Theorem on the differentiation of a composite function with a vector argument
|
|
- Laurence Brown
- 5 years ago
- Views:
Transcription
1 Poceedings of the Estonian Academy of Sciences doi:.376/poc..3. Avaiae onine at Theoem on the diffeentiation of a composite function with a vecto agument Vadim Kapain and Üe Kotta Institute of Cyenetics at Tainn Univesity of Technoogy Akadeemia tee 68 Tainn Estonia; kotta@cc.ioc.ee Received 9 Noveme 9 evised 5 Januay accepted 8 Januay Astact. The pape povides a theoem on the diffeentiation of a composite function with a vecto agument. The theoem shows how the patia deivative of the tota deivative of the composite function can e expessed though the tota deivative of the patia deivative of the composite function. The poof of the theoem is ased on Mishkov s fomua which is the geneaization of the we-known Faà di Buno s fomua fo a composite function with a vecto agument. Key wods: diffeentia cacuus patia deivative tota deivative composite function.. INTRODUCTION The theoem poved in this pape was equied as an intemediate esut in soving the poem of the tansfomation of the noninea conto system descied y state equations into the oseve fom and finding the necessay conditions fo the possiiity of such tansfomation []. The deduction of the necessay conditions invoves fequent appication of the diffeentiation of the composite functions with espect to time agument and taking the patia deivatives of the diffeentiated composite function with espect to one of the vaiaes o its deivatives. The goa of this pape is to pesent and pove a fomua commutation ue which aows changing the ode of taking the tota highe-ode deivatives of the composite function and thei patia deivatives with espect to one of the vaiaes o its deivative. Since this esut may e usefu in the soution of othe noninea conto poems we popose it as a sepaate contiution. Fo exampe poay the main esut povided in the pape can e appied fo oseve design in [4]. The main too fo poving the theoem commutation ue is Mishkov s theoem [3] which povides the expicit fomua fo the nth deivative of a composite function with a vecto agument. Mishkov s fomua is a staightfowad geneaization of the we-known Faà di Buno s fomua [] which gives an expicit equation fo the nth-ode deivative of the composite function with a scaa agument. Coesponding autho vkapain@cc.ioc.ee
2 96 Poceedings of the Estonian Academy of Sciences MAIN RESULT The foowing theoem shows how the patia deivative of the tota deivative of the composite function can e expessed though the tota deivative of the patia deivative of this function. The composite function with the vecto agument with an aitay nume of components is consideed. Theoem. Assume that f ξ tξ t...ξ t is a composite function fo which deivatives up to ode a + ae defined; then f ξ tξ t...ξ t a+ f ξ a = Ca+ ξ tξ t...ξ t t ξ t whee =... Ca+ is the inomia coefficient and a ae nonnegative integes. Poof. In the poof we omit the vaiae t of ξ i t i.e. use instead of ξ i t a shote notation ξ i which aows the uky fomuas to e witten in a moe compact fom. Accoding to Mishkov s fomua [3] the a+th deivative of the composite function with a vecto agument can e computed y the fomua f ξ ξ...ξ a+ = a+ a+ i! k i a +! a+ q i j! k f ξ p ξ p ξ p a+ j= ξ i qi j j whee the espective sums ae taken ove a nonnegative intege soutions of the Diophantine equations as foows: k + k + + a + k a+ = a + q i + q i + + q i = k i 3 i fo i =...a + and p j and k on the ight-hand side of satisfy the eations p j = q j + q j + + q a+ j j =... k = p + p + + p = k + k + + k a+. 4 In taking the patia deivative of sum with espect to ξ a ony addends of sum with q a wi matte. Denote y h and g the pats of sum coesponding to q a and q a = espectivey; then f ξ ξ...ξ a+ = h + g. 5 Note that it is possie to state that h equas the expession in the ight-hand side of whee in addition to the estictions expessed y 3 and 4 the condition q a has to e satisfied. Note aso that if q a then k a. We pove the fomua sepaatey fo the cases a > and a. Fist conside the case when a >. Since k a and q a in ode to satisfy and 3 the foowing must hod k a = k i = < i a + i a q a = q a j = j =... j q i j = < i a + i a j =... As a esut unde the condition q a one can ewite as foows: and in 3 now i =... k + k + + k = 7 6
3 V. Kapain and Ü. Kotta: Diffeentiation of a composite function 97 Using 6 and changing the notations taking p j = p j fo j =... j p = p and k = k equations 4 may e ewitten as p j = q j + q j + + q j j =... k = p + p + + p = k + k + + k. 8 Note aso that unde conditions 6 a+ i! k i = a! i! k i a+ q i j! = q i j! and a+ ξ i qi j j = ξ a ξ i qi j j. Taking into account the aove equations and the fact that the patia deivative of g in 5 with espect to equas we otain in new vaiaes p j and k: ξ a f ξ ξ...ξ a+ ξ a = a! a +! i! k i q i j! k+ f ξ p ξ p ξ p + ξ p + + ξ p ξ i qi j j. 9 Note that in 9 a the patia deivatives with espect to ξ j ae of ode p j except with espect to ξ when the ode of the patia deivative is p +. In ode to unify the odes denote f := f ξ. We aso mutipy the ight-hand side of equation 9 y!! to otain f ξ ξ...ξ a+ ξ a = C a+ i! k i! q i j! k f ξ p ξ p ξ p ξ i qi j j. It is easy to oseve now that accoding to Mishkov s fomua the sum on the ight-hand side of togethe with the conditions 3 fo i =... 7 and 8 is the th-ode tota deivative of the function f. Consequenty f ξ ξ...ξ a+ ξ a = Ca+ f = Ca+ ξ f ξ ξ...ξ. Second conside the case a. Since k a in ode to satisfy and 3 the foowing must hod: k i = < i a + q i j = < i a + j =...
4 98 Poceedings of the Estonian Academy of Sciences Theefoe it is possie to ewite condition as k + + a k a + ak a + a + k a+ + + k = 3 and in 3 now i =... Again in ode to unify the notation in 3 one can take k i = k i fo i =... i a and k a = k a. This aows 3 to e ewitten as foows: k + k + + k = 4 and 3 as i a q i + q i + + q i = k i i =... i a q a + q a + + q a = k a +. Since q a we can denote q a := q a and the emaining q s as q i j := q i j. Theey 5 can e ewitten in unified notation as i 5 q i + q i + + q i = k i 6 fo i =... Changing notations taking p j = p j fo j =... j p = p and k = k equations 4 may e ewitten as p j = q j + q j + + q j j =... k = p + p + + p = k + k + + k. 7 Taking into account and using vaiaes k i and q i j we have a+ a+ i! k i = a! = ξ ξ i qi j j q i! k i a+ ξ a qa q i j! = q a + ξ a qa + q i j! ξ a+ qa+ ξ q j= j 8 ξ i qi j j. Futhemoe on the asis of 8 and the fact that the patia deivative of g in 5 with espect to ξ a equas we otain in new vaiaes p j and k f ξ ξ...ξ a+ ξ a = a! a +! i! k i q i j! k+ f ξ p ξ p ξ p + ξ p + + ξ p ξ i qi j j.
5 V. Kapain and Ü. Kotta: Diffeentiation of a composite function 99 Like in case a > we denote f = f ξ and mutipy the ight-hand side of the equaity given aove y! otain f ξ ξ...ξ a+ ξ a = C a+ i! k i! q i j! k f ξ p ξ p ξ p! to ξ i qi j j. 9 Again it is not difficut to oseve that accoding to Mishkov s fomua the sum on the ight-hand side of equation 9 togethe with the conditions 4 6 and 7 is the th-ode tota deivative of the function f. Consequenty hods again and this competes the poof. Some usefu cooaies of the theoem ae given eow. Cooay. Unde the assumptions of Theoem f ξ tξ t...ξ t m+n f ξ tξ t...ξ t m n = ξ t ξ t whee m and n ae nonnegative integes. Cooay. Unde the assumptions of Theoem f ξ tξ t...ξ t n f ξ tξ t...ξ t n = ξ t ξ t whee n is a nonnegative intege. 3. EXAMPLE The exampe in this section iustates the statement of Theoem. Conside the composite function f xtyt and assume that we need to take the patia deivative with espect to ÿt of the 3d-ode tota deivative of the function. Diect computations yied f xtyt 3 ÿt = 3 f xtyt yt ẏt + 3 f xtyt ẋt. xt yt On the othe hand taking the patia deivative of f xtyt with espect to yt and the tota deivative of the otained esut one gets f xtyt = f xtyt yt yt ẏt + f xtyt ẋt. xt yt Mutipying oth sides of the aove equaity y C3 we have f xtyt C3 = 3 f xtyt yt yt ẏt + 3 f xtyt ẋt. xt yt It is not difficut to check that f xtyt 3 f xtyt = C3. ÿt yt
6 Poceedings of the Estonian Academy of Sciences CONCLUSIONS The pape shows how to commute the opeations of taking highe-ode tota and patia deivatives of composite functions with vecto aguments. The fomua povided in the pape may e appicae not ony in diffeentia cacuus. As aeady mentioned in the intoduction the theoem was a usefu too in deiving sovaiity conditions of a cetain poem in noninea conto theoy. With high poaiity it may e usefu in deaing with othe noninea conto poems whee the deivatives of the composite functions with a vecto agument often show up. ACKNOWLEDGEMENT This wok was patiay suppoted y the Estonian Science Foundation though gant No. 69. REFERENCES. Johnson W. P. The cuious histoy of Faa di Buno s Fomua. Ame. Math. Monthy Kapain V. and Kotta Ü. Necessay and sufficient conditions in tems of diffeentia-foms fo ineaization of the state equations up to input-output injections. UKACC Intenationa Confeence on CONTROL 7 Septeme Coventy UK sumitted fo puication. 3. Mishkov R. L. Geneaization of the fomua of Faa di Buno fo a composite function with a vecto agument. Intenat. J. Math. Math. Sci Mishkov R. L. Noninea oseve design y educed geneaized oseve canonica fom. Intenat. J. Conto Teoeem vektoagumendiga iitfunktsiooni difeentseeimisest Vadim Kapain ja Üe Kotta On tõestatud teoeem vektoagumendiga iitfunktsiooni difeentseeimise kohta. Teoeemis esitatud vaem näita kuidas iitfunktsiooni täistuetise osatuetist saa väjendada tema osatuetise täistuetise kaudu. Teoeemi tõestus põhine Mishkovi vaemi mis omakoda kujuta endast tuntud Faà di Buno vaemi üdistust vektoagumendiga iitfunktsiooni jaoks. Näide iusteei teoeetiist tuemust.
Objectives. We will also get to know about the wavefunction and its use in developing the concept of the structure of atoms.
Modue "Atomic physics and atomic stuctue" Lectue 7 Quantum Mechanica teatment of One-eecton atoms Page 1 Objectives In this ectue, we wi appy the Schodinge Equation to the simpe system Hydogen and compae
More informationOn the differentiation of a composite function with a generalized vector argument on homogeneous time scales
Proceedings of the Estonian Academy of Sciences 207 66 3 309 322 https://doi.org/0.376/proc.207.3.04 Avaiae onine at www.eap.ee/proceedings On the differentiation of a composite function with a generaized
More informationJournal of Inequalities in Pure and Applied Mathematics
Jounal of Inequalities in Pue and Applied Mathematics COEFFICIENT INEQUALITY FOR A FUNCTION WHOSE DERIVATIVE HAS A POSITIVE REAL PART S. ABRAMOVICH, M. KLARIČIĆ BAKULA AND S. BANIĆ Depatment of Mathematics
More informationPressure in the Average-Atom Model
Pessue in the Aveage-Atom Moe W.. Johnson Depatment of Physics, 225 Nieuwan Science Ha Note Dame Univesity, Note Dame, IN 46556 Febuay 28, 2002 Abstact The (we-known) quantum mechanica expession fo the
More informationOn the Comparison of Stability Analysis with Phase Portrait for a Discrete Prey-Predator System
Intenational Jounal of Applied Engineeing Reseach ISSN 0973-4562 Volume 12, Nume 24 (2017) pp. 15273-15277 Reseach India Pulications. http://www.ipulication.com On the Compaison of Staility Analysis with
More informationPROBLEM SET #1 SOLUTIONS by Robert A. DiStasio Jr.
POBLM S # SOLUIONS by obet A. DiStasio J. Q. he Bon-Oppenheime appoximation is the standad way of appoximating the gound state of a molecula system. Wite down the conditions that detemine the tonic and
More informationOn a quantity that is analogous to potential and a theorem that relates to it
Su une quantité analogue au potential et su un théoème y elatif C R Acad Sci 7 (87) 34-39 On a quantity that is analogous to potential and a theoem that elates to it By R CLAUSIUS Tanslated by D H Delphenich
More informationSeidel s Trapezoidal Partitioning Algorithm
CS68: Geometic Agoithms Handout #6 Design and Anaysis Oigina Handout #6 Stanfod Univesity Tuesday, 5 Febuay 99 Oigina Lectue #7: 30 Januay 99 Topics: Seide s Tapezoida Patitioning Agoithm Scibe: Michae
More informationHomework 1 Solutions CSE 101 Summer 2017
Homewok 1 Soutions CSE 101 Summe 2017 1 Waming Up 1.1 Pobem and Pobem Instance Find the smaest numbe in an aay of n integes a 1, a 2,..., a n. What is the input? What is the output? Is this a pobem o a
More informationGeneralized net model of the process of ordering of university subjects
eventh Int. okshop on Gs, ofia, 4- Juy 006, -9 Geneaized net mode of the pocess of odeing of univesity subjects A. hannon, E. otiova, K. Atanassov 3, M. Kawczak 4, P. Meo-Pinto,. otiov, T. Kim 6 KvB Institute
More informationϕ direction (others needed boundary conditions And we have following boundary conditions in
Exact Analytical 3-D Steady-State Solution fo Cylindical Fin ANITA PILIKSERE, ANDRIS BUIKIS Institute of Mathematics and Compute Science Univesity of Latvia Raina ulv 9, Riga, LV459 LATVIA saulite@latnetlv,
More information2000 Conference on Decision and Control Sydney, Australia
Confeence on Decision and Conto Sydney, Austaia A Geometic Pespective on Bifucation Conto Yong Wang yongwang@cds.catech.e Richad. uay muay@cds.catech.e Conto and Dynamica Systems Caifonia Institute of
More informationWITH DEPENDENCE. Bo Lindqvist. Abstract: Klotz (1972,1973) develops a model for Bernoulli trials. p1,...,pr and a dependence parameter
STATISTICAL RESEARCH REPORT Institute of Mathematics Univesity of Oso June 1978 A MODEL FOR MULTINOMIAL TRIALS WITH DEPENDENCE by Bo Lindqvist Abstact: Kotz (1972,1973) deveops a mode fo Benoui tias with
More informationUnobserved Correlation in Ascending Auctions: Example And Extensions
Unobseved Coelation in Ascending Auctions: Example And Extensions Daniel Quint Univesity of Wisconsin Novembe 2009 Intoduction In pivate-value ascending auctions, the winning bidde s willingness to pay
More informationKOEBE DOMAINS FOR THE CLASSES OF FUNCTIONS WITH RANGES INCLUDED IN GIVEN SETS
Jounal of Applied Analysis Vol. 14, No. 1 2008), pp. 43 52 KOEBE DOMAINS FOR THE CLASSES OF FUNCTIONS WITH RANGES INCLUDED IN GIVEN SETS L. KOCZAN and P. ZAPRAWA Received Mach 12, 2007 and, in evised fom,
More informationMATH 417 Homework 3 Instructor: D. Cabrera Due June 30. sin θ v x = v r cos θ v θ r. (b) Then use the Cauchy-Riemann equations in polar coordinates
MATH 417 Homewok 3 Instucto: D. Cabea Due June 30 1. Let a function f(z) = u + iv be diffeentiable at z 0. (a) Use the Chain Rule and the fomulas x = cosθ and y = to show that u x = u cosθ u θ, v x = v
More informationUsing Laplace Transform to Evaluate Improper Integrals Chii-Huei Yu
Available at https://edupediapublicationsog/jounals Volume 3 Issue 4 Febuay 216 Using Laplace Tansfom to Evaluate Impope Integals Chii-Huei Yu Depatment of Infomation Technology, Nan Jeon Univesity of
More informationMechanics Physics 151
Mechanics Physics 5 Lectue 5 Centa Foce Pobem (Chapte 3) What We Did Last Time Intoduced Hamiton s Pincipe Action intega is stationay fo the actua path Deived Lagange s Equations Used cacuus of vaiation
More informationDepartment of Applied Mathematics, National Chiao Tung University, Hsinchu 30050, Taiwan.
Gaphs and Cobinatoics (1999) 15 : 95±301 Gaphs and Cobinatoics ( Spinge-Veag 1999 T-Cooings and T-Edge Spans of Gaphs* Shin-Jie Hu, Su-Tzu Juan, and Gead J Chang Depatent of Appied Matheatics, Nationa
More informationMechanics Physics 151
Mechanics Physics 5 Lectue 5 Centa Foce Pobem (Chapte 3) What We Did Last Time Intoduced Hamiton s Pincipe Action intega is stationay fo the actua path Deived Lagange s Equations Used cacuus of vaiation
More informationNew problems in universal algebraic geometry illustrated by boolean equations
New poblems in univesal algebaic geomety illustated by boolean equations axiv:1611.00152v2 [math.ra] 25 Nov 2016 Atem N. Shevlyakov Novembe 28, 2016 Abstact We discuss new poblems in univesal algebaic
More informationA generalization of the Bernstein polynomials
A genealization of the Benstein polynomials Halil Ouç and Geoge M Phillips Mathematical Institute, Univesity of St Andews, Noth Haugh, St Andews, Fife KY16 9SS, Scotland Dedicated to Philip J Davis This
More informationON INDEPENDENT SETS IN PURELY ATOMIC PROBABILITY SPACES WITH GEOMETRIC DISTRIBUTION. 1. Introduction. 1 r r. r k for every set E A, E \ {0},
ON INDEPENDENT SETS IN PURELY ATOMIC PROBABILITY SPACES WITH GEOMETRIC DISTRIBUTION E. J. IONASCU and A. A. STANCU Abstact. We ae inteested in constucting concete independent events in puely atomic pobability
More informationJackson 3.3 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell
Jackson 3.3 Homewok Pobem Soution D. Chistophe S. Baid Univesity of Massachusetts Lowe POBLEM: A thin, fat, conducting, cicua disc of adius is ocated in the x-y pane with its cente at the oigin, and is
More informationPerturbation to Symmetries and Adiabatic Invariants of Nonholonomic Dynamical System of Relative Motion
Commun. Theo. Phys. Beijing, China) 43 25) pp. 577 581 c Intenational Academic Publishes Vol. 43, No. 4, Apil 15, 25 Petubation to Symmeties and Adiabatic Invaiants of Nonholonomic Dynamical System of
More informationResearch Article On Alzer and Qiu s Conjecture for Complete Elliptic Integral and Inverse Hyperbolic Tangent Function
Abstact and Applied Analysis Volume 011, Aticle ID 697547, 7 pages doi:10.1155/011/697547 Reseach Aticle On Alze and Qiu s Conjectue fo Complete Elliptic Integal and Invese Hypebolic Tangent Function Yu-Ming
More informationA NEW VARIABLE STIFFNESS SPRING USING A PRESTRESSED MECHANISM
Poceedings of the ASME 2010 Intenational Design Engineeing Technical Confeences & Computes and Infomation in Engineeing Confeence IDETC/CIE 2010 August 15-18, 2010, Monteal, Quebec, Canada DETC2010-28496
More informationGROWTH ESTIMATES THROUGH SCALING FOR QUASILINEAR PARTIAL DIFFERENTIAL EQUATIONS
Annales Academiæ Scientiaum Fennicæ Mathematica Volumen 32, 2007, 595 599 GROWTH ESTIMATES THROUGH SCALING FOR QUASILINEAR PARTIAL DIFFERENTIAL EQUATIONS Teo Kilpeläinen, Henik Shahgholian and Xiao Zhong
More informationCompactly Supported Radial Basis Functions
Chapte 4 Compactly Suppoted Radial Basis Functions As we saw ealie, compactly suppoted functions Φ that ae tuly stictly conditionally positive definite of ode m > do not exist The compact suppot automatically
More informationarxiv: v1 [math.co] 4 May 2017
On The Numbe Of Unlabeled Bipatite Gaphs Abdullah Atmaca and A Yavuz Ouç axiv:7050800v [mathco] 4 May 207 Abstact This pape solves a poblem that was stated by M A Haison in 973 [] This poblem, that has
More informationA Sardinas-Patterson Characterization Theorem for SE-codes
A Sadinas-Patteson Chaacteization Theoem fo SE-codes Ionuţ Popa,Bogdan Paşaniuc Facuty of Compute Science, A.I.Cuza Univesity of Iaşi, 6600 Iaşi, Romania May 0, 2004 Abstact The aim of this pape is to
More informationOn decompositions of complete multipartite graphs into the union of two even cycles
On decompositions of complete multipatite gaphs into the union of two even cycles A. Su, J. Buchanan, R. C. Bunge, S. I. El-Zanati, E. Pelttai, G. Rasmuson, E. Spaks, S. Tagais Depatment of Mathematics
More informationON THE INVERSE SIGNED TOTAL DOMINATION NUMBER IN GRAPHS. D.A. Mojdeh and B. Samadi
Opuscula Math. 37, no. 3 (017), 447 456 http://dx.doi.og/10.7494/opmath.017.37.3.447 Opuscula Mathematica ON THE INVERSE SIGNED TOTAL DOMINATION NUMBER IN GRAPHS D.A. Mojdeh and B. Samadi Communicated
More informationOn the integration of the equations of hydrodynamics
Uebe die Integation de hydodynamischen Gleichungen J f eine u angew Math 56 (859) -0 On the integation of the equations of hydodynamics (By A Clebsch at Calsuhe) Tanslated by D H Delphenich In a pevious
More informationConservative Averaging Method and its Application for One Heat Conduction Problem
Poceedings of the 4th WSEAS Int. Conf. on HEAT TRANSFER THERMAL ENGINEERING and ENVIRONMENT Elounda Geece August - 6 (pp6-) Consevative Aveaging Method and its Application fo One Heat Conduction Poblem
More informationOn the Poisson Approximation to the Negative Hypergeometric Distribution
BULLETIN of the Malaysian Mathematical Sciences Society http://mathusmmy/bulletin Bull Malays Math Sci Soc (2) 34(2) (2011), 331 336 On the Poisson Appoximation to the Negative Hypegeometic Distibution
More informationCOORDINATE TRANSFORMATIONS - THE JACOBIAN DETERMINANT
COORDINATE TRANSFORMATIONS - THE JACOBIAN DETERMINANT Link to: phsicspages home page. To leave a comment o epot an eo, please use the auilia blog. Refeence: d Inveno, Ra, Intoducing Einstein s Relativit
More informationDonnishJournals
DonnishJounals 041-1189 Donnish Jounal of Educational Reseach and Reviews. Vol 1(1) pp. 01-017 Novembe, 014. http:///dje Copyight 014 Donnish Jounals Oiginal Reseach Pape Vecto Analysis Using MAXIMA Savaş
More informationPositivity results for cluster algebras from surfaces
Positivity esuts fo custe agebas fom sufaces Gegg Musike (MSRI/MIT) (Joint wok with Raf Schiffe (Univesity of Connecticut) and Lauen Wiiams (Univesity of Caifonia, Bekeey)) AMS 009 Easten Sectiona Octobe,
More informationA NOTE ON ROTATIONS AND INTERVAL EXCHANGE TRANSFORMATIONS ON 3-INTERVALS KARMA DAJANI
A NOTE ON ROTATIONS AND INTERVAL EXCHANGE TRANSFORMATIONS ON 3-INTERVALS KARMA DAJANI Abstact. Wepove the conjectue that an inteval exchange tansfomation on 3-intevals with coesponding pemutation (1; 2;
More informationEfficiency Loss in a Network Resource Allocation Game: The Case of Elastic Supply
Efficiency Loss in a Netwok Resouce Allocation Game: The Case of Elastic Supply axiv:cs/0506054v1 [cs.gt] 14 Jun 2005 Ramesh Johai (johai@stanfod.edu) Shie Manno (shie@mit.edu) John N. Tsitsiklis (jnt@mit.edu)
More informationThe Solutions of the Classical Relativistic Two-Body Equation
T. J. of Physics (998), 07 4. c TÜBİTAK The Soutions of the Cassica Reativistic Two-Body Equation Coşkun ÖNEM Eciyes Univesity, Physics Depatment, 38039, Kaysei - TURKEY Received 3.08.996 Abstact With
More informationDeflection of light due to rotating mass a comparison among the results of different approaches
Jounal of Physics: Confeence Seies OPEN ACCESS Deflection of light due to otating mass a compaison among the esults of diffeent appoaches Recent citations - Gavitational Theoies nea the Galactic Cente
More informationFunctions Defined on Fuzzy Real Numbers According to Zadeh s Extension
Intenational Mathematical Foum, 3, 2008, no. 16, 763-776 Functions Defined on Fuzzy Real Numbes Accoding to Zadeh s Extension Oma A. AbuAaqob, Nabil T. Shawagfeh and Oma A. AbuGhneim 1 Mathematics Depatment,
More informationA THREE CRITICAL POINTS THEOREM AND ITS APPLICATIONS TO THE ORDINARY DIRICHLET PROBLEM
A THREE CRITICAL POINTS THEOREM AND ITS APPLICATIONS TO THE ORDINARY DIRICHLET PROBLEM DIEGO AVERNA AND GABRIELE BONANNO Abstact. The aim of this pape is twofold. On one hand we establish a thee citical
More informationConstruction and Analysis of Boolean Functions of 2t + 1 Variables with Maximum Algebraic Immunity
Constuction and Analysis of Boolean Functions of 2t + 1 Vaiables with Maximum Algebaic Immunity Na Li and Wen-Feng Qi Depatment of Applied Mathematics, Zhengzhou Infomation Engineeing Univesity, Zhengzhou,
More informationOn absence of solutions of a semi-linear elliptic equation with biharmonic operator in the exterior of a ball
Tansactions of NAS of Azebaijan, Issue Mathematics, 36, 63-69 016. Seies of Physical-Technical and Mathematical Sciences. On absence of solutions of a semi-linea elliptic euation with bihamonic opeato
More informationMerging to ordered sequences. Efficient (Parallel) Sorting. Merging (cont.)
Efficient (Paae) Soting One of the most fequent opeations pefomed by computes is oganising (soting) data The access to soted data is moe convenient/faste Thee is a constant need fo good soting agoithms
More informationSPECTRAL SEQUENCES. im(er
SPECTRAL SEQUENCES MATTHEW GREENBERG. Intoduction Definition. Let a. An a-th stage spectal (cohomological) sequence consists of the following data: bigaded objects E = p,q Z Ep,q, a diffeentials d : E
More informationQuestion 1: The dipole
Septembe, 08 Conell Univesity, Depatment of Physics PHYS 337, Advance E&M, HW #, due: 9/5/08, :5 AM Question : The dipole Conside a system as discussed in class and shown in Fig.. in Heald & Maion.. Wite
More informationChapter 2: Basic Physics and Math Supplements
Chapte 2: Basic Physics and Math Supplements Decembe 1, 215 1 Supplement 2.1: Centipetal Acceleation This supplement expands on a topic addessed on page 19 of the textbook. Ou task hee is to calculate
More informationAPPENDIX. For the 2 lectures of Claude Cohen-Tannoudji on Atom-Atom Interactions in Ultracold Quantum Gases
APPENDIX Fo the lectues of Claude Cohen-Tannoudji on Atom-Atom Inteactions in Ultacold Quantum Gases Pupose of this Appendix Demonstate the othonomalization elation(ϕ ϕ = δ k k δ δ )k - The wave function
More informationEfficiency Loss in a Network Resource Allocation Game
Efficiency Loss in a Netwok Resouce Allocation Game Ramesh Johai johai@mit.edu) John N. Tsitsiklis jnt@mit.edu) June 11, 2004 Abstact We exploe the popeties of a congestion game whee uses of a congested
More information6 PROBABILITY GENERATING FUNCTIONS
6 PROBABILITY GENERATING FUNCTIONS Cetain deivations pesented in this couse have been somewhat heavy on algeba. Fo example, detemining the expectation of the Binomial distibution (page 5.1 tuned out to
More informationGreen s Identities and Green s Functions
LECTURE 7 Geen s Identities and Geen s Functions Let us ecall The ivegence Theoem in n-dimensions Theoem 7 Let F : R n R n be a vecto field ove R n that is of class C on some closed, connected, simply
More informationDuality between Statical and Kinematical Engineering Systems
Pape 00, Civil-Comp Ltd., Stiling, Scotland Poceedings of the Sixth Intenational Confeence on Computational Stuctues Technology, B.H.V. Topping and Z. Bittna (Editos), Civil-Comp Pess, Stiling, Scotland.
More informationAsymptotically Lacunary Statistical Equivalent Sequence Spaces Defined by Ideal Convergence and an Orlicz Function
"Science Stays Tue Hee" Jounal of Mathematics and Statistical Science, 335-35 Science Signpost Publishing Asymptotically Lacunay Statistical Equivalent Sequence Spaces Defined by Ideal Convegence and an
More informationMeasure Estimates of Nodal Sets of Polyharmonic Functions
Chin. Ann. Math. Se. B 39(5), 08, 97 93 DOI: 0.007/s40-08-004-6 Chinese Annals of Mathematics, Seies B c The Editoial Office of CAM and Spinge-Velag Belin Heidelbeg 08 Measue Estimates of Nodal Sets of
More informationChapter Eight Notes N P U1C8S4-6
Chapte Eight Notes N P UC8S-6 Name Peiod Section 8.: Tigonometic Identities An identit is, b definition, an equation that is alwas tue thoughout its domain. B tue thoughout its domain, that is to sa that
More informationAST 121S: The origin and evolution of the Universe. Introduction to Mathematical Handout 1
Please ead this fist... AST S: The oigin and evolution of the Univese Intoduction to Mathematical Handout This is an unusually long hand-out and one which uses in places mathematics that you may not be
More informationarxiv: v1 [physics.pop-ph] 3 Jun 2013
A note on the electostatic enegy of two point chages axiv:1306.0401v1 [physics.pop-ph] 3 Jun 013 A C Tot Instituto de Física Univesidade Fedeal do io de Janeio Caixa Postal 68.58; CEP 1941-97 io de Janeio,
More informationB da = 0. Q E da = ε. E da = E dv
lectomagnetic Theo Pof Ruiz, UNC Asheville, doctophs on YouTube Chapte Notes The Maxwell quations in Diffeential Fom 1 The Maxwell quations in Diffeential Fom We will now tansfom the integal fom of the
More informationFINSLER SPACE SUBJECTED TO A KROPINA CHANGE WITH AN. 1. Introduction
FACTA UNIVERSITATIS (NIŠ) Se.Math.Infom. Vol. 30 No 4 (2015) 513 525 FINSER SPACE SUBJECTE TO A KROPINA CHANGE WITH AN h-vector M. K. Gupta and P. N. Pandey Abstact. In this pape we discuss the Finsle
More informationAn Application of Bessel Functions: Study of Transient Flow in a Cylindrical Pipe
Jouna of Mathematics and System Science 6 (16) 7-79 doi: 1.1765/159-591/16..4 D DAVID PUBLISHING An Appication of Besse Functions: Study of Tansient Fow in a Cyindica Pipe A. E. Gacía, Lu Maía Gacía Cu
More informationSurveillance Points in High Dimensional Spaces
Société de Calcul Mathématique SA Tools fo decision help since 995 Suveillance Points in High Dimensional Spaces by Benad Beauzamy Januay 06 Abstact Let us conside any compute softwae, elying upon a lage
More informationOn the Quasi-inverse of a Non-square Matrix: An Infinite Solution
Applied Mathematical Sciences, Vol 11, 2017, no 27, 1337-1351 HIKARI Ltd, wwwm-hikaicom https://doiog/1012988/ams20177273 On the Quasi-invese of a Non-squae Matix: An Infinite Solution Ruben D Codeo J
More informationASTR415: Problem Set #6
ASTR45: Poblem Set #6 Cuan D. Muhlbege Univesity of Mayland (Dated: May 7, 27) Using existing implementations of the leapfog and Runge-Kutta methods fo solving coupled odinay diffeential equations, seveal
More informationOn the ratio of maximum and minimum degree in maximal intersecting families
On the atio of maximum and minimum degee in maximal intesecting families Zoltán Lóánt Nagy Lale Özkahya Balázs Patkós Máté Vize Septembe 5, 011 Abstact To study how balanced o unbalanced a maximal intesecting
More informationTransverse Wakefield in a Dielectric Tube with Frequency Dependent Dielectric Constant
ARDB-378 Bob Siemann & Alex Chao /4/5 Page of 8 Tansvese Wakefield in a Dielectic Tube with Fequency Dependent Dielectic Constant This note is a continuation of ARDB-368 that is now extended to the tansvese
More informationSyntactical content of nite approximations of partial algebras 1 Wiktor Bartol Inst. Matematyki, Uniw. Warszawski, Warszawa (Poland)
Syntactical content of nite appoximations of patial algebas 1 Wikto Batol Inst. Matematyki, Uniw. Waszawski, 02-097 Waszawa (Poland) batol@mimuw.edu.pl Xavie Caicedo Dep. Matematicas, Univ. de los Andes,
More informationA Bijective Approach to the Permutational Power of a Priority Queue
A Bijective Appoach to the Pemutational Powe of a Pioity Queue Ia M. Gessel Kuang-Yeh Wang Depatment of Mathematics Bandeis Univesity Waltham, MA 02254-9110 Abstact A pioity queue tansfoms an input pemutation
More informationJournal of Number Theory
Jounal of umbe Theoy 3 2 2259 227 Contents lists available at ScienceDiect Jounal of umbe Theoy www.elsevie.com/locate/jnt Sums of poducts of hypegeometic Benoulli numbes Ken Kamano Depatment of Geneal
More informationA Three-Dimensional Magnetic Force Solution Between Axially-Polarized Permanent-Magnet Cylinders for Different Magnetic Arrangements
Poceedings of the 213 Intenational Confeence on echanics, Fluids, Heat, Elasticity Electomagnetic Fields A Thee-Dimensional agnetic Foce Solution Between Axially-Polaied Pemanent-agnet Cylindes fo Diffeent
More informationMDS Array Codes with Optimal Rebuilding
0 IEEE Intenational Symposium on Infomation Theoy Poceedings MDS Aay Codes with Optimal Reuilding Itzhak Tamo, Zhiying Wang and Jehoshua Buck Electical Engineeing Depatment, Califonia Institute of Technology,
More informationAnalytic Evaluation of two-electron Atomic Integrals involving Extended Hylleraas-CI functions with STO basis
Analytic Evaluation of two-electon Atomic Integals involving Extended Hylleaas-CI functions with STO basis B PADHY (Retd.) Faculty Membe Depatment of Physics, Khalikote (Autonomous) College, Behampu-760001,
More informationRelating Branching Program Size and. Formula Size over the Full Binary Basis. FB Informatik, LS II, Univ. Dortmund, Dortmund, Germany
Relating Banching Pogam Size and omula Size ove the ull Binay Basis Matin Saueho y Ingo Wegene y Ralph Wechne z y B Infomatik, LS II, Univ. Dotmund, 44 Dotmund, Gemany z ankfut, Gemany sauehof/wegene@ls.cs.uni-dotmund.de
More informationNumerical Inversion of the Abel Integral Equation using Homotopy Perturbation Method
Numeical Invesion of the Abel Integal Equation using Homotopy Petubation Method Sunil Kuma and Om P Singh Depatment of Applied Mathematics Institute of Technology Banaas Hindu Univesity Vaanasi -15 India
More informationA Multivariate Normal Law for Turing s Formulae
A Multivaiate Nomal Law fo Tuing s Fomulae Zhiyi Zhang Depatment of Mathematics and Statistics Univesity of Noth Caolina at Chalotte Chalotte, NC 28223 Abstact This pape establishes a sufficient condition
More informationarxiv: v2 [math.ag] 4 Jul 2012
SOME EXAMPLES OF VECTOR BUNDLES IN THE BASE LOCUS OF THE GENERALIZED THETA DIVISOR axiv:0707.2326v2 [math.ag] 4 Jul 2012 SEBASTIAN CASALAINA-MARTIN, TAWANDA GWENA, AND MONTSERRAT TEIXIDOR I BIGAS Abstact.
More informationPhysics 2A Chapter 10 - Moment of Inertia Fall 2018
Physics Chapte 0 - oment of netia Fall 08 The moment of inetia of a otating object is a measue of its otational inetia in the same way that the mass of an object is a measue of its inetia fo linea motion.
More informationFI 2201 Electromagnetism
F Eectomagnetism exane. skana, Ph.D. Physics of Magnetism an Photonics Reseach Goup Magnetostatics MGNET VETOR POTENTL, MULTPOLE EXPNSON Vecto Potentia Just as E pemitte us to intouce a scaa potentia V
More informationA matrix method based on the Fibonacci polynomials to the generalized pantograph equations with functional arguments
A mati method based on the Fibonacci polynomials to the genealized pantogaph equations with functional aguments Ayşe Betül Koç*,a, Musa Çama b, Aydın Kunaz a * Coespondence: aysebetuloc @ selcu.edu.t a
More informationHua Xu 3 and Hiroaki Mukaidani 33. The University of Tsukuba, Otsuka. Hiroshima City University, 3-4-1, Ozuka-Higashi
he inea Quadatic Dynamic Game fo Discete-ime Descipto Systems Hua Xu 3 and Hioai Muaidani 33 3 Gaduate School of Systems Management he Univesity of suuba, 3-9- Otsua Bunyo-u, oyo -0, Japan xuhua@gssm.otsua.tsuuba.ac.jp
More information= ρ. Since this equation is applied to an arbitrary point in space, we can use it to determine the charge density once we know the field.
Gauss s Law In diffeentia fom D = ρ. ince this equation is appied to an abita point in space, we can use it to detemine the chage densit once we know the fied. (We can use this equation to ve fo the fied
More informationLax pairs and Fourier analysis: The case of sine- Gordon and Klein-Gordon equations
Jounal of Physics: Confeence Seies Lax pais and Fouie analysis: The case of sine- Godon and Klein-Godon equations To cite this aticle: Pavle Saksida 22 J. Phys.: Conf. Se. 343 29 View the aticle online
More informationStanford University CS259Q: Quantum Computing Handout 8 Luca Trevisan October 18, 2012
Stanfod Univesity CS59Q: Quantum Computing Handout 8 Luca Tevisan Octobe 8, 0 Lectue 8 In which we use the quantum Fouie tansfom to solve the peiod-finding poblem. The Peiod Finding Poblem Let f : {0,...,
More information3.1 Random variables
3 Chapte III Random Vaiables 3 Random vaiables A sample space S may be difficult to descibe if the elements of S ae not numbes discuss how we can use a ule by which an element s of S may be associated
More informationStudy on Application of New Theorem of Kinetic Energy
Intenational Jounal of pplied Physics and Mathematics Study on pplication of New Theoem of Kinetic Enegy Yong Yang * Tongchuan Banch of Shanxi Radio and TV Univesity, Tongchuan Polytechnics, Tongchuan
More informationMath 2263 Solutions for Spring 2003 Final Exam
Math 6 Solutions fo Sping Final Exam ) A staightfowad appoach to finding the tangent plane to a suface at a point ( x, y, z ) would be to expess the cuve as an explicit function z = f ( x, y ), calculate
More informationDo not turn over until you are told to do so by the Invigilator.
UNIVERSITY OF EAST ANGLIA School of Mathematics Main Seies UG Examination 2015 16 FLUID DYNAMICS WITH ADVANCED TOPICS MTH-MD59 Time allowed: 3 Hous Attempt QUESTIONS 1 and 2, and THREE othe questions.
More informationOn the global uniform asymptotic stability of time-varying dynamical systems
Stud. Univ. Babeş-Bolyai Math. 59014), No. 1, 57 67 On the global unifom asymptotic stability of time-vaying dynamical systems Zaineb HajSalem, Mohamed Ali Hammami and Mohamed Mabouk Abstact. The objective
More information3D-Central Force Problems I
5.73 Lectue #1 1-1 Roadmap 1. define adial momentum 3D-Cental Foce Poblems I Read: C-TDL, pages 643-660 fo next lectue. All -Body, 3-D poblems can be educed to * a -D angula pat that is exactly and univesally
More informationH.W.GOULD West Virginia University, Morgan town, West Virginia 26506
A F I B O N A C C I F O R M U L A OF LUCAS A N D ITS SUBSEQUENT M A N I F E S T A T I O N S A N D R E D I S C O V E R I E S H.W.GOULD West Viginia Univesity, Mogan town, West Viginia 26506 Almost eveyone
More informationMath 124B February 02, 2012
Math 24B Febuay 02, 202 Vikto Gigoyan 8 Laplace s equation: popeties We have aleady encounteed Laplace s equation in the context of stationay heat conduction and wave phenomena. Recall that in two spatial
More informationAn upper bound on the number of high-dimensional permutations
An uppe bound on the numbe of high-dimensional pemutations Nathan Linial Zu Luia Abstact What is the highe-dimensional analog of a pemutation? If we think of a pemutation as given by a pemutation matix,
More informationIntegral operator defined by q-analogue of Liu-Srivastava operator
Stud. Univ. Babeş-Bolyai Math. 582013, No. 4, 529 537 Integal opeato defined by q-analogue of Liu-Sivastava opeato Huda Aldweby and Maslina Daus Abstact. In this pape, we shall give an application of q-analogues
More informationONE-POINT CODES USING PLACES OF HIGHER DEGREE
ONE-POINT CODES USING PLACES OF HIGHER DEGREE GRETCHEN L. MATTHEWS AND TODD W. MICHEL DEPARTMENT OF MATHEMATICAL SCIENCES CLEMSON UNIVERSITY CLEMSON, SC 29634-0975 U.S.A. E-MAIL: GMATTHE@CLEMSON.EDU, TMICHEL@CLEMSON.EDU
More informationIntroduction to Mathematical Statistics Robert V. Hogg Joeseph McKean Allen T. Craig Seventh Edition
Intoduction to Mathematical Statistics Robet V. Hogg Joeseph McKean Allen T. Caig Seventh Edition Peason Education Limited Edinbugh Gate Halow Essex CM2 2JE England and Associated Companies thoughout the
More informationHE DI ELMONSER. 1. Introduction In 1964 H. Mink and L. Sathre [15] proved the following inequality. n, n N. ((n + 1)!) n+1
-ANALOGUE OF THE ALZER S INEQUALITY HE DI ELMONSER Abstact In this aticle, we ae inteested in giving a -analogue of the Alze s ineuality Mathematics Subject Classification (200): 26D5 Keywods: Alze s ineuality;
More informationWhy Professor Richard Feynman was upset solving the Laplace equation for spherical waves? Anzor A. Khelashvili a)
Why Pofesso Richad Feynman was upset solving the Laplace equation fo spheical waves? Anzo A. Khelashvili a) Institute of High Enegy Physics, Iv. Javakhishvili Tbilisi State Univesity, Univesity St. 9,
More information