Differential Geometry and Matrices

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1 Differetial Geometry ad Matrices Nead Vesić Abstract A geeralizatio of curves ad surfaces is preseted i this paper This geeralizatio ca be used i various algorithms Some algorithms, preseted o cofereces, use basic terms from this paper Papers with these algorithms are idea for a geeralizatio i this paper Programs for calculatig of the sizes preseted i this paper ad some graphic presetatios will be preseted i the special sectio of this paper Key words: curve, surface, graphic, matrix 200 AMS Classificatio: Primary: 47A56; Secodary: 4H05, 4J29 Itroductio The mai goal of this paper is to itroduce ecessary terms for algorithms i cryptography ad pharmacology That will be realized with geometric view o matrix valued fuctios Some of these models were used i previous papers Let start with defiitios kow from the papers [, 2, 3] Defiitio [, 2] Matrix valued fuctio m (t) m q((t) M = M(t) = () m p (t) mp q(t) of the type p q, where is m i j : D R, D R, i =, p, j =, q, is matrix curve o the set D of the type p q Fuctios m i j ( coordiates) of the curve M are elemets Faculty of Sciece ad Mathematics, Niš Paper is fiacially supported by project 7402 of Serbia Miistry of Educatio ad Sciece

2 Defiitio 2 [3] Matrix valued fuctio m M = M(u,, u K (u) m q(u) ) = M(u) = (2) m p(u) m p q(u) of the type p q, where is m i j : D K R, i =, p, j =, q is K-matrix surface o the set D K R K of the type p q Defiitio 3 Coordiate K-matrix surface of the type p q is K-matrix surface m M = M(u) (u) m q(u) = (3) m p(u) m p q(u) of the type p q with elemets m i j : D i j R, where is D i j RK Set D = (( D D ( q) D p )) Dp q (4) is domai of the K-matrix surface (3) Remark Matrix curve is -matrix surface Remark 2 All followig defiitios i itroductio will be defiitios for matrix surface Remark 3 Matrix surfaces are cotiuous ad eough time (cotiuous) differetiable by ay of variables Remark 4 Coordiate matrix curves of type p q with domais D = (D D q ) p := (D D q ) (5) will be useful i applicatios i pharmacology 2

3 Followig defiitios are importat i applicatios of matrix valued fuctios as previous defied geometric terms Defiitio 4 If M = M(u) = M(u,, u K ) is (coordiate) K-matrix surface ad v = {i,, i d } is ordered set cosisted of elemets from the set {,, K}, (coordiate) K-matrix surface M v (u) = is i i d derivative of the surface M d M(u) (u i ) (u i d) (6) Defiitio 5 If P p = {v p,, vp } is set of all subsets of p differet ( K p) elemets from the set {,, K}, (p K), space is p-taget space of the surface (3) t p (M) = L(M,, M ) (7) v p v p ( K p) Next defiitios are importat for statistical applicatios of matrix valued fuctios as geometric terms Defiitio 6 K-matrix surface M i i r = M i i r (u) = m i (u) m i q (u) m i 2 (u) m i 2 q (u) m ir is i i r -surface of the matrix surface M (u) mir q (u) (8) Defiitio 7 K-matrix surface P(M) = P(M(u)) = P M = is para-surface of the matrix surface M p M i (u) (9) i= Defiitio 8 K-matrix surface M(M) = M(M(u)) = M M = p is mid-para-surface of the matrix surface M 3 p M i (u) (0) i=

4 Remark 5 If M is a matrix curve, correspodig para-surface ad midpara-surface are para-curve ad mid-para-curve Followig defiitios are defiitios of Christoffel symbols of matrix surfaces Defiitio 9 K-matrix surfaces Γ i jk = [ rγ i jk ] () of the types p, where Γ r i jk, r =, p, are Christoffel symbols of r-surfaces of the K-matrix surface (2) Defiitio 0 K-matrix surfaces Γ i jk = [ r Γs i jk ] (2) of the type p q, r =, p, s =, q, where r Γ s i jk are Christoffel symbols of the surface (u,, u k, m r s(u)), are Christoffel Symbols of the K-matrix surfaces (2,3) 2 Basic properties of matrix surfaces Some basic, but very useful theorems ad corollaries will be preseted i this part of the paper Before these properties, i pharmacology will be useful the followig propositio Propositio 2 If real umbers satisfies the coditio a,, a tha a + + a = a α R, ( i 0, i 0 )a i0 α 4

5 2 satisfies the coditio tha a + + a = a α R, ( i 0, i 0 ) a i0 α 3 satisfies the coditio tha a + + a = a > α R, ( i 0, i 0 )a i0 > α 4 satisfies the coditio tha a + + a = a < α R, ( i 0, i 0 )a i0 < α Proof Let prove Other proves are aalogously If all a i, i =,, satisfies coditio a i < α, tha is what is cotradictio a = a + + a Theorem 2 Let M = M (u ) = < α = α, [ m [ i m j (u )],, M = M (u ) = i j (u )] be k i -matrix surfaces of the types k p q, k =, Para-surface of the matrix surface m (u ) m q(u ) m p (u ) m p q (u ) M = M(u,, u ) = m (u ) m q(u ) m p (u ) m p q (u ) 5

6 is Proof P M = P M i (3) i= P M = ) ( ) ((M ) + + M p + + M + + M = p Corollary 2 Mid-para-surface of the matrix surface () is M M = p + + p P M i i= i p M M i (4) Corollary 22 Mid-para-surface of the matrix surface () satisfies the followig coditio M M max M M i, (5) i {,,} for ay orm 3 Matrix curves Matrix curves will be detailed preseted i this part of the paper Defiitio 3 Let m (t) m 2 (t) m 3 (t) M = M(t) = m p (t) mp 2 (t) mp 3 (t) i= be a matrix curve of the type p 3 Matrix curve c K M = c K M (t) = [K Mi (t)] (6) of the type p, where K Mi are curvatures of curves ( m i (t), mi 2 (t), mi 3 (t)), i =, p, is curvature of the matrix curve M 6

7 Defiitio 32 Let m (t) m q(t) M = M(t) = m p (t) mp q(t) be a (coordiate) matrix curve of the type p q Matrix curve c K M = c [ ] K M (t) = Km i (t) j (7) of the type p q, where K m i j (t) is siged curvature of a curve r i j = r i j(t) = (t, m i j(t), 0), is siged coordiate curvature of the curve M Defiitio 33 Let m (t) m 2 (t) m 3 (t) M = M(t) = m p (t) mp 2 (t) mp 3 (t) be a matrix curve of the type p 3 Matrix curve c τ M = c τ M (t) = [τ Mi (t)] (8) of the type p, where τ Mi are torsios of curves ( m i (t), mi 2 (t), mi 3 (t)), i =, p, is torsio of matrix curve M Remark 3 Coordiate torsio of a matrix curve is equal to 0-matrix Coectio betwee coordiate curvature ad matrix curve is preseted i the followig theorem Theorem 3 If matrix curve c K = c [ ] K(t) = Km i (t) is siged coordiate curvature of a j M = M(t) = [ m i j(t) ] o the set D of the type p q, tha is ( m i j(t) = sih (tah )) K m i (t)dt j dt (9) 7

8 Proof Let be f C 2 (D) a itegrable fuctio ad F (t) = f (t) ( + (f (t)) 2) 3/2 It is F (t)dt = f (t) ( + (f (t)) 2) 3/2 dt = u = f (t) du = f (t)dt = That implies du ( + u 2 ) 3/2 = u = sih v du = cosh vdv = dv cosh 2 v = tah v = tah ( sih ( f (t) )) ( f (t) = sih (tah )) F (t)dt f(t) = ( sih (tah )) F (t)dt dt Fuctio F (t) = K m i j (t) proves the theorem 4 Programs Some programs, ecessary for matrix surfaces realized i software package MATHEMATICA 70, are preseted i this sectio Para-surface ParaSurface[m ] := Module[{pc = m[[]]}, For[i = 2, i <= Dimesios[m][[]], i++, pc += m[[i]]]; pc]; This program calculates para-surface of a matrix surface m 2 Mid-para-surface midparasurface[m ] := ParaSurface[m]/Dimesios[m][[]]; Result of this program applied o a matrix surface m is mid-parasurface of this matrix surface 8

9 3 Graphic presetatio CoordiateMatrixCurveGraphics[m, ymi, ymax, Domai ] := Module[{cmcg = ParametricPlot[{{0, ymi + (ymax - ymi)*t}, {T[[]] + (T[[-]] - T[[]])*t}}, {t, 0, }, AspectRatio -> ], AxesLabel->{Times,Evaluate}}, For[i =, i <= Dimesios[m][[]], i++, For[j =, j <= Dimesios[m][[2]], j++, cmcg = Apped[cmcg, Plot[m[[i, j]], {t, Domai[[i, j, ]], Domai[[i, j, 2]]}, PlotStyle -> Black]]]]; Show[cmcg]]; Graphical presetatio of coordiate matrix curves is result of this program This is ecessary for aalyzig of results i pharmacology 5 Possible applicatios Results of applicatios of matrix surfaces i cryptography ad pharmacology is preseted i this part of paper 9

10 Cryptography 3-matrix surface M = M(x, y, z) = {{ x x y xy x 2 y y xy x 2 y z xz x 2 z yz xyz x 2 yz y 2 z xy 2 z 7800x 2 y 2 z z xz x 2 z yz xyz x 2 yz y 2 z 2 424xy 2 z 2 + 5x 2 y 2 z 2, x x y xy x 2 y y xy x 2 y z xz x 2 z yz xyz x 2 yz y 2 z xy 2 z 9402x 2 y 2 z z xz x 2 z yz xyz 2 920x 2 yz y 2 z 2 248xy 2 z 2 + 5x 2 y 2 z 2 }, { x x y xy x 2 y y xy x 2 y z xz x 2 z yz xyz x 2 yz y 2 z xy 2 z 6292x 2 y 2 z z xz x 2 z yz xyz x 2 yz y 2 z 2 53xy 2 z 2 + 7x 2 y 2 z 2, x x y xy x 2 y y xy x 2 y z xz x 2 z yz xyz x 2 yz y 2 z xy 2 z 228x 2 y 2 z z xz x 2 z yz xyz x 2 yz y 2 z xy 2 z 2 + 7x 2 y 2 z 2 }} is public key of a ecrypted text Message is (x, z, y) 0

11 Decrypted text is ZLATIBOR Pharmacology Expectatio of evaluated results i patiets is tested here Coditio is expectatio less tha The first graphic tests evaluates of all patiets Fig Expected results of all patiets It is evidet that coditio is ot satisfied i ay momet Let fid i which group is problem Expected evaluates i group are

12 Fig2 Expected results of patiets from group Coditio is ot satisfied at the start of experimet, oly Expected evaluates i group 2 are 2

13 Fig3 Expected results of patiets from group 2 Coditio i this group of patiet is ot satisfied at ay time So, i this group of patiets drug does ot satisfies the coditio ad problem is localized i this group of patiets Refereces [] Nead Vesic, Dusica Ilic, Testig Effects of a Drug: Drugs for Chroic Diseases, Proceedigs of ICTIC 202, -9 [2] Nead Vesić, Testig Effects of Drugs: Drugs for Chroic Diseases, Semiar for Geometry, Educatio ad Visualizatio with Applicatios, Serbia Academy of Sciece ad Arts 202, abstract, [i Serbia] [3] Nead O Vesic, Dusa J Simjaovic, Tesors ad Cryptography, Axepted for publicatio i IfoTech 202 coferece [4] Nead O Vesić, Dušica Ilić, Differetial Geometry ad Pharmacology: New Approach to Testig Effects of Drugs for Chroic Diseases, Paper i preparatio 3

1.3 Convergence Theorems of Fourier Series. k k k k. N N k 1. With this in mind, we state (without proof) the convergence of Fourier series.

1.3 Convergence Theorems of Fourier Series. k k k k. N N k 1. With this in mind, we state (without proof) the convergence of Fourier series. .3 Covergece Theorems of Fourier Series I this sectio, we preset the covergece of Fourier series. A ifiite sum is, by defiitio, a limit of partial sums, that is, a cos( kx) b si( kx) lim a cos( kx) b si(