Volume3 PROGRESSINPHYSICS July,2012 Generalizations of the Distance and Dependent Function − in Extenics to 2D, 3D, and n D FlorentinSmarandache UniversityofNewMexico,MathematicsandScienceDepartment,705GurleyAve.,Gallup,NM87301,USA E-mail:[email protected] Dr.CaiWendefinedinhis1983paper:—thedistanceformulabetweenapointx and 0 aone-dimensional (1D)interval<a,b>; —andthedependence functionwhichgives the degree of dependence of a point with respect to a pair of included 1D-intervals. HispaperinspiredustogeneralizetheExtensionSettotwo-dimensions, i.e. inplane of real numbers R2 where one has a rectangle (instead of a segment of line), deter- minedbytwoarbitrarypointsA(a ,a )and B(b ,b ). AndsimilarlyinR3,whereone 1 2 1 2 hasaprismdeterminedbytwoarbitrarypoints A(a ,a ,a )and B(b ,b ,b ). Wege- 1 2 3 1 2 3 ometricallydefinethelinearand non-linear distance betweenapoint andthe2D and 3D-extension set and the dependent function for a nest of two included 2D and 3D- extensionsets.Linearlyandnon-linearlyattractionpointprinciplestowardstheoptimal pointarepresentedaswell. Thesameprocedurecanbethenusedconsidering,instead ofarectangle,anybounded2D-surfaceandsimilarlyanybounded 3D-solid,andany bounded (n−D)-bodyinRn. Thesegeneralizations arevery important sincetheEx- tensionSetisgeneralizedfromone-dimensionto2,3andevenn-dimensions,therefore moreclassesofapplicationswillresultinconsequence. 1 Introduction ExtensionTheory(or Extenics) was developedby Professor CaiWenin1983bypublishingapapercalledExtensionSet Fig.1: andNon-CompatibleProblems.Itsgoalistosolvecontradic- toryproblemsandalsononconventional,nontraditionalideas in many fields. Extenics is at the confluence of three disci- plines: philosophy, mathematics, and engineering. A con- Fig.2: tradictoryproblemisconvertedbyatransformationfunction intoanon-contradictoryone.Thefunctionsoftransformation are: extension, decomposition, combination, etc. Extenics oritsminimumrangevalue− b−a dependsontheintervalX has many practical applications in Management, Decision- 2 extremitiesaandb,anditoccurswhenthepointx coincides Making,StrategicPlanning,Methodology,DataMining,Ar- withthemidpointoftheinterv(cid:16)alX(cid:17),i.e. x = a+b.0Thecloser tificial Intelligence, Information Systems, Control Theory, 0 2 istheinteriorpoint x tothemidpointoftheinterval<a,b>, 0 etc. Extenicsisbasedonmatter-element,affair-element,and thenegativelylargerisρ(x ,X). 0 relation-element. InFig.1,forinteriorpoint x betweenaand a+b,theex- 0 2 tensiondistanceρ(x ,X)=a−x isthenegativelengthofthe 2 ExtensionDistancein1D-space 0 0 brownlinesegment[leftside]. Whereasforinteriorpoint x 0 Let’susethenotation<a,b>foranykindofclosed,open,or between a+b andb,theextensiondistanceρ(x ,X) = x −b half-closedinterval[a,b],(a,b),(a,b],[a,b). Prof. CaiWen 2 0 0 is the negative length of the blue line segment [right side]. has defined the extension distance between a point x and a 0 Similarly, the furtheris exterior point x with respectto the 0 realintervalX =<a,b>,by closestextremityoftheinterval<a,b>toit(i.e.toeitheraor a+b b−a b),thepositivelylargerisρ(x0,X). ρ(x ,X)= x − − , (1) 0 0 In Fig.2, for exterior point x <a, the extension distance 2 2 0 (cid:12)(cid:12) (cid:12)(cid:12) ρ(x0,X) = a − x0 is the positive length of the brown line whereingeneral: (cid:12) (cid:12) (cid:12) (cid:12) segment[leftside]. Whereasforexteriorpoint x >b,theex- (cid:12) (cid:12) 0 ρ:(R,R2)→(−∞,+∞). (2) tensiondistanceρ(x0,X)= x0−bisthepositivelengthofthe bluelinesegment[rightside]. Algebraically studying this extension distance, we find thatactuallytherangeofitis: 3 PrincipleoftheExtension1D-Distance b−a Geometrically studying this extension distance, we find the ρ(x ,X)∈ − ,+∞ (3) 0 2 following principle that Prof. Cai Wen has used in 1983 " # 54 FlorentinSmarandache.GeneralizationsoftheDistanceandDependentFunctioninExtenicsto2D,3D,andn−D July,2012 PROGRESSINPHYSICS Volume3 definingit: ρ(x ,X)isthegeometricdistancebetweenthepointx 0 0 andtheclosestextremitypointoftheinterval<a,b>to it(goinginthedirectionthatconnects x withtheop- 0 timalpoint),distancetakenasnegativeif x ∈<a,b>, 0 andaspositiveif x ⊂<a,b>. 0 Thisprincipleisveryimportantinordertogeneralizethe extension distance from 1D to 2D (two-dimensional real space), 3D (three-dimensional real space), and n−D (n-dimensionalrealspace). Fig.3: PisaninteriorpointtotherectangleAMBNandtheoptimal The extremity points of interval <a,b> are the point a pointOisinthecenterofsymmetryoftherectangle. and b, which are also the boundary(frontier)of the interval <a,b>. 4 DependentFunctionin1D-Space Prof.CaiWendefinedin1983in1DtheDependentFunction K(y). If one considerstwo intervals X and X, that have no 0 commonendpoint,andX ⊂ X,then: 0 ρ(y,X) K(y)= . (4) ρ(y,X)−ρ(y,X ) 0 Since K(y) was constructedin 1D in termsof the exten- Fig.4:PisanexteriorpointtotherectangleAMBNandtheoptimal siondistanceρ(.,.),wesimplygeneralizeittohigherdimen- pointOisinthecenterofsymmetryoftherectangle. sionsbyreplacingρ(.,.)withthegeneralizedinahigherdi- mension. Thisstep canbedoneinthefollowingway: considering P′astheintersectionpointbetweenthelinePOandthefron- 5 ExtensionDistancein2D-Space tieroftherectangle,andtakenamongtheintersectionpoints Insteadofconsideringasegmentofline ABrepresentingthe that point P′ which is the closest to P; this case is entirely interval <a,b> in 1R, we consider a rectangle AMBN rep- consistentwithDr.Cai’sapproachinthesensethatwhenre- resentingallpointsofitssurfacein2D. Similarlyasfor1D- ducingfroma2D-spaceproblemtotwo1D-spaceproblems, space,therectanglein2D-spacemaybeclosed(i.e.allpoints oneexactlygetshisresult. lyingonitsfrontierbelongtoit),open(i.e. nopointlyingon TheExtension2D-Distance,forP,O,willbe: itsfrontierbelongtoit),orpartiallyclosed(i.e. somepoints ρ (x ,y ),AMBN =d pointP, rectangleAMBN = lyingonitsfrontierbelongtoit, while otherpointslyingon 0 0 itsfrontierdonotbelongtoit). =(cid:0)|PO|−|P′O|=(cid:1)±|PP′(cid:0)|, (cid:1) (5) Let’sconsidertwoarbitrarypointsA(a ,a )andB(b ,b ). 1 2 1 2 i) which is equal to the negative length of the red seg- ThroughthepointsAandBonedrawsparallelstotheaxesof ment|PP′|inFig.3,whenPisinteriortotherectangle theCartesiansystem XY andonethusoneformsarectangle AMBN; AMBNwhoseoneofthediagonalsisjustAB. ii) orequaltozero,whenPliesonthefrontieroftherect- Let’s note by O the midpointof the diagonal AB, but O angleAMBN(i.e.onedgesAM,MB,BN,orNA)since isalsothecenterofsymmetry(intersectionofthediagonals) PcoincideswithP′; oftherectangleAMBN. Thenonecomputesthedistancebe- iii) orequaltothepositivelengthofthebluesegment|PP′| tweenapointP(x ,y )andtherectangleAMBN. Onecando 0 0 in Fig.4, when P is exterior to the rectangle AMBN, thatfollowingthesameprincipleasDr.CaiWendid: where |PO| means the classical 2D-distance between — computethedistancein2D(twodimensions)between thepointPandO,andsimilarlyfor|P′O|and|PP′|. thepointPandthecenterOoftherectangle(intersec- The Extension 2D-Distance, for the optimal point, i.e. tionofrectangle’sdiagonals); P=O,willbe — nextcomputethedistancebetweenthepointPandthe ρ(O,AMBN)=d(pointO, rectangleAMBN)= closestpoint(let’snoteitbyP′)toitonthefrontier(the rectangle’sfouredges)oftherectangleAMBN. =−maxd pointO, pointMonthefrontierofAMBN . (6) (cid:0) (cid:1) FlorentinSmarandache.GeneralizationsoftheDistanceandDependentFunctioninExtenicsto2D,3D,andn−D 55 Volume3 PROGRESSINPHYSICS July,2012 ThelaststepistodevisetheDependentFunctionin2D- where a +b −2y spacesimilarlyasDr.Cai’sdefinedthedependentfunctionin y =y + 2 2 0 (a −x ). (19) 1D. Themidpoint(orcenterofsymmetry)Ohasthecoordi- P′ 0 a1+b1−2x0 1 0 nates a +b a +b 6 Properties O 1 1, 2 2 . (7) 2 2 Asfor1D-distance,thefollowingpropertiesholdin2D: ! Let’scomputethe 6.1 Property1 |PO|−|P′O|. (8) a) (x,y) ∈ Int(AMBN) if ρ[(x,y),AMBN] < 0, where Inthiscase,weextendthelineOPtointersectthefrontier Int(AMBN)meansinteriorofAMBN; oftherectangleAMBN. P′ iscloserto Pthan P′′,therefore b) (x,y) ∈ Fr(AMBN) if ρ[(x,y),AMBN] = 0, where we consider P′. Theequationofthe line PO, thatof course Fr(AMBN)meansfrontierofAMBN; passesthroughthepointsP(x ,y )andO a1+b1,a2+b2 ,is: 0 0 2 2 c) (x,y)< AMBNifρ[(x,y),AMBN]>0. a2+b2 −y (cid:16) (cid:17) y−y = 2 0 (x−x ). (9) 6.2 Property2 0 a1+b1 −x 0 2 0 LetA0M0B0N0andAMBNbetworectangleswhosesidesare Sincethex-coordinateofpointP′isa becauseP′lieson parallel to the axes of the Cartesian system of coordinates, 1 therectangle’sedgeAM,onegetsthey-coordinateofpointP′ suchthattheyhavenocommonendpoints,andA0M0B0N0 ⊂ byasimplesubstitutionofx =a intotheaboveequality: AMBN. We assume they have the same optimal points P′ 1 O ≡ O ≡ O located in the center of symmetry of the two a +b −2y 1 2 y =y + 2 2 0 (a −x ). (10) rectangles. Then for any point (x,y) ⊂ R2 one has P′ 0 a +b −2x 1 0 1 1 0 ρ[(x,y),A M B N ]>ρ[(x,y),AMBN].SeeFig.5. 0 0 0 0 ThereforeP′hasthecoordinates a +b −2y P′ x =a , y =y + 2 2 0 (a −x ) . (11) P′ 1 P′ 0 a +b −2x 1 0 " 1 1 0 # Thedistance a +b 2 a +b 2 d(PQ)=|PQ|= x − 1 1 + y − 2 2 , (12) 0 0 s 2 2 ! ! whilethedistance d(P′,Q)=|P′Q|= = s a1− a1+2b1 2+ yP′ − a2+2b2 2 = OFi2g.≡5O: Tlwocoatiendcliundtehdeirreccotamngmleosnwceitnhtetrheofsasymmemopettrimy.alpointsO1 ≡ ! ! a −b 2 a +b 2 = 1 1 + y − 2 2 . (13) 7 Dependent2D-Function P′ s 2 2 ! ! LetA M B N andAMBNbetworectangleswhosesidesare 0 0 0 0 Also,thedistance parallel to the axes of the Cartesian system of coordinates, suchthattheyhavenocommonendpoints,andA M B N ⊂ 0 0 0 0 d(PP′)=|PP′|= (a1−x0)2+(yP′ −y0)2. (14) AMBN. q TheDependent2D-Functionformulais: WhencetheExtension2D-distanceformula ρ[(x,y),AMBN] ρ (x ,y ), AMBN = K = . (20) 0 0 2D(x,y) ρ[(x,y),AMBN,]−ρ[(x,y),A M B N ] 0 0 0 0 =d P(x ,y ), A(a ,a )MB(b ,b )N = (cid:2) 0 0 (cid:3)1 2 1 2 =|P(cid:2)Q|−|P′Q| (cid:3) (15) 7.1 Property3 = x0−a1+2b1 2+y0−a2+2b2 2− a1−2b1 2+yP′−a2+2b2 2 (16) Again, similarly to the Dependent Function in 1D-space, r r onehas: =±(cid:16)|PP′| (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (17) a) If(x,y)∈Int(A M B N ),thenK >1; 0 0 0 0 2D(x,y) =± (a1−x0)2+(yP′ −y0)2, (18) b) If(x,y)∈Fr(A0M0B0N0),thenK2D(x,y)=1; q 56 FlorentinSmarandache.GeneralizationsoftheDistanceandDependentFunctioninExtenicsto2D,3D,andn−D July,2012 PROGRESSINPHYSICS Volume3 c) If(x,y)∈Int(AMBN−A M B N ), 0 0 0 0 then0< K <1; 2D(x,y) d) If(x,y)∈Fr(AMBN),thenK =0; 2D(x,y) e) If(x,y)< AMBN,thenK2D(x,y)<0. 8 GeneralCasein2D-Space Onecanreplacetherectanglesbyanyfinitesurfaces,bounded byclosedcurvesin2D-space,andonecanconsideranyop- timalpointO (notnecessarilythe symmetrycenter). Again, weassumetheoptimalpointsarethesameforthisnestoftwo surfaces. SeeFig.6. Fig.7:TheoptimalpointOasanattractionpointforallotherpoints P ,P ,...,P intheuniverseofdiscourseR2. 1 2 8 10 Remark1 Another possible way, for computing the distance between the point P and the closestpoint P′ to iton the frontier(the rectangle’sfouredges)oftherectangleAMBN,wouldbeby drawingaperpendicular(orageodesic)fromPontotheclos- est rectangle’sedge, anddenotingby P′ the intersectionbe- tweentheperpendicular(geodesic)andtherectangle’sedge. And similarly if one has an arbitrary set S in the 2Dspace, boundedbyaclosedurve.Onecomputes Fig.6:Twoincludedarbitraryboundedsurfaceswiththesameopti- malpointssituatedintheircommoncenterofsymmetry. d(P,S)=Inf |PQ| (21) Q∈S asintheclassicalmathematics. 9 LinearAttractionPointPrinciple WeintroducetheAttractionPointPrinciple,whichisthefol- 11 ExtensionDistancein3D-Space lowing: Wefurthergeneralizeto3D-spacetheExtensionSetandthe LetS be a given set in the universeof discourse U, and DependentFunction. Assumewehavetwopoints(a ,a ,a ) 1 2 3 the optimalpointO ⊂ S. Then each point P(x ,x ,...,x ) 1 2 n and(b ,b ,b )inD. DrawingthroughAendBparallelplanes 1 2 3 fromthe universeof discourse tendstowards, or is attracted totheplanes’axes(XY,XZ,YZ)intheCartesiansystemXYZ by, the optimal point O, because the optimal point O is an we get a prism AM M M BN N N (with eight vertices) 1 2 3 1 2 3 ideal of each point. That’s why one computes the exten- whoseoneofthetransversaldiagonalsisjustthelinesegment sion (n−D)-distance between the point P and the set S as AB. Let’snotebyOthemidpointofthetransversediagonal ρ[(x ,x ,...,x ),S]onthedirectiondeterminedbythepoint 1 2 n AB,butOisalsothecenterofsymmetryoftheprism. PandtheoptimalpointO,oronthelinePO,i.e.: Therefore, from the line segment AB in 1D-space, to a) ρ[(x ,x ,...,x ),S] is the negative distance between a rectangle AMBN in 2D-space, and now to a prism 1 2 n Pandthesetfrontier,ifPisinsidethesetS; AM M M BN N N in3D-space. Similarlyto1D-and2D- 1 2 3 1 2 3 b) ρ[(x ,x ,...,x ),S]=0,ifPliesonthefrontierofthe space, the prism may be closed (i.e. all points lying on its 1 2 n setS; frontierbelongtoit),open(i.e. nopointlyingonitsfrontier belongtoit),orpartiallyclosed(i.e. somepointslyingonits c) ρ[(x ,x ,...,x ),S]isthepositivedistancebetweenP 1 2 n frontier belong to it, while other points lying on its frontier andthesetfrontier,ifPisoutsidetheset. donotbelongtoit). It is a king of convergence/attraction of each point to- Then one computes the distance between a point wardstheoptimalpoint.Thereareclassesofexampleswhere P(x ,y ,z )andtheprismAM M M BN N N . Onecando suchattractionpointprincipleworks.Ifthisprincipleisgood 0 0 0 1 2 3 1 2 3 thatfollowingthesameprincipleasDr.Cai’s: inallcases,thenthereisnoneedtotakeintoconsiderationthe — computethedistancein3D(twodimensions)between centerofsymmetryofthesetS,sinceforexampleifwehave thepointPandthecenterOoftheprism(intersection a 2D piece which has heterogeneous material density, then its center of weight (barycenter)is differentfrom the center ofprism’stransversediagonals); ofsymmetry. Let’sseebelowsuchexampleinthe2D-space: — nextcomputethedistancebetweenthepointPandthe Fig.7. closestpoint(let’snoteitbyP′)toitonthefrontierof FlorentinSmarandache.GeneralizationsoftheDistanceandDependentFunctioninExtenicsto2D,3D,andn−D 57 Volume3 PROGRESSINPHYSICS July,2012 theprismAM M M BN N N (theprism’slateralsur- AM M M BN N N . We assume they have the same opti- 1 2 3 1 2 3 1 2 3 1 2 3 face);consideringP′ astheintersectionpointbetween malpointsO ≡O ≡Olocatedinthecenterofsymmetryof 1 2 the line OP and the frontier of the prism, and taken thetwoprisms. amongtheintersectionpointsthatpointP′whichisthe Thenforanypoint(x,y,z)∈R3onehas closest to P; this case is entirely consistent with Dr. Cai’s approach in the sense that when reducing from ρ[(x,y,z),A0M01M02M03B0N01N02N]03 > 3D-spaceto1D-spaceonegetsexactlyDr.Cai’sresult; ρ[(x,y,z)AM M M BN N N ]. 1 2 3 1 2 3 — the Extension 3D-Distance d(P,AM M M BN N N ) 1 2 3 1 2 3 isd(P,AM M M BN N N ) = |PO|−|P′O| = ±|PP′|, 1 2 3 1 2 3 14 TheDependent3D-Function where |PO| means the classical distance in 3D-space betweenthepointPandO,andsimilarlyfor|P′O|and ThelaststepistodevisetheDependentFunctionin3D-space |PP′|. SeeFig.8. similarly to Dr. Cai’s definition of the dependent function in1D-space. LettheprismsA M M M B N N N and 0 01 02 03 0 01 02 03 AM M M BN N N be two prisms whose faces are paral- 1 2 3 1 2 3 lel to the axes of the Cartesian system of coordinates XYZ, suchthattheyhavenocommonendpointsinsuchawaythat A M M M B N N N ⊂ AM M M3BN N N . Weas- 0 01 02 03 0 01 02 03 1 2 1 2 3 sumetheyhavethesameoptimalpointsO ≡O ≡Olocated 1 2 inthecenterofsymmetryofthesetwoprisms. TheDependent3D-Functionformulais: K = ρ[(x,y,z),AM M M BN N N ] × 3D(x,y,z) 1 2 3 1 2 3 × ρ[(x,y,(cid:16)z),AM M M BN N N ,]− (cid:17) 1 2 3 1 2 3 −ρ(cid:16) [(x,y,z),A M M M BN N N ] −1. (22) 0 01 02 03 01 02 03 (cid:17) 15 Property6 Again, similarly to the DependentFunctionin 1D- and 2D- spaces,onehas: a) If(x,y,z)∈Int(A M M M B N N N ), 0 01 02 03 0 01 02 03 thenK (x,y,z)>1; 3D Fig.8: Extension3D-Distancebetweenapointandaprism,where b) If(x,y,z)∈Fr(A M M M B N N N ), 0 01 02 03 0 01 02 03 Oistheoptimalpointcoincidingwiththecenterofsymmetry. thenK (x,y,z)=1; 3D c) If(x,y,z)∈Int(AM M M BN N N − 1 2 3 1 2 3 −A M M M B N N N ), 12 Property4 0 01 02 03 0 01 02 03 then0<K (x,y,z)<1; 3D a) (x,y,z)∈Int(AM M M BN N N ) 1 2 3 1 2 3 d) If(x,y,z)∈Fr(AM M M BN N N ), ifρ[(x,y,z),AM M M BN N N ]<0, 1 2 3 1 2 3 1 2 3 1 2 3 thenK (x,y,z)=0; whereInt(AM M M BN N N )meansinterior 3D 1 2 3 1 2 3 e) If(x,y,z)< AM M M BN N N , ofAM M M BN N N ; 1 2 3 1 2 3 1 2 3 1 2 3 thenK (x,y,z)<0. b) (x,y,z)∈Fr(AM M M BN N N ) 3D 1 2 3 1 2 3 ifρ[(x,y,z),AM M M BN N N ]=0 1 2 3 1 2 3 16 GeneralCasein3D-Space meansfrontierofAM M M BN N N ; 1 2 3 1 2 3 Onecanreplacetheprismsbyanyfinite3D-bodies,bounded c) (x,y,z)< AM M M BN N N 1 2 3 1 2 3 by closed surfaces, and one considers any optimal point O ifρ[(x,y,z),AM M M BN N N ]>0. 1 2 3 1 2 3 (not necessarily the centers of surfaces’ symmetry). Again, we assume the optimal points are the same for this nest of 13 Property5 two3D-bodies. Let A M M M B N N N and AM M M BN N N 0 01 02 03 0 01 02 03 1 2 3 1 2 3 17 Remark2 be two prisms whose sides are parallel to the axes of the Cartesian system of coordinates, such that they have no Another possible way, for computing the distance between common end points, and A M M M B N N N ⊂ thepointPandtheclosestpointP′toitonthefrontier(lateral 0 01 02 03 0 01 02 03 58 FlorentinSmarandache.GeneralizationsoftheDistanceandDependentFunctioninExtenicsto2D,3D,andn−D July,2012 PROGRESSINPHYSICS Volume3 surface) of the prism AM M M BN N N is by drawing a 1 2 3 1 2 3 perpendicular(orageodesic)fromPontotheclosestprism’s face,anddenotingbyP′theintersectionbetweentheperpen- dicular(geodesic)andtheprism’sface. Andsimilarlyifonehasanarbitraryfinitebody Binthe 3D-space,boundedbysurfaces.Onecomputesasinclassical mathematics: d(P,B)=Inf |PB|. (23) Q∈B 18 LinearAttractionPointPrinciplein3D-Space Fig.10:Non-LinearAttractionPointPrincipleforanybounded3D- body. two points P and P′; Fr(S) meansthefrontierofsetS; and |OP′| means the line segment between the points O and P′ (theextremitypointsOandP′included),thereforeP∈|OP′| means that P lies on the line OP′, in between the points O andP′. For P coinciding with O, one defined the distance be- tweentheoptimalpointOandthesetS asthenegativelymax- imumdistance(tobeinconcordancewiththe1D-definition). Fig.9:LinearAttractionPointPrincipleforanybounded3D-body. AndtheExtensionNon-Linear(n−D)-Distancebetween pointPandsetS,is: 19 Non-LinearAttractionPointPrinciplein3D-Space, −d (P,P′), P,0, P∈c(OP′) andin(n−D)-Space c P′∈Fr(S) TdLeherteg’rsoesnmeoeitgblheintloebwaerlssypuabcchyeseuxpwaomhnepsrleoemtfheoersappteotrcianicfittscioPnnopnwh-hleionnseoeamrtrecanujaervcutenos--. ρc(P,S)= d−cPm(′P∈Fa,r(xPS)d′)c,(P,M), PP,=00, P′ ∈c(OP) (25) lriineseaorf3aDtt-rcauctrivoens.towardstheoptimalpointfiollowsomenon- where meansthePe′x∈Ftre(Sn),sMi∈oc(nO)distance as measured along the curve c; O is the optimal point (or non-linearly attraction 20 (n−D)-Space point); the points are attracting by the optimal point on tra- Ingeneral,inauniverseofdiscourseU,let’shavean(n−D)- jectories described by an injective curve c; dc(P,P′) means set S and a point P. Then the Extension Linear (n−D)- the non-linearly (n−D)-distance between two points P and DistancebetweenpointPandsetS,is: P′,orthearclengthofthecurvecbetweenthepointsPand P′;Fr(S)meansthefrontierofsetS;andc(OP′)meansthe −d(P,P′), P,0, P∈|OP′| curve segment between the points O and P′ (the extremity P′∈Fr(S) ρ(P,S)= d(P,P′), P,0, P′ ∈|OP| (24) pointsOandP′ included),thereforeP ∈ (OP′)meansthatP  −Pm′∈Far(xS)d(P,M), P=0 liesFoonrthPeccouirnvceidciinngbwetiwtheeOn,thoenpeodinetfisnOedanthdePd′.istance be- where O is the optimP′a∈lFr(pS)oint (or linearly attraction point); mtwaexeinmtuhmecouprtvimilianleparodinisttaOncaen(dtothbeeisnetcoSncaosrdthaencneewgaitthivtehlye d(P,P′)meanstheclassicallinearly(n−D)-distancebetween 1D-definition). FlorentinSmarandache.GeneralizationsoftheDistanceandDependentFunctioninExtenicsto2D,3D,andn−D 59 Volume3 PROGRESSINPHYSICS July,2012 Ingeneral,inauniverseofdiscourseU,let’shaveanest 1) ρ[(x ,x ,...,x ),S] is the negative distance between 1 2 n of two (n−D)-sets, S ⊂ S , with no common end points, Pandthesetfrontier,ifPisinsidethesetS; 1 2 andapointP. ThentheExtensionLinearDependent(n−D)- 2) ρ[(x ,x ,...,x ),S]=0,ifPliesonthefrontierofthe 1 2 n FunctionreferringtothepointP(x ,x ,...,x )is: 1 2 n setS; K (P)= ρ(P,S2) , (26) 3) ρ[(x1,x2,...,xn),S]isthepositivedistancebetweenP nD ρ(P,S )−ρ(P,S ) andthesetfrontier,ifPisoutsidetheset. 2 1 Wegotthefollowingproperties: where is the previous extension linear (n−D)-distance be- tweenthepointPandthe(n−D)-setS . 4) Itisobviousfromtheabovedefinitionoftheextension 2 And the Extension Non-LinearDependent(n−D)-Func- (n−D)-distance between a point P in the universe of tionreferringtopointP(x ,x ,...,x )alongthecurvecis: discourseandtheextension(n−D)-setS that: 1 2 n i) PointP(x ,x ,...,x )∈Int(S) ρ (P,S ) 1 2 n KnD(P)= ρ (P,Sc)−ρ2(P,S ), (27) ifρ[(x1,x2,...xn),S]<0; c 2 c 1 ii) PointP(x ,x ,...,x )∈Fr(S) 1 2 n where is the previous extension non-linear (n−D)-distance ifρ[(x ,x ,...x ),S]=0; 1 2 n betweenthepointPandthe(n−D)-setS alongthecurvec. 2 iii) PointP(x1,x2,...,x )<S n ifρ[(x ,x ,...x ),S]>0. 21 Remark3 1 2 n Particularcasesofcurvesc couldbeinterestingtostudying, 5) Let S1 and S2 be two extension sets, in the universe forexampleifcareparabolas,orhaveellipticforms,orarcs of discourse U, such that they have no common end ofcircle,etc. Especiallyconsideringthegeodesicswouldbe points, and S1 ⊂ S2. We assume they have the same for manypracticalapplications. Tremendousnumberof ap- optimal points O1 ≡ O2 ≡ O located in their center plicationsof Extenicscouldfollowin alldomainswhereat- ofsymmetry. ThenforanypointP(x1,x2,...,xn)∈U tractionpointswouldexist;theseattractionpointscouldbein onehas: physics(forexample,theearthcenterisanattractionpoint), economics(attractiontowardsa specific product),sociology ρ[(x1,x2,...xn),S2]>ρ[(x1,x2,...xn),S1]. (28) (forexampleattractiontowardsaspecificlifestyle),etc. Then we proceed to the generalization of the dependent 22 Conclusion function from 1D-space to Linear (or Non-Linear) (n−D)- spaceDependentFunction,usingthepreviousnotations. In this paper we introduced the Linear and Non-Linear At- TheLinear(orNon-Linear)Dependent(n−D)-Function tractionPointPrinciple,whichisthefollowing: ofpointP(x ,x ,...,x )alongthecurvec,is: Let S be an arbitrary set in the universe of discourse U 1 2 n of any dimension, and the optimal pointO ∈ S. Then each K (x ,x ,...,x )= ρ [(x ,x ,...x ),S ] × pointP(x ,x ,...,x ), n>1,fromtheuniverseofdiscourse nD 1 2 n c 1 2 n 2 1 2 n (linearlyornon-linearly)tendstowards,orisattractedby,the (cid:16) (cid:17) −1 × ρ [(x ,x ,...x ),S ]−ρ [(x ,x ,...x ),S ] (29) optimal point O, because the optimal point O is an ideal of c 1 2 n 2 c 1 2 n 1 eachpoint. (cid:16) (cid:17) (wherecmaybeacurveorevenaline)whichhasthefollow- It is a king of convergence/attraction of each point to- ingproperty: wards the optimalpoint. There are classes of examplesand applicationswheresuchattractionpointprinciplemayapply. 6) IfpointP(x1,x2,...,xn)∈Int(S1), Ifthisprincipleisgoodinallcases,thenthereisnoneed thenKnD(x1,x2,...,xn)>1; to take into consideration the center of symmetry of the set 7) IfpointP(x ,x ,...,x )∈Fr(S1), 1 2 n S, since for example if we have a 2D factory piece which thenK (x ,x ,...,x )=1; nD 1 2 n hasheterogeneousmaterialdensity,thenitscenterofweight 8) IfpointP(x ,x ,...,x )∈Int(S2−S1), (barycenter)isdifferentfromthecenterofsymmetry. 1 2 n thenK (x ,x ,...,x )∈(0,1); nD 1 2 n Then we generalized in the track of Cai Wen’s idea 9) IfpointP(x ,x ,...,x )∈Int(S2), to extend 1D-set to an extension (n−D)-set, and thus de- 1 2 n thenK (x ,x ,...,x )=0; finedtheLinear (orNon-Linear)Extension(n−D)-Distance nD 1 2 n between a point P(x ,x ,...,x ) and the (n−D)-set S as 10) IfpointP(x ,x ,...,x )<Int(S2), 1 2 n 1 2 n ρ[(x ,x ,...,x ),S] on the linear (or non-linear) direction thenK (x ,x ,...,x )<0. 1 2 n nD 1 2 n determinedby the point P and the optimalpointO (the line PO,orrespectivelythecurvilinearPO)inthefollowingway: SubmittedonJuly15,2012/AcceptedonJuly18,2012 60 FlorentinSmarandache.GeneralizationsoftheDistanceandDependentFunctioninExtenicsto2D,3D,andn−D July,2012 PROGRESSINPHYSICS Volume3 References 1. 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