European Journal of Mathematics and Computer Science Vol. 3 No. 1, 2016 ISSN ISSN

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1 Euroe Jour of Mthemtcs d omuter Scece Vo. No. 6 ISSN ISSN ON AN INVESTIGATION O THE MATRIX O THE SEOND PARTIA DERIVATIVE IN ONE EONOMI DYNAMIS MODE S. I. Hmdov Bu Stte Uverst ABSTRAT The rtce des wth the mode of ecoomc dmcs of the eotef te wth sector. We stud the chge rte of the sectors sttes deedg o chges the rce vector. Ths equbrum vector of the mode s souto of some fucto equto. We stud the roertes of the mtr of the secod rt dervtves of the eft hd sde of ths equto. A estmte s derved for the orm of the growth rte of the dustres rtcur. or ths urose the roertes of Metzer mtr s used. ewords: Ecoomc dmcs equbrum stte mtr. AMS Subject ssfcto: 9B55 INTRODUTION The rtce [] cosders mode of ecoomc dmcs of the eotev te wth sectors.e. t s ssumed tht ech sector roduces sge roduct d vce vers ech roduct s mde b o oe sector. Producto ctvtes of the sectors re descrbed b the roducto fuctos for whch. It s so ssumed tht these fuctos re twce cotuous dfferetbe d strct suerer. Ths mes tht the fuctos re cocve oom uform d stsf the equt -dmeso vector f s dsroortote to. Stte of the -th sector s gve b -th eemet of whch dctes the mout of - th roduct t the dsos of ths sector. As resut of roducto ctvt the vector rt turs to the vector B v v. Dgo mtr B wth dgo vector v v v s ced reservto mtr. Thus eedg t the begg of the tme erod vector to ts ed the sector w j hve the vector v v. Stte of the whoe modeed sstem s reseted b the vector X R the th sector. I stud of the gve mode the mode s used wth fed budget [] U R s stte of Progressve Acdemc Pubshg U Pge 59

2 Euroe Jour of Mthemtcs d omuter Scece Vo. No. 6 ISSN ISSN s some eemet of the coe R U U U U s utt fucto defed b the reto U f f B f. Note tht the fucto U resets the cost of fuds beogg to the corresodg sector wth rocess f f f. the mode stds for the vector gve budget of the th sector. the coordte of whch s These of the vectors forms the equbrum stte the mode f the vectors re soutos of the robem U f m wth the codtos [ ]. et be equbrum stte of the mode. urther the cove rogrmmg robem ~ U. f m Is cosdered b some ~ ~ d f d f s chose suosg tht t s ossbe. The ~ f ~ U f f. The fucto s troduced r f U f. The t s suosed tht q t q f q A the eemet e of the mtr A hs the foowg form Progressve Acdemc Pubshg U Pge 6

3 Euroe Jour of Mthemtcs d omuter Scece Vo. No. 6 ISSN ISSN Progressve Acdemc Pubshg U Pge 6 ~ e e. Some sttemets re gve beow cocerg the mtr. emm. [] B the mtr s vertbe d e q osequece.or o the m dgo of the mtr o zeros. osequece. or the eemets o the m dgo of the mtr re ostve d other eemets re egtve. M resuts et us cosder the cse det. The the mtrces te the forms q d so s vd. rom these retos t s es to deduce the foowg sttemet.

4 Euroe Jour of Mthemtcs d omuter Scece Vo. No. 6 ISSN ISSN et ~ ~. emm. If the the sgs of the coordtes of the vectors cosde wth the sgs of the coordtes of the corresodg vectors ~ = emm. et. The the sgs of the coordtes of the vectors d cosde wth the sgs of the coordtes of the corresodg vectors d The sgs of the coordtes of the vectors m dffer from the sgs of the coordtes of the vectors ~. Oe c mmedte chec the vdt of the foowg sttemet. Theorem.The sgs d vues of the frst coordtes of the vectors re comete defed b the mtr umber d vector ot deed o d d do Note. B the smr theorem s ot true. It resets terest the orm of the growth rte of ech sector serte. To estmte ths w orm some fcts re requred. To rove the sttemet Metzer mtr [] j be used.e. the mtr for whch j. et us setch the roof of the foow emm. j emm 4.et be smmetrc Metzer mtr. The t s vd the equt j A Re for the Metzer mtr t s stsfed A I s some ostve umber A. Note tht ths cse the mtr A w be smmetrc. et be sectr rdus for the mtr. The Progressve Acdemc Pubshg U Pge 6

5 Euroe Jour of Mthemtcs d omuter Scece Vo. No. 6 ISSN ISSN A I A I A I A.. rom ths d the fct tht the orm of the smmetrc oertor cocde wth the sectr rdus t foows the roof of the equt. j emm 5.et the mtr j be gve d j j j j Where s vector wth ostve coordtes. The for the oegtve mtr A I m d vector s true A. O the bss of the we-ow theorem o the estmto of the sectr rdus4 we cocude tht A ξ. Ths mes the estmte for the orm of the mtr. rom ths equt d from emm [] foows the estmte for the orm of the mtr et us cosder the t mode. emm 6. or d s vd the estmto q Progressve Acdemc Pubshg U Pge 6

6 Euroe Jour of Mthemtcs d omuter Scece Vo. No. 6 ISSN ISSN Progressve Acdemc Pubshg U Pge 64 m m j j Here j re the eemets of the mtr A costructed smr to A from emm 5. Theorem.or s vd the estmto q Proof. Substtutg the reto for B to the formu 5 [] d mg some evdet trsformtos we obt the equt B rom ths oe c get the estmto ] [ m. et m G. The we chose the eemets from the coes g g g d te. m m q q

7 Euroe Jour of Mthemtcs d omuter Scece Vo. No. 6 ISSN ISSN Thus for rge eough t s vd the estmto G q cost therefore s vd. REERENES [] Hmdov S.I. Equbrum Mechsms Modes of Reroducto wth ed Budget Amerc Jour of Aed Mthemtcs 5; [] Poterovch V. O the stbt of some of the rocesses of dstrbuto fuds d requto of rces. Mthemtc Ecoomcs d ucto Ass M. Nu 974. [] Gtmher.R. Mtr theor. M. Nu 967 [4] Ootcev V.I. Noer sstemsttstcs. M. Nu 986 Progressve Acdemc Pubshg U Pge 65