terms of Bessel functions.
8 8 mier.nb An Eamle Consider the integral According to the formula (), we have sinhl W= þ H + L Ç W= þþþþ + sinj þþþ ƒ N þþþ Ç = þþþ É where f * HsL and f * HsL are Mellin transforms of È  and sinh þþþþ L: s f = þþþ + Ç If i jrehsl > Ä ReHsL <, cscj þþþ s N, s þþþþ + Ç y { g+é gé f * HsL f * HsL Ç s Therefore, f = SinA þþþþ E s Ç If i s jrehsl > Ä ReHsL <, GHsL sinj þþþ N, s sin i j y Ç y { { W=þþþ É g+é gé GHsL Ç s where g is defined by conditions of the eistence of Mellin transforms <g=rehsl < As it follows from the revious section we can transform the integration contour Hg , g+él into the right loo L +. Then, using the residue theorem we evaluate the integral as a sum of residues at simle oles s =,,.... W=þþþþ È = H ÈL  þ È HL þþþþ!
9 mier.nb 9 Meijer GFunction We did not intend to give a comlete icture of the Meijer Gfunction here, but only necessary facts. In Version 3. the Gfunction is defined by a,..., a n <, 8a n+, a n+,..., a <<, 88b, b,..., b m <, 8b m+, b m+,..., b q <<, D = þþþ É L m,n G i,q j a, a, b, b,..., a y =..., b q { À m n i= GHb i + sl À i= GH  a i  sl þþ þþ q þ þþþþ À i=n+ GHa i + sl À i=m+ GH  b i  sl s Ç s and the contour L is a left (or right) loo L  (or L + ) searated oles of G Hb j + sl from G H  a j  sl. The current imlementation of the Meijer function has two noticeable features that differ it from the classical G function. First, you are not allowed to choose or move the contour of integration, and, second, MeijerG has an otional arameter (the classical Gfunction does not) that regulates the branch cut. The Meijer function is suorted symbolically and numerically. 8<<, 88<, 8<<, D þþþþ + MeijerGA88<, 8, <<, 99, þþþþ =, 8<=, þþþþ E MeijerG i j88<, 8, <<, ::, >, 8<>, y { Only in very trivial cases MeijerG is simlified automatically to the lower level secial functions. Beyond that all further transformations are assigned to FunctionEand: !!!! I HL þþþþ !!!! þ þ I HL È È MeijerG is interlaced with indefinite and definite integration, and with solving linear differential equations with olynomial coefficients. Here is an eamle related to definite integration:
10 mier.nb D þþþ!!!!!!!!!!!!!!  Ç!!!! MeijerG i j:8<, : >>, 88,, <, 8<<, y { The first element of the second argument of MeijerG, which is {,,}, indicates that the integrand of the corresondent contour integral contains the roduct of three Gamma functions GHs  L GHsL GHsL and has an infinite number of trile oles at s=, =,,,.... From the design oint of view it is definitely an advantage for Integrate to return a short object, MeijerG, rather than enormous infinite sums involving derivatives of the Gamma function. Hyergeometric Functions The essential art of integration is the generalied hyergeometric function, which is defined by a <, 8b,..., b q <, D = F i q j a, a, b, b,..., a y =..., b q { À j= Ha j L È þ q þ À j= Hb j L = þþ! where HaL is a Pochhammer symbol Ha j L = É l= Ha j + l  L = G Ha j + L þ G Ha j L Conditions of convergence can be easily obtained by alying the d Alembert test. It follows, F q converges for all finite if ˆ q, and for < if = q +. Additional conditions are required on the circle of convergence =. The above series definition of F q has an ecetional case, when = and q =  such HyergeometricPFQ is defined via the confluent hyergeometric function HyergeometricU b<, 8<, D a i j y U i { ja, a  b +,  y { For r and = q +, the hyergeometric function is defined as an analytic continuation via the Mellin Barnes integral, with the branch cut [, ): q+ F i q j a, a, b, b,..., a q+ y =..., b q { q À = GHb L þ þ q+ GHa L À = g+é þþþ É gé G HsL À q+ j= G Ha j  sl þþ q þþþ G Hb j  sl À j= HL s Ç s
11 mier.nb where the straight line Hg É, g+él searates oles of G(s) from G Ha j  sl. Almost all symbolic simlifications of F q to the lower level functions are done automatically HyergeometricFA þþþþ 3, þþþþ 3, þþþþ 3, þþþþ E 9!!!! 3 I!!!! 3 +3 loghlm HyergeometricPFQA9, þþþþ 3, þþþþ 3 =, 9 þþþþ 3, þþþþ 3 =,E 9 yhli j y 3 { <, 8,, <, D 7 þþþ H loghl + 3 H3LL The ecetion is the Gauss function HyergeometricF. Since it has so many different transformation and simlification rules some of them are assigned to FunctionEand: HyergeometricFA, þþþþ, þþþþ 5, þþþþ E F i j, ; 5 ; y { cot  I!!!! M log I  þþþ!!!! M log I + þþþ!!!! M þ!!!! þ  þ!!!! + þ!!!! PrincialValue Integrals Consider the integral  þþþþ Ç Integrate::idiv : Integral of  Ç does not converge on 8, <.
12 mier.nb which does not eist in the Riemann sense, since it has a nonintegrable singularity at =. However, if we isolate = by e > from the left and by e > from the right and tae a limit of corresondent integrals when e and e, we obtain lim e e i j  e Ç + e Ç y { = lim HlogHL + loghe L  loghe LL The double limit eists, and so it is a given integral, if and only if e =e. Such understanding of divergent integrals is called the rincialvalue or the Cauchy rincialvalue. It is easy to see that if an integral eists in the Riemann sense, it eists in the Cauchy sense. Thus, the class of Cauchy integrals is larger than the class of Riemann integrals. In Version 3., definite integrals in the Riemann sense and rincialvalue integrals are searated by the new otion PrincialValue. If you want to evaluate an integral in the Cauchy sense, set the otion PrincialValue to True (the default setting is False). Here is an eamle e e IntegrateA þþþþ, 8, , <, PrincialValue TrueE loghl IntegrateA þ þ, 8, , <, PrincialValue TrueE + D loghcothhl tanhh3ll 5 IntegrateA 9,  þþþþ 3 H3 L, þþþ =, PrincialValue TrueE C þ 9 If the integrand contains highorder olynomials, Integrate returns RootSum objects IntegrateA, 8, , <, PrincialValue TrueE É þþ þþ RootH# 7 + # + &,L 6 RootSum i logh#  L j#7 + # + &, þ þþþþ & y 7# 6 + { + RootSumi logh  #L j#7 + # + &, þþþ 7# 6 + &y { Here are eamles of integrals with movable singularities
13 mier.nb 3 IntegrateA a  9,, þþþþ =, PrincialValue TrueE If i j a >, þ a +, þþþ a  tan HL Ç y { IntegrateA l þ  a, 8,, <, PrincialValue TrueE If i ja > Ä ReHlL > Ä ReHlL <, a l cothll,  a Ç y { Here Integrate detects that the integrand has a singular oint along the ath of integration if the arameter a is real ositive. l New Classes of Integrals In this section we give a short overview of secific classes of definite integrals, which were essentially imroved in the new version. Mathematica now is able to calculate almost all indefinite and about half of definite integrals from the wellnown collection of integrals comiled by Gradshteyn and Ryhi. Moreover, the Version 3. maes it ossible to calculate thousands of new integrals not included in any ublished handboos. integrals of rational functions The RootSum object has been lined to Integrate to dislay the result of integration in a more elegant and shorter way  þþ Ç RootSum i j# 7 + # 3 log H  #L #  log H  #L + &, þþ þ & y  7# 6 + 3# { RootSum i j#7 + # 3 log H#L #  log H#L + &, þþþþ & y 7# 6 + 3# { logarithmic and Polylogarithm integrals + D þþþþ + Ç 8 H loghl H3LL
14 mier.nb  D 3 H  L Ç  6Li 3H þþþ  þþþ L H  L ellitic integrals The acage, conducted an ellitic integration, is now autoloaded (as are as all other integration acages), so you don t need to worry about its loading anymore þþþ  þþþ þþ!!!!!!!!!!!!!!!!! FJ 8 N Ç þþþ þþ!!!!!!!!!!!!!!!!! Ç 3 F i j, 3,; 3, 3 ;y { integrals involving Bessel functions t D td td Çt 3 +!!!! i È K j y { D þþþ þþþ D Ç J HL K HL Ç GH þþþþ 6  þþ L þþþ!!!! 3!!!! 6 3 3ƒ
15 mier.nb 5 integrals involving nonanalytic functions + I Ç I H3 + 3 ÉL i!!!! j þþþ 3  cosi!!!!!!!! M cosi M y þþþþ 3!!!!  þþþ 3!!!! { 3 þ DD Ç 3!!!! 3 þ Additional Features Mathematica s caability for definite integration gained substantial ower in the new version. Other essential features of Integrate not discussed in the revious sections are convergence tests and the assumtions mechanism. Conditions In Version 3. Integrate is "conditional." In most cases, if the integrand or limits of integration contains symbolic arameters, Integrate returns an If statement of the form If[ conditions, answer, held integral ] which gives necessary conditions for the eistence of the integral. For eamle l D Ç IfJReHaL > Ä ReHlL >, a l GHlL, È  a l Ç N Setting the otion GenerateConditions to False revents Integrate from returning conditional results (as in Version.): l D, 8,, <, GenerateConditions FalseD a l GHlL If a given definite integral has symbolic arameters, then the result of integration essentially always deends on certain secific conditions on those arameters. In this eamle the restrictions ReHaL > and ReHlL >
16 6 mier.nb came from conditions of the convergence. Even when a definite integral is convergent, some other conditions on arameters might aear. For instance, the resence of singularities on the integration ath could lead to essential changes when the arameters vary. The net section is devoted to the convergence of definite integrals. Convergence The new integration code contains criteria for the convergence of definite integrals. Each time Integrate eamines the integrand for convergence þ  Ç Integrate::idiv : Integral of coshl does not converge on 8, <. þþþ þþþ þþþþ coshl þþþ Ç  This integral has a nonintegrable singularity at =. Thus, Integrate generates a warning message and returns unevaluated. However, the integral eists in the Cauchy sense. Setting the otion PrincialValue to True, we obtain IntegrateA þ 8, , <, PrincialValue TrueE ƒƒfullsimlify Consider another integral with a symbolic arameter a r Ç If i j ReHrL >, y HL I þþþ r+ M yhl I þþþ r+3 M + þþþ, r tan  HL Ç y r { The integral has a singular oint at =, which is integrable only if ReHrL >. If you are sure that a articular integral is convergent or you don t care about the convergence, you can avoid testing the convergence by setting the otion GenerateConditions to False. It will mae Integrate return an answer a bit faster. Setting GenerateConditions to False also lets you evaluate divergent integrals IntegrateA þþþþ, 8,, <, GenerateConditions FalseE loghl
17 mier.nb 7 Assumtions The new otion Assumtion is used to secify articular assumtions on arameters in definite integrals. Consider the integral with the arbitrary arameter y Here we set the otion Assumtions to ReHyL > È   y þ!!! þ Ç IntegrateA  yd þ, 8,, <, Assumtions > E!!!! È þþ!!!! i y 8 yk þþþþ j þþ y y 8 { Setting Assumtions to ReHyL <, we get a different form of the answer IntegrateA  yd þ, 8,, <, Assumtions < E!!!! þþ È y 8!!!!!!!  y II  þþþþ I y 8 M + I þþþþ I y 8 MM þþ þþ!!!! though the integral is a continuous function with resect to the arameter y PlotANIntegrateA  yd þ, 8,, <E, 8y, , <E!!!! ú Grahics ú The net integral is discontinuous with resect to the arameter g
18 8 mier.nb IntegrateA þþ H  þ DL þ, 8,, <, Assumtions g>e Hg L H  DL IntegrateA þþþ, 8,, <, Assumtions  <g<e H  DL IntegrateA þþþ, 8,, <, Assumtions g<e i y þ  j "####### g { If the given assumtions eactly match the generated assumtions, then the latter don t show u in the outut; otherwise, Integrate roduces assumtions comlementary to ones given: n H  L m, 8,, <, Assumtions > D If i jrehml >, GHmL GHnL GHm+nL, H  L m n Ç y { References. O.I. Marichev, Handboo of integral transforms of higher transcendental functions: theory and algorithmic tables, Ellis Horwood, V.S. Adamchi, K.S. Kolbig, A definite integral of a roduct of two olylogarithms, SIAM J. Math. Anal. (988), V.S. Adamchi, O.I. Marichev, The algorithm for calculating integrals of hyergeometric tye functions and its realiation in REDUCE system, Proc. Conf. ISSAC 9, Toyo (99), .. V. Adamchi, The evaluation of integrals of Bessel functions via Gfunction identities, J. Com. Alied Math. 6(995), L.J. Slater, Generalied Hyergeometric Functions, Cambridge Univ. Press, Y.L. Lue, The Secial Functions and Their Aroimations, Vol., Academic Press, 969.
More Properties of Limits: Order of Operations
math 30 day 5: calculating its 6 More Proerties of Limits: Order of Oerations THEOREM 45 (Order of Oerations, Continued) Assume that!a f () L and that m and n are ositive integers Then 5 (Power)!a [ f
More informationClassical Fourier Series Introduction: Real Fourier Series
Math 344 May 1, 1 Classical Fourier Series Introduction: Real Fourier Series The orthogonality roerties of the sine and cosine functions make them good candidates as basis functions when orthogonal eansions
More informationUnited Arab Emirates University College of Sciences Department of Mathematical Sciences HOMEWORK 1 SOLUTION. Section 10.1 Vectors in the Plane
United Arab Emirates University College of Sciences Deartment of Mathematical Sciences HOMEWORK 1 SOLUTION Section 10.1 Vectors in the Plane Calculus II for Engineering MATH 110 SECTION 0 CRN 510 :00 :00
More informationComplex Conjugation and Polynomial Factorization
Comlex Conjugation and Polynomial Factorization Dave L. Renfro Summer 2004 Central Michigan University I. The Remainder Theorem Let P (x) be a olynomial with comlex coe cients 1 and r be a comlex number.
More informationAs we have seen, there is a close connection between Legendre symbols of the form
Gauss Sums As we have seen, there is a close connection between Legendre symbols of the form 3 and cube roots of unity. Secifically, if is a rimitive cube root of unity, then 2 ± i 3 and hence 2 2 3 In
More informationPRIME NUMBERS AND THE RIEMANN HYPOTHESIS
PRIME NUMBERS AND THE RIEMANN HYPOTHESIS CARL ERICKSON This minicourse has two main goals. The first is to carefully define the Riemann zeta function and exlain how it is connected with the rime numbers.
More informationNAVAL POSTGRADUATE SCHOOL THESIS
NAVAL POSTGRADUATE SCHOOL MONTEREY CALIFORNIA THESIS SYMMETRICAL RESIDUETOBINARY CONVERSION ALGORITHM PIPELINED FPGA IMPLEMENTATION AND TESTING LOGIC FOR USE IN HIGHSPEED FOLDING DIGITIZERS by Ross
More informationLecture 21 and 22: The Prime Number Theorem
Lecture and : The Prime Number Theorem (New lecture, not in Tet) The location of rime numbers is a central question in number theory. Around 88, Legendre offered eerimental evidence that the number π()
More informationSOME PROPERTIES OF EXTENSIONS OF SMALL DEGREE OVER Q. 1. Quadratic Extensions
SOME PROPERTIES OF EXTENSIONS OF SMALL DEGREE OVER Q TREVOR ARNOLD Abstract This aer demonstrates a few characteristics of finite extensions of small degree over the rational numbers Q It comrises attemts
More informationlecture 25: Gaussian quadrature: nodes, weights; examples; extensions
38 lecture 25: Gaussian quadrature: nodes, weights; examles; extensions 3.5 Comuting Gaussian quadrature nodes and weights When first aroaching Gaussian quadrature, the comlicated characterization of the
More informationPoint Location. Preprocess a planar, polygonal subdivision for point location queries. p = (18, 11)
Point Location Prerocess a lanar, olygonal subdivision for oint location ueries. = (18, 11) Inut is a subdivision S of comlexity n, say, number of edges. uild a data structure on S so that for a uery oint
More informationThe impact of metadata implementation on webpage visibility in search engine results (Part II) q
Information Processing and Management 41 (2005) 691 715 www.elsevier.com/locate/inforoman The imact of metadata imlementation on webage visibility in search engine results (Part II) q Jin Zhang *, Alexandra
More informationThe fast Fourier transform method for the valuation of European style options inthemoney (ITM), atthemoney (ATM) and outofthemoney (OTM)
Comutational and Alied Mathematics Journal 15; 1(1: 16 Published online January, 15 (htt://www.aascit.org/ournal/cam he fast Fourier transform method for the valuation of Euroean style otions inthemoney
More informationCBus Voltage Calculation
D E S I G N E R N O T E S CBus Voltage Calculation Designer note number: 3121256 Designer: Darren Snodgrass Contact Person: Darren Snodgrass Aroved: Date: Synosis: The guidelines used by installers
More informationENFORCING SAFETY PROPERTIES IN WEB APPLICATIONS USING PETRI NETS
ENFORCING SAFETY PROPERTIES IN WEB APPLICATIONS USING PETRI NETS Liviu Grigore Comuter Science Deartment University of Illinois at Chicago Chicago, IL, 60607 lgrigore@cs.uic.edu Ugo Buy Comuter Science
More informationConfidence Intervals for CaptureRecapture Data With Matching
Confidence Intervals for CatureRecature Data With Matching Executive summary Caturerecature data is often used to estimate oulations The classical alication for animal oulations is to take two samles
More informationDAYAHEAD ELECTRICITY PRICE FORECASTING BASED ON TIME SERIES MODELS: A COMPARISON
DAYAHEAD ELECTRICITY PRICE FORECASTING BASED ON TIME SERIES MODELS: A COMPARISON Rosario Esínola, Javier Contreras, Francisco J. Nogales and Antonio J. Conejo E.T.S. de Ingenieros Industriales, Universidad
More informationMultistage Human Resource Allocation for Software Development by Multiobjective Genetic Algorithm
The Oen Alied Mathematics Journal, 2008, 2, 9503 95 Oen Access Multistage Human Resource Allocation for Software Develoment by Multiobjective Genetic Algorithm Feng Wen a,b and ChiMing Lin*,a,c a Graduate
More informationMinimizing the Communication Cost for Continuous Skyline Maintenance
Minimizing the Communication Cost for Continuous Skyline Maintenance Zhenjie Zhang, Reynold Cheng, Dimitris Paadias, Anthony K.H. Tung School of Comuting National University of Singaore {zhenjie,atung}@com.nus.edu.sg
More informationTRANSCENDENTAL NUMBERS
TRANSCENDENTAL NUMBERS JEREMY BOOHER. Introduction The Greeks tried unsuccessfully to square the circle with a comass and straightedge. In the 9th century, Lindemann showed that this is imossible by demonstrating
More information2.1 Simple & Compound Propositions
2.1 Simle & Comound Proositions 1 2.1 Simle & Comound Proositions Proositional Logic can be used to analyse, simlify and establish the equivalence of statements. A knowledge of logic is essential to the
More information4 Perceptron Learning Rule
Percetron Learning Rule Objectives Objectives  Theory and Examles  Learning Rules  Percetron Architecture 3 SingleNeuron Percetron 5 MultileNeuron Percetron 8 Percetron Learning Rule 8 Test Problem
More informationA Modified Measure of Covert Network Performance
A Modified Measure of Covert Network Performance LYNNE L DOTY Marist College Deartment of Mathematics Poughkeesie, NY UNITED STATES lynnedoty@maristedu Abstract: In a covert network the need for secrecy
More informationOn the predictive content of the PPI on CPI inflation: the case of Mexico
On the redictive content of the PPI on inflation: the case of Mexico José Sidaoui, Carlos Caistrán, Daniel Chiquiar and Manuel RamosFrancia 1 1. Introduction It would be natural to exect that shocks to
More informationAn important observation in supply chain management, known as the bullwhip effect,
Quantifying the Bullwhi Effect in a Simle Suly Chain: The Imact of Forecasting, Lead Times, and Information Frank Chen Zvi Drezner Jennifer K. Ryan David SimchiLevi Decision Sciences Deartment, National
More informationIntroduction to NPCompleteness Written and copyright c by Jie Wang 1
91.502 Foundations of Comuter Science 1 Introduction to Written and coyright c by Jie Wang 1 We use timebounded (deterministic and nondeterministic) Turing machines to study comutational comlexity of
More informationUniversiteitUtrecht. Department. of Mathematics. Optimal a priori error bounds for the. RayleighRitz method
UniversiteitUtrecht * Deartment of Mathematics Otimal a riori error bounds for the RayleighRitz method by Gerard L.G. Sleijen, Jaser van den Eshof, and Paul Smit Prerint nr. 1160 Setember, 2000 OPTIMAL
More informationChapter 3. Special Techniques for Calculating Potentials. r ( r ' )dt ' ( ) 2
Chater 3. Secial Techniues for Calculating Potentials Given a stationary charge distribution r( r ) we can, in rincile, calculate the electric field: E ( r ) Ú Dˆ r Dr r ( r ' )dt ' 2 where Dr r 'r. This
More informationConcurrent Program Synthesis Based on Supervisory Control
010 American Control Conference Marriott Waterfront, Baltimore, MD, USA June 30July 0, 010 ThB07.5 Concurrent Program Synthesis Based on Suervisory Control Marian V. Iordache and Panos J. Antsaklis Abstract
More informationThe Online Freezetag Problem
The Online Freezetag Problem Mikael Hammar, Bengt J. Nilsson, and Mia Persson Atus Technologies AB, IDEON, SE3 70 Lund, Sweden mikael.hammar@atus.com School of Technology and Society, Malmö University,
More informationMATH 132: CALCULUS II SYLLABUS
MATH 32: CALCULUS II SYLLABUS Prerequisites: Successful completion of Math 3 (or its equivalent elsewhere). Math 27 is normally not a sufficient prerequisite for Math 32. Required Text: Calculus: Early
More informationAbout the Gamma Function
About the Gamma Function Notes for Honors Calculus II, Originally Prepared in Spring 995 Basic Facts about the Gamma Function The Gamma function is defined by the improper integral Γ) = The integral is
More informationTHE ROTATION INDEX OF A PLANE CURVE
THE ROTATION INDEX OF A PLANE CURVE AARON W. BROWN AND LORING W. TU The rotation index of a smooth closed lane curve C is the number of comlete rotations that a tangent vector to the curve makes as it
More informationImplementation of Statistic Process Control in a Painting Sector of a Automotive Manufacturer
4 th International Conference on Industrial Engineering and Industrial Management IV Congreso de Ingeniería de Organización Donostia an ebastián, etember 8 th  th Imlementation of tatistic Process Control
More informationAlgorithms for Constructing ZeroDivisor Graphs of Commutative Rings Joan Krone
Algorithms for Constructing ZeroDivisor Grahs of Commutative Rings Joan Krone Abstract The idea of associating a grah with the zerodivisors of a commutative ring was introduced in [3], where the author
More informationMultiperiod Portfolio Optimization with General Transaction Costs
Multieriod Portfolio Otimization with General Transaction Costs Victor DeMiguel Deartment of Management Science and Oerations, London Business School, London NW1 4SA, UK, avmiguel@london.edu Xiaoling Mei
More informationLectures on the Dirichlet Class Number Formula for Imaginary Quadratic Fields. Tom Weston
Lectures on the Dirichlet Class Number Formula for Imaginary Quadratic Fields Tom Weston Contents Introduction 4 Chater 1. Comlex lattices and infinite sums of Legendre symbols 5 1. Comlex lattices 5
More informationSQUARE GRID POINTS COVERAGED BY CONNECTED SOURCES WITH COVERAGE RADIUS OF ONE ON A TWODIMENSIONAL GRID
International Journal of Comuter Science & Information Technology (IJCSIT) Vol 6, No 4, August 014 SQUARE GRID POINTS COVERAGED BY CONNECTED SOURCES WITH COVERAGE RADIUS OF ONE ON A TWODIMENSIONAL GRID
More informationA MOST PROBABLE POINTBASED METHOD FOR RELIABILITY ANALYSIS, SENSITIVITY ANALYSIS AND DESIGN OPTIMIZATION
9 th ASCE Secialty Conference on Probabilistic Mechanics and Structural Reliability PMC2004 Abstract A MOST PROBABLE POINTBASED METHOD FOR RELIABILITY ANALYSIS, SENSITIVITY ANALYSIS AND DESIGN OPTIMIZATION
More informationAssignment 9; Due Friday, March 17
Assignment 9; Due Friday, March 17 24.4b: A icture of this set is shown below. Note that the set only contains oints on the lines; internal oints are missing. Below are choices for U and V. Notice that
More informationMath 5330 Spring Notes Prime Numbers
Math 5330 Sring 206 Notes Prime Numbers The study of rime numbers is as old as mathematics itself. This set of notes has a bunch of facts about rimes, or related to rimes. Much of this stuff is old dating
More informationThe Natural Logarithmic Function: Integration. Log Rule for Integration
6_5.qd // :58 PM Page CHAPTER 5 Logarithmic, Eponential, and Other Transcendental Functions Section 5. EXPLORATION Integrating Rational Functions Earl in Chapter, ou learned rules that allowed ou to integrate
More informationCABRS CELLULAR AUTOMATON BASED MRI BRAIN SEGMENTATION
XI Conference "Medical Informatics & Technologies"  2006 Rafał Henryk KARTASZYŃSKI *, Paweł MIKOŁAJCZAK ** MRI brain segmentation, CT tissue segmentation, Cellular Automaton, image rocessing, medical
More informationA Simple Model of Pricing, Markups and Market. Power Under Demand Fluctuations
A Simle Model of Pricing, Markus and Market Power Under Demand Fluctuations Stanley S. Reynolds Deartment of Economics; University of Arizona; Tucson, AZ 85721 Bart J. Wilson Economic Science Laboratory;
More informationCOST CALCULATION IN COMPLEX TRANSPORT SYSTEMS
OST ALULATION IN OMLEX TRANSORT SYSTEMS Zoltán BOKOR 1 Introduction Determining the real oeration and service costs is essential if transort systems are to be lanned and controlled effectively. ost information
More informationChapter 7 Outline Math 236 Spring 2001
Chapter 7 Outline Math 236 Spring 2001 Note 1: Be sure to read the Disclaimer on Chapter Outlines! I cannot be responsible for misfortunes that may happen to you if you do not. Note 2: Section 7.9 will
More informationMeasuring relative phase between two waveforms using an oscilloscope
Measuring relative hase between two waveforms using an oscilloscoe Overview There are a number of ways to measure the hase difference between two voltage waveforms using an oscilloscoe. This document covers
More informationMachine Learning with Operational Costs
Journal of Machine Learning Research 14 (2013) 19892028 Submitted 12/11; Revised 8/12; Published 7/13 Machine Learning with Oerational Costs Theja Tulabandhula Deartment of Electrical Engineering and
More informationRAJALAKSHMI ENGINEERING COLLEGE MA 2161 UNIT I  ORDINARY DIFFERENTIAL EQUATIONS PART A
RAJALAKSHMI ENGINEERING COLLEGE MA 26 UNIT I  ORDINARY DIFFERENTIAL EQUATIONS. Solve (D 2 + D 2)y = 0. 2. Solve (D 2 + 6D + 9)y = 0. PART A 3. Solve (D 4 + 4)x = 0 where D = d dt 4. Find Particular Integral:
More informationFourier Analysis of Stochastic Processes
Fourier Analysis of Stochastic Processes. Time series Given a discrete time rocess ( n ) nz, with n :! R or n :! C 8n Z, we de ne time series a realization of the rocess, that is to say a series (x n )
More informationFree Software Development. 2. Chemical Database Management
Leonardo Electronic Journal of Practices and echnologies ISSN 15831078 Issue 1, JulyDecember 2002. 6976 Free Software Develoment. 2. Chemical Database Management Monica ŞEFU 1, Mihaela Ligia UNGUREŞAN
More informationMath 201 Lecture 23: Power Series Method for Equations with Polynomial
Math 201 Lecture 23: Power Series Method for Equations with Polynomial Coefficients Mar. 07, 2012 Many examples here are taken from the textbook. The first number in () refers to the problem number in
More informationPOISSON PROCESSES. Chapter 2. 2.1 Introduction. 2.1.1 Arrival processes
Chater 2 POISSON PROCESSES 2.1 Introduction A Poisson rocess is a simle and widely used stochastic rocess for modeling the times at which arrivals enter a system. It is in many ways the continuoustime
More informationarxiv:0711.4143v1 [hepth] 26 Nov 2007
Exonentially localized solutions of the KleinGordon equation arxiv:711.4143v1 [heth] 26 Nov 27 M. V. Perel and I. V. Fialkovsky Deartment of Theoretical Physics, State University of SaintPetersburg,
More informationSTATISTICAL CHARACTERIZATION OF THE RAILROAD SATELLITE CHANNEL AT KUBAND
STATISTICAL CHARACTERIZATION OF THE RAILROAD SATELLITE CHANNEL AT KUBAND Giorgio Sciascia *, Sandro Scalise *, Harald Ernst * and Rodolfo Mura + * DLR (German Aerosace Centre) Institute for Communications
More informationApproximate Guarding of Monotone and Rectilinear Polygons
Aroximate Guarding of Monotone and Rectilinear Polygons Erik Krohn Bengt J. Nilsson Abstract We show that vertex guarding a monotone olygon is NPhard and construct a constant factor aroximation algorithm
More informationStochastic Derivation of an Integral Equation for Probability Generating Functions
Journal of Informatics and Mathematical Sciences Volume 5 (2013), Number 3,. 157 163 RGN Publications htt://www.rgnublications.com Stochastic Derivation of an Integral Equation for Probability Generating
More informationRisk in Revenue Management and Dynamic Pricing
OPERATIONS RESEARCH Vol. 56, No. 2, March Aril 2008,. 326 343 issn 0030364X eissn 15265463 08 5602 0326 informs doi 10.1287/ore.1070.0438 2008 INFORMS Risk in Revenue Management and Dynamic Pricing Yuri
More informationPythagorean Triples and Rational Points on the Unit Circle
Pythagorean Triles and Rational Points on the Unit Circle Solutions Below are samle solutions to the roblems osed. You may find that your solutions are different in form and you may have found atterns
More informationStatic and Dynamic Properties of Smallworld Connection Topologies Based on Transitstub Networks
Static and Dynamic Proerties of Smallworld Connection Toologies Based on Transitstub Networks Carlos Aguirre Fernando Corbacho Ramón Huerta Comuter Engineering Deartment, Universidad Autónoma de Madrid,
More informationA Brief Introduction to Design of Experiments
J. K. TELFORD D A Brief Introduction to Design of Exeriments Jacqueline K. Telford esign of exeriments is a series of tests in which uroseful changes are made to the inut variables of a system or rocess
More informationEffect Sizes Based on Means
CHAPTER 4 Effect Sizes Based on Means Introduction Raw (unstardized) mean difference D Stardized mean difference, d g Resonse ratios INTRODUCTION When the studies reort means stard deviations, the referred
More informationInduction Problems. Tom Davis November 7, 2005
Induction Problems Tom Davis tomrdavis@earthlin.net http://www.geometer.org/mathcircles November 7, 2005 All of the following problems should be proved by mathematical induction. The problems are not necessarily
More informationAgain, the limit must be the same whichever direction we approach from; but now there is an infinity of possible directions.
Chapter 4 Complex Analysis 4.1 Complex Differentiation Recall the definition of differentiation for a real function f(x): f f(x + δx) f(x) (x) = lim. δx 0 δx In this definition, it is important that the
More information1. [2.3] Techniques for Computing Limits Limits of Polynomials/Rational Functions/Continuous Functions. Indeterminate FormEliminate the Common Factor
Review for the BST MTHSC 8 Name : [] Techniques for Computing Limits Limits of Polynomials/Rational Functions/Continuous Functions Evaluate cos 6 Indeterminate FormEliminate the Common Factor Find the
More informationStorage Basics Architecting the Storage Supplemental Handout
Storage Basics Architecting the Storage Sulemental Handout INTRODUCTION With digital data growing at an exonential rate it has become a requirement for the modern business to store data and analyze it
More informationWeb Application Scalability: A ModelBased Approach
Coyright 24, Software Engineering Research and Performance Engineering Services. All rights reserved. Web Alication Scalability: A ModelBased Aroach Lloyd G. Williams, Ph.D. Software Engineering Research
More informationLearning Objectives for Math 165
Learning Objectives for Math 165 Chapter 2 Limits Section 2.1: Average Rate of Change. State the definition of average rate of change Describe what the rate of change does and does not tell us in a given
More informationCoin ToGa: A CoinTossing Game
Coin ToGa: A CoinTossing Game Osvaldo Marrero and Paul C Pasles Osvaldo Marrero OsvaldoMarrero@villanovaedu studied mathematics at the University of Miami, and biometry and statistics at Yale University
More informationChapter 2 Limits Functions and Sequences sequence sequence Example
Chapter Limits In the net few chapters we shall investigate several concepts from calculus, all of which are based on the notion of a limit. In the normal sequence of mathematics courses that students
More informationTOWARDS REALTIME METADATA FOR SENSORBASED NETWORKS AND GEOGRAPHIC DATABASES
TOWARDS REALTIME METADATA FOR SENSORBASED NETWORKS AND GEOGRAPHIC DATABASES C. Gutiérrez, S. Servigne, R. Laurini LIRIS, INSA Lyon, Bât. Blaise Pascal, 20 av. Albert Einstein 69621 Villeurbanne, France
More informationStability Improvements of Robot Control by Periodic Variation of the Gain Parameters
Proceedings of the th World Congress in Mechanism and Machine Science ril ~4, 4, ianin, China China Machinery Press, edited by ian Huang. 868 Stability Imrovements of Robot Control by Periodic Variation
More informationA Complete Operational Amplifier Noise Model: Analysis and Measurement of Correlation Coefficient
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATION, VOL. 47, NO. 3, MARCH 000 40 for ractical alication, oening the ath for widesread adotion of the clockgating technique
More informationLecture Notes: Discrete Mathematics
Lecture Notes: Discrete Mathematics GMU Math 125001 Sring 2007 Alexei V Samsonovich Any theoretical consideration, no matter how fundamental it is, inevitably relies on key rimary notions that are acceted
More informationFDA CFR PART 11 ELECTRONIC RECORDS, ELECTRONIC SIGNATURES
Document: MRM1004GAPCFR11 (0005) Page: 1 / 18 FDA CFR PART 11 ELECTRONIC RECORDS, ELECTRONIC SIGNATURES AUDIT TRAIL ECO # Version Change Descrition MATRIX 449 A Ga Analysis after adding controlled documents
More informationJoint Production and Financing Decisions: Modeling and Analysis
Joint Production and Financing Decisions: Modeling and Analysis Xiaodong Xu John R. Birge Deartment of Industrial Engineering and Management Sciences, Northwestern University, Evanston, Illinois 60208,
More informationThe MagnusDerek Game
The MagnusDerek Game Z. Nedev S. Muthukrishnan Abstract We introduce a new combinatorial game between two layers: Magnus and Derek. Initially, a token is laced at osition 0 on a round table with n ositions.
More informationBranchandPrice for Service Network Design with Asset Management Constraints
BranchandPrice for Servicee Network Design with Asset Management Constraints Jardar Andersen Roar Grønhaug Mariellee Christiansen Teodor Gabriel Crainic December 2007 CIRRELT200755 BranchandPrice
More informationHOMEWORK (due Fri, Nov 19): Chapter 12: #62, 83, 101
Today: Section 2.2, Lesson 3: What can go wrong with hyothesis testing Section 2.4: Hyothesis tests for difference in two roortions ANNOUNCEMENTS: No discussion today. Check your grades on eee and notify
More informationSome Notes on Taylor Polynomials and Taylor Series
Some Notes on Taylor Polynomials and Taylor Series Mark MacLean October 3, 27 UBC s courses MATH /8 and MATH introduce students to the ideas of Taylor polynomials and Taylor series in a fairly limited
More informationComparing Dissimilarity Measures for Symbolic Data Analysis
Comaring Dissimilarity Measures for Symbolic Data Analysis Donato MALERBA, Floriana ESPOSITO, Vincenzo GIOVIALE and Valentina TAMMA Diartimento di Informatica, University of Bari Via Orabona 4 76 Bari,
More informationAn Analysis of Reliable Classifiers through ROC Isometrics
An Analysis of Reliable Classifiers through ROC Isometrics Stijn Vanderlooy s.vanderlooy@cs.unimaas.nl Ida G. SrinkhuizenKuyer kuyer@cs.unimaas.nl Evgueni N. Smirnov smirnov@cs.unimaas.nl MICCIKAT, Universiteit
More informationAutomatic Search for Correlated Alarms
Automatic Search for Correlated Alarms KlausDieter Tuchs, Peter Tondl, Markus Radimirsch, Klaus Jobmann Institut für Allgemeine Nachrichtentechnik, Universität Hannover Aelstraße 9a, 0167 Hanover, Germany
More informationDiscrete Math I Practice Problems for Exam I
Discrete Math I Practice Problems for Exam I The ucoming exam on Thursday, January 12 will cover the material in Sections 1 through 6 of Chater 1. There may also be one question from Section 7. If there
More informationHYPOTHESIS TESTING FOR THE PROCESS CAPABILITY RATIO. A thesis presented to. the faculty of
HYPOTHESIS TESTING FOR THE PROESS APABILITY RATIO A thesis resented to the faculty of the Russ ollege of Engineering and Technology of Ohio University In artial fulfillment of the requirement for the degree
More informationBeyond the F Test: Effect Size Confidence Intervals and Tests of Close Fit in the Analysis of Variance and Contrast Analysis
Psychological Methods 004, Vol. 9, No., 164 18 Coyright 004 by the American Psychological Association 108989X/04/$1.00 DOI: 10.1037/108989X.9..164 Beyond the F Test: Effect Size Confidence Intervals
More informationThe risk of using the Q heterogeneity estimator for software engineering experiments
Dieste, O., Fernández, E., GarcíaMartínez, R., Juristo, N. 11. The risk of using the Q heterogeneity estimator for software engineering exeriments. The risk of using the Q heterogeneity estimator for
More informationCHAPTER 13. Definite Integrals. Since integration can be used in a practical sense in many applications it is often
7 CHAPTER Definite Integrals Since integration can be used in a practical sense in many applications it is often useful to have integrals evaluated for different values of the variable of integration.
More informationMathematics 31 Precalculus and Limits
Mathematics 31 Precalculus and Limits Overview After completing this section, students will be epected to have acquired reliability and fluency in the algebraic skills of factoring, operations with radicals
More informationLocal Connectivity Tests to Identify Wormholes in Wireless Networks
Local Connectivity Tests to Identify Wormholes in Wireless Networks Xiaomeng Ban Comuter Science Stony Brook University xban@cs.sunysb.edu Rik Sarkar Comuter Science Freie Universität Berlin sarkar@inf.fuberlin.de
More informationX How to Schedule a Cascade in an Arbitrary Graph
X How to Schedule a Cascade in an Arbitrary Grah Flavio Chierichetti, Cornell University Jon Kleinberg, Cornell University Alessandro Panconesi, Saienza University When individuals in a social network
More informationPrecalculus Prerequisites a.k.a. Chapter 0. August 16, 2013
Precalculus Prerequisites a.k.a. Chater 0 by Carl Stitz, Ph.D. Lakeland Community College Jeff Zeager, Ph.D. Lorain County Community College August 6, 0 Table of Contents 0 Prerequisites 0. Basic Set
More informationPressure Drop in Air Piping Systems Series of Technical White Papers from Ohio Medical Corporation
Pressure Dro in Air Piing Systems Series of Technical White Paers from Ohio Medical Cororation Ohio Medical Cororation Lakeside Drive Gurnee, IL 600 Phone: (800) 4480770 Fax: (847) 855604 info@ohiomedical.com
More informationA Novel Architecture Style: Diffused Cloud for Virtual Computing Lab
A Novel Architecture Style: Diffused Cloud for Virtual Comuting Lab Deven N. Shah Professor Terna College of Engg. & Technology Nerul, Mumbai Suhada Bhingarar Assistant Professor MIT College of Engg. Paud
More informationMean shiftbased clustering
Pattern Recognition (7) www.elsevier.com/locate/r Mean shiftbased clustering KuoLung Wu a, MiinShen Yang b, a Deartment of Information Management, Kun Shan University of Technology, YungKang, Tainan
More informationOn the (in)effectiveness of Probabilistic Marking for IP Traceback under DDoS Attacks
On the (in)effectiveness of Probabilistic Maring for IP Tracebac under DDoS Attacs Vamsi Paruchuri, Aran Durresi 2, and Ra Jain 3 University of Central Aransas, 2 Louisiana State University, 3 Washington
More informationFailure Behavior Analysis for Reliable Distributed Embedded Systems
Failure Behavior Analysis for Reliable Distributed Embedded Systems Mario Tra, Bernd Schürmann, Torsten Tetteroo {tra schuerma tetteroo}@informatik.unikl.de Deartment of Comuter Science, University of
More informationTHE WELFARE IMPLICATIONS OF COSTLY MONITORING IN THE CREDIT MARKET: A NOTE
The Economic Journal, 110 (Aril ), 576±580.. Published by Blackwell Publishers, 108 Cowley Road, Oxford OX4 1JF, UK and 50 Main Street, Malden, MA 02148, USA. THE WELFARE IMPLICATIONS OF COSTLY MONITORING
More informationCourse Notes for Math 162: Mathematical Statistics Approximation Methods in Statistics
Course Notes for Math 16: Mathematical Statistics Approximation Methods in Statistics Adam Merberg and Steven J. Miller August 18, 6 Abstract We introduce some of the approximation methods commonly used
More informationPredicate Encryption Supporting Disjunctions, Polynomial Equations, and Inner Products
Predicate Encrytion Suorting Disjunctions, Polynomial Equations, and Inner Products Jonathan Katz Amit Sahai Brent Waters Abstract Predicate encrytion is a new aradigm for ublickey encrytion that generalizes
More information