Application of Quaternion Neural Network to Time Reversal Based Nonlinear Elastic Wave Spectroscopy
aa r X i v : . [ phy s i c s . g e n - ph ] D ec Application of Quaternion Neural Network to Time Reversal Based Nonlinear ElasticWave Spectroscopy
Sadataka Furui (Formerly) Graduate School of Science and Engineering, Teikyo University, Utsunomiya, Japan ∗ Serge Dos Santos
INSA Centre Val de Loire, Blois, Inserm U1253, Universit´e de Tours,Imagerie et Cerveau, imaging and brain : iBrain, France † (Dated: December 17, 2020)An application of quaternion neural network to identification of crack positions in materials usingthe time reversal based nonlinear elastic wave spectroscopy (TR-NEWS) is proposed.Transducers which emit forward propagating solitonic waves and time-reversed backward propa-gating solitonic waves are produced by memristers, are placed on a line of the left wall and scatteredby cracks in the material and received by receivers which are placed on the right wall on the lineparallel to that of transducers.By minimizing the difference of the scattered forward propagating wave and the scattered back-ward propagating wave, we get information of the position of the crack by using the neural networktechnique. Route of the solitons are expressed by 2 dimensional projective quaternion functions,and parameters for getting the optimal route from signals are expected to be reduced.When the soliton is expressed by conformal waves, symmetry protected topological inpurities andgravitational effects would manifest itself due to Atiyah-Patodi-Singer’s theorem. I. INTRODUCTION
In non destructive testing (NDT), time reversal (TR)based nonlinear elastic wave spectroscopy (NEWS) [1, 2]is an efficient method to detect scatterers of ultra highfrequency phonons in materials.At present, position of scatterers are defined by man-ually looking for an angle of the receiver relative tothe transducer where interference of original waves andTR waves produce a peak. If there are several trans-ducers which emit the wave pairs, and several receiversthat measure convolutions of various pair waves, one canimagine getting interference patterns of beam pairs fromdifferent transducers. The information one obtains islarge, but using techniques of artificial inteligence(AI),it may be possible to detect scattering positions. An aimof this paper is to present a technique for this purpose.Propagation of elastic phonetic waves with the direc-tion [ ] described by w = w exp [ √− K x x + K y y − ωt )]in materials, and effects of nonlinearity are discussed in[3–5]. Propagation of cylindrical phonetic waves are dis-cussed in [6, 7].Inelastic scattering of phonons with the wave vector K and K can construct a phonon K through nonlin-ear effects. Interaction of phonons with lattices of mate-rials yields X n exp [ √− K − K − K ) · r n ] ∗ [email protected] † [email protected] where r n is the coordinate of the lattice vertex, and scat-tering conditions K = K + K , or K = K + K + G , where G is the inverse lattice vector, emerge.The quantized momentum p = ~ K in normal met-als, the thermal conductivity from electrons and phononswere discussed. The interference of phonons reflectedfrom the boundary of the material creates Lamb waves[9].Approximate solutions of nonlinear acoustic waveequation in materials with axial symmetry similar to theequation of phonons with the direction [ ] which isequivalent to the Khokhlov-Zabolotskaya(KhZa) equa-tion were considered in[10–12]. (In order to evade con-fusion with the Knizhnik-Zamolchikov(KZ) equation[14],which is relevant to the SU (2) Wess-Zumino-Wittenmodel, we use the abbreviation differnt from that of [13].)They considered the particle velocity along the propaga-tion direction u , and across the propagation direction v ,using dimensionless variables V = u/u , U = v · (2 l d /u a ) , θ = ω ( t − x/c s ) ,z = x/l s , R = r/a (1)where c s is the sound velocity, x and r are the axial andthe transverse coordinates, l d = ωa / c s is the diffrac-tion length, l s = c s /ǫωu , ǫ is the nonlinearity param-eter, N = l s /l d , u , ω, a are characteristic values of theamplitude, frequency, and beam radius. The equation is ∂V∂z − V ∂V∂θ = − N R ∂∂R ( RU ) ∂U∂θ + ∂V∂R = 0 . (2)An exact solution of the KhZa equation is obtained byrewriting the equation ∂ V∂T ∂z − N ∂ V∂R + 1 R ∂V∂R ) = 0 f ′ ( z ) f ( z ) V ∂V∂T + ∂V∂z ∂V∂T − N ∂V∂R ) = 0 (3)where T = θ + zV in the case of a plane wave propagation.A simple complex solution of equation (2) and (3) givenin [12] is V = C − √− N z (1 − √− b ) × exp [ √− T − R − √− b − √− N z (1 − √− b ) ] f ( z ) = 1 − √− N z (1 − √− b ) , (4)where C and b are constant. The solution can be checkedby using Mathematica[16].The function V is complex, but the real part de-scribes the propagation of a focused harmonic wave witha Gaussian transverse distribution. ReV indicates theˆ u − component of the wave front coordinate, and ImV indicates the ˆ v − component of the wave front, that prop-agates on the (ˆ u, ˆ v ) plane.In the limit of z = x/l s = 0, V = Cexp [ √− ω ( t − x/c s )] exp [ − r (1 − bN )] × (cos( − br ) + √− − br )) . (5)In the case of Burger’s equation, soliton solutions canbe obtained by performing Cole-Hopf transformation andthe Lie group transformation of the initial boundaryvalue problem[15], and there is a similar aproach to theKhZa equation[13, 17]. The stability of sound beams ex-pressed by a more generalized Khokhlov-Zabolotskaya-Kuznetsov equation is discussed in [18].In the analysis of propagation of phonons on 2 dimen-sional (2 D ) plane, we take the state vector in quaternionprojected space. A quaternion h ∈ H is described as h = α I + β i + γ j + δ k = ασ + β √− σ + γ √− σ + δ √− σ = (cid:18) α + √− δ γ + √− β − γ + √− β α − √− δ (cid:19) , (6)where σ i ( i = 1 , ,
3) are the Pauli matrices, and i = j = k = − ij = − ji = k , jk = − kj = i , ki = − ik = j .Using complex coordinates z, w ∈ C , one can write H = { (cid:18) z w − ¯ w ¯ z (cid:19) } . Quaternions p and p are said to be equivalent if thereexists h ∈ H \ { H } such that hp = p h . We considerpropagation of solitons in a 2 D plane. Structure of this presentation is as follows. In sect..II.,we summarize the principle of TR-NEWS: time-reversalbased nonlinear elastic wave spectroscopy. In sect. III,we explain setup of transducers and receivers. In sect.IVconvolution of the KhZa wave function and its TR wavefunction is explained. Quaternion neural network andits topological properties are explained in sect. V. Insect.VI, we present mathematical bases of QuaternionFourier Transforms (QFT), using the fact that quater-nions are Clifford numbers[19]. Discussion and conclu-sions are given in sect. VII. II. SYMMETRIES IN PROPAGATION OFSOLITARY WAVES IN MATTERS
Time reversal symmetry based nonlinear elastic wavespectroscopy, in which one optimizes the convolution ofthe scattered wave from defects in materials and its timereversed wave show peaks was an effective method fornon-destructive testing (NDT)[21].
A. A simulation of (1 + 1) D phonetic waves byKhelli et al. In the following, we review numerical simulations ofacoustic waves in a cylinder performed by Khelil et al.[8]. Khelil et al. considered linearized Euler equation ∂ρ∂t + ρ ∇ · V = 0 , ∂ V ∂t + 1 ρ ∇ p = 0 , dpdρ = c (7)where ρ, p, V and c are respectively, the density, pressure, fluid velocity, and sound velocity in the medium.The velocity c is time dependent C ( t ) and the eq.(7)reduces to1 ρ c ∂p∂t + c ∇ · V = 0 , ∂ V ∂t + c [ 1 ρ c ∇ p ] = 0 (8)One introduces θ = p − p p c , where p is the uniform steady pressure of the mediumat rest. The eq.(8) becomes ∂θ∂t + c ∇ · V = − θc ∂c∂t , ∂ V ∂t + c ∇ θ = 0 . (9)Khelil et al. assumed C ( t ) = c (1 + m cos(Ω t + Ψ)) , (10)where m ≪ ∂θ∂t + c ∇ · V = m Ω θ t + Ψ) ∂ V ∂t + c ∇ θ = 0 (11)In (1 + 1) D , a numerical solution of ∂θ∂t + c ∂v∂x = m Ω θ t + Ψ) ∂v∂t + c ∂θ∂x = 0 (12)Introducing new variables w = v + θ and w = v − θ ,the eq.(12 ) becomes ∂w ∂t + c ∂w ∂x = m Ω4 ( w − w ) sin(Ω t + Ψ) ,∂w ∂t + c ∂w ∂x = − m Ω4 ( w − w ) sin(Ω t + Ψ) . (13)The system of eq.(13) can be written as ∂U∂t + ∂F ( U ) ∂x = SU ( x,
0) = U ( x ) (14)where x ∈ R and F ( U ) = (cid:18) c w − c w (cid:19) , U = (cid:18) w w (cid:19) ,S = (cid:18) m Ω4 ( w − w ) sin(Ω t + Ψ) − m Ω4 ( w − w ) sin(Ω t + Ψ) (cid:19) . (15)The numerical solution U ni = U ( i ∆ x, n ∆ t ) at time ( n +1)∆ t is, in the first order approximation U n +1 i = U ni − ∆ x ∆ t [ f i +1 / ( U n ) − f i − / ( U n )] + ∆ tS ( U ni ) . (16)Including higher orders, Khelil took f i +1 / ( U ni +1 , U ni ) = α F ( U ni ) + α ( U ni +1 ) (17)where α = 12 (1 + ν ) , α = 12 (1 − ν ) (18)and ν = c ∆ t/ ∆ x is the Courant number associated withthe velocity c [70].In order to satisfy the stability condition Khelil et al.proposed modification of α and α as¯ α = α + (1 − B i +1 / ) α , ¯ α = α B i +1 / (19)where B i +1 / is defined by r i = ( U ni − U ni − ) / ( U ni +1 − U ni )as follows. B i +1 / = (1 − − | ν | ) / | ν | , r i ≥ − r i (1 − | ν | )) / | ν | , ≤ r i ≤ , / ≤ r i ≤ − r i (1 − | ν | )) / | ν | , ≤ r i ≤ / / | ν | , r i ≤ ν was chosen to be 0.6.The initial condition is defined by the wave length λ =2 πc ω and x L which satisfies (0 . − x L ) = 3 λ as follows. w = 2 sin(2 π ( x − x L ) /λ, w = 0 when x L < x < .
5, and w = w = 0 otherwise. Khelil et al. took ω = 2 π s − , c = 1500 m/s , Ω = 2 ω , Ψ = π and thesmall parameter m = 4 . × − and m = 3 . × − .Envelopes of the pressure are shown for t = 0 . e − ,3 . e − , 7 . e − , 14 . e − , 21 . e − , 26 . e − and30 e − s .We will calculate the envelope of pressure p in the ma-terial successively in time, and obtain parameters to fitthe KhZa soliton in the future. B. Elastodynamics in (2 + 1) D spatial cylindricalsystems Propagation of phonetic waves in solids in cylindricalgeometries was studied by Rose[6]. He defined the elasticdisplacement u whose components are u n n = 1 , , x whose components are x n , and u n ( x , t ) = Z V c ijkl ǫ Tkl ( ξ, t ) ∗ ∂G nl ∂ξ j ( x ; ξ ; t ) dV ( ξ ) (21)where ∗ means taking a convolution in t . He assumedthat the source of finite extent can be aproximated by apoint source, and wrote u n ( x , t ) = M ij ( t ) ∗ ∂G ni ∂ξ j ( x ; ; t ) ,M ij ( t ) = Z V c ijkl ǫ Tkl ( ξ, t ) dV ( ξ ) (22)When material is isotropic, elastic response is expectedto be described by the shear modulus µ and the Pois-son ratio ν . The longitudinal phonon velocity c L andtransverse phonon velocity c T are c L = 2 µ (1 − ν ) / [ ρ (1 − ν )] , c T = µ/ρc T /c L = (1 − ν ) / (2 − ν ) (23)where ρ is the material density. In case of alminium ν = 0 . c L = 6 . φ ( r, z, t ) , ψ ( r, z, t )where r = ( x + x ) / , z = x . The displacement components u z and u r , and the stresstensor components σ zz and σ zr are written by using acomma to denote a partial derivetive with respect to thecoordinates shown as subscripts following the comma, u z = φ, z − ∆ r ψ,u r = φ, r + ψ, rz σ zz = 2 µ [(1 − ν )(1 − ν ) − φ, zz + ν (1 − ν )∆ r φ − (∆ r ψ ) , z ] ,σ zr = µ (2 φ, z + ψ, zz − ∆ r ψ ) , r , ∆ r ψ = r − ( rψ, r ) , r . (24)The solution of the equation φ ( r, z, t ) was transformed asfollows. Φ( ζ, z, s ) = Z ∞ ˆ φ ( r, z, s ) J ( sζ ) rdr, ˆ φ ( r, z, s ) = Z ∞ φ ( r, z, t ) e − st dt. (25)After performing changes of variables, he obtained theSCOE: Φ( ζ, z, s ) = 2Λ s − ζ β R e − sαz , Ψ( ζ, z, s ) = λs − γ R e − sβz . (26)The solution of these equations can be expressed by R = ( r + z ) / , θ = tan − ( r/z ) . On points of the z-axis ( r = 0), the axial displacement g Hz ( r, z, t ) is given g Hz (0 , z, t ) = Λ z − [ ˙ F L ( t, z ) + ˙ F T ( t, z )] ,F L ( t, z ) = − H ( t − az )( ζ α β R ) ζ =( t /z − a ) / ,F T ( t, z ) = H ( t − bz )( ζ βγ R ζ =( t /z − b ) / . (27)where H is the Heaviside step function, and ˙ F mean adifferential of F with respect to t . C. (1 + 1) D TR-NEWS
Memristor[26] is the fourth electronic circuit elementsafter resister, inductor and capacitor. In Muthuswamy-Chua’s circuit[23], one can construct chaotic memristiccircuits[24].For nonlinear acoustic wave spectroscopy in mediawith hysteresis, one can use memristor that creates for-ward propagating and backward propagating waves, andmeasure scattered waves. When transducers and re-ceivers are displayed on a 2 D plane, quaternion neuralnetwork may help optimization of NDT[30, 33–35]. Inpropagation of acoustic waves in non linearly oscilatingmedia, solitary wave property appears. In the analysis ofTR-NEWS in NDT, a proper choice of strengths and in-tervals of impulse from transducers may allow detectionof gravitational effects.In generalized pulse inversion methods of TR-NEWS[32], excitation function bases are taken as x E = x ( t ) , x ǫ = x ( t ) e i π/ , x ǫ ∗ = x ( t ) e − i π/ and their response are y E , y ǫ , y ǫ ∗ respectively.In neural networks, function on imput layers are x A ( t ) = − x ( t ) , x B ( t ) = √ x ( t ) , x B ( t ) = − √ x ( t ) , and that on output layers are y A , y B , y B , respectively.Nonlinear responses are parametrized as y ( t ) = N L [ x ( t )] = N x ( t ) + N x ( t ) + N x ( t ) . The nonlinear responses on output layers are expressedas N x ( t ) = 43 [ y E ( t ) + 2 y A ( t ) − y B ( t ) − y B ( t )] = s ( t ) ,N x ( t ) = 23 [ y B ( t ) + y B ( t )] = s ( t ) ,N x ( t ) = y E ( t ) − s ( t ) − s ( t ) = s ( t ) . The Fourier transform F s ( ω ) has a peak around sev-eral hundred kHz dependent on chirp coded excitation c ( t ) = A · sin( ψ ( t ))where ψ ( t ) is linearly changing instantenious phase of theorder of a few MHz.When the impulse response of the medium is expressedby h ( t − t ′ , T ) where T is the time duration, the responseis expressed by convolution y ( t, T ) = h ( t ) ∗ c ( t ) = Z R h ( t − t ′ , T ) c ( t ′ ) dt ′ The response of time reversed impuls that memoducersemit is given by the convolution y T R ( t, T ) = Z ∆ t y ( T − t ′ − t ) c ( T − t ′ ) dt ′ ∗ h ( t ) ∼ δ ( t − T ) , which is a linear combination of s ( t ) , s ( t ) , s ( t ).In the calculation of convolutions we use semigroupsplitting method of Trotter [37]. The differential equation˙ y = f ( y ) in R n is split as ˙ y = f [1] ( y ) + f [2] ( y ), andexact flows ϕ [1] t and ϕ [2] t of ˙ y = f [1] ( y ) and ˙ y = f [2] ( y ),respectively are calculated. The functions Φ ∗ h = ϕ [ ] h ◦ ϕ [ ] h , Φ h = ϕ [ ] h ◦ ϕ [ ] h connects initial value y and final value y , via differentpaths. h ( ) h ( ) y0 y2y1 FIG. 1. Path dependence of outputs.
The formula e A + B = lim n →∞ ( e A/n e B/n ) n does not fol-low if A, B are non-commutative. However Trotter[37]showed that T t f ( x ) = f ( x − t ) and T ′ t f ( x ) = f ( x + t ) formsemi-groups, and due to the Hille-Yosida’s theorem[38–41] S t f ( x ) = lim h → ( T h T ′ h ) [ t/h ] f ( x ) . h ( ) h ( ) y0 y2y1 FIG. 2. Path dependence of outputs. can be defined in discrete time steps.If the semi-group of linear operators T t satisfy1) T = I, T t T s = T t + s , s · lim T t x = T t x, ( x ∈ X ) , k T t k≤ e β | t | ( β > s · lim means strong convergence limit, then oper-ators { T t } form a group in Banach space X .Hille-Yosida theorem[38–41] says that for finite opertor T ′ t for t > | ν |→∞ k (( λ + √− ν ) I − A ) − k < ∞ if λ > β is satisfied the operator T ′ t can be extended to aregular operator T t + √− s in the cone area { t + √− s ; | s | ≤ ct, c, is , real , postive } . Operators { T t } form a semi-group with infiniesimalgenerator A s · lim h → h − ( T h − I ) x = Ax.
The i th layer has connection to j th hidden layer h ( i, j ) = h ( i − , · h ( i − , ∀ j ∈ { , } . If the semi-group of linear operators T t satisfy1) T = I, T t T s = T t + s , s · lim T t x = T t x, ( x ∈ X ) , k T t k≤ e β | t | ( β > s · lim means strong convergence limit, then oper-ators { T t } form a group in Banach space X . III. SETUP OF TRANSDUCERS ANDRECEIVERS
Let us consider memoducers or transducers which havehysteresis and emits sonic beams and TR sonic waves.On a closed curve C = { z ( t ) = u ( t ) + √− v ( t ) } de-fined by ABA of Fig,3, one can define tangential vectors
A BC C'A'B' - - FIG. 3. Holonomic curves that consist of parallel transforma-tions of tangential vectors. parallel to holizontal u axis along ACB ′ B and BC ′ A ′ A .Tangential vectors at A and at B that goes to + ∞ and to −∞ respectively are parallel ( T A + = T B − , dashed lines).Tangential vectors at C and C ′ are parallel, and one candefine parallel transformation of ACB ′ to A ′ C ′ B . Thevector AA ′ and BB ′ are defined by parallel vectors of CC ′ , but they are not parallel to horizontal axis, henceclosed curve ACB ′ BC ′ A ′ after identifying A and A ′ , B and B ′ due to parallel transformations becomes a closedcurve ABA . The vector CC ′ is ortogonal to the tangentvector at C and at C ′ , but the vector AA ′ is not orthogo-nal to the tangent vector at A , and hence the closed curveis not holonomic. Non holonomicity yields hysteresis.Paralleltransformations have special a meaning inClifford algebra. Lounesto[69] defined the basis { , e , e , e ∧ e } for ∧ V , that satisfy the following mul-tiplication table ∧ e e e ∧ e e e ∧ e e − e ∧ e e ∧ e ∧ whose multiplication table isthe following.˙ ∧ e e e ∧ e e e ∧ e + b − be e − e ∧ e − b − be e ∧ e − be − be − b − be ∧ e where b > ∧ e e e ∧ e + be e ∧ e + b e − e ∧ e − b e ∧ e + b D , shifts in e ∧ e can be treatedby a simple coordinate transformation.Effects that appear by identifying parallel lines alongthe horizontal axis from A and B are regarded as instan-tons, although the presence of the global symmetry is notevident.We arrange memoducers on the left wall of a metalequally spaced and receivers on the right wall equally sep-arated. By adjusting memoducers, solitonic wave froma transducer T i and time reversed (TR) solitonic wavepropagate on a 2 D plane (ˆ u, ˆ v ). At t = 0, the wave frontis at x = 0, and propagate within the cone in t > t < x =- =- =- =- =- = = = = = = =- =- =- =- =- = = = = = = = = = = = FIG. 4. Wave front of solitonic waves.
We consider phonons produced at x = 0 at time t =0 and propagate forward and backward with a scaledvelocity, as shown in Fig. 4. In order to reduce effectsof boundary conditions we add padding layers at t = ± x = ±
5. The number of padding layers is to bechanged according to accumulated data.Using the notation of [53], we choose for Transducerson a line and receivers on a line L = 5 , B = 1, and forforward and backward propagation d = 2, and the filter F = 2. L = L − F + d = 4, B = B − F + d = 0.The filter has the size 2 × × T , · · · , T N on the leftwalls are received by receivers R , · · · , R N on the rightwall. Sound waves are scattered by an object shown bya black disk between the walls and on a plane on whichtransducers and receivers are placed. Dashed lines be-tween transducers and receivers have longer paths thansolid lines, whose information is contained in larger timedelay τ i in signals of receivers R i .The waves from T i are scattered by cracks in the metalif they exist, and solitonic waves are disturbed, and theyare received by receiver R j . We consider the situation of5 transducers and 5 receivers T1T2T3T4T5 R1R2R3R4R5 t yx z
FIG. 5. Networks of 5 transducers T , · · · , T and 5 receivers R , · · · , R . Trajectories are expressed by quaternions H = τ I + x i + y j , where τ, x, y are real and can be mapped to M (2 , C ). R = = = = = = = = = = FIG. 6. Disturbance of beams between transducers T i and re-ceivers R j on the 2 D plane shown in Fig. 4. Black disks cor-respond to strong disturbances and yellow disks correspondto weak disturbances. Taking the length x in the unit of c s t , where c s is thesound velocity in the material, the wave front which wasat x = 0 at t = 0 propagates to x ≥ t ≥
1. For TRwaves the wave front propagates to x ≥ t ≤ − D plane on which transducers and receivers areplaced h = τ I + x i + y j , q = ( t + l mm /c s ) I + x i + y j , q = ( t − l mm /c s ) I + x i + y j . hq − q h = l mm τc − l mm √− x − τc s + l mm yc s + ˜ ty l mm √− x − τc s + l mm yc s − ˜ ty l mm τ + √− xc s ! where ˜ ty = τ y − ty .A real trajectory outputs y j has the local partialderivative z ( m, n ) = ∂y m ∂y n , whose products gives varia-tion of real outputs o with respect to weight functions: ∂o∂w = X P ∈P Y ( m,n ) ∈P z ( m, n ) . Here the P is the aggregates of paths.Relativistic dynamics of the Maxwell-Einstein equa-tion can be represented by instant form, front form andpoint form[25]. The usual parametrization t in instantform is defined by parallel transformation from the sys-tem that satisfy p σ p σ −M = 0 at the Lorentz coordinate u = 0, while parametrization τ = t − x/c s in KhZa dy-namics is front form.When a (2 + 1) D front form wave function is expressedby quaternions q = 0, equivalent quaternions that satisfy q q = qq have the periodicity in τ direction by 2 l mn /c s .When phonons have effective mass due to scattering inmedia, the wave front in instatnt form changes to thepoint form. The point form comes from taking a branchof hyperboloid u ρ u ρ = κ , u > FIG. 7. The 3D wave front of massless particles in the frontform .FIG. 8. The 3D wave front of massive particle in the pointform inside the cone of massless particles.
Actions depending on paths yields hysteresis which isrelated to transformation of holonomy groups under par-allel displacements.We use instead of t , a step parameter τ = t ± l m,n /c , FIG. 9. The 2 D wave propagation direction dependence onthe transducer position T m at (0 , , , (0 , ,
2) and receiverposition R n : ( m, n ). - - - - FIG. 10. The propagation direction dependence of pointform wave fronts with non-zero effective mass. Ordinate is τ ,abscissa is the distance from the origin. where l mj is the length of the path between the trans-ducer T m and receiver R n on the ( X, Y ) plane.In order to get positions of scatterers, we define theLoss function for input X , output YL = X ( X,Y ) ∈ D ( Y − ˆ Y ) + λ d X i =0 w i where d = 3 is the degree of bases function, ˆ Y = P di =0 w i X i , and λ is a constant. Following Aggarwal[53], we denote ( w , · · · , w d ) = ¯ W .Consider g ( · ) , g ( · ) , · · · , g N ( · ) computed in layer m , and composition function in layer m + 1 is f ( g ( · ) , · · · , g N ( · )), and for an input X , weight w , con-sider input f ( w ) and split paths from an input layer toan output layer Y and hidden layer Z by Y = f ( w ) , p = g ( Y ) and Z = f ( w ) , q = h ( Z ) . In our application X and Y are descibed by a KhZasoliton wave functions propagating in the definite direc-tion.The partial derivative of output with respect to w is ∂o∂w = ∂o∂p · ∂p∂w + ∂o∂q · ∂q∂w = ∂o∂p · ∂p∂y · ∂y∂w + ∂o∂q · ∂q∂z · ∂z∂w = ∂K ( p, q ) ∂p · g ′ ( y ) · f ′ ( w ) + ∂K ( p, q ) ∂q · h ′ ( z ) · f ′ ( w )(28)We define w ( X m , Y n ) defined by the path from inputposition X m of the transducer to the output position Y n of the receiver. When there is a scatterer in the 2 D planeas shown in the Fig.III, among functions w ( X m , Y n ), w (3 ,
3) and w (4 ,
3) are expected to have large distur-bances. With wider range of filters of | X m − X n | ≤ w (1 ,
4) and w (5 ,
2) will have large disturbances.Since there are X E , x ǫ and x ǫ ∗ bases we take p = 3.One of the aim of this research is to find optimal func-tions f ( w ) , g ( y ) , h ( z ) and z ( i, j ), such that the loss func-tion L becomes small.We define outputs Y n in the forward phase, using hid-den layer variables h ( i, q ), where q = 1 , i definesthe recurrence order. The hidden layer variable h ( i, q )and h T R ( i, q ) distinguish interference with original or TRphonon beams. Relation between outputs and hiddenlayer variables are h ( i, q ) = α ( W hh ⊗ h ( i − , q ) + W hx ⊗ X i + b h ) ,α ( Q ) = f ( r ) I + f ( x ) i + f ( y ) j ,Y i = β ( W hy ⊗ h ( i, q )) , (29)where α and β are split activation functions. b h is thebias of the hidden state.For calculation of ¯ Y = [ y , · · · , y n ] T , we choose train-ing sets ˆ Y = [ˆ y , · · · , ˆ y n ], that satisfy ˆ y i = ¯ H i ¯ W T = P nj =1 w j Φ( ¯ X i ). Here ¯ H i is m dimensional and repre-sents in the hidden layer, yields ˆ y for the i -th trainingpoint ¯ X i .The hidden layer weight function for original waves¯ h ( i, q ) and for TR waves can be taken as¯ h ( i, q ) = tanh( W xh ¯ X i + W hh ¯ h ( i − , q ))¯ h T R ( i, q ) = tanh( W T Rxh ¯ X i + W T Rhh ¯ h T R ( i + 1 , q ))¯ Y ( i, q ) = W hy ¯ h ( i, q ) + W T Rhy ¯ h T R ( i + 1 , q )ˆ Y i = ¯ W · ¯ X i The partial derivatives of the loss function are givenby the trained hidden layer function h ( i, q ). ∂L i ∂W rhy = ∂L i ∂y ri ∂y ri ∂W rhy + ∂L i ∂p i i ∂p i i ∂W rhy + ∂L i ∂p j i ∂p j i ∂W rhy = ( p ri − y ri ) h r ( i, q ) + ( p i i − y i i ) h i ( i, q ) + ( p j i − y j i ) h j ( i, q ) ∂L i ∂W i hy = ( p ri − y ri )( − h i ( i, q )) + ( p i i − y i i ) h r ( i, q ) ∂L i ∂W j hy = ( p ri − y ri )( − h j ( i, q )) + ( p j i − y j i ) h r ( i, q )Here p i = β ( W hy ⊗ h ( i, q )) is calculated by split ac-tivation functions and hidden weight matrix for K timesteps, ∂L∂W hh = K X i =1 ∂L i ∂W hh . (30)At each i th time step, ∂L i ∂W hh = i X m =1 ( ∂L m ∂W rhh + ∂L m ∂W i hh i + ∂L m ∂W j hh j ) . (31)Input weight matrix is ∂L∂W hx = K X i =1 ∂L i ∂W hx .Hidden biases is ∂L∂b h = K X i =1 ∂L i ∂b h .The update of ¯ W can be written as ¯ W ⇐ ¯ W (1 − α · λ ) + α ( Y i − ˆ Y i ) ¯ X .Mapping of metric data ϕ ( x , x , t ) → ϕ ( x , x , N , N , τ ) ( transformation from the in-stant form to the front form) allows decomposition ofwave function as ϕ = ϕ + + ϕ − , where ϕ + = 12 ( ϕ + √− N ⊥ · ϕ ) , ϕ − = 12 ( ϕ −√− N ⊥ · ϕ ) (32)where N ⊥ is orthnormal to the wave front defined on the( x , x ) plane.We calculate propagation of a pulse f ( x , x ) and theTR pulse ˆ f ( x , x ), expressed by inputs X, ˆ X of T m , andexpressed by outputs Y, ˆ Y of R n .In the case of propagation of phonons in hystereticmedia, the output at time t is given by extending thedouble integral to that in Preisach-Mayergoyz space. IV. CONVOLUTION OF THE KHZA WAVEFUNCTION AND ITS TR WAVE FUNCTION
In this section we explain the mechanism of TR-NEWS using the soliton wave function of Lapidus andRudenko[11, 12]. They showed a spectral decompositionof the fuction V of eq. (1-3) as V = ∞ X n =1 nz J n [ nz √ N z exp ( − R N z )] × sin[ n ( θ + tan − ( N z ) − R N z N z )] , (33)where R is a constant defined by N which characterisesnonlinearity of the material, and the shock formation co-ordinate z s .As a test, we take n = 1 , , N = 0 . , ≤ z ≤ R = 1 and c s = 1. For smaller R , z s becomes smaller.If there are singularities in finite dimensional wavefun-cions, the convolution of distributions[55, 56] is a usefultool. Since we know analytical solutions of KhZa non-linear differential equation, we first consider (1 + 1) D convolution, before (2 + 1) D convolutions of real func-tions. A. Convolution of (1 + 1) D real wave functions The variable t = 0 yields τ = t − z , θ = ωτ = ωt − ωz .We consider cases of θ = ωτ = 0 , π/ , π/ , π/ ω and z are dependent on phonetic beamsof transducers and directions of the beams relative toreceivers.
10 15 20 - - FIG. 11. V − wave function of τ = t − z (red) and V +123 wavefunction of τ = t + z (blue) for ωτ = 0.
10 15 20 - - FIG. 12. V − wave function of τ = t − z (red) and V +123 wavefunction of τ = t + z (blue) for ωτ = π/
10 15 20 - - FIG. 13. V − wave function of τ = t − z (red) and V +123 wavefunction of τ = t + z (blue) . for ωτ = π/
10 15 20 - - FIG. 14. V − wave function of τ = t − z (red) and V +123 wavefunction of τ = t + z (blue) . for ωτ = 3 π/ When ωτ = 0 there appears hysteretic effects between V − defined by τ = t − z and V +123 defined by τ = t + z ,but when ωτ = π/ V − and V +123 are identical.The Fourier transform of V − contains lower andbroader peak spectra than those in the convolution of V − and V +123 . The convolution of V +123 and V − for ωτ = 0 and π/ ωτ = π/ π/
10, the height of peakes are almost the same, butthey the shape of sidelobes are different. Discrete fouriertransformations were than by using Mathematica[16] anda computer at the RCNP.
50 100 150 200 250510152025
FIG. 15. Convolution of V +123 and V − for ωτ = 0. The reduced velocity V = V (1) + V (2) are parametrizedas[11] V = B sin ψ + B sin 2 ψ − A cos 2 ψ, (34)whose maximum is assumed to occur at ψ = π/ δ ,0
50 100 150 200 25024 FIG. 16. Convolution of V +123 and V − for ωτ = 3 π/ V + given by the eq. (35) as a functionof N and z . where δ is a small quantity, and within error of δ , V + = B + A + 4 B / ( B + 4 A ) , (35)and the value of the peak satisfies V + = B sin( A B + 2 B + zV + )+ B sin[2( A B + 2 B + zV + )] − A cos[2( A B + 2 B + zV + )] (36)Dependence of B , B , A on N and z are given in [11].In Fig. 17, V + of eq. (35) as a funcfion of n and z areplotted, and in the Fig. 18, the right hand side of eq. (36)with V + as a funcion of n and z , which is tangential tothe surface of V + of eq. (36) are presented. (0 ≤ N ≤ .
8, 1 ≤ z ≤ N, z ) plane willbe obtained by choosing l mm such that ωτ becomes 0 or π/ FIG. 18. The surface V + given by the eq. (35) as a functionof N and z and the surface of V + given by eq. (36), that istangential to the surface of V + given by the eq. (35) at itspeak. passes the middle of ordinary wave source and TR wavesource, and the distance between the point and the re-ceiver is z . N is a parameter that represents the ratio ofthe distance between the position of shock formation andthe diffraction length. They are normalized as 0 < z < < N <
1. The following figure is the logarithm
FIG. 19. 2D convolution of KhZa ordinary and TR solitonsas a function of z and N . 72 ×
72 lattice points. (2 + 1) D convolution on a 72 ×
72 lattice (blue) , 48 × ×
36 lattice (red) at N = 0 . z . There appears local regions where log-arithm of convolution is almost linear. Near z = 0 and1, there appears a point where the (2 + 1) D convolutionof KhZa solitons is negative. These singular points aredropped in the figureIV A. The figureIV A shows the sin-gular point tends to conv = 0 as the mesh size increases.1 - - (cid:13)(cid:14)(cid:15) - - (cid:18)(cid:19)(cid:20) - (cid:21)(cid:22)(cid:23)L(cid:24)(cid:25) _ [ c(cid:26)(cid:27)(cid:28) ] FIG. 20. Logarithm of 2D convolution of KhZa ordinary andTR solitons as a function of z and at N = 0 . (cid:29)(cid:30)(cid:31) !" - -./3 - FIG. 21. 2D convolution of KhZa ordinary and TR solitonsas a function of z and at N = 0 . B. The APS index and the convolution of (2 + 1) D real wave functions Since logarithm of the negative convolution value is acomplex number, the (2 + 1) D convolution is related tothe APS index, which is formulated for systems in heatbath[48].Schwartz defined set of all distribution in R as D ( R )and distribution in R with compact support as E ( R ).The convolution of 2 D real distributions u ( x, y ) ∈D ′ ( R ) and u ( x, y ) ∈ E ( R ) which is denoted as u ∗ u is numerically calculated as p ( k x , k y ) = m x − X i x =0 m y − X i y =0 u ( i x , i y ) u ( k x − i x , k y − i y )= m x − X j x =0 m y − X j y =0 u ( j x , j y ) u ( k x − j x , k y − j y ) , (37)where k x = 0 , · · · , m x − k y = 0 , · · · , m y − u ( x, y ) and u ( x, y ) are de- noted asˆ u ( ξ, ζ ) = Z dx Z dy u ( x, y ) e √− h x,ξ i e √− h y,ζ i ˆ u ( ξ, ζ ) = Z dx Z dy u ( x, y ) e √− h x,ξ i e √− h y,ζ i . (38)The symbol h x, ξ i means for a function φ ∈ C ∞ ( R ) andcoordinates x, ξ , h x, ξ i = x ( ξ ) = x ( φξ ) = ( ξx )( φ ) = ( ξx )(1) (39)when φ is 1 near the overlapping region of the support of x and that of ξ [56].The Fourier transform of p ( k x , k y ) is the product ofˆ u ( ξ, ζ ) and ˆ u ( ξ, ζ ).When u and u are analytic function of z ∈ C , theCauchy-Riemann system ∂u/∂ ¯ z j = g j , ( j = 1 , , (40)where g j satisfies the compatibility condition[57] ∂g j /∂ ¯ z k − ∂g k /∂ ¯ z j = 0 , (41)should be constructed.When there are no cracks, u k are proportional to thesolution V ( z, t ) and one can check that g k = 0. Thesituation is the same for the TR solution V ( z, − t ).Hoermander[56, 58] defined principal symbols p ( γ )that transforms the coordinate system from E labeledby x to F labeled by ξ ( x ), and p ( γ ) is defined on thesection of a cotangent bundle T ∗ ( X ). On T ∗ ( X ) one candefine one form ω and at γ ∈ T ∗ ( X ) choose a tangentvector t , such that h t, ω i = h π ∗ t, γ i . On T ∗ ( X ) one can define the symplectic form σ = dω which is expressed by the standard coordinate system x, ξ as σ = n X j =1 dξ j ∧ dx j . For an element A of general linear group and for n di-mensional manifold, σ ( A (Ξ))( A ( X )) = σ (Ξ)( X ), definesthe symplectic group Sp ( n ), which can be identified witha subgroup of Sp ( n, C )[19].At a receiver convolution of phonons from differenttranducers need to be measured and analyzed. Diferentbeam directions can be expressed by mapping of differ-ent coordinate system p and by different sound velocity c . We define the local coodinate system κ and κ suchthat κ κ − : κ ( X κ ∩ X κ ) → κ ( X κ ∩ X κ ) , (42)where X κ and X κ are open sets in X .In X κ ∩ X κ , u = u κ ◦ κ − = u κ ◦ κ . When f = du is an n − form on X and coordinates are κ ( x ) =2( x , · · · , x n ) and a C ∞ map ψ : X → Y , one can define ψ = κ ◦ κ and ψ ∗ f ,( κ − ) ∗ f = f κ dx ∧ · · · ∧ dx n . (43)Using a map ψ : Y → X , f κ = ( detψ ) ψ ∗ f κ (44)in κ ( X κ ∩ X κ ).When κ system is defined as the beam line T → R in Fig. 5 as the real axis of the complex plane S , and κ system is defined as the beam line T → R in Fig.5 as the real axis of the complex plane S , a Jacobianbecomes dependent on the angle between the two realaxis.The variable of KhZa wave functions is complex z , butwave functions are R . In a simple situation in whicha receiver is on the average height of the transducerswhich emits the KhZa wave and the TR-KhZa wave, thestrength of the convolution of the KhZa wave and theTR-KhZa wave can be calculated by the 2 D convolutionprogram which exists in the library ASL of SX-ACE su-percomputer. The 2 D consists of the distance along thebeam z and the parameter N in the equation (33). Wechoose lattice points N z = N N = 9. We compare casesof dx = 0 . . R ischosen to be 1, phase θ is chosen to be 0 and the series ofthe Bessel function is truncated at n = 3 in the Figs.19and 20.When dx is large, the distance between transducer andreceiver is large and the peak is reduced, The short ridgeof the convolution value is along the z axis, which meansthat the variation along z axis is steeper than that along N axis.Soliton wave functions are disturbed by singularitiesand change their structure. V. QUATERNION NEURAL NETWORK ANDTOPOLOGICAL PROPERTIES
In convolutional neural networks[53], input and out-put layers are defined on a 2 D plane L q × B q , where L q denotes the height of the wall and B q denotes thewidth of the wall, where the suffix q indicates the depthof the q th layer. The system contains hidden layers hav-ing F q × F q × d q parameters for filtering. L q +1 = L q − F q + 1 , B q +1 = B q − F q + 1 . (45)For simplicity, we choose F q = 3 as in [53].We first consider the case of 1 D convolution and con-sider the case of 2 D .We assume that the phonon can be approximated byKhZa solitons which is holomorphic on R when1 − bN z + (1 + b ) n z = 0 . (46)The condition can be checked by the initial condition. The p -th filter in the q -th layer has parameters W ( p,q ) = [ w ( p,q ) ij ], ( i, j ) = (1 , q -th layer areparametrized by 2 D tensor [ h ( q ) ij ]. Convolutional oper-ations from q th layer to the ( q + 1)th layer are definedas h ( q +1) ij = F q X r =1 F q X s =1 w ( p,q ) rs h qi + r − j + s − , (47) ∀ i ∈ { , · · · , L q − F q + 1 } , ∀ j ∈ { , · · · , B q − F q + 1 } .Each transducers and receivers exchange informationon quaternion bases t, x, y, z .Transducers T , · · · , T M and receivers R , · · · , R M areconnected by weight function w mn ≤ m ≤ M, ≤ n ≤ M . By padding, L q and B q increase from M to M + F q − D plane on which transducers and receivers areplaced h = τ I + x i + y j , q = ( t + l mn /c ) I + x i + y j , q = ( t − l mn /c ) I + x i + y j .Here l mn is the distance between the transducer T m and the receiver R n . A. Convolution of complex wave functions in D spatial space In the case of the input phonon beam and the TRphonon beam are parallel, we take m = n , and take intoaccount the quaternion equivalence[66]. hq − q h = l mm τc s − l mm √− x − τc s + l mm yc s + ˜ ty l mm √− x − τc s + l mm yc s − ˜ ty l mm τ + √− xc s ! , where ˜ ty = τ y − ty .If the path length is taken M od [2 l mm /c s ], i.e. backscattering is ignored, it becomes l mm c s − t ) y + τ y ( l mm c s + t ) y − τ y ! , (48) q becomes equivalent to q , via a choice of y = y , τ = t ± l mm /c s , the matrix becomes 0 H . Since we imposedno back scattering condition, − l mm /c s = l mm /c s , M od. [2 l mm /c s ] . (49)In calculations of the loss function, it is necessary totake into account that the soliton beam line is defined ina finite region of ( τ, z ⊥ ) plane, where z ⊥ is the 2 D planeperpendicular to the beam direction. B. Convolution of complex waves in D spacialspace In the case of convolution of 2 D complex wave func-tions, one can construct a model based on symplecticgroup Sp ( n, C ), which yields a Jacobian.3The coordinate whose input phonon beam direction isparallel to the x-axis of a complex plane S is defined as κ and the coordinate whoseTR phonon beam directionis parallel to the x-axis of a complex plane S is definedas κ . Detailed calculations including Jacobians are leftfor the future.Conformal wave functions in finite regions and sym-metry protected topological bosonic phase of matter wasdiscussed by Witten[48]. He proposed the symmetry pro-tected topological (SPT) bosonic phase, which is dis-turbed by anomalies. In the three-manifold X = R × M ,where R is the parameter of time, there is Chern-Simonscoupling of fields A : CS ( A ) = e π Z Y d xǫ ijk A i ∂ j A k , k ∈ Z . (50)Topological insurators and superconductors indicatethat interaction of fermions and bosons are important.Fermion topological phases are characterised by the in-dex theory.Atiyah-Singer index theorem[42] states that for an el-liptic differential operator on compact manifold, the an-alytical index is equal to the topological index. Thetheorem was extended by Atiyah-Patodi-Singer[43] to bemathcalm applicable to an elliptic differential operatorson manifolds with boundary. As in Fig. 4, we restictedthe variable z = τ I + xi + yj to be on a complex plane withfinite boundaries. In the process of arbitrary paddingnear the boundary, analytical continuation of the KhZasoliton wave function inside the area before padding ofFig. 4 to whole area after padding such that the bound-ary value becomes 0 will become possible.The APS index for the Dirac equation of fermion withmass M D Ψ = D µ γ µ − √− M −√− M !! ψ ψ ! = 0(51)where ψ ( t, x , y ) and ψ ( − t, x , y ) are time reversedstates that satisfy T ψ ( t, x , x ) = γ ψ ( − t, x , x ) (52)where T is the time-reversal operator, is Index ( D ) = dimKer ( D ) − dimCoker ( D ) . (53)In order to characterize the index, one defines metric g and gauge field A and a spectral flow characterised bya parameter s ; ( A , g ) → ( A φ , g φ ) (54)such that ( A s , g s ) coincides with ( A , g ) at s = 0 and( A φ , g φ ) at s = 1.When a system of Dirac fermions loses T symmetry,and satisfies ( √− γ ν D ν + √− µ ) ψ = 0, the regularizedpartition function becomes Z ψ,reg = Y k λ k λ k + √− µ . (55) For large µ >
0, each eigenvalue λ k contributes √− −√− Z ψ , and Z ψ = | Z ψ | exp ( − √− π X k sign ( λ k )) . (56)Thus Z ψ = | Z ψ | exp ( ∓√− πη/
2) (57)where η = lim s → P k sign ( λ k ) | λ k | − s The APS index theorem says that when boundaryfermions give a T conserving results. exp ( ∓√− πη/ exp ( ±√− π ( P − ˆ A ( R )) = ( − I (58)where P is the instanton number, ˆ A ( R ) = R X ˆ A ( R ) is thegravitational correction due to spacial curvatures.The formulae are valid also for Majorana fermions, andthe relation η CS ( A )2 π − CS grav π , mod Z (59)was proposed. Gravitational anomalies are written in[52].In (2 + 1) D massless fermions Ψ, Z Ψ = | P f ( D ) | exp ( − π √− η R /
2) (60)where
P f is the Paffian [49].The heat equation and the index theorem is discussedin [44, 45].Gravitational anomaly is discussed by Witten in [50]and [52]. Witten considered metrics ds = dt + [(1 − λ ( t )) g µν + λ ( t ) g πµν ] dx µ dx ν (61)where g µν is the Euclidean metric on R n − , g πµν is itsconjugate under diffeomorphism π , and √− is a chi-larity projection operator.It is interesting that the t dependence of the metric hassimilarity to v dependence of activation functions Φ( v ) ofneural network[53].In this metrics, the APS invariant η is defined as[50] η = lim ǫ → X E A =0 ( signE A ) exp ( − ǫ | E A | )where E A are the eigenvalues of the Dirac operator on( M × S ) π .In order to derive APS index for Majorana-Weylfermions in 8 k + 2 dimensional space, that satisfy √− ψ = ψ , metric tensor ds = dt + ǫg tµν ( x α ) dx µ dx ν on ( M × S ) π manifold, where α, µ, ν indicate coordinatesof M , and the suffix π indicates a diffeomorphism from4the boundary circle of upperhalf hemisphere S n ∈ M tothe boundary circle of lowerhalf hemisphere is done.Optimum values of input values of hidden layers F q × F q × d q in the padding area would be defined by solvingthe Atiyah-Patodi-Singer’s boundary problem[43, 46–49,51].We replace A , A by Re [ V ( z, r, T )] , Im [ V ( z, r, T )],and seek solutions which minimizes CS ( A ) and effec-tively V ( z, r, T ) becomes 0 at the boundary of the paddedarea and the square of differences of phonons beaming atreceivers and TR-phonons beaming at the same receiverbecomes minimum. VI. QUATERNION FOURIER TRANSFORM
Fourier transformation of complex functions in thetopological vector space (TVS) [54] was establishedby Schwartz[55]. In his theory, the Bochner-Minlostheorem[60] plays an essential role. It says that if thereis a nuclear space A , a characteristic functional C andfor any z j ∈ C and x j ∈ A , n X j =1 n X k =1 z j ¯ z k C ( x j − x k ) ≥ µ and the dual space A ′ , C ( y ) = Z A ′ e √− h x,y i dµ ( x ) . (63)In engineering, quaternion functions apperar in colorimage processings [61] and recurrent neural networks[62].The two-sided quaternionic Fourier transform (QFT)was introduced by Hitzer and Sangwine[63] extending thepioneering work of Ell [64]. Georgiev et al.[65] consid-ered, due to noncommutativity of quaternions, a left-sided QFT, a right-sided QFT and a two-sided QFT,using the finite integral F r ( f )( ω , ω ) = Z ba Z ba f ( x , x ) e i ω x e j ω x dx dx (64)where F r is right-handed transform, and f ( x , x ) = [ f ( x , x )] + [ f ( x , x )] i +[ f ( x , x )] j + [ f ( x , x )] k , (65)where [ f ] ℓ : [ a, b ] × [ a, b ] → R ( ℓ = 0 , , , f are real quaternion functions.Quaternion Fourier transform can also be studied in5 D Clifford Algebra following works of Garling[67] andAtiyah-Bott-Shapiro[68].The algebra A +4 , is isomorphic to M ( H ) which havethe representation j ( e ) = √− Q ⊗ Q, j ( e ) = √− J ⊗ Q,j ( e ) = U ⊗ Q, j ( e ) = √− I ⊗ J,j ( e ) = I ⊗ U, (66) where Q = ! , J = −
11 0 ! , U = − ! (67)and I is the unit matrix.For a A , , there exists a Clifford algebra B , . If π : B , ⊕ B , → B , is the projection onto the firstcoordinate, φ = π ◦ j is a projection of A , to B , On the 5 D Euclidean space R , one can define Cliffordmapping k : R , → M ( H ), by choosing bases k ( X i =1 x i e i ) = " x i + x j + x k − x + x x + x − x i − x j − x k (68)One can regard A , as a 5 D holographic space and B , as quaternion projected spaces, in which T a and T b areidentified if there exists a quaternion h = 0 that satisfy hT a = T b h in B , , and π is a projection on ordinaryand time reversed space.In (2+1)D subspaces T = x + x i + x j = " x + x √− x − x x − √− x ∈ M ( H )(69)and its pair¯ T = x + ¯ x i + ¯ x j = " x − x √− x − x x + √− x (70)satisfies T ¯ T = " x + x x x x + x . (71)It means that the gaussian structure remains afterquaternion Fourier transformations.The KhZa complex soliton wave functions fit the (2 +1) D Euclidian subspace wave functions in B , of positivechirality and negative chirality. VII. DISCUSSION AND CONCLUSION
The TR-NEWS searches singularity on the border ofcone of propagating sound by convolutions of regular andtime reversed waves. We want to maximize the convolu-tion of the output from original sound G ϕ f ( ω , ω ; b , b )and output from the TR sound d G ϕ f ( ω , ω ; ˆ b , ˆ b ) assmall as possible. That is minimize Z C [ G ϕ f ( ω , ω ; b , b ) − d G ϕ f ( ω ω ; ˆ b , ˆ b )] dω dω (72)by proper choices of b , b , ˆ b , ˆ b . We observed a neg-ative 2 D convolution value near boundary of compact5Riemann space, which is expected to be related to theAPS index.The (2+1) D quaternion representation of instant formcan be transformed to that of front form, and time stepscan be selected as τ , · · · , τ K .Chua showed that in memristic circuits, output fre-quency shows Devil’s staircase structure[27, 28] thatthere are stable output frequency regions. Each stepsmay correspond to emergences of equivalent quaternionwave functions in the projective space. We showed a pos-sible method of applying the quaternion neural networkto non destructive testing. In order to realize the project,it is necessary to optimize the phonetic pulse shape andminimize the difference of convolutions of original waveand that of TR wave.For optimization of getting positions of cracks in a rect-angular media, quaternion neural network is a promissingmethod. Effects of quantum gravity through metrics ingauge theories could be analysed. • The Lie-Trotter formula and parametrization oftime by τ = t ± l mn c , and extension of regular func-tions in the cone area due to Hille-Yosida theory are importance for achieving the convergence of evolu-tion equations. • The choice of quaternion projective space on 2 D planes is expected to reduce number of training pa-rameters, which needs further study.Although detection of the gravitational anomalies fromthe 2 D convolution of phonetic waves is not guaranteed,a program inspired by AI to search parameters that re-produce patterns of phonons emitted from transducerson a wall, scattered by cracks inside materials, and de-tected by receivers on a wall on the other side, are underinvestigation. Acknowledgment
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