Isofunctions and their properties - The RM Santilli Foundation

Isofunctions and their propertiesS. GeorgievSorbonne University, Paris, France and Department of Differential Equations, University of Sofia, Sofia,Bulgaria. E-mail: sgg2000bg@yahoo.comAbstract. The main purpose of this talk is to be made lift of continuous and continuous - differentiable real - valued functionsin Santilli isomathematics. Here we give and prove some of their basic properties.Keywords: iso-calculus, iso-continuous isofunction, iso-derivative.PACS: 03.65.-w, 45.20.JjINTRODUCTIONLet ˆF R be the field of the isonumbers â for which a ∈ R and basic unit Î 1 , with ˆF N we will denote the field of theisonumbers â for which a ∈ N and basic unit Î 1 , With ˆF C we will denote the field of the isonumbers â for which a ∈ Cand basic unit Î 1 , with ˆF K we will denote either ˆF C or ˆF R and basic unit Î 1 .In [1] was introduced the term isofunction with corresponding isounit Î = 1ˆT (x) , as we will suppose that ˆT (x) isenough time continuous- differentiable function. Here we will consider in details the term isonumber isofunction.Let ˆD and Ŷ be given isosets.Definition. We will tell that in the isoset ˆD is defined the isofunction ˆf if of every isoelement ˆx is compared uniqueisoelement ŷ ∈ Ŷ . In this case we will use the notationorŷ = ˆf ∧ ( ˆx) := ˆf (T 1 ˆx) = f (x)T (x) ,ˆf : ˆD −→Ŷ ˆ .The isoelement ˆx will be called isoargument of the isofunction ˆf or isoindepenedent isovariable, and its isoimageŷ = ˆf ∧ ( ˆx) will be called isodependent isovariable or isovalue of the isofunction ˆf . The isosetˆf ∧ ( ˆD) = { ˆf ∧ ( ˆx) : ˆx ∈ ˆD}will be called isocodomain of isovalues of the isofunction ˆf . The isoset ˆD will be called isodomain of the isofunctionˆf .Every isobijection ˆf : ˆD −→ Ŷ has an isoinverse isofunction f −1 ˆ : Ŷ −→ ˆ ˆD and we haveIfˆx =ˆ f −1∧ (ŷ) ⇐⇒ ŷ = ˆf ∧ ( ˆx) ∀ ˆx ∈ ˆD,ŷ ∈ Ŷ . (1)f −1 ˆ is the isoinverse of ˆf then ˆf is the isoinverse of f −1 ˆ and inverse. From (1) we haveandDepending on the isodomain ˆD we have the following cases1) isofunctions of ˆn isoreal isovariables if ˆD ˆ⊆ ˆF R n,2) isofunctions of ˆn isocomplex isovariables if ˆD ˆ⊆ ˆF C n.ˆ f −1∧ ( ˆf ∧ ( ˆx)) = ˆx ∀ ˆx ∈ ˆD (2)ˆf ∧ ( ˆ f −1∧ (ŷ)) = ŷ ∀ŷ ∈ Ŷ . (3)

followsWe will writeor| ˆf ∧ ( ˆx) − â| ˆˆ0 there exists ˆδ = ˆδ(ˆε) such that from| ˆx − ˆx 0 | ˆ< ˆδ, ˆx ˆ> ˆx 0followsWe will writeor| ˆf ∧ ( ˆx) − â| ˆ ˆx 0 , ˆx ˆn −→ ˆ ˆn −→ ˆ ˆ∞ ˆx 0 , follows thatlim ˆ ˆn −→ ˆ ˆ∞ ˆf ∧ ( ˆx ˆn ) = â.IsoHeine and isoCauchy definitions for isoright isolimts are equivalent. Theorem. Let ˆx 0 ∈ ˆF R . Then there existsif and only if there exist ˆf ∧ ( ˆx 0 + ˆ0), ˆf ∧ ( ˆx 0 − ˆ0) andˆ lim ˆx−→ ˆx0 , ˆx∈ ˆD ˆf ∧ ( ˆx) = âˆf ∧ ( ˆx 0 − ˆ0) = ˆf ∧ ( ˆx 0 + ˆ0) = â.ISODIFFERENTIABLE ISOFUNCTIONSIn [1] was introduced the term isodifferentiable isofunction. The definition given there isDefinition. Let ˆx ∈ ˆD. First isoderivative of the isofunctions ˆf at the isopoint ˆx will be called( ˆf ) ⊛ := limĥ ˆ ( −→ˆ0 ˆˆf ∧ ( ˆx + ĥ) − ˆf ∧ ( ˆx)) ⋌ ĥ.In this case we will say that ˆf is isodifferentiable at the isopoint ˆx.The main connection between isoderivative and the classical one is given with( ˆf ) ⊛ = ˆf ′ − ˆf ˆ× ˆ T ′ ˆ×ˆ(Î). (4)

Below we will use the above isoform of the isoderivative.If the isofunction ˆf has isoderivative at every isopoint of ˆD we will say that the isofunction ˆf is isodifferentiable inˆD. The isoclass of such isofunctions will be denoted with FC ˆ1ˆD.In [1] was deducted the following properties of the isoderivative. Namely1. Let ˆf ,ĝ ∈ ˆ FC 1 D be isodifferentiable at the point ˆx ∈ ˆD. Then ˆf ± ĝ are isodifferentiable at the point ˆx ∈ ˆD and( ˆf ± ĝ) ⊛ = ( ˆf ) ⊛ ± (ĝ) ⊛ .2. Let ˆf ,ĝ ∈ ˆFC 1 D be isodifferentiable at the point ˆx ∈ ˆD. Then ˆf xĝ are isodifferentiable at the point ˆx ∈ ˆD and( ˆf xĝ) ⊛ = ˆf ⊛ xĝ + ˆf xĝ ⊛ + ˆf xĝ ˆ×x T ˆ ′ ˆ×ˆ(Î).3. Let ˆf ∈ FC ˆ1 D be isodifferentiable at the point ˆx ∈ ˆD. Then Î ⋌ ˆf is isodifferentiable at the point ˆx ∈ ˆD and(Î ⋌ ˆf ) ⊛ = − ˆf ′ ˆ×(Î ⋌ ˆf ) ˆ×(Î ⋌ ˆf ) − (Î ⋌ ˆf ) ˆ× T ˆ ′ ˆ×ˆ(Î).4. Let ˆf ,ĝ ∈ FC ˆ1 D be isodifferentiable at the point ˆx ∈ ˆD. Then ˆf ⋌ ĝ is isodifferentiable at the point ˆx and( ˆf ⋌ ĝ) ⊛ = ˆf ⊛ ⋌ ĝ − ˆf ˆ×ĝ ⊛ ˆ×(Î ⋌ ĝ) ˆ×(Î ⋌ ĝ)− ˆf ˆ×ĝ ˆ× ˆ T ′ ˆ×ˆ(Î) ˆ×(Î ⋌ ĝ) ˆ×(Î ⋌ ĝ).Theorem. Let ˆf be isodifferentiable at the isopoint ˆx 0 ∈ ˆD. Then ˆf is isocontionuous at ˆx 0 .Definition. We will say that the isofunction ˆf has isolocal isomaximum at the isopoint ˆx 0 ∈ ˆD if there existsisoneighbourhood Û( ˆx 0 ) such that for every ˆx ∈ Û( ˆx 0 ) ˆ∩ ˆD we haveˆf ∧ ( ˆx) ˆ≤ ˆf ∧ ( ˆx 0 ).Definition. We will say that the isofunction ˆf has isolocal isominimum at the isopoint ˆx 0 ∈ ˆD if there existsisoneighbourhood Û( ˆx 0 ) such that for every ˆx ∈ Û( ˆx 0 ) ˆ∩ ˆD we haveˆf ∧ ( ˆx) ˆ≥ ˆf ∧ ( ˆx 0 ).Definition. We will say that the isofunction ˆf has isolocal isoextremum at the isopoint ˆx 0 ∈ ˆD if it has isolocalisominimum or isolocal isomaximum.Theorem. Let the isofunction ˆf is isodifferentiable at the isopoint ˆx 0 ∈ ˆD and has isolocal isoextremum at it. Then( ˆf ⊛ ) ∧ ( ˆx 0 ) = ˆ0.ACKNOWLEDGEMENTSThis research has been supported in part by a grant from the R. M. Santilli Foundation.REFERENCES1. S. G. Georgiev. Introduction in the Iso-Time Scales Calculus, to appear.2. R. M. Santilli, Hadronic Mathematics, Mechanics and Chemistry, Volume I: Limitations of EinsteinâĂŹs Special andGeneral Relativities, Quantum Mechanics and Quantum Chemistry, available as a free download in pdf from the websitehttp://www.i-b-r.org/Hadronic-Mechanics.htm3. R. M. Santilli, Hadronic Mathematics, Mechanics and Chemistry, Volume II: Isodual Theory of Antimatter, Antigravity andSpacetime Machines, available as a free download in pdf from the website http://www.i-b-r.org/Hadronic-Mechanics.htm4. R. M. Santilli, Hadronic Mathematics, Mechanics and Chemistry, Volume III: Iso-, Geno-, Hyper-Formulations for Matterand Their Isoduals for Antimatter, available as a free download in pdf from the website http://www.i-b-r.org/Hadronic-Mechanics.htm5. R. M. Santilli, Hadronic Mathematics, Mechanics and Chemistry, Volume IV: Experimental Verifications, Theoretical Advancesand Industrial Applications in Particle Physics, Nuclear Physics and Astrophysics, available as a free download in pdf from thewebsite http://www.i-b-r.org/Hadronic-Mechanics.htm