-Pareto Optimality for Nondifferentiable Multiobjective Programming via Penalty Function
|
|
- Felicity Matthews
- 5 years ago
- Views:
Transcription
1 JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 198, ARTICLE NO Pareto Otalty for Nodfferetable Multobectve Prograg va Pealty Fucto J. C. Lu Secto of Matheatcs, Natoal Uersty Prearatory School, Natoal Oerseas Chese Studet Uersty, Lkou, Tae, Hse, Tawa, 24402, Reublc of Cha Subtted by Koch Mzuka Receved Jauary 9, 1995 Necessary ad suffcet codtos wthout a costrat qualfcato for -Pareto otalty of ultobectve rograg are derved. The ecessary KuhTucker codto suggests the establshet of a Wolf-tye dualty theore for odfferetable, covex, ultobectve zato robles. The geeralzed -saddle ot for Pareto otalty of the vector Lagraga s studed Acadec Press, Ic. 1. INTRODUCTION Several authors have bee terested recetly -otal solutos olear rograg. For detals, the readers are advsed to cosult 15. Lorda 4 derved soe roertes of -effcet ots soluto for vector zato robles ad used the Ekelad s varatoal rcle 6 to establsh the -Pareto otalty ad -quas Pareto otalty. I 7, Lu also adated the sae aroach to obta the -dualty theore of odfferetable ocovex ultobectve rograg. Recetly, several authors 812 have used a exact ealty fucto to trasfor the olear scalar rograg roble to a ucostraed roble ad derved the -otalty. I 14, Yokoyaa was cocered wth the -aroxate solutos ad exteded soe results of 13 to the vector zato robles. Slar to 13, Yokoyaa trasfored the vector robles to the ucostraed robles by usg the exact ealty fuctos ad showed the -otalty crtera by estatg the ealty araeter ters of -aroxate solutos for the assocated dual robles X96 $18.00 Coyrght 1996 by Acadec Press, Ic. All rghts of reroducto ay for reserved. 248
2 -PARETO OPTIMALITY 249 I ths aer, we are sred to use the exact ealty fucto to trasfor the equalty ultobectve rograg roble to a scalar ucostraed roble ad to derve the KuhTucker codtos whch Lagrage ultlers of obectve fuctos are oe. Soe deftos ad otatos are gve Secto 2. I Secto 3, we use scalar ealty fuctos to establsh the ecessary ad suffcet codtos of -Pareto soluto. Usg ths result, we forulate a dual roble of the Wolfe-tye for ultobectve rograg. I Secto 4, we gve soe relatoshs betwee the ral roble ad the dual roble. The geeralzed -Pareto saddle ot of the vector Lagraga s dscussed Secto PRELIMINARIES We cosder the followg covex ultobectve rograg roble: Ž P. ze fž x. subect to g Ž x. 0, 1,...,, where f Ž f, f,..., f. 1 2 ad each cooet fucto s a covex cotuous real-valued fucto defed o R ad where g are covex cotu- ous real-valued fuctos defed o R, 1. We deote the feasble set x R g Ž x. 0, 1 4 by F ad assue the feasble set F s oety. Let be a eleet of R. We troduce the feasble set F, ½ 5 F x R g x,1. For coveece, let. To trasfor the roble Ž P. to a scalar ucostraed roble, we use the exact ealty fucto troduced by Zagwll 8 : Ž x,. f Ž x. ax 0, g Ž x., 1 Ž. where 0. The assocated ucostraed roble whch ze Ž x,.
3 250 J. C. LIU s called a ealzed roble wth resect to the ealty araeter. For coveece, let Ž t. axž 0, t.. Clearly, we have x, f x Žg Ž x.. 1. DEFINITION 2.1. A ot x R s called a -Pareto soluto of Ž P. f x F ad there s o x F such that fž x. fž x. ad fž x. fž x., R. DEFINITION 2.2. A ot x R s called a alost -Pareto soluto of P f x F ad there s o x F such that fž x. fž x. ad fž x. fž x., R. DEFINITION 2.3. Let 0. A ot x R s called a -soluto of f the scalar roble Ž x,. Ž x,., for all x R. DEFINITION 2.4. Let h: R R 4 be a covex fucto, fte at x. The -subdfferetal of h at x s the set hž x. defed by 4 ² : hž x. x*r hž y. hž x. x*, y x for ay y R. 3. NECESSARY AND SUFFICIENT CONDITIONS I ths secto, we reset soe KuhTucker codtos for -Pareto otalty. THEOREM 3.1. If there exsts such that x s a -soluto for Ž. 0 for ay, the x s a -Pareto soluto for Ž P. 0 ad there exst scalars 0 Ž 1., 0 Ž 1., 0 Ž 1. such that: Ž. 0 f Ž x. Ž g.ž x., Ž 1. Ž. g Ž x. 0. Ž 2.
4 -PARETO OPTIMALITY 251 Proof. If x s a -soluto of Proble Ž., Ž x,. Ž x,., for all x R. Ž 3. Clearly, Thus, we have Ž x,. f Ž x., for all x F. f Ž x. Ž x,. f Ž x., for all x F. Ž 4. If x F, Žg Ž x Choose ay feasble ot ˆx whch s also F ad let ž / ax f Ž x. f Ž x. g Ž x.,. We the have the cocluso ½ 5 ˆ Ž. 0 1 f Ž ˆx. Ž ˆx,. Ž x,. f Ž ˆx.. Ths cocluso gves a cotradcto ad hece x F. If x s ot a -Pareto soluto of Ž P., the there exsts x1 F such that fž x1. fž x., 1, wth at least oe strct equalty. Therefore, we have f Ž x. f Ž x., 1 whch cotradcts Ž. 4. Thus x s a -Pareto soluto of Ž. P. Wth Ž. 3 ad the result of 15, we have Ž ž./ 1 0 f Ž. g Ž. Ž x.. The, there exst scalars 0 Ž 1., 0 Ž 1., 0 Ž 1., ad 0 Ž 1. such that, Ž f Ž x. Ž g.ž x., Ž 6. 1
5 252 J. C. LIU where 1, g Ž x. g Ž x. for 1,...,. Ž 7. Ž. Ž. By 5 ad 7, we have 1 1 g Ž x. 0. Ž. Ž. Fally, we obta the results 1 ad 2 by settg,, 1, 1. REMARK 3.1. If 1, the the ecessary codto of Theore 3.1 reduces to Theore 4.1 of 13. THEOREM 3.2. If x s a feasble soluto of Ž P. ad there exst 0 Ž 1., 0 Ž 1., 0 Ž 1. such that: Ž. 0 f Ž x. Ž g.ž x., Ž 8. Ž. g Ž x. 0. Ž 9. The, xsa-pareto soluto of Ž P.. Proof. If x s a feasble soluto of Ž P. ad there exst 0 Ž 1., 0 Ž 1., 0 Ž 1. whch satsfy relatos Ž. 8 ad Ž. 9, the there exst x f Ž x.,1, y Ž g.ž x.,1, such that x y 0. By usg the characterzato of the -subgradet, we obta 1 f Ž x. f Ž x. x, xx, 1, g x g x y, xx, 1. ² : ² :
6 -PARETO OPTIMALITY 253 Thus, we have f Ž x. g Ž x. 1 1 Ž. Wth 9, we have f Ž x. g Ž x. f Ž x., for all x F. 1 f Ž x. f Ž x., for all x F. Ž 10. If x s ot a -Pareto soluto of P, there exsts x F such that 1 fž x1. fž x., 1, wth at least oe strct equalty. Therefore, we have f Ž x. f Ž x., 1 whch cotradcts 10. Thus x s a -Pareto soluto of P. THEOREM 3.3. If for suffcetly large, xsa-soluto for Ž., the x s a alost -Pareto soluto for Ž P. ad there exst scalars 0 Ž 1., 0 Ž 1., 0 Ž 1. such that: Ž. 0 f Ž x. g Ž x., 1 Ž. g Ž x Proof. If x s a -soluto of Proble, f Ž x. Ž x,. f Ž x., for all x F.
7 254 J. C. LIU Sce f f Ž x. f Ž x., xr we have g Ž x. f f Ž x. f f Ž x.. 1 xf xr Let Ž. f f Ž x. f f Ž x. 0 xf xr The, there exsts such that 0 Ž. g Ž x., 1. Hece, we have x F. Ths cocludes the roof of the theore. 4. -DUALITY THEOREM OF THE WOLFE TYPE The result of Theore 3.1 s used to forulate such a dual roble of the Wolfe tye for ultobectve rograg as follows: 4 Ž D. axze LŽ x,. Ž x,. F D ; here D ½ 1 gž x. 0, 1 1 0, 1, 0, 1 5, F Ž x,. R R 0 f Ž x. g Ž x.,
8 -PARETO OPTIMALITY 255 ad the vector Lagraga fucto L x, s defed by LŽ x,. fž x. ²², gž x. :: 1 ; 1 1 f x 1 g x,..., f x 1 g x, for all x R, R, 1. to of Ž D. f Ž x,. F ad there s o Ž x,. F, such that DEFINITION 4.1. A ot x, R R s called a -Pareto solu- D f Ž x. Ž 1. g Ž x. f Ž x. Ž 1. g Ž x., 1, 1 1 wth at least oe strct equalty. THEOREM 4.1 Ž Dualty.. If there exsts 0 such that x s a -soluto for Ž. for ay, the x s a -Pareto soluto for Ž P. 0 ad there exst scalars R such that Ž x,. s a -Pareto soluto of Ž D,for. all, 1. Proof. Wth Theore 3.1, we coclude that Ž x,. s a feasble soluto of D. Let x, R R be ay feasble soluto of Ž D.. The, there exst x f Ž x.,1, y Ž g.ž x.,1, such that 1 x y 0, g Ž x By usg the characterzato of the -subgradet, we obta Thus, we have f Ž x. f Ž x. x, xx, 1, ² : g Ž x. g Ž x. y, xx, 1. ² : f Ž x. g Ž x. f Ž x. g Ž x f Ž x. f Ž x. g Ž x.. 1 D
9 256 J. C. LIU Sce ad g x 0, for all 1, 1 g Ž x. g Ž x.. 1 We the deduce that f Ž x. g Ž x. 1 f Ž x. g Ž x., for all Ž x,. F. Ž 11. D 1 If Ž x,. s ot a -Pareto soluto of the dual roble Ž D., there exsts Ž x*, *. F such that D 1 f x* 1 g x* f Ž x. Ž 1. g Ž x., 1, 1 wth at least oe strct equalty. Thus, we have 1 1 f x* g x* f x g x whch cotradcts 11. THEOREM 4.2 Ž Coverse Dualty.. Let x be a feasble soluto of Ž P.. If Ž x,. s a feasble soluto of Ž D,. xsa-pareto soluto of Ž P.. Proof. It follows fro Theore VECTOR LAGRANGIAN AND ITS -PARETO SADDLE POINT I ths secto, we cosder the -Pareto saddle ot of the vector Lagraga fucto.
10 -PARETO OPTIMALITY 257 ot of the vector Lagraga LŽ x,. f the followg codtos hold: DEFINITION 5.1. A ot x, R R s called a -Pareto saddle Ž. L x, Ž. L x,, for all R ; L x, Ž. L x,, for all x R. That s to say, there exst ether R or x R such that: Ž. f Ž x. Ž 1. g Ž x. 1 wth at least oe strct equalty, Ž. f Ž x. Ž 1. g Ž x. 1 wth at least oe strct equalty. f Ž x. Ž 1. g Ž x., 1, 1 f Ž x. Ž 1. g Ž x., 1, 1 THEOREM 5.1. If there exsts such that x s a -soluto for Ž. 0 for ay 0, the x s a -Pareto soluto for P ad there exst scalars R such that Ž x,. s a -Pareto-saddle ot of the ector Lagraga. Proof. Wth Theore 3.1, there exst R such that 0 f Ž x. g Ž x., Ž g Ž x. 0. Ž The, there exst x f Ž x.,1, y Ž g.ž x.,1, such that 1 x y 0.
11 258 J. C. LIU By usg the characterzato of the -subgradet, we obta Thus, we have f Ž x. f Ž x. x, xx, 1, g x g x y, xx, 1. ² : ² : fž x. gž x. fž x. gž x f Ž x. fž x. gž x., for all x R. 1 Assue that there s a x* R such that 1 1 f x* 1 g x* f Ž x. Ž 1. g Ž x., 1, wth at least oe strct equalty, we obta 1 1 f Ž x*. g Ž x*. f Ž x. g Ž x., Ž 14. whch cotradcts Ž 14.. Ths gves the frst codto of the defto for -Pareto saddle ot. Wth Ž 13., we deduce that 1 1 g x 0. Sce x F, 1g x 0, for all R. Thus, we have 1 1 g x g x for all R.
12 -PARETO OPTIMALITY 259 Therefore, we obta f Ž x. g Ž x. 1 f x g x, for all R If there s a * R such that 1 f x 1 g x f Ž x. Ž 1. g Ž x., 1, 1 wth at least oe strct equalty, we obta 1 1 f x g x f x g x whch cotradcts 15. Ths coletes the roof. THEOREM 5.2. If Ž x,. s a -Pareto saddle ot of the ector Lagraga L ad g Ž x. g Ž x.,1, for all x F, the x s a alost -Pareto soluto of Ž P.. Proof. If x, s a -Pareto saddle ot of the vector Lagraga L, there s o R such that f Ž x. 1 g Ž x. f Ž x. 1 g Ž x., 1, 1 1 wth at least oe strct equalty. If x F, for soe k ad all. Thus, we have for all. g Ž x. k gkž x. kgkž x. kgkž x. Ž 16.
13 260 J. C. LIU Choose 1 ad for all k for L x, ; we obta k k f Ž x. 1 g Ž x. f Ž x. 1 1 g Ž x. 1 g Ž x. k k 1 k f Ž x. 1 g Ž x. 1 f Ž x. 1 g Ž x. 1 g Ž x. k k k for all whch cotradcts Ž 16.. We coclude that x F. Now, we use the other equalty for a -Pareto saddle ot. The, there s o x R such that f Ž x. Ž 1. g Ž x. f Ž x. Ž 1. g Ž x., 1, 1 1 wth at least oe strct equalty. Fro g Ž x. g Ž x., 1, for all x F, we coclude that there s o x F such that fž x. fž x., 1, wth at least oe strct equalty. Ths cocludes the roof of the theore. ACKNOWLEDGMENTS The author s thakful to Dr. Koch Mzuka of Hrosak Uversty for hs coets ad suggestos o a earler verso of the aer, esecally o the roof of Theore 5.2. REFERENCES 1. S. S. Kutateladze, Covex -rograg, Soet. Math. Dokl 20 Ž 1979., P. Lorda, Necessary codtos for -otalty, Math. Prograg Study 19 Ž 1982., P. Lorda ad J. Morga, Pealty fuctos -rograg ad -ax robles, Math. Prograg 26 Ž 1983., P. Lorda, -Soluto vector zato robles, J. Ot. Theory Al. 43 Ž 1984.,
14 -PARETO OPTIMALITY J. J. Strodot, V. H. Nguye, ad N. Heukees, -Otal solutos odfferetable covex rograg ad soe related questos, Math. Prograg 25 Ž 1983., I. Ekelad, O the varatoal rcle, J. Math. Aal. Al. 47 Ž 1974., J. C. Lu, -Dualty theore of odfferetable ocovex ultobectve rograg, J. Ot. Theory Al. 69 Ž 1991., W. I. Zagwll, Nolear rograg va ealty fuctos, Maageet Sc. 13 Ž 1967., D. P. Bertsekas, Necessary ad suffcet codtos for a ealty ethod to be exact, Math. Prograg 9 Ž 1975., S. P. Ha ad O. L. Magasara, Exact ealty fuctos olear rograg, Math. Prograg 17 Ž 1979., S. P. Ha ad O. L. Magasara, A dual dfferetable exact ealty fucto, Math. Prograg 25 Ž 1983., O. L. Magasara, Suffcecy of exact ealty zato, SIAM J. Cotrol Ot. 23 Ž 1985., K. Yokoyaa, -Otalty crtera for covex rograg robles va exact ealty fuctos, Math. Prograg 56 Ž 1992., K. Yokoyaa, -Otalty crtera for vector zato robles va exact ealty fuctos, J. Math. Aal. Al. 187 Ž 1994., J. B. Hrart Urruty, -Subdfferetal calculus, Covex Aalyss ad Otzato, Research Notes Matheatcs Seres, Vol. 57 Ž J. P. Aub ad R. B. Vter, Eds..,. 4392, Pta, Bosto, MA, P. Wolfe, A dualty theore for olear rograg, Quart. Al. Math. 19 Ž 1961., H. C. La ad C. P. Ho, Dualty theore of odfferetable covex ultobectve rograg, J. Ot. Theory Al. 50 Ž 1986., J. C. Lu, Dualty for odfferetable ultobectve rograg wthout a costrat, rert. 19. T. Tao ad Y. Sawarag, Dualty theore ultobectve rograg, J. Ot. Theory Al. 27 Ž 1979., J. Jah, Dualty vector otzato, Math. Prograg 25 Ž 1983., R. R. Egudo ad M. A. Haso, Multobectve dualty wth vexty, J. Math. Aal. Al. 126 Ž 1987., T. Wer, Dualty for odfferetable ultle obectve fractoal rograg robles, Utltas Math. 36 Ž 1989., T. Wer, Proer effcecy ad dualty for vector valued otzato robles, J. Austral. Math. Soc. Ser. A 43 Ž 1987., B. Mod, I. Hsa, ad M. V. Durga Prassad, Dualty for a class of odfferetable ultobectve rogras, Utltas Math. 39 Ž 1991., R. N. Kual, S. K. Suea, ad M. K. Srvastava, Otalty crtera ad dualty ultle-obectve otzato volvg geeralzed vexty, J. Ot. Theory Al. 80 Ž 1994., J. W. Neuwehus, Soe ax theores vector-valued fuctos, J. Ot. Theory Al. 40 Ž 1983.,
MULTIOBJECTIVE NONLINEAR FRACTIONAL PROGRAMMING PROBLEMS INVOLVING GENERALIZED d - TYPE-I n -SET FUNCTIONS
THE PUBLIHING HOUE PROCEEDING OF THE ROMANIAN ACADEMY, eres A OF THE ROMANIAN ACADEMY Volue 8, Nuber /27,.- MULTIOBJECTIVE NONLINEAR FRACTIONAL PROGRAMMING PROBLEM INVOLVING GENERALIZED d - TYPE-I -ET
More informationA Penalty Function Algorithm with Objective Parameters and Constraint Penalty Parameter for Multi-Objective Programming
Aerca Joural of Operatos Research, 4, 4, 33-339 Publshed Ole Noveber 4 ScRes http://wwwscrporg/oural/aor http://ddoorg/436/aor4463 A Pealty Fucto Algorth wth Obectve Paraeters ad Costrat Pealty Paraeter
More informationA New Method for Solving Fuzzy Linear. Programming by Solving Linear Programming
ppled Matheatcal Sceces Vol 008 o 50 7-80 New Method for Solvg Fuzzy Lear Prograg by Solvg Lear Prograg S H Nasser a Departet of Matheatcs Faculty of Basc Sceces Mazadara Uversty Babolsar Ira b The Research
More informationA Family of Non-Self Maps Satisfying i -Contractive Condition and Having Unique Common Fixed Point in Metrically Convex Spaces *
Advaces Pure Matheatcs 0 80-84 htt://dxdoorg/0436/a04036 Publshed Ole July 0 (htt://wwwscrporg/oural/a) A Faly of No-Self Mas Satsfyg -Cotractve Codto ad Havg Uque Coo Fxed Pot Metrcally Covex Saces *
More informationUnique Common Fixed Point of Sequences of Mappings in G-Metric Space M. Akram *, Nosheen
Vol No : Joural of Facult of Egeerg & echolog JFE Pages 9- Uque Coo Fed Pot of Sequeces of Mags -Metrc Sace M. Ara * Noshee * Deartet of Matheatcs C Uverst Lahore Pasta. Eal: ara7@ahoo.co Deartet of Matheatcs
More informationDUALITY FOR MINIMUM MATRIX NORM PROBLEMS
HE PUBLISHING HOUSE PROCEEDINGS OF HE ROMNIN CDEMY, Seres, OF HE ROMNIN CDEMY Vole 6, Nber /2005,. 000-000 DULIY FOR MINIMUM MRI NORM PROBLEMS Vasle PRED, Crstca FULG Uverst of Bcharest, Faclt of Matheatcs
More informationOPTIMALITY CONDITIONS FOR LOCALLY LIPSCHITZ GENERALIZED B-VEX SEMI-INFINITE PROGRAMMING
Mrcea cel Batra Naval Acadey Scetfc Bullet, Volue XIX 6 Issue he joural s dexed : PROQUES / DOAJ / Crossref / EBSCOhost / INDEX COPERNICUS / DRJI / OAJI / JOURNAL INDEX / IOR / SCIENCE LIBRARY INDEX /
More informationSome Different Perspectives on Linear Least Squares
Soe Dfferet Perspectves o Lear Least Squares A stadard proble statstcs s to easure a respose or depedet varable, y, at fed values of oe or ore depedet varables. Soetes there ests a deterstc odel y f (,,
More informationSUBCLASS OF HARMONIC UNIVALENT FUNCTIONS ASSOCIATED WITH SALAGEAN DERIVATIVE. Sayali S. Joshi
Faculty of Sceces ad Matheatcs, Uversty of Nš, Serba Avalable at: http://wwwpfacyu/float Float 3:3 (009), 303 309 DOI:098/FIL0903303J SUBCLASS OF ARMONIC UNIVALENT FUNCTIONS ASSOCIATED WIT SALAGEAN DERIVATIVE
More informationTHE PROBABILISTIC STABILITY FOR THE GAMMA FUNCTIONAL EQUATION
Joural of Scece ad Arts Year 12, No. 3(2), pp. 297-32, 212 ORIGINAL AER THE ROBABILISTIC STABILITY FOR THE GAMMA FUNCTIONAL EQUATION DOREL MIHET 1, CLAUDIA ZAHARIA 1 Mauscrpt receved: 3.6.212; Accepted
More information( x) min. Nonlinear optimization problem without constraints NPP: then. Global minimum of the function f(x)
Objectve fucto f() : he optzato proble cossts of fg a vector of ecso varables belogg to the feasble set of solutos R such that It s eote as: Nolear optzato proble wthout costrats NPP: R f ( ) : R R f f
More information7.0 Equality Contraints: Lagrange Multipliers
Systes Optzato 7.0 Equalty Cotrats: Lagrage Multplers Cosder the zato of a o-lear fucto subject to equalty costrats: g f() R ( ) 0 ( ) (7.) where the g ( ) are possbly also olear fuctos, ad < otherwse
More informationDecomposition of Hadamard Matrices
Chapter 7 Decomposto of Hadamard Matrces We hae see Chapter that Hadamard s orgal costructo of Hadamard matrces states that the Kroecer product of Hadamard matrces of orders m ad s a Hadamard matrx of
More informationSOME ASPECTS ON SOLVING A LINEAR FRACTIONAL TRANSPORTATION PROBLEM
Qattate Methods Iqres SOME ASPECTS ON SOLVING A LINEAR FRACTIONAL TRANSPORTATION PROBLEM Dora MOANTA PhD Deartet of Matheatcs Uersty of Ecoocs Bcharest Roaa Ma blshed boos: Three desoal trasort robles
More informationDuality Theory for Interval Linear Programming Problems
IOSR Joural of Matheatcs (IOSRJM) ISSN: 78-578 Volue 4, Issue 4 (Nov-Dec, ), 9-47 www.osrourals.org Dualty Theory for Iterval Lear Prograg Probles G. Raesh ad K. Gaesa, Departet of Matheatcs, Faculty of
More informationAlgorithms behind the Correlation Setting Window
Algorths behd the Correlato Settg Wdow Itroducto I ths report detaled forato about the correlato settg pop up wdow s gve. See Fgure. Ths wdow s obtaed b clckg o the rado butto labelled Kow dep the a scree
More informationA Characterization of Jacobson Radical in Γ-Banach Algebras
Advaces Pure Matheatcs 43-48 http://dxdoorg/436/ap66 Publshed Ole Noveber (http://wwwscrporg/joural/ap) A Characterzato of Jacobso Radcal Γ-Baach Algebras Nlash Goswa Departet of Matheatcs Gauhat Uversty
More informationJournal Of Inequalities And Applications, 2008, v. 2008, p
Ttle O verse Hlbert-tye equaltes Authors Chagja, Z; Cheug, WS Ctato Joural Of Iequaltes Ad Alcatos, 2008, v. 2008,. 693248 Issued Date 2008 URL htt://hdl.hadle.et/0722/56208 Rghts Ths work s lcesed uder
More informationCapacitated Plant Location Problem:
. L. Brcker, 2002 ept of Mechacal & Idustral Egeerg The Uversty of Iowa CPL/ 5/29/2002 page CPL/ 5/29/2002 page 2 Capactated Plat Locato Proble: where Mze F + C subect to = = =, =, S, =,... 0, =, ; =,
More informationDebabrata Dey and Atanu Lahiri
RESEARCH ARTICLE QUALITY COMPETITION AND MARKET SEGMENTATION IN THE SECURITY SOFTWARE MARKET Debabrata Dey ad Atau Lahr Mchael G. Foster School of Busess, Uersty of Washgto, Seattle, Seattle, WA 9895 U.S.A.
More informationGeneralized Convex Functions on Fractal Sets and Two Related Inequalities
Geeralzed Covex Fuctos o Fractal Sets ad Two Related Iequaltes Huxa Mo, X Su ad Dogya Yu 3,,3School of Scece, Bejg Uversty of Posts ad Telecommucatos, Bejg,00876, Cha, Correspodece should be addressed
More informationSTRONG CONSISTENCY FOR SIMPLE LINEAR EV MODEL WITH v/ -MIXING
Joural of tatstcs: Advaces Theory ad Alcatos Volume 5, Number, 6, Pages 3- Avalable at htt://scetfcadvaces.co. DOI: htt://d.do.org/.864/jsata_7678 TRONG CONITENCY FOR IMPLE LINEAR EV MODEL WITH v/ -MIXING
More informationQ-analogue of a Linear Transformation Preserving Log-concavity
Iteratoal Joural of Algebra, Vol. 1, 2007, o. 2, 87-94 Q-aalogue of a Lear Trasformato Preservg Log-cocavty Daozhog Luo Departmet of Mathematcs, Huaqao Uversty Quazhou, Fua 362021, P. R. Cha ldzblue@163.com
More informationNon-uniform Turán-type problems
Joural of Combatoral Theory, Seres A 111 2005 106 110 wwwelsevercomlocatecta No-uform Turá-type problems DhruvMubay 1, Y Zhao 2 Departmet of Mathematcs, Statstcs, ad Computer Scece, Uversty of Illos at
More informationD KL (P Q) := p i ln p i q i
Cheroff-Bouds 1 The Geeral Boud Let P 1,, m ) ad Q q 1,, q m ) be two dstrbutos o m elemets, e,, q 0, for 1,, m, ad m 1 m 1 q 1 The Kullback-Lebler dvergece or relatve etroy of P ad Q s defed as m D KL
More informationTHE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY, Series A, OF THE ROMANIAN ACADEMY Volume 9, Number 3/2008, pp
THE PUBLISHIN HOUSE PROCEEDINS OF THE ROMANIAN ACADEMY, Seres A, OF THE ROMANIAN ACADEMY Volume 9, Number 3/8, THE UNITS IN Stela Corelu ANDRONESCU Uversty of Pteşt, Deartmet of Mathematcs, Târgu Vale
More informationStrong Convergence of Weighted Averaged Approximants of Asymptotically Nonexpansive Mappings in Banach Spaces without Uniform Convexity
BULLETIN of the MALAYSIAN MATHEMATICAL SCIENCES SOCIETY Bull. Malays. Math. Sc. Soc. () 7 (004), 5 35 Strog Covergece of Weghted Averaged Appromats of Asymptotcally Noepasve Mappgs Baach Spaces wthout
More information18.413: Error Correcting Codes Lab March 2, Lecture 8
18.413: Error Correctg Codes Lab March 2, 2004 Lecturer: Dael A. Spelma Lecture 8 8.1 Vector Spaces A set C {0, 1} s a vector space f for x all C ad y C, x + y C, where we take addto to be compoet wse
More informationfor each of its columns. A quick calculation will verify that: thus m < dim(v). Then a basis of V with respect to which T has the form: A
Desty of dagoalzable square atrces Studet: Dael Cervoe; Metor: Saravaa Thyagaraa Uversty of Chcago VIGRE REU, Suer 7. For ths etre aer, we wll refer to V as a vector sace over ad L(V) as the set of lear
More informationStrong Laws of Large Numbers for Fuzzy Set-Valued Random Variables in Gα Space
Advaces Pure Matheatcs 26 6 583-592 Publshed Ole August 26 ScRes http://wwwscrporg/oural/ap http://dxdoorg/4236/ap266947 Strog Laws of Large Nubers for uzzy Set-Valued Rado Varables G Space Lae She L Gua
More informationNONDIFFERENTIABLE MATHEMATICAL PROGRAMS. OPTIMALITY AND HIGHER-ORDER DUALITY RESULTS
HE PUBLISHING HOUSE PROCEEDINGS OF HE ROMANIAN ACADEMY, See A, OF HE ROMANIAN ACADEMY Volue 9, Nube 3/8,. NONDIFFERENIABLE MAHEMAICAL PROGRAMS. OPIMALIY AND HIGHER-ORDER DUALIY RESULS Vale PREDA Uvety
More informationOn Convergence a Variation of the Converse of Fabry Gap Theorem
Scece Joural of Appled Matheatcs ad Statstcs 05; 3(): 58-6 Pulshed ole Aprl 05 (http://www.scecepulshggroup.co//sas) do: 0.648/.sas.05030.5 ISSN: 376-949 (Prt); ISSN: 376-953 (Ole) O Covergece a Varato
More informationPROJECTION PROBLEM FOR REGULAR POLYGONS
Joural of Mathematcal Sceces: Advaces ad Applcatos Volume, Number, 008, Pages 95-50 PROJECTION PROBLEM FOR REGULAR POLYGONS College of Scece Bejg Forestry Uversty Bejg 0008 P. R. Cha e-mal: sl@bjfu.edu.c
More informationSebastián Martín Ruiz. Applications of Smarandache Function, and Prime and Coprime Functions
Sebastá Martí Ruz Alcatos of Saradache Fucto ad Pre ad Core Fuctos 0 C L f L otherwse are core ubers Aerca Research Press Rehoboth 00 Sebastá Martí Ruz Avda. De Regla 43 Choa 550 Cadz Sa Sarada@telele.es
More informationDuality for a Control Problem Involving Support Functions
Appled Matheatcs, 24, 5, 3525-3535 Pblshed Ole Deceber 24 ScRes. http://www.scrp.org/oral/a http://d.do.org/.4236/a.24.5233 Dalty for a Cotrol Proble volvg Spport Fctos. Hsa, Abdl Raoof Shah 2, Rsh K.
More informationASYMPTOTIC STABILITY OF TIME VARYING DELAY-DIFFERENCE SYSTEM VIA MATRIX INEQUALITIES AND APPLICATION
Joural of the Appled Matheatcs Statstcs ad Iforatcs (JAMSI) 6 (00) No. ASYMPOIC SABILIY OF IME VARYING DELAY-DIFFERENCE SYSEM VIA MARIX INEQUALIIES AND APPLICAION KREANGKRI RACHAGI Abstract I ths paper
More informationLINEARLY CONSTRAINED MINIMIZATION BY USING NEWTON S METHOD
Jural Karya Asl Loreka Ahl Matematk Vol 8 o 205 Page 084-088 Jural Karya Asl Loreka Ahl Matematk LIEARLY COSTRAIED MIIMIZATIO BY USIG EWTO S METHOD Yosza B Dasrl, a Ismal B Moh 2 Faculty Electrocs a Computer
More information= lim. (x 1 x 2... x n ) 1 n. = log. x i. = M, n
.. Soluto of Problem. M s obvously cotuous o ], [ ad ], [. Observe that M x,..., x ) M x,..., x ) )..) We ext show that M s odecreasg o ], [. Of course.) mles that M s odecreasg o ], [ as well. To show
More informationKURODA S METHOD FOR CONSTRUCTING CONSISTENT INPUT-OUTPUT DATA SETS. Peter J. Wilcoxen. Impact Research Centre, University of Melbourne.
KURODA S METHOD FOR CONSTRUCTING CONSISTENT INPUT-OUTPUT DATA SETS by Peter J. Wlcoxe Ipact Research Cetre, Uversty of Melboure Aprl 1989 Ths paper descrbes a ethod that ca be used to resolve cossteces
More informationJournal of Mathematical Analysis and Applications
J. Math. Aal. Appl. 365 200) 358 362 Cotets lsts avalable at SceceDrect Joural of Mathematcal Aalyss ad Applcatos www.elsever.com/locate/maa Asymptotc behavor of termedate pots the dfferetal mea value
More informationOn the Subdifferentials of Quasiconvex and Pseudoconvex Functions and Cyclic Monotonicity*
Ž. Joural of Mathematcal Aalyss ad Applcatos 237, 3042 1999 Artcle ID jmaa.1999.6437, avalable ole at http:www.dealbrary.com o O the Subdfferetals of Quascovex ad Pseudocovex Fuctos ad Cyclc Mootocty*
More informationComplete Convergence and Some Maximal Inequalities for Weighted Sums of Random Variables
Joural of Sceces, Islamc Republc of Ira 8(4): -6 (007) Uversty of Tehra, ISSN 06-04 http://sceces.ut.ac.r Complete Covergece ad Some Maxmal Iequaltes for Weghted Sums of Radom Varables M. Am,,* H.R. Nl
More informationFactorization of Finite Abelian Groups
Iteratoal Joural of Algebra, Vol 6, 0, o 3, 0-07 Factorzato of Fte Abela Grous Khald Am Uversty of Bahra Deartmet of Mathematcs PO Box 3038 Sakhr, Bahra kamee@uobedubh Abstract If G s a fte abela grou
More informationEntropy ISSN by MDPI
Etropy 2003, 5, 233-238 Etropy ISSN 1099-4300 2003 by MDPI www.mdp.org/etropy O the Measure Etropy of Addtve Cellular Automata Hasa Aı Arts ad Sceces Faculty, Departmet of Mathematcs, Harra Uversty; 63100,
More informationPartition Optimization for a Random Process Realization to Estimate its Expected Value
SERBIAN JOURNAL OF ELECTRICAL ENGINEERING Vol 4, No, Octoer 7, -4 UDC: 698:49]:59 DOI: htts://doorg/98/sjee7m Partto Otzato for a Rado Process Realzato to Estate ts Exected Value Vladr Marchu, Igor Shrafel,
More informationResearch Article A New Iterative Method for Common Fixed Points of a Finite Family of Nonexpansive Mappings
Hdaw Publshg Corporato Iteratoal Joural of Mathematcs ad Mathematcal Sceces Volume 009, Artcle ID 391839, 9 pages do:10.1155/009/391839 Research Artcle A New Iteratve Method for Commo Fxed Pots of a Fte
More informationInternational Journal of Mathematical Archive-5(8), 2014, Available online through ISSN
Iteratoal Joural of Mathematcal Archve-5(8) 204 25-29 Avalable ole through www.jma.fo ISSN 2229 5046 COMMON FIXED POINT OF GENERALIZED CONTRACTION MAPPING IN FUZZY METRIC SPACES Hamd Mottagh Golsha* ad
More information1 Lyapunov Stability Theory
Lyapuov Stablty heory I ths secto we cosder proofs of stablty of equlbra of autoomous systems. hs s stadard theory for olear systems, ad oe of the most mportat tools the aalyss of olear systems. It may
More informationApplication of Generating Functions to the Theory of Success Runs
Aled Mathematcal Sceces, Vol. 10, 2016, o. 50, 2491-2495 HIKARI Ltd, www.m-hkar.com htt://dx.do.org/10.12988/ams.2016.66197 Alcato of Geeratg Fuctos to the Theory of Success Rus B.M. Bekker, O.A. Ivaov
More informationCOMPROMISE HYPERSPHERE FOR STOCHASTIC DOMINANCE MODEL
Sebasta Starz COMPROMISE HYPERSPHERE FOR STOCHASTIC DOMINANCE MODEL Abstract The am of the work s to preset a method of rakg a fte set of dscrete radom varables. The proposed method s based o two approaches:
More informationNeural Networks for Nonlinear Fractional Programming
Iteratoal Joural o Scetc & Egeerg Research, Volue, Issue, Deceber- ISSN 9-558 Neural Networks or Nolear Fractoal Prograg S.K Bso, G. Dev, Arabda Rath Abstract - Ths paper presets a eural etwork or solvg
More informationTESTS BASED ON MAXIMUM LIKELIHOOD
ESE 5 Toy E. Smth. The Basc Example. TESTS BASED ON MAXIMUM LIKELIHOOD To llustrate the propertes of maxmum lkelhood estmates ad tests, we cosder the smplest possble case of estmatg the mea of the ormal
More informationTHE TRUNCATED RANDIĆ-TYPE INDICES
Kragujeac J Sc 3 (00 47-5 UDC 547:54 THE TUNCATED ANDIĆ-TYPE INDICES odjtaba horba, a ohaad Al Hossezadeh, b Ia uta c a Departet of atheatcs, Faculty of Scece, Shahd ajae Teacher Trag Uersty, Tehra, 785-3,
More informationA Conventional Approach for the Solution of the Fifth Order Boundary Value Problems Using Sixth Degree Spline Functions
Appled Matheatcs, 1, 4, 8-88 http://d.do.org/1.4/a.1.448 Publshed Ole Aprl 1 (http://www.scrp.org/joural/a) A Covetoal Approach for the Soluto of the Ffth Order Boudary Value Probles Usg Sth Degree Sple
More informationRelations to Other Statistical Methods Statistical Data Analysis with Positive Definite Kernels
Relatos to Other Statstcal Methods Statstcal Data Aalyss wth Postve Defte Kerels Kej Fukuzu Isttute of Statstcal Matheatcs, ROIS Departet of Statstcal Scece, Graduate Uversty for Advaced Studes October
More informationThe Bijectivity of the Tight Frame Operators in Lebesgue Spaces
Iteratoal Joural o Aled Physcs ad atheatcs Vol 4 No Jauary 04 The Bectvty o the Tght Frae Oerators Lebesgue Saces Ka-heg Wag h-i Yag ad Kue-Fag hag Abstract The otvato o ths dssertato aly s whch ae rae
More informationStandard Deviation for PDG Mass Data
4 Dec 06 Stadard Devato for PDG Mass Data M. J. Gerusa Retred, 47 Clfde Road, Worghall, HP8 9JR, UK. gerusa@aol.co, phoe: +(44) 844 339754 Abstract Ths paper aalyses the data for the asses of eleetary
More informationHájek-Rényi Type Inequalities and Strong Law of Large Numbers for NOD Sequences
Appl Math If Sc 7, No 6, 59-53 03 59 Appled Matheatcs & Iforato Sceces A Iteratoal Joural http://dxdoorg/0785/as/070647 Háje-Réy Type Iequaltes ad Strog Law of Large Nuers for NOD Sequeces Ma Sogl Departet
More informationRademacher Complexity. Examples
Algorthmc Foudatos of Learg Lecture 3 Rademacher Complexty. Examples Lecturer: Patrck Rebesch Verso: October 16th 018 3.1 Itroducto I the last lecture we troduced the oto of Rademacher complexty ad showed
More informationDerived Limits in Quasi-Abelian Categories
Prépublcatos Mathématques de l Uversté Pars-Nord Derved Lmts Quas-Abela Categores by Fabee Prosmas 98-10 March 98 Laboratore Aalyse, Géométre et Applcatos, UMR 7539 sttut Gallée, Uversté Pars-Nord 93430
More informationON WEIGHTED INTEGRAL AND DISCRETE OPIAL TYPE INEQUALITIES
M atheatcal I equaltes & A pplcatos Volue 19, Nuber 4 16, 195 137 do:1.7153/a-19-95 ON WEIGHTED INTEGRAL AND DISCRETE OPIAL TYPE INEQUALITIES MAJA ANDRIĆ, JOSIP PEČARIĆ AND IVAN PERIĆ Coucated by C. P.
More information{ }{ ( )} (, ) = ( ) ( ) ( ) Chapter 14 Exercises in Sampling Theory. Exercise 1 (Simple random sampling): Solution:
Chapter 4 Exercses Samplg Theory Exercse (Smple radom samplg: Let there be two correlated radom varables X ad A sample of sze s draw from a populato by smple radom samplg wthout replacemet The observed
More informationChannel Models with Memory. Channel Models with Memory. Channel Models with Memory. Channel Models with Memory
Chael Models wth Memory Chael Models wth Memory Hayder radha Electrcal ad Comuter Egeerg Mchga State Uversty I may ractcal etworkg scearos (cludg the Iteret ad wreless etworks), the uderlyg chaels are
More informationSemi-Riemann Metric on. the Tangent Bundle and its Index
t J Cotem Math Sceces ol 5 o 3 33-44 Sem-Rema Metrc o the Taet Budle ad ts dex smet Ayha Pamuale Uversty Educato Faculty Dezl Turey ayha@auedutr Erol asar Mers Uversty Art ad Scece Faculty 33343 Mers Turey
More informationGrowth of a Class of Plurisubharmonic Function in a Unit Polydisc I
Issue, Volue, 7 5 Growth of a Class of Plursubharoc Fucto a Ut Polydsc I AITASU SINHA Abstract The Growth of a o- costat aalytc fucto of several coplex varables s a very classcal cocept, but for a fte
More informationThe Mathematical Appendix
The Mathematcal Appedx Defto A: If ( Λ, Ω, where ( λ λ λ whch the probablty dstrbutos,,..., Defto A. uppose that ( Λ,,..., s a expermet type, the σ-algebra o λ λ λ are defed s deoted by ( (,,...,, σ Ω.
More informationSolving Constrained Flow-Shop Scheduling. Problems with Three Machines
It J Cotemp Math Sceces, Vol 5, 2010, o 19, 921-929 Solvg Costraed Flow-Shop Schedulg Problems wth Three Maches P Pada ad P Rajedra Departmet of Mathematcs, School of Advaced Sceces, VIT Uversty, Vellore-632
More informationON THE LOGARITHMIC INTEGRAL
Hacettepe Joural of Mathematcs ad Statstcs Volume 39(3) (21), 393 41 ON THE LOGARITHMIC INTEGRAL Bra Fsher ad Bljaa Jolevska-Tueska Receved 29:9 :29 : Accepted 2 :3 :21 Abstract The logarthmc tegral l(x)
More informationOrder Nonlinear Vector Differential Equations
It. Joural of Math. Aalyss Vol. 3 9 o. 3 39-56 Coverget Power Seres Solutos of Hgher Order Nolear Vector Dfferetal Equatos I. E. Kougas Departet of Telecoucato Systes ad Networs Techologcal Educatoal Isttute
More information2SLS Estimates ECON In this case, begin with the assumption that E[ i
SLS Estmates ECON 3033 Bll Evas Fall 05 Two-Stage Least Squares (SLS Cosder a stadard lear bvarate regresso model y 0 x. I ths case, beg wth the assumto that E[ x] 0 whch meas that OLS estmates of wll
More informationExtend the Borel-Cantelli Lemma to Sequences of. Non-Independent Random Variables
ppled Mathematcal Sceces, Vol 4, 00, o 3, 637-64 xted the Borel-Catell Lemma to Sequeces of No-Idepedet Radom Varables olah Der Departmet of Statstc, Scece ad Research Campus zad Uversty of Tehra-Ira der53@gmalcom
More informationInternational Journal of Mathematical Archive-3(5), 2012, Available online through ISSN
Iteratoal Joural of Matheatcal Archve-(5,, 88-845 Avalable ole through www.a.fo ISSN 9 546 FULLY FUZZY LINEAR PROGRAMS WITH TRIANGULAR FUZZY NUMERS S. Mohaaselv Departet of Matheatcs, SRM Uversty, Kattaulathur,
More informationA COMPARATIVE STUDY OF THE METHODS OF SOLVING NON-LINEAR PROGRAMMING PROBLEM
DAODIL INTERNATIONAL UNIVERSITY JOURNAL O SCIENCE AND TECHNOLOGY, VOLUME, ISSUE, JANUARY 9 A COMPARATIVE STUDY O THE METHODS O SOLVING NON-LINEAR PROGRAMMING PROBLEM Bmal Chadra Das Departmet of Tetle
More informationELEMENTS OF NUMBER THEORY. In the following we will use mainly integers and positive integers. - the set of integers - the set of positive integers
ELEMENTS OF NUMBER THEORY I the followg we wll use aly tegers a ostve tegers Ζ = { ± ± ± K} - the set of tegers Ν = { K} - the set of ostve tegers Oeratos o tegers: Ato Each two tegers (ostve tegers) ay
More informationINTEGRATION THEORY AND FUNCTIONAL ANALYSIS MM-501
INTEGRATION THEORY AND FUNCTIONAL ANALYSIS M.A./M.Sc. Mathematcs (Fal) MM-50 Drectorate of Dstace Educato Maharsh Dayaad Uversty ROHTAK 4 00 Copyrght 004, Maharsh Dayaad Uversty, ROHTAK All Rghts Reserved.
More informationExpanding Super Edge-Magic Graphs
PROC. ITB Sas & Tek. Vol. 36 A, No., 00, 7-5 7 Exadg Suer Edge-Magc Grahs E. T. Baskoro & Y. M. Cholly, Deartet of Matheatcs, Isttut Tekolog Badug Jl. Gaesa 0 Badug 03, Idoesa Eals : {ebaskoro,yus}@ds.ath.tb.ac.d
More informationAn Algorithm for Capacitated n-index Transportation Problem
Iteratoal Joural of Coutatoal Scece a Matheatcs ISSN 974-389 Volue 3, Nuber 3 2), 269-275 Iteratoal Research Publcato House htt://wwwrhouseco A Algorth for Caactate -Ie Trasortato Proble SC Shara a 2 Abha
More informationInvestigation of Partially Conditional RP Model with Response Error. Ed Stanek
Partally Codtoal Radom Permutato Model 7- vestgato of Partally Codtoal RP Model wth Respose Error TRODUCTO Ed Staek We explore the predctor that wll result a smple radom sample wth respose error whe a
More informationThe Arithmetic-Geometric mean inequality in an external formula. Yuki Seo. October 23, 2012
Sc. Math. Japocae Vol. 00, No. 0 0000, 000 000 1 The Arthmetc-Geometrc mea equalty a exteral formula Yuk Seo October 23, 2012 Abstract. The classcal Jese equalty ad ts reverse are dscussed by meas of terally
More informationBivariate Vieta-Fibonacci and Bivariate Vieta-Lucas Polynomials
IOSR Joural of Mathematcs (IOSR-JM) e-issn: 78-78, p-issn: 19-76X. Volume 1, Issue Ver. II (Jul. - Aug.016), PP -0 www.osrjourals.org Bvarate Veta-Fboacc ad Bvarate Veta-Lucas Polomals E. Gokce KOCER 1
More informationIMPROVED GA-CONVEXITY INEQUALITIES
IMPROVED GA-CONVEXITY INEQUALITIES RAZVAN A. SATNOIANU Corresodece address: Deartmet of Mathematcs, Cty Uversty, LONDON ECV HB, UK; e-mal: r.a.satoau@cty.ac.uk; web: www.staff.cty.ac.uk/~razva/ Abstract
More informationOn Optimal Termination Rule for Primal-Dual Algorithm for Semi- Definite Programming
Avalable ole at wwwelagareearchlbrarco Pelaga Reearch Lbrar Advace Aled Scece Reearch 6:4-3 ISSN: 976-86 CODEN USA: AASRFC O Otal Terato Rule or Pral-Dual Algorth or Se- Dete Prograg BO Adejo ad E Ogala
More informationLINEAR RECURRENT SEQUENCES AND POWERS OF A SQUARE MATRIX
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 6 2006, #A12 LINEAR RECURRENT SEQUENCES AND POWERS OF A SQUARE MATRIX Hacèe Belbachr 1 USTHB, Departmet of Mathematcs, POBox 32 El Ala, 16111,
More informationSolving the fuzzy shortest path problem on networks by a new algorithm
Proceedgs of the 0th WSEAS Iteratoal Coferece o FUZZY SYSTEMS Solvg the fuzzy shortest path proble o etworks by a ew algorth SADOAH EBRAHIMNEJAD a, ad REZA TAVAKOI-MOGHADDAM b a Departet of Idustral Egeerg,
More informationThe Number of the Two Dimensional Run Length Constrained Arrays
2009 Iteratoal Coferece o Mache Learg ad Coutg IPCSIT vol.3 (20) (20) IACSIT Press Sgaore The Nuber of the Two Desoal Ru Legth Costraed Arrays Tal Ataa Naohsa Otsua 2 Xuerog Yog 3 School of Scece ad Egeerg
More informationFunctions of Random Variables
Fuctos of Radom Varables Chapter Fve Fuctos of Radom Varables 5. Itroducto A geeral egeerg aalyss model s show Fg. 5.. The model output (respose) cotas the performaces of a system or product, such as weght,
More informationUniform asymptotical stability of almost periodic solution of a discrete multispecies Lotka-Volterra competition system
Iteratoal Joural of Egeerg ad Advaced Research Techology (IJEART) ISSN: 2454-9290, Volume-2, Issue-1, Jauary 2016 Uform asymptotcal stablty of almost perodc soluto of a dscrete multspeces Lotka-Volterra
More informationX ε ) = 0, or equivalently, lim
Revew for the prevous lecture Cocepts: order statstcs Theorems: Dstrbutos of order statstcs Examples: How to get the dstrbuto of order statstcs Chapter 5 Propertes of a Radom Sample Secto 55 Covergece
More informationMAX-MIN AND MIN-MAX VALUES OF VARIOUS MEASURES OF FUZZY DIVERGENCE
merca Jr of Mathematcs ad Sceces Vol, No,(Jauary 0) Copyrght Md Reader Publcatos wwwjouralshubcom MX-MIN ND MIN-MX VLUES OF VRIOUS MESURES OF FUZZY DIVERGENCE RKTul Departmet of Mathematcs SSM College
More informationChapter 5 Properties of a Random Sample
Lecture 6 o BST 63: Statstcal Theory I Ku Zhag, /0/008 Revew for the prevous lecture Cocepts: t-dstrbuto, F-dstrbuto Theorems: Dstrbutos of sample mea ad sample varace, relatoshp betwee sample mea ad sample
More informationD. L. Bricker, 2002 Dept of Mechanical & Industrial Engineering The University of Iowa. CPL/XD 12/10/2003 page 1
D. L. Brcker, 2002 Dept of Mechacal & Idustral Egeerg The Uversty of Iowa CPL/XD 2/0/2003 page Capactated Plat Locato Proble: Mze FY + C X subject to = = j= where Y = j= X D, j =, j X SY, =,... X 0, =,
More informationOn the Solution of a Special Type of Large Scale. Linear Fractional Multiple Objective Programming. Problems with Uncertain Data
Appled Mathematcal Sceces, Vol. 4, 200, o. 62, 3095-305 O the Soluto of a Specal Type of Large Scale Lear Fractoal Multple Obectve Programmg Problems wth Ucerta Data Tarek H. M. Abou-El-Ee Departmet of
More informationChapter 2 - Free Vibration of Multi-Degree-of-Freedom Systems - II
CEE49b Chapter - Free Vbrato of Mult-Degree-of-Freedom Systems - II We ca obta a approxmate soluto to the fudametal atural frequecy through a approxmate formula developed usg eergy prcples by Lord Raylegh
More informationBERNSTEIN COLLOCATION METHOD FOR SOLVING NONLINEAR DIFFERENTIAL EQUATIONS. Aysegul Akyuz Dascioglu and Nese Isler
Mathematcal ad Computatoal Applcatos, Vol. 8, No. 3, pp. 293-300, 203 BERNSTEIN COLLOCATION METHOD FOR SOLVING NONLINEAR DIFFERENTIAL EQUATIONS Aysegul Ayuz Dascoglu ad Nese Isler Departmet of Mathematcs,
More informationThe Necessarily Efficient Point Method for Interval Molp Problems
ISS 6-69 Eglad K Joural of Iformato ad omputg Scece Vol. o. 9 pp. - The ecessarly Effcet Pot Method for Iterval Molp Problems Hassa Mshmast eh ad Marzeh Alezhad + Mathematcs Departmet versty of Ssta ad
More informationA nonsmooth Levenberg-Marquardt method for generalized complementarity problem
ISSN 746-7659 Egla UK Joural of Iformato a Computg Scece Vol. 7 No. 4 0 pp. 67-7 A osmooth Leveberg-Marquart metho for geeralze complemetarty problem Shou-qag Du College of Mathematcs Qgao Uversty Qgao
More informationAbout k-perfect numbers
DOI: 0.47/auom-04-0005 A. Şt. Uv. Ovdus Costaţa Vol.,04, 45 50 About k-perfect umbers Mhály Becze Abstract ABSTRACT. I ths paper we preset some results about k-perfect umbers, ad geeralze two equaltes
More informationNumerical Study for the Fractional Differential Equations Generated by Optimization Problem Using Chebyshev Collocation Method and FDM
Appl. Math. If. Sc. 7, No. 5, 2011-2018 (2013) 2011 Appled Matheatcs & Iforato Sceces A Iteratoal Joural http://dx.do.org/10.12785/as/070541 Nuercal Study for the Fractoal Dfferetal Equatos Geerated by
More informationArithmetic Mean and Geometric Mean
Acta Mathematca Ntresa Vol, No, p 43 48 ISSN 453-6083 Arthmetc Mea ad Geometrc Mea Mare Varga a * Peter Mchalča b a Departmet of Mathematcs, Faculty of Natural Sceces, Costate the Phlosopher Uversty Ntra,
More informationGlobal Optimization for Solving Linear Non-Quadratic Optimal Control Problems
Joural of Appled Matheatcs ad Physcs 06 4 859-869 http://wwwscrporg/joural/jap ISSN Ole: 37-4379 ISSN Prt: 37-435 Global Optzato for Solvg Lear No-Quadratc Optal Cotrol Probles Jghao Zhu Departet of Appled
More information