Unifying vectors and matrices of different dimensions through nonlinear embeddings
UUnifying vectors and matrices of different dimensionsin smooth generalized structures
Vladimir Garc´ıa-MoralesDepartament de Termodin`amica, Universitat de Val`encia,E-46100 Burjassot, [email protected]
Physical theories address different numbers of degrees of freedom depending on the scale underconsideration. In this work generalized mathematical structures (nonlinear B κ -embeddings)are constructed that encompass objects with different dimensionality as the continuous scaleparameter κ ∈ R is varied. Based on this method, a new approach to compactification inunified physical theories (e.g. supergravity in 10 or 11-dimensional spacetimes) is pointed out.We also show how B κ -embeddings can be used to connect all cellular automata (CAs) to coupledmap lattices (CMLs) and nonlinear partial differential equations, deriving a class of nonlineardiffusion equations. Finally, by means of nonlinear embeddings we introduce CA connections,a class of CMLs that connect any two arbitrary CAs in the limits κ → κ → ∞ of theembedding. Applications to biophysics and fundamental physics are discussed.1 a r X i v : . [ n li n . C G ] M a r . INTRODUCTION Physical theories often involve relationships between mathematical objects with a varyingnumber of dimensions (degrees of freedom) that depend on the scale considered [1]. For example,macroscopic systems and classical fields involve a much lower number of degrees of freedom thanmicroscopic systems or quantum fields on which the former are based. Furthermore, the latterreveal the operator character of physical observables and fields compared to the former (inwhich observables are given by real-valued scalar quantities).Scalars, vectors, matrices and tensors are mathematical objects with different numbers ofdimensions (degrees of freedom) which generally store different amounts of information. Thequestion whether these different mathematical structures can be encompassed by more generalones may be interesting in physical applications. To that aim, we construct in this article math-ematical structures that are able to behave as scalars, vectors or matrices when a parameter κ ∈ R is being continuously tuned from 0 to ∞ . These structures are specific instances ofnonlinear B κ -embeddings that have been very recently introduced and applied to the problemof finding all roots of a complex polynomial [2].We can apply nonlinear B κ -embeddings to any situation involving connections among math-ematical objects with different dimensions and degrees of freedom. Examples of these situationsare provided by conformational changes and phase transitions in Statistical Mechanics, irre-versible processes involving information loss (e.g. coarse-graining of the microscopic dynamics)and the holographic principle (which relates degrees of freedom of quantum field theories indifferent dimensions).Another specific example of a situation involving objects with different dimensions is pro-vided by the quest for a unified theory of general relativity and quantum mechanics in theframework of, e.g. supergravity. These proceed by extending the 4-dimensional metric ofpure gravity to higher dimensions [3–7]. Although the observable universe is described by a4-dimensional metric tensor, this latter object needs to have 10 or 11 dimensions (depending onwhether supergravity arises as the low energy limit of string theory or M-theory, respectively [3])in order to consistently accomodate the gauge groups describing the standard model of particlephysics [5, 8]. Since there are only 4 observable dimensions of spacetime, a mechanism calledcompactification is needed to explain why the 6 (respectively 7) additional extra dimensionsare not observable (see [6] for a review on compactification mechanisms). Compactification canthus be viewed as a connection of a metric tensor in four dimensions with another one in 10 or11 dimensions. Usually, it involves Fourier transforming the D extra dimensions and truncatingthe resulting infinite Fourier series so that the manifold over these dimensions is folded to a tinyunobservable scale. With help of appropriate nonlinear B κ -embeddings we point out anotherpossible alternative scheme that involves no Fourier expansion and no truncation and whichleads to a κ -deformed formalism of gravity.Nonlinear B κ -embeddings can also be applied to dynamical systems, as cellular automata(CAs) [11–22] and coupled map lattices (CMLs) [13, 23–28]. These models of complex phys-ical systems are popular in biophysics [15], have given rise to novel approaches to quantummechanics [21, 22, 29, 30] and have been conjectured to play a crucial role in unified field theo-ries, giving rise to the concept of chaotic strings in the framework of stochastic quantization [8].However, natural physical systems display a great deal of variability and their evolution departs2rom the specification of a few rigid, deterministic rules perfectly operating on finite amountsof information. It is, therefore, interesting to study how these models can be embedded in moresophisticated ones [13, 16, 31–34]. In a previous recent work [13], we have presented a generalmechanism that allows any CA to be embedded in a CML in terms of a control parameter κ that governs the embedding. We display here new constructions and we show how these yieldnonlinear B κ -embeddings that are able to connect CAs to nonlinear partial differential equations(PDEs) and derive from these connections certain nonlinear diffusion equations. Furthermore,we construct B κ -embeddings ( κ -deformed structures) corresponding to CMLs that are able toglue together several different CAs. In similar ways, CMLs can be glued together to form morecomplicated structures. The mathematical methods presented here may be of interest in bio-physics (dynamics of multicellular ensembles) and in fundamental physics (extended formalismsof gravity and the embedding of different unified theories of physics related by dualities). Quiteinterestingly, a certain class of CMLs have been used to simulate quantum field theories on anappropriate scaling limit [8, 9] and it has been shown that there are 6 different such unifiedtheories in terms of chaotic strings that are somehow analogous to the six different models ofa string considered in string theory and which are embedded in M-theory [8, 10].The outline of this article is as follows. First, in Section II we introduce the method toconstruct the nonlinear B κ -embeddings involved in connecting structures with different dimen-sions (thus generalizing the concept of vectors and matrices) and illustrate these structureswith examples. In Section III we apply this method to the problem of finding a nonlinear B κ -embedding that is able to connect a 4-dimensional and a 11-dimensional metric tensors, asthose found in theories of supergravity, in the appropriate limits of the κ parameter. In thisapplication, κ is related to the characteristic scale at which spacetime is probed compared tothe Planck length. In Section IV we construct B κ -embeddings that have CAs in the κ → κ large to certain nonlinear partialdifferential equations. We derive through this method a class of nonlinear diffusion equations,and discuss the parameter values that lead to linear diffusion equations. Finally, in SectionV we show how two arbitrary CAs in rule space can be connected in the κ → κ → ∞ limits by means of appropriate nonlinear B κ -embeddings. We discuss some potential physicalapplications and present some conclusions. II. NONLINEAR B κ -EMBEDDINGS: CONNECTING SCALARS, VECTORS ANDMATRICES The method to construct B κ -embeddings connecting objects with different dimensions beginsby noting that the Kronecker delta δ nj ( δ nj = 1 if n = j and δ nj = 0 otherwise) admits a simplerepresentation in terms of the boxcar function [11, 12] B ( x, y ) ≡ (cid:18) x + y | x + y | − x − y | x − y | (cid:19) = sign( y ) if | x | < | y | sign( y ) / | x | = | y | , y (cid:54) = 00 otherwise (1)3here x, y ∈ R . Indeed, we have δ nj = B (cid:18) n − j, (cid:19) (2)We now note that, by means of convolution, any N -tuple v = ( v , v , . . . , v N − ) can bewritten in terms of its components v n ∈ C as v = v j δ nj (3)where n = 0 , , . . . , N − v = v j B (cid:18) n − j, (cid:19) (4)The outer product out( v , w ) of two vectors v and w with dimensions N and M , respectively,leads to a matrix of size N × M with two free indicesout( v , w ) ≡ v j w k B (cid:18) n − j, (cid:19) B (cid:18) m − k, (cid:19) (5)If M = N the inner product (cid:104) . . . (cid:105) of two vectors v and w is given by (cid:104) v , w (cid:105) ≡ v n w n (6)and can be obtained from the outer product by contracting the free indices (cid:104) v , w (cid:105) = v j w k B (cid:18) n − j, (cid:19) B (cid:18) m − k, (cid:19) B (cid:18) n − m, (cid:19) (7)We define a ket vector | ψ (cid:105) over a finite-dimensional vector space in terms of B -functions as | ψ (cid:105) = ψ j B (cid:18) n − j, (cid:19) (8)where the ψ j are complex numbers. A bra vector is defined as (cid:104) ψ | = ψ j B (cid:18) n − j, (cid:19) (9)where the overline denotes complex conjugation. The inner product of a bra and a ket is thendefined as (cid:104) φ | ψ (cid:105) = φ j ψ k B (cid:18) n − j, (cid:19) B (cid:18) m − k, (cid:19) B (cid:18) n − m, (cid:19) = φ n ψ n (10)Let a and b be complex numbers. We clearly have the following properties4. | aψ (cid:105) = a | ψ (cid:105) (cid:104) aψ | = a (cid:104) ψ | (cid:104) aφ + bφ | ψ (cid:105) = a (cid:104) φ | ψ (cid:105) + b (cid:104) φ | ψ (cid:105) (cid:104) φ | aψ + bψ (cid:105) = a (cid:104) φ | ψ (cid:105) + b (cid:104) φ | ψ (cid:105) (cid:104) ψ | ψ (cid:105) ≥
06. Let d ( φ, ψ ) ≡ (cid:112) (cid:104) ψ − φ | ψ − φ (cid:105) . Then:6.1. d ( ψ, ψ ) = 0;6.2. d ( φ, ψ ) = d ( ψ, φ );6.3. d ( φ, ψ ) ≤ d ( φ, η ) + d ( η, ψ )Because of these properties, the vector space with inner product defined above is a Hilbertspace.A matrix A can be written in terms of its elements A nm as A = A A . . . A ,N − A A . . . A ,N − . . . . . . A nm . . .A ,N − A ,N − . . . A N − ,N − = A jk B (cid:18) n − j, (cid:19) B (cid:18) m − k, (cid:19) (11)We note that an N -dimensional matrix can equivalently be written in a more compact formas a vector of indexed N -tuples. If we define A h = A k + Nj ≡ A jk j, k ∈ [0 , N −
1] (12)where h ≡ k + N j then Eq. (11) becomes A = N − (cid:88) h =0 A h B (cid:18) h − m − N n, (cid:19) n, m ∈ [0 , N −
1] (13)Note that there are two free indices m and n in this expression and that there are N coefficients A h .The B -function (boxcar function) is a suitable representation of the Kronecker delta whenits first argument x is an integer and its second argument y has value | y | ≤ . The crucialinterest of this representation of the Kronecker delta is that it can be easily embedded in thereal numbers by means of a one-parameter deformation function B κ ( x, y ) which (pointwise)converges to B ( x, y ) as κ → κ → ∞ [2, 13] B κ ( x, y ) ≡ (cid:20) tanh (cid:18) x + yκ (cid:19) − tanh (cid:18) x − yκ (cid:19)(cid:21) (14)with x, y ∈ R . It is straightforward to observe that B κ ( − x, y ) = B κ ( x, y ) (15) B κ ( x, − y ) = −B κ ( x, y ) (16) B − κ ( x, y ) = −B κ ( x, y ) (17)5nd to check all following identities0 < B κ ( x, y ) < sign y ∀ κ (18)0 < B κ ( x, y ) B κ (0 , y ) < ∀ κ (19)lim κ → B κ ( x, y ) = B ( x, y ) (20)lim κ → B κ ( x, y ) B κ (0 , y ) = B ( x, y ) B (0 , y ) = B ( x, | y | ) (21)lim κ →∞ B κ ( x, y ) = 0 (22)lim κ →∞ B κ ( x, y ) B κ (0 , y ) = 1 (23)by noting that, after some manipulations, we have B κ ( x, y ) = e y/κ − e − y/κ e y/κ + e x/κ + e − x/κ + e − y/κ (24) B κ ( x, y ) B κ (0 , y ) = e y/κ + 2 + e − y/κ e y/κ + e x/κ + e − x/κ + e − y/κ (25)We also observe the fact that, while B κ ( x, y ) is an even function of x and an odd function of κ and y , the ratio B κ ( x, y ) / B κ (0 , y ) is an even function of all x , y and κ . We note, furthermore,that [13] B κ ( x, y + z ) = B κ ( x + y, z ) + B κ ( x − z, y ) (26) N − (cid:88) k =0 B κ ( x − ky, y ) = B κ ( x − ( N − y, N y ) (27)Interestingly, we have the following expansions e x/κ = 1 + B κ (cid:0) , x (cid:1) − B κ (cid:0) , x (cid:1) = (cid:104) B κ (cid:16) , x (cid:17)(cid:105) ∞ (cid:88) j =0 (cid:104) B κ (cid:16) , x (cid:17)(cid:105) j = 1 + 2 ∞ (cid:88) j =1 (cid:104) B κ (cid:16) , x (cid:17)(cid:105) j (28)cosh xκ = 1 + 2 ∞ (cid:88) j =1 (cid:104) B κ (cid:16) , x (cid:17)(cid:105) j (29)sinh xκ = 2 ∞ (cid:88) j =1 (cid:104) B κ (cid:16) , x (cid:17)(cid:105) j − (30)The B κ -function is an infinitely differentiable function of x, y and κ . The n -th derivatives6re given by ∂ n ∂x n B κ ( x, y ) = (2 κ ) n n − (cid:88) j =0 (cid:28) nj (cid:29) (cid:34) e x + y )(1+ j ) /κ (1 + e x + y ) /κ ) n +1 − e x − y )(1+ j ) /κ (1 + e x − y ) /κ ) n +1 (cid:35) (31) ∂ n ∂y n B κ ( x, y ) = (2 κ ) n n − (cid:88) j =0 (cid:28) nj (cid:29) (cid:34) e x + y )(1+ j ) /κ (1 + e x + y ) /κ ) n +1 + e − x − y )(1+ j ) /κ (1 + e − x − y ) /κ ) n +1 (cid:35) (32) ∂ n ∂ (1 /κ ) n B κ ( x, y ) = 2 n n − (cid:88) j =0 (cid:28) nj (cid:29) (cid:34) ( x + y ) n e x + y )(1+ j ) /κ (1 + e x + y ) /κ ) n +1 − ( x − y ) n e x − y )(1+ j ) /κ (1 + e x − y ) /κ ) n +1 (cid:35) (33)where we have introduced the Eulerian number (cid:28) nj (cid:29) ≡ j +1 (cid:88) m =0 ( − m (cid:18) n + 1 m (cid:19) ( j + 1 − m ) n (34)We also have (cid:90) ba dx B κ ( x, y ) = κ (cid:34) cosh b + yκ cosh b − yκ cosh a − yκ cosh a + yκ (cid:35) (35)Therefore, independently of κ , 12 y (cid:90) ∞−∞ dx B κ ( x, y ) = 1 (36)For κ sufficiently large ( κ > | x | + | y | ) π ) the hyperbolic tangents in the definition of the B κ -function can be expanded in their convergent Maclaurin series, and we have B κ ( x, y ) = 12 ∞ (cid:88) j =1 j (2 j − B j (2 j )! κ j − (cid:104) ( x + y ) j − − ( x − y ) j − (cid:105) = 12 ∞ (cid:88) j =1 j (2 j − B j (2 j )! κ j − j − (cid:88) h =0 (cid:18) j − h (cid:19) x j − − h (1 − ( − h ) y h = ∞ (cid:88) j =1 j (2 j − B j (2 j )! κ j − j (cid:88) h =1 (cid:18) j − h − (cid:19) x j − h ) y h − = B κ (0 , y ) + ∞ (cid:88) j =2 j (2 j − B j (2 j )! κ j − j − (cid:88) h =1 (cid:18) j − h − (cid:19) y h − x j − h ) = yκ − y + 3 x y κ + O (cid:0) κ − (cid:1) (37) B κ ( x, y ) B κ (0 , y ) = 1 + 1 B κ (0 , y ) ∞ (cid:88) j =2 j (2 j − B j (2 j )! κ j − j − (cid:88) h =1 (cid:18) j − h − (cid:19) y h − x j − h ) = 1 − x κ + O (cid:0) κ − (cid:1) (38)7here the B m denote the even Bernoulli numbers: B = 1, B = , B = − B = , etc.In the above expression we have also used that B κ (0 , y ) = ∞ (cid:88) j =1 j (2 j − B j y j − (2 j )! κ j − = tanh yκ (39)The main trick introduced in this manuscript is to replace any B -function in Eqs. (4)or (11), e.g. B (cid:0) n − j, (cid:1) , by a κ -deformed counterpart with the form either B κ (cid:0) n − j, (cid:1) or B κ (cid:0) n − j, (cid:1) / B κ (cid:0) , (cid:1) depending on the application under consideration. We shall call thefirst kind of replacement Mode I and the latter Mode II. If we simply replace all B -functionsfollowing Mode I we obtain from Eqs. (4) or (11), respectively, v ( I ) κ ≡ v j B κ (cid:18) n − j, (cid:19) (40) A ( I ) κ ≡ A jk B κ (cid:18) n − j, (cid:19) B κ (cid:18) m − k, (cid:19) (41)These nonlinear B κ -embeddings constitute κ -deformed structures that generalize those of avector and of a matrix respectively as follows: In the limit κ →
0, by using Eq. (20) in Eqs.(40) and (41), we regain Eqs. (4) and (11), respectively. However, in the limit κ → ∞ , fromEq. (22), we obtain lim κ →∞ v ( I ) κ = 0 and lim κ →∞ A ( I ) κ = 0.If, instead, we replace all B -functions following Mode II we obtain from Eqs. (4) or (11),respectively v ( II ) κ ≡ v j B κ (cid:0) n − j, (cid:1) B κ (cid:0) , (cid:1) (42) A ( II ) κ ≡ A jk B κ (cid:0) n − j, (cid:1) B κ (cid:0) , (cid:1) B κ (cid:0) m − k, (cid:1) B κ (cid:0) , (cid:1) (43)In the limit κ →
0, by using Eq. (20) in Eqs. (40) and (41), we regain Eqs. (4) and (11),respectively. However, in the limit κ → ∞ , from Eq. (23), we now obtainlim κ →∞ v ( II ) κ = N − (cid:88) j =0 v j ≡ v (44)lim κ →∞ A ( II ) κ = N − (cid:88) j =0 N − (cid:88) k =0 A jk ≡ A (45)i.e., the vector and the matrix collapse, respectively, to the scalars v and A formed by summingover all their entries.Note that, in both modes, the dimensionality of the mathematical object is reduced from N to 0 as κ is varied from 0 to ∞ when all B -functions are being replaced. If one choosesto replace only a finite subset of the B -functions and combines both modes of replacement,the dimensionality of the object can be tuned to any integer value between 0 and N . Which8eplacement mode to choose and which entries of the matrix are to be deformed by the κ parameter depend on the application at hand. For example, let us imagine that we want toconstruct a nonlinear B κ -embedding that connects any arbitrary operator A with its trace Tr A .Then we begin by noting that we can equivalently write A as A = A jk B (cid:18) n − j, (cid:19) B (cid:18) m − k, (cid:19) (46)= N − (cid:88) j =0 A jj B (cid:18) n − j, (cid:19) B (cid:18) m − j, (cid:19) + N − (cid:88) j =0 (cid:88) k (cid:54) = j A jk B (cid:18) n − j, (cid:19) B (cid:18) m − k, (cid:19) (47)From this, we can combine replacement Modes I and II to construct a nonlinear B κ -embedding A κ = N − (cid:88) j =0 A jj B κ (cid:0) n − j, (cid:1) B κ (cid:0) , (cid:1) B κ (cid:0) m − j, (cid:1) B κ (cid:0) , (cid:1) + N − (cid:88) j =0 (cid:88) k (cid:54) = j A jk B κ (cid:18) n − j, (cid:19) B κ (cid:18) m − k, (cid:19) (48)In the limit κ →
0, by using Eq. (20) in Eq. (48), we regain Eq. (46). However, in the limit κ → ∞ , from Eq. (23), we now obtainlim κ →∞ A κ = N − (cid:88) j =0 A jj = Tr A (49)In this way, the nonlinear B κ -embedding given by Eq. (46) smoothly connects a matrix operatorwith its (scalar) trace.As another example, we can construct a nonlinear B κ -embedding that connects the outerand inner product of two vectors. To that end, we can apply replacement Mode II to Eq. (7)to define the κ -deformed outer product as (cid:104) v , w (cid:105) κ ≡ v j w k B (cid:18) n − j, (cid:19) B (cid:18) m − k, (cid:19) B κ (cid:0) n − m, (cid:1) B κ (cid:0) , (cid:1) (50)We have lim κ →∞ (cid:104) v , w (cid:105) κ = v j w k B (cid:18) n − j, (cid:19) B (cid:18) m − k, (cid:19) = out( v , w ) (51)lim κ → (cid:104) v , w (cid:105) κ = v j w k B (cid:18) n − j, (cid:19) B (cid:18) m − k, (cid:19) B (cid:18) n − m, (cid:19) = (cid:104) v , w (cid:105) (52)We note thus that B κ -embeddings can be used to continuously tune the degree to which twofree indices can be contracted.We note that in any nonlinear B κ -embedding, the limits κ → κ → ∞ can be exchangedby making the transformation κ → /κ . Thus, for any nonlinear B κ -embedding, there exists a‘dual’ structure obtained by making the latter transformation.9 II. COMPACTIFICATION IN UNIFIED PHYSICAL THEORIES
We now show how the ideas in the previous section can be applied to model compactificationin the passage from supergravity to pure 4-dimensional gravity. The main object of generalrelativity is the metric tensor with elements g µν ( µ, ν ∈ { , , , } ) that governs the geometryof spacetime. It is a 4 × g µν = g νµ ) with 10 different components. In ournotation, the metric tensor can be specified as g D ≡ g g g g g g g g g g g g g g g g = (cid:88) j =0 3 (cid:88) k =0 g jk B (cid:18) n − j, (cid:19) B (cid:18) m − k, (cid:19) (53)In 11-dimensional supergravity (10-dimensional supergravity would proceed on analogouslines), the metric tensor has the form g D ≡ g g . . . g g , g g . . . g g , . . .g g . . . g g , g , g , . . . g , g , = (cid:88) j =0 10 (cid:88) k =0 g jk B (cid:18) n − j, (cid:19) B (cid:18) m − k, (cid:19) (54)The problem that leads to introduce the idea of compactification is that 11 dimensions arenecessary in supergravity (seen as the low energy limit of M-theory) to consistently bringtogether general relativity and quantum mechanics and, hence, it may be described by a tensorof the form of Eq. (54). However, the observable universe has 4 dimensions and, hence, it isdescribed by an object of the form of Eq. (53). If we regard the extra 7 dimensions as true,physical dimensions, on a par with the four observed dimensions [5] this suggests to embedboth the 4-dimensional metric tensor of pure gravity and the 11-dimensional tensor in a singlenonlinear B κ -embedding. Let (cid:96) be the scale at which spacetime is probed. If we identify κ ∝ (cid:96)L p (55)(where L p = (cid:113) G (cid:126) c = 1 . × − cm is the Planck length, with G being the gravitationalconstant, c the speed of light and (cid:126) Planck’s constant), we can translate the problem of com-pactification to the description of a mathematical mechanism involved in the passage from g D to g D as κ → ∞ . This suggests the construction of a nonlinear B κ -embedding that yields a4-dimensional metric tensor g D in the limit κ → ∞ (thus being able to reproduce generalrelativity in this limit) and the 11-dimensional metric tensor g D in the limit κ → g D = (cid:88) j =0 10 (cid:88) k =0 g jk B (cid:18) n − j, (cid:19) B (cid:18) m − k, (cid:19) (56)= (cid:88) j =0 3 (cid:88) k =0 g jk B (cid:18) n − j, (cid:19) B (cid:18) m − k, (cid:19) + (cid:88) j =4 3 (cid:88) k =0 g jk B (cid:18) n − j, (cid:19) B (cid:18) m − k, (cid:19) + (cid:88) j =0 10 (cid:88) k =4 g jk B (cid:18) n − j, (cid:19) B (cid:18) m − k, (cid:19) + (cid:88) j =4 10 (cid:88) k =4 g jk B (cid:18) n − j, (cid:19) B (cid:18) m − k, (cid:19) We can now apply the replacement Mode I to every B -function involving a dummy index higheror equal than four. In this way, we construct a nonlinear B κ -embedding that includes Eq. (54)in the limit κ →
0. We obtain g κ = (cid:88) j =0 3 (cid:88) k =0 g jk B (cid:18) n − j, (cid:19) B (cid:18) m − k, (cid:19) + (cid:88) j =4 3 (cid:88) k =0 g jk B κ (cid:18) n − j, (cid:19) B (cid:18) m − k, (cid:19) (57)+ (cid:88) j =0 10 (cid:88) k =4 g jk B (cid:18) n − j, (cid:19) B κ (cid:18) m − k, (cid:19) + (cid:88) j =4 10 (cid:88) k =4 g jk B κ (cid:18) n − j, (cid:19) B κ (cid:18) m − k, (cid:19) We have, by using Eqs. (20) and (23)lim κ → g κ = (cid:88) j =0 10 (cid:88) k =0 g jk B (cid:18) n − j, (cid:19) B (cid:18) m − k, (cid:19) = g D (58)lim κ →∞ g κ = (cid:88) j =0 3 (cid:88) k =0 g jk B (cid:18) n − j, (cid:19) B (cid:18) m − k, (cid:19) = g D (59)If we denote by ( dx , dx , . . . , dx ) any 11-dimensional spacetime infinitesimal displacementwe can write the nonlinear B κ -embedding of the squared differential of the arc-length ds κ as ds κ = (cid:88) n =0 10 (cid:88) m =0 dx m dx n (cid:34) (cid:88) j =0 3 (cid:88) k =0 g jk B (cid:18) n − j, (cid:19) B (cid:18) m − k, (cid:19) (60)+ (cid:88) j =4 3 (cid:88) k =0 g jk B κ (cid:18) n − j, (cid:19) B (cid:18) m − k, (cid:19) + (cid:88) j =0 10 (cid:88) k =4 g jk B (cid:18) n − j, (cid:19) B κ (cid:18) m − k, (cid:19) + (cid:88) j =4 10 (cid:88) k =4 g jk B κ (cid:18) n − j, (cid:19) B κ (cid:18) m − k, (cid:19)(cid:35) κ → ds κ = (cid:88) n =0 10 (cid:88) m =0 g mn dx m dx n (61)lim κ →∞ ds κ = (cid:88) n =0 3 (cid:88) m =0 g mn dx m dx n (62)These equations provide the right differential of the arc-lengths for the respective metric tensors.We thus see that the κ -deformed structure, Eq. (57) contains the metric tensors of 4-dimensionalgravity and 11-dimensional supergravity as specific limiting cases, regardless of the specific formof their elements g µν . Although we have assumed a simple dependence of the parameter κ onthe Planck length, Eq. (55), this dependence can be more involved and might be derived fromfirst-principles in terms of the local curvature of spacetime, etc. It should be noted that, for κ nonvanishing, the B κ -function is a smooth and infinitely differentiable function of κ and thatthe nonlinear B κ -embeddings given by Eq. (57) and (60) are smooth, differentiable and welldefined for any value of κ .We note that, from Eq. (13) we can more concisely write Eqs. (53) and (54) as g D = (cid:88) h =0 g h B (cid:18) h − m − n, (cid:19) n, m ∈ [0 ,
3] (63) g D = (cid:88) h =0 g h B (cid:18) h − m − n, (cid:19) n, m ∈ [0 ,
10] (64)By using replacement Mode I on these expressions we arrive to an alternative nonlinear B κ -embedding with the same asymptotic properties as Eq. (57) g κ = (cid:88) j =0 3 (cid:88) k =0 g jk B (cid:18) k + 11 j − m − n, (cid:19) + (cid:88) j =4 3 (cid:88) k =0 g jk B κ (cid:18) k + 11 j − m − n, (cid:19) + (cid:88) j =0 10 (cid:88) k =4 g jk B κ (cid:18) k + 11 j − m − n, (cid:19) n, m ∈ [0 ,
10] (65)= (cid:88) h =0 g h B κ h (cid:18) h − m − n, (cid:19) n, m ∈ [0 ,
10] (66)where we have introduced an h -dependent deformation parameter κ h so that κ h = 0 + is van-ishingly small for h ∈ { , , , , , , , , , , , , , , , } and equal to κ > h , being h ∈ [0 , h ∈ Z . IV. CONNECTING CELLULAR AUTOMATA AND (NONLINEAR) PARTIAL DIF-FERENTIAL EQUATIONS THROUGH NONLINEAR B κ -EMBEDDINGS In Section III we have shown how nonlinear B κ -embeddings can be used to construct gen-eralized structures that connect mathematical objects with different dimensionality. In this12nd the following section, we show how they can be used indeed to smoothly connect differentqualitative dynamical behaviors governed by different dynamical (evolution) rules. Nonlinear B κ -embeddings can therefore be used to construct ‘cartographies’ of physical theories and bi-furcation scenarios where qualitative changes in dynamical behavior are induced by tuning thecontinuous parameter κ . In these cartographies, theories (specific models) are encompassed bymore advanced theories in a hierarchical manner.CAs [11, 12, 14–22], CMLs [13, 23–28] and (nonlinear) PDEs [35, 36] constitute the differentmathematical approaches to model spatiotemporal pattern formation outside of equilibrium,as found in experimental physical systems [15]. CAs are fully discrete coupled maps in whichspace and time are discrete and the local phase space is both discrete and finite. CAs serveas toy models for the overall observed features of complex physical systems [14]. An exampleof this is spatiotemporal intermittency [14, 37, 38]. CMLs [23, 24] are discrete maps rulingthe evolution of a dynamical system on a discrete spacetime but for which the local phasespace is continuous. Finally, PDEs constitute continuous models of dynamical system evolvingon continuous and differentiable spacetimes and with a continuous local phase space. In thissection we show how all CAs can be encompassed by means of B κ -embeddings that connectthem to certain CMLs and nonlinear PDEs. In particular, we show how some CAs lead tononlinear diffusion equations.We define the alphabet A p ≡ { , , . . . , p − } , p ≥ p ∈ N , as the set of integers in theinterval [0 , p − A Np for the Cartesian product of N copies of A p . Let x jt ∈ A p bea dynamical variable at time t ∈ Z and position j ∈ Z on a ring of N s sites, j ∈ [0 , N s − l , r be non-negative integers. A CA, with rule vector ( a , a , . . . , a p l + r +1 − ), a n ∈ Z ∈A p , ∀ n ∈ [0 , p l + r +1 − N = l + r + 1 and Wolfram code R = (cid:80) p l + r +1 − n =0 a n p n , is a map A N s p → A N s p acting locally at each site j as A Np → A p and synchronously at every t accordingto the universal map [11] x jt +1 = p r + l +1 − (cid:88) n =0 a n B (cid:32) n − l (cid:88) k = − r p k + r x j + kt , (cid:33) j = 0 , , . . . , N s − j + k = j + k mod N s . We note that the Wolfram code R is an integer R ∈ [0 , p p l + r +1 ].All parameters specifying any CA rule can be given in a compact notation by means ofthe code l R rp [11]. For example, all 256 Wolfram elementary CA R are obtained by taking p = 2, l = r = 1 in Eq. (67). Thus, Wolfram rule 30 is denoted by and has rule vector( a , a , . . . a ) = (0 , , , , , , , a n of any CA can be directlyobtained from the Wolfram code R by means of the following expression [21] a n = (cid:22) Rp n (cid:23) − p (cid:22) Rp n +1 (cid:23) (68)where (cid:98) . . . (cid:99) denotes the lower closest integer (floor) function.Specially interesting for physical applications are those CAs that are locally isotropic so thatthe CA output does not depend on the particular arrangement of the dynamical states withina neighborhood, but on the sum of the cell values. These are called totalistic CAs and are a13ubset of those described by Eq. (67). Totalistic CAs are given by the map x jt +1 = ( r + l +1)( p − (cid:88) n =0 σ n B (cid:32) n − l (cid:88) k = − r x j + kt , (cid:33) j = 0 , , . . . , N s − σ n ∈ A p . The Wolfram code of a totalistic CA is given by RT ≡ (cid:80) ( r + l +1)( p − n =0 σ n p n andis an integer number satisfying RT ∈ [0 , p r + l +1)( p − ].Let the real line excluding all half-integer numbers ( . . . , − , − , , , . . . ) be denoted by R \ Z / . From the definition of the B -function, it is clear that the function B (cid:0) n − x, (cid:1) for n ∈ Z and x ∈ R \ Z / provides a surjective application Z × R \ Z / → { , } . Thus, if we relax any x jt in Eq. (67) to be a real number so that n jt ≡ (cid:80) lk = − r p k + r x j + kt ∈ R \ Z / , we find that x jt +1 ∈ A p .Therefore, for all initial conditions x j for which n j = (cid:80) lk = − r p k + r x j + k ∈ R \ Z / ( ∀ j ) the localdynamics provided by Eq. (67) has the form A Np → A p for any t >
0. This fact allowed us togeneralize in [13] the universal map for CA, Eq. (67), to real-valued deterministic CA in termsof the B κ -function, Eq. (14) x jt +1 = p r + l +1 − (cid:88) n =0 a n B κ (cid:32) n − l (cid:88) k = − r p k + r x j + kt , (cid:33) j = 0 , , . . . , N s − κ →
0. Inthe limit κ → ∞ one has, from Eq. (37), x jt +1 ∼ κ [13] so that, if the limit is strictly taken x jt = 0 , ∀ j and ∀ t > x jt +1 = p r + l +1 − (cid:88) n =0 a n B κ (cid:16) n − (cid:80) lk = − r p k + r x j + kt , (cid:17) B κ (cid:0) , (cid:1) j = 0 , , . . . , N s − κ → κ → ∞ is x jt +1 = p r + l +1 − (cid:88) n =0 a n j = 0 , , . . . , N s − B κ -embeddings connect CA and certain CMLs.If we consider the replacement Mode II on totalistic CA, we have, from Eqs. (69) and (39) x jt +1 = ( r + l +1)( p − (cid:88) n =0 σ n B κ (cid:16) n − (cid:80) lk = − r x j + kt , (cid:17) B κ (cid:0) , (cid:1) j = 0 , , . . . , N s − ( r + l +1)( p − (cid:88) n =0 σ n − (cid:16) n − (cid:80) lk = − r x j + kt (cid:17) κ + O (cid:0) κ − (cid:1) j = 0 , , . . . , N s − κ large. Let us consider, morespecifically, local rules for which l = r = 1. We obtain x jt +1 = p − (cid:88) n =0 σ n (cid:34) − (cid:0) n − x j +1 t − x jt − x j − t (cid:1) κ + O (cid:0) κ − (cid:1)(cid:35) j = 0 , , . . . , N s − p − (cid:88) n =0 σ n (cid:34) − (cid:0) n − x j +1 t + 2 x jt − x j − t − x jt (cid:1) κ + O (cid:0) κ − (cid:1)(cid:35) j = 0 , , . . . , N s − x jt + P (cid:0) x jt (cid:1) + D ∆ L x jt − F (cid:0) x jt , ∆ L x jt (cid:1) (75)where we have defined ∆ L x jt ≡ x j +1 t − x jt + x j − t (76) D ≡ κ p − (cid:88) n =0 σ n n (77) P (cid:0) x jt (cid:1) ≡ − x jt + p − (cid:88) n =0 σ n (cid:34) − (cid:0) n − x jt (cid:1) κ (cid:35) (78) F (cid:0) x jt , ∆ x jt (cid:1) ≡ (cid:0) x jt ∆ x jt + (∆ x jt ) (cid:1) (cid:80) p − n =0 σ n κ + O (cid:0) κ − (cid:1) (79)By further introducing ∆ T x jt ≡ x jt +1 − x jt (80)we, finally, obtain ∆ T x jt = P (cid:0) x jt (cid:1) + D ∆ L x jt − F (cid:0) x jt , ∆ L x jt (cid:1) (81)This is a difference equation involving a discretized Laplacian ∆ L x jt and the first-order timedifference ∆ T x jt . By making the following transformations t → N T τ (82) t + 1 → ( N T + 1) τ (83) j → N L (cid:96) (84) j + 1 → ( N L + 1) (cid:96) (85) j − → ( N L − (cid:96) (86) x ≡ j = N L (cid:96) (87) u ( t, x ) ≡ x jt (88)and by taking the limits N T → ∞ , N L → ∞ , τ → (cid:96) →
0, so that N T τ = t , N L (cid:96) = x remainconstant, we obtain from Eq. (81) τ ∂u∂t = P ( u ) + (cid:96) D ∂ u∂x − F (cid:18) u, (cid:96) ∂ u∂x (cid:19) (89)15he function P (cid:0) x jt (cid:1) is a quadratic polynomial governing the homogeneous dynamics. The func-tion F (cid:0) x jt , ∆ L x jt (cid:1) is nonlinearly dependent on the discretized Laplacian and the cell dynamicalstate x jt and can be thought as a kind of ‘nonlinear diffusion’ term. If this last term vanishes,Eq. (89) is a Fisher-Kolmogorov equation. Furthermore, a linear diffusion equation is alwaysobtained in the limit κ large if the following conditions are met p − (cid:88) n =0 σ n = 0 (90) p − (cid:88) n =0 nσ n > ∂u∂t = f ( u ) + D ∂ u∂x (92)where f ( u ) = (3 D − uτ − τ κ p − (cid:88) n =0 n σ n (93) D ≡ (cid:96) τ κ p − (cid:88) n =0 nσ n (94)If 3 D < κ sufficiently large) the homogeneous dynamics ofEq. (92) converges to the stable homogeneous fixed point given by f ( u ∗ ) = 0, i.e. u ∗ = 1(3 D − κ p − (cid:88) n =0 n σ n (95)We note that the conditions expressed by Eqs. (90) and (91) are curiously the same as thosefound in the construction of appropriate difference operators for generalized logarithms andgroup entropies (see Eq. (2) in [39]). V. CELLULAR AUTOMATA CONNECTIONS
We now show how any two CAs in rule space can be connected by means of a nonlinear B κ -embedding so that in the limits κ → κ → ∞ each of the CAs entering in theconnection is obtained. We call such nonlinear B κ -embedding, generally behaving as a CML, aCA connection. The embedded CAs are called the CA limits of the CA connection.For intermediate κ values, we derive a mean-field model of the connection that can quali-tatively capture many of its dynamical features, as observed in its spatiotemporal evolution.In general, a finite non-vanishing value of the parameter κ weakens the ‘pure’ behavior of the16A limits and we believe that the general concept of CA connections introduced here may beuseful in, e.g. biophysical models of multicellular ensembles [40], where variability and networkheterogeneity need to be taken into account and may incorporate several kinds of typical CAdynamics. In these applications κ may be related to biological time/or and to the connectivityof the multicellular ensemble mediated by gap junctions [41]. The strength of the couplingis given by 1 /κ so that, when κ → /κ leading to conformational changes that,in turn, may lead to dynamical changes so that the network is loosened. Finally, the presenceof other agents in the network, facilitated by the decreased network connectivity may induce adifferent CA dynamics that is qualitatively different than the one obtained in the limit κ → κ → ∞ is anothercellular automaton of the form of Eq. (67) for all initial conditions x j ∈ A p ( ∀ j ) but with agenerally different rule vector ( b , b , . . . , b p l + r +1 − ), b n ∈ Z ∈ A p , ∀ n ∈ [0 , p l + r +1 − x jt +1 = p r + l +1 − (cid:88) n =0 b n B (cid:32) n − l (cid:88) k = − r p k + r x j + kt , (cid:33) j = 0 , , . . . , N s − κ -deformed formula so that two such CA are connected in the limits κ → κ → ∞ . We note that x jt +1 ∈ A p is any of the integers m in the interval [0 , p − x jt +1 = p − (cid:88) m =0 mδ mx jt +1 = p − (cid:88) m =0 m B (cid:0) m − x jt +1 (cid:1) = p − (cid:88) m =0 m B (cid:0) m − x jt +1 (cid:1) B (cid:0) m − x jt +1 (cid:1) (97)By using the replacement Mode II and the κ → /κ transformation, we construct from here anonlinear B κ -embedding f κ ( x jt +1 ) = p − (cid:88) m =0 m B κ (cid:0) m − x jt +1 , (cid:1) B κ (0 , ) B /κ (cid:0) m − x jt +1 , (cid:1) B /κ (0 , ) j = 0 , , . . . , N s − a jt ( κ ) ≡ p l + r +1 − (cid:88) n =0 a n B κ (cid:32) n − l (cid:88) k = − r p k + r x j + kt , (cid:33) (99) b jt ( κ ) ≡ p l + r +1 − (cid:88) n =0 b n B /κ (cid:32) n − l (cid:88) k = − r p k + r x j + kt , (cid:33) (100)17e can construct from Eq. (98) a CA connection involving two CAs l [ R ] rp and l [ R ] rp in theCA limits of the connection x jt +1 = p − (cid:88) m =0 m B κ (cid:0) m − a jt ( κ ) , (cid:1) B κ (0 , ) B /κ (cid:0) m − b jt ( κ ) , (cid:1) B /κ (0 , ) j = 0 , , . . . , N s − l [ R ] rp κ −→ l [ R ] rp . In the limit κ →
0, Eq. (101) becomes Eq.(67) sincelim κ → x jt +1 = lim κ → p − (cid:88) m =0 m B κ (cid:0) m − a jt ( κ ) , (cid:1) B κ (0 , ) B /κ (cid:0) m − b jt ( κ ) , (cid:1) B /κ (0 , )= p − (cid:88) m =0 m B (cid:18) m − a jt (0) , (cid:19) = a jt (0) = p l + r +1 − (cid:88) n =0 a n B (cid:32) n − l (cid:88) k = − r p k + r x j + kt , (cid:33) (102)However, in the limit κ → ∞ Eq. (101) becomes Eq. (96)lim κ →∞ x jt +1 = lim κ →∞ p − (cid:88) m =0 m B κ (cid:0) m − a jt ( κ ) , (cid:1) B κ (0 , ) B /κ (cid:0) m − b jt ( κ ) , (cid:1) B /κ (0 , )= p − (cid:88) m =0 m B (cid:18) m − b jt ( ∞ ) , (cid:19) = b jt ( ∞ ) = p l + r +1 − (cid:88) n =0 b n B (cid:32) n − l (cid:88) k = − r p k + r x j + kt , (cid:33) (103)We note that the transformation κ → /κ merely reverses the connection, i.e. l [ R ] rp κ −→ l [ R ] rp changes to l [ R ] rp κ −→ l [ R ] rp .For Boolean CA connections ( p = 2) Eq. (101) simplifies to x jt +1 = B κ (cid:0) − a jt ( κ ) , (cid:1) B κ (0 , ) B /κ (cid:0) − b jt ( κ ) , (cid:1) B /κ (0 , ) j = 0 , , . . . , N s − B κ function we find that 0 ≤ x jt ≤ κ , t and j .To get insight in the complex spatiotemporal dynamics of Eq. (101) it proves useful toconsider the mean field model obtained by taking x jt ≡ u t . Then Eq. (101) reduces to u t +1 = p − (cid:88) m =0 m B κ (cid:0) m − a t ( κ ) , (cid:1) B κ (0 , ) B /κ (cid:0) m − b t ( κ ) , (cid:1) B /κ (0 , ) (105)where a t ( κ ) ≡ p l + r +1 − (cid:88) n =0 a n B κ (cid:18) n − p l + r +1 − p − u t , (cid:19) (106) b t ( κ ) ≡ p l + r +1 − (cid:88) n =0 b n B /κ (cid:18) n − p l + r +1 − p − u t , (cid:19) (107)18 IG. 1:
The cell state x jt +1 ∈ [0 ,
1] vs. the neighborhood sum value n jt = (cid:80) k = − k + r x j + kt for the CA connection κ −→ and for the values of κ indicated on the curves. The panels are separated to better show the changes obtained in Eq. (104) withincreasing κ values. This reduced model describes the behavior of homogeneous initial conditions and is also a meanfield approximation for arbitrary (inhomogeneous) initial conditions. Of special interest are the ω -limit sets of Eq. (105) and their change with the control parameter κ . These yield thebifurcation diagram of the reduced model that can be used to interpret certain results obtainedwith the full model. The ω -limit sets of CMLs can be used to model the vacuum fluctuations ofchaotic strings (which are specific one-dimensional CMLs underlying the Parisi-Wu approachof stochastic quantization on a small scale) [9].As an example to illustrate the above concepts, we consider a CA connection between thetwo elementary CA provided by the Boolean CA rules and . The former is knownto be a random number generator and the latter is capable of universal computation [14]. Wethus consider the CA connection κ −→ described by Eq. (104) with p = 2, l = r = 1and rule vectors ( a , a , . . . , a ) = (0 , , , , , , ,
0) and ( b , b , . . . , b ) = (0 , , , , , , , x jt +1 in Eq. (104) as a function of the neighborhood value n jt = (cid:80) k = − p k + r x j + kt for different values of κ and for the above CA connection: Note that CA rule is obtainedin the limit κ → is obtained in the limit κ → ∞ (Fig. 1 right).For intermediate values of κ the curves always lie within the unit interval.The spatiotemporal evolution of x jt obtained from Eq. (104) for the CA connection κ −→ is shown in Fig. 2 j ∈ [0 , t ∈ [0 , κ values indicated over thepanels for a simple initial condition consisting of a single site with value ’1’ surrounded bysites with value ’0’. For κ = 0 .
001 the behavior typical of the limit κ → κ = 15 the limit κ → ∞ isalready well approximated and the CA connection coincides with Wolfram’s CA rule 110. Forintermediate values of κ the behavior is highly nontrivial and gliders and coherent structurestypical of Class 4 CA [14, 17] are observed (e.g. for κ = 4 . κ finitesomehow interpolates between the limits κ → κ → ∞ , although there is a wide varietyof qualitatively different behavior. At κ = 1 a coexistence is observed between a travelingturbulent patch at high values of x jt and homogeneous period-2 oscillations that form domains(clusters) connecting two different low values of x jt . Most remarkably, it is observed that in theinterval 2 . (cid:46) κ (cid:46)
4, the homogeneous quiescent state loses stability to aperiodic oscillations,19
IG. 2:
Spatiotemporal evolution of x jt obtained from Eq. (104) for the CA connection κ −→ starting from a singlesite with value 1 surrounded by sites with zero value and for the values of κ indicated on the panels. In each panel, time flows fromtop to bottom. The system size is N s = 240 and 240 iteration steps are shown. In the gray scale used, black and white correspondto values zero and one, respectively. the system being highly sensitive to small perturbations.Insight on the results of Fig. 2 can be gained by means of the mean-field reduction of theCA connection, Eq. (105) which, in this particular case, takes the form u t +1 = B κ (cid:0) − a t ( κ ) , (cid:1) B κ (0 , ) B /κ (cid:0) − b t ( κ ) , (cid:1) B /κ (0 , ) (108)where a t ( κ ) ≡ (cid:88) n =0 a n B κ (cid:18) n − u t , (cid:19) (109) b t ( κ ) ≡ (cid:88) n =0 b n B /κ (cid:18) n − u t , (cid:19) (110)20 IG. 3:
Bifurcation diagram calculated from the asymptotic behavior of Eq. (108) for the CA connection κ −→ . Thevalues that the dynamical variable u ∞ takes on the orbit at large times are shown as a function of the parameter κ . with ( a , a , . . . , a ) = (0 , , , , , , ,
0) and ( b , b , . . . , b ) = (0 , , , , , , , u ∈ [0 ,
1] uniformly filling the unit interval. In this way,the ω -limit sets u ∞ of the values of u t as t → ∞ are numerically obtained. The bifurcationdiagram is shown in Fig. 3 in which u ∞ is plotted vs. κ .Some observations made in Fig. 2 can be qualitatively understood by means of the bifurca-tion diagram in Fig. 3: • In the limits κ → κ → ∞ we have u ∞ →
0. This corresponds to the CA limits andboth rules and in the connection fix the quiescent state and no other homoge-neous configuration. The mean-field model is not able to capture the complex dynamicsof Eq. (104) in these CA limits because the dynamical behavior is highly correlatedthrough the CA dynamics and the correlations, that are affected by the neighborhoodconfigurations, are lost in the mean-field model. • For κ ≈ u ∞ and a chaotic stripe at high values of u ∞ is observed. A gap in u ∞ is seen separatingboth behaviors. This qualitatively captures the observation made in Fig. 2 for this valueof κ . • In the interval 2 . (cid:46) κ (cid:46) u ∞ displays a wide variety of possible states that are allreached in an aperiodic, chaotic, manner. The branch at low u ∞ is now fused with thechaotic stripe and there is no gap. This matches the observation made on the turbulent21ehavior described in Fig. 2 and the mean-field model allows to relate that turbulencein the full model to low-dimensional chaos in the reduced one. • In the interval 5 (cid:46) κ (cid:46) u ∞ ) and a chaotic stripe at high values of u ∞ . Again, thesebehaviors are separated by a wide gap in the possible values of u ∞ . VI. CONCLUSIONS
In this work nonlinear B κ -embeddings have been constructed that are able to yield mathe-matical objects with different dimensionality (scalars, vectors, matrices) and dynamical classesof models (CAs, CMLs, nonlinear PDEs, CA connections) as a continuous parameter κ ∈ R isvaried. We note that κ should have a wide physical significance. If one considers, for exam-ple, many particle systems governed by statistical laws, κ can be thought as a coarse-grainingparameter, an increased value of it leading to fuzzier descriptions involving a lower number ofdegrees of freedom. κ can also be considered as a scale parameter in unified field theories, in-volved in the connection of objects (tensors) of different dimensions at different scales in whichspacetime is probed.Based on appropriate nonlinear B κ -embeddings [2], a new approach to compactificationin unified physical theories (e.g. supergravity in 10 or 11-dimensional spacetimes) has beensuggested. The method involves no Fourier expansion and no truncation, as is usually performedon extra dimensions to account for the observable 4-dimensional universe. The limits κ → κ → ∞ of the B κ -embeddings are robust and yield the metric tensors of the spacetimewith extra dimensions and the one of the observable universe, respectively.We have also shown how B κ -embeddings can be used to asymptotically connect CA withnonlinear PDEs through appropriate CMLs, all these structures being particular instances of theembedding. In particular, we have shown how (nonlinear) diffusion equations naturally emergeasymptotically from this construction. This mathematical approach sheds light, therefore,on why the Laplacian operator has such a tremendous importance in physical theories, sinceit already emerges from the most elementary dynamical systems and interactions when thecontinuum limit is performed.In this article, we have also introduced the concept of CA connections. These are CMLsobtained from nonlinear B κ -embeddings, that depend on a control parameter κ such that inthe limits κ → κ → ∞ the CML collapses to a CA. We have shown that any two CAsin rule space can be connected in this way. A mean-field, reduced model allows a bifurcationdiagram to be calculated that qualitatively captures the features observed in the spatiotemporalevolution of the connection (in those parameter regimes where the neighborhood dynamicsis approximately homogeneous). We have illustrated these general results with the specificexample of Wolfram elementary Boolean CA rules 30 and 110 [14] constructing a connectionbetween both rules. At intermediate κ values, a wide variety of dynamical behavior has beenobserved ranging from coherent to seemingly chaotic behavior, as well as the coexistence ofcoherence and disorder for simple initial conditions. These behaviors have been qualitativelyinvestigated by means of a mean-field model derived from the connection. The results presentedin this article can be easily generalized to more dimensions and arbitrary order in time [11].22f the parameter κ in a CA connection is interpreted as the coupling strength on the lattice,we suggest that modulations introduced through CA connections can be used in biophysicalapplications to model changes in dynamical behavior induced by fuzziness or the coarseningof network connectivity [41]. If one considers physical models of networks governed by a finiteset of strict rules (CA-like), a non-vanishing value for the parameter κ may incorporate theoverall effect of the network heterogeneity, as well as the weakening of cooperative phenomenaas κ is increased. If κ is made explicitly dependent on time, specific CA connections may alsoaccount for the effect of aging in the evolution of a system dynamics. We, therefore, believethat the structures here introduced can be helpful to model the long time evolution of biologicalorganisms [40]. For example, a CA connection can be designed such that as κ is varied, thedynamics of a multicellular ensemble is mostly governed by cell division ( κ low) and then,gradually, cell differentiation ( κ intermediate) and apoptosis ( κ large). [1] A. N. Vasil’ev, The field theoretic renormalization group in critical behavior theory and stochasticdynamics (CRC Press, Boca Raton, FL, 2004).[2] V. Garc´ıa-Morales, Nonlinear embeddings: Applications to analysis, fractals and polynomial rootfinding. Chaos Sol. Fract. 99 (2017) 312.[3] K. Krasnov and R. Percacci, Gravity and unification: a review, Class. Quantum Grav. 35 (2018)143001.[4] J. M. Overduin and P. S. Wesson, Kaluza-Klein gravity, Phys. Rep. 283 (1997) 303-380.[5] E. Witten, Search for a realistic Kaluza-Klein theory, Nuclear Physics B186 (1981) 412-428.[6] D. Bailin and A. Love, Kaluza-Klein theories, Rep. Prog. Phys. 50 (1987) 1087-1170.[7] M. J. Duff, B. E. W. Nilsson and C. N. Pope, Kaluza-Klein supergravity, Phys. Rep. 130 (1986)1-142.[8] C. Beck,
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