Unifying vectors and matrices of different dimensions through nonlinear embeddings

Complex systems may morph between structures with different dimensionality and degrees of freedom. As a tool for their modelling, nonlinear embeddings are introduced that encompass objects with different dimensionality as a continuous parameter κ∈R is being varied, thus allowing the unification of vectors, matrices and tensors in single mathematical structures. This technique is applied to construct warped models in the passage from supergravity in 10 or 11-dimensional spacetimes to four-dimensional ones. We also show how nonlinear embeddings can be used to connect cellular automata (CAs) to coupled map lattices (CMLs) and to nonlinear partial differential equations, deriving a class of nonlinear diffusion 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 → 0 and κ → ∞ of the embedding.


Introduction
Scalars, vectors and matrices are mathematical objects with different numbers of dimensions (degrees of freedom) which generally store different amounts of information. The question whether these different mathematical structures can be encompassed by more general ones may be interesting in the modelling of complex systems (since these are often able to morph as continuous parameters are being varied [1]). To that aim, we construct in this article mathematical 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 of nonlinear B κ -embeddings that have been very recently introduced and applied to the problem of finding all roots of a complex polynomial [2].
We can apply nonlinear B κ -embeddings to any situation involving connections among mathematical objects with different dimensions and degrees of freedom. Examples of these situations are provided by conformational changes and phase transitions in Statistical Mechanics, irreversible 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 in different dimensions).
Another specific example of a situation involving objects with different dimensions is provided by the quest for a unified theory of general relativity and quantum mechanics in the framework of, e.g. supergravity. These proceed by extending the four-dimensional metric of pure gravity to higher dimensions [3][4][5][6][7]. Although the observable universe is described by a four-dimensional metric tensor, this latter object needs to have 10 or 11 dimensions (depending on whether supergravity arises as the low energy limit of string theory or Mtheory, respectively [3]) in order to consistently accomodate the gauge groups describing the standard model of particle physics [5,8]. Since there are only four observable dimensions of spacetime, warped models for the metric are considered to be able to change the dimensionality of the metric [9,10]. With help of appropriate nonlinear B κ -embeddings we point out how new warped models and a κ-deformed formalism of gravity can be constructed.
Nonlinear B κ -embeddings can also be applied to dynamical systems, as cellular automata (CAs) [11][12][13][14][15][16][17][18][19][20][21][22] and coupled map lattices (CMLs) [13,[23][24][25][26][27][28]. These models of complex physical systems are popular in biophysics [15], have given rise to novel approaches to quantum mechanics [21,22,29,30] and have been conjectured to play a crucial role in unified field theories, 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 departs from the specification of a few rigid, deterministic rules perfectly operating on finite amounts of information. It is, therefore, interesting to study how these models can be embedded in more sophisticated ones [13,16,[31][32][33][34]. In a previous recent work [13], we have presented a general mechanism 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 yield nonlinear 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 to glue together several different CAs. In similar ways, CMLs can be glued together to form more complicated structures. The mathematical methods presented here may be of interest in biophysics (dynamics of multicellular ensembles) and in fundamental physics (extended formalisms of gravity and the embedding of different unified theories of physics related by dualities). Quite interestingly, a certain class of CMLs have been used to simulate quantum field theories on an appropriate scaling limit [8,35] and it has been shown that there are six different such unified theories in terms of chaotic strings that are somehow analogous to the six different models of a string considered in string theory and which are embedded in M-theory [8,36].
The outline of this article is as follows. First, in section 2 we introduce the method to construct the nonlinear B κ -embeddings involved in connecting structures with different dimensions (thus generalizing the concept of vectors and matrices). In section 3 we give several specific examples of nonlinear embeddings. In section 4 we apply nonlinear embeddings to the problem of connecting a four-dimensional and an 11-dimensional metric tensors, as those found in theories of supergravity, at appropriate limits of the κ parameter. In this application, κ is related to the characteristic scale at which spacetime is probed when compared to the Planck length. In section 5 we construct B κ -embeddings that have CAs in the κ → 0 limit and we show how they are connected in the asymptotic limit κ large to certain nonlinear partial differential 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 section 6 we show how two arbitrary CAs in rule space can be connected in the κ → 0 and κ → ∞ limits by means of appropriate nonlinear B κ -embeddings. We discuss some potential physical applications and present some conclusions.

Nonlinear B κ -embeddings: connecting scalars, vectors and matrices
The method to construct B κ -embeddings connecting objects with different dimensions begins by noting that the Kronecker delta δ nj (δ nj = 1 if n = j and δ nj = 0 otherwise) admits a simple representation in terms of the boxcar function [11,12] B(x, y) ≡ 1 2 where x, y ∈ R. Indeed, we have We now note that, by means of convolution, any N- where n is a free index labelling the component. We note that equation (3) can equivalently be written by using equation (2) as It is straightforward to prove that the definition of a vector given by the r.h.s. of equation (4) is indeed consistent with that of an element of a vector space V of N dimensions. All eight following properties are satisfied: (a) Associative property, for any three vectors u, v and w ∈ V Conmutativity of addition, for any two vectors u and v ∈ V (c) Identity of addition: there exists 0, 0 n ≡ N−1 j=0 0B n − j, 1 2 such that for any vector u ∈ V Inverse elements of addition: for any vector u, (u) Compatibility of scalar multiplication with field multiplication. For any two a, b scalars in a field F, we have (f) Identity element of scalar multiplication. Let 1 denote the multiplicative unit in the field F. Then, we have for n = 0, 1, . . . , N − 1. (g) Distributivity of scalar multiplication with respect to vector addition. For any two vectors u, v and a scalar a ∈ F we have (h) Distributivity of scalar multiplication with respect to field addition. For any vector u, v and any two scalars The Kronecker product 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 indices v n w n (6) and can be obtained from the outer product by contracting the free indices v, w = We define a ket vector |ψ over an N-dimensional vector space in terms of B-functions as where the ψ j are complex numbers. A bra vector is defined as where the overline denotes complex conjugation. The inner product of a bra and a ket is then obtained as Let a and b be complex numbers. We clearly have the following properties Because of these properties, the vector space with inner product defined above is a Hilbert space. An N × N matrix A can be written in terms of its elements A nm as We note that an N-dimensional matrix can equivalently be written in a more compact form as a vector of indexed N-tuples. If we define where h ≡ k + Nj then equation (11) becomes Note that there are two free indices m and n in this expression and that there are N 2 coefficients A h . The B-function (boxcar function) is a suitable representation of the Kronecker delta when its first argument x is an integer and its second argument y has value |y| 1 2 . The crucial interest of this representation of the Kronecker delta is that it can be easily embedded in the real numbers by means of a one-parameter deformation function B κ (x, y) which (pointwise) converges to B(x, y) as κ → 0 and (uniformly) converges to 0 as κ → ∞ [2,13] with x, y ∈ R. It is straightforward to observe that and to check all following identities by noting that, after some manipulations, we have B κ x, y = e 2y/κ − e −2y/κ e 2y/κ + e 2x/κ + e −2x/κ + e −2y/κ (24) B κ x, y B κ 0, y = e 2y/κ + 2 + e −2y/κ e 2y/κ + e 2x/κ + e −2x/κ + e −2y/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] Interestingly, we have the following expansions The B κ -function is an infinitely differentiable function of x, y and κ.
where we have introduced the Eulerian number For κ sufficiently large (κ > 2(|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 where the B 2m denote the even Bernoulli numbers: In the above expression we have also used that The main trick introduced in this manuscript is to replace any B-function in equation (4) or (11), e.g. B n − j, 1 2 , by a κ-deformed counterpart with the form either B κ n − j, 1 2 or B κ n − j, 1 2 /B κ 0, 1 2 depending on the application under consideration. We shall call the first kind of replacement mode I and the latter mode II. If we simply replace all B-functions following mode I we obtain from equation (4) or (11), respectively, These nonlinear B κ -embeddings constitute κ-deformed structures that generalize those of a vector and of a matrix, respectively, as follows: in the limit κ → 0, by using equation (20) in equations (40) and (41), we regain equations (4) and (11), respectively. However, in the limit κ → ∞, from equation (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 equations (4) or (11) In the limit κ → 0, by using equation (20) in equations (40) and (41), we regain equations (4) and (11), respectively. However, in the limit κ → ∞, from equation (23), we now obtain i.e., the vector and the matrix collapse, respectively, to the scalars v and A formed by summing over 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 chooses to 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. Which replacement 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 to construct a nonlinear B κ -embedding that connects any arbitrary operator A with its trace TrA. Then we begin by noting that we can equivalently write A in terms of its elements as From this, we can combine replacement modes I and II to construct a nonlinear B κ -embedding In the limit κ → 0, by using equation (20) in equation (47), we regain equation (46). However, in the limit κ → ∞, from equation (23), we now obtain In this way, the nonlinear B κ -embedding given by equation (46) smoothly connects a matrix operator with its (scalar) trace.
We note that in any nonlinear B κ -embedding, the limits κ → 0 and κ → ∞ can be exchanged by making the transformation κ → 1/κ. Thus, for any nonlinear B κ -embedding, there exists a 'dual' structure obtained by making the latter transformation.

Examples of nonlinear B κ -embeddings
We now give more specific examples of nonlinear B κ -embeddings. The following one is inspired in the process of cell division found in biological systems [37]. We can model the space occupied by a cell by means of the function which is approximately equal to one inside a circle of unit radius and zero outside. Note that κ = 0.1 in the B κ -function indicates that the border that separates the interior of the circle from its outside is smooth [38].
We can take C 0 (x, y) as a part at κ ≈ 0 of an embedding so that at the opposite limit, κ → ∞, we have rather two cells with the centers displaced a certain distance along the x axis We can now construct a B κ -embedding interpolating between these two limiting cases, introducing replacement modes I and II as From this expression we clearly obtain In figure 1, the function C κ (x, y), given by equation (51), is represented in the plane for six different values of the parameter κ indicated over the panels. We observe that for κ = 0.1 the limit κ ≈ 0 of the embedding, equation (49), is approached, while for κ = 6 we obtain the limit κ → ∞ of the embedding, equation (50). For intermediate κ ∈ R values we obtain a continuous, smooth transition that mimics the process of cell division (cytokinesis) found in many biological cells.
Let us now consider another example. The following embedding connects an N-tuple of natural numbers to their sum This structure is typical of the replacement mode II described above, which interpolates between the limits The embedding N κ given by equation (54) is plotted in figure 2 as a function of κ. We note the difference in the limiting behaviors κ → 0 and κ → ∞. At the former limit, the embedding has different branches, each indexed by different values of the variable n. At the latter limit the embedding collapses to an scalar value insensitive to n. Following any of the branches of the embedding, from κ ≈ 0 to κ → ∞, information is lost. If we think in κ as equivalent to time, the embedding collapses to a fixed point at κ → ∞ and information on the starting branch at κ ≈ 0 cannot be recovered. Nonlinear B κ -embeddings can thus be used to model irreversible processes found in natural physical systems.  We note that, along the branches n = 0, 1, 2 we obtain, from equation (54) where we have used equation (15). Let p(x) denote the total number of unrestricted partitions of the natural number x. It is clear that x = 10 is obtained from equation (54) at κ → ∞ starting from parts 1, 3 and 6 at κ → 0, but we could have considered other parts as well (e.g. 1, 2, 2 and 5) that would yield a different embedding with the same limiting behavior at κ → ∞. Indeed, there are p(10) = 42 possible embeddings whose parts at κ → 0 are consistent with the sum of the parts being equal to 10 at κ → ∞. Let x 0 ∈ N be the part of the embedding at κ ≈ 0 and x ∞ ∈ N the part at κ → ∞. We can quantify the irreversibility in going from κ ≈ 0 to κ → ∞ by means of the entropy change For the embedding in equation (54)  The above entropy of the embedding between natural-valued parts exactly matches the entropy of a quantum mechanical nonrelativistic string with fixed endpoints (see, e.g. [39], pp 498-500). This shows that nonlinear B κ -embeddings between natural-valued parts may find application in string theory. We note that, and, for any n ∈ Z and y = 1/2, This is a crucial property of all nonlinear B κ -embeddings obtained by means of replacement mode II. To understand it let us consider two functions f(x) and g(x) of a real variable x. We can homotopically connect both by the following embedding for which we have the limits Because of equation (60), we have The functions f(x) and g(x) are homotopically connected by the embedding in a way that it is possible to obtain them to arbitrary precision for κ nonvanishing and finite (f(x) for κ sufficiently small and g(x) for κ sufficiently large but finite). The functions f(x) and g(x) behave as fixed points of the embedding (if we look at the latter dynamically, with κ playing the role of a time variable). This strongly contrasts with a linear homotopy in the unit interval with q ∈ R, q ∈ [0, 1]. We have so that, at the vicinity of q ≈ 0 and q ≈ 1, h q (x) can be very different to f(x) and g(x), respectively. This contrasts with the nonlinear B κ -embeddings here introduced. The latter approach the function f(x) for sufficiently small (but nonvanishing) κ and the function g(x) for sufficiently large (but finite) κ. The parts at zero and infinity are being homotopically connected by sigmoid-like functions. This is apparent in figure 2.
The crucial property of the nonlinear embeddings described above make them potentially interesting in the mathematical modeling of complex phenomena which are usually described by e.g. partial differential equations. Because, if one interprets κ as time, any arbitrary initial condition can be carried to the same stationary fixed point as time goes to infinity. If there is no fixed point, transient and stationary behaviors can be modeled with nonlinear embeddings as well, although there will also necessarily be a time dependence on the parts carried by the embedding.
The B κ -function has remarkable analytic properties in its other arguments as well, and can also be used in the design of continuous differentiable functions with prescribed asymptotic behavior. Let f(x) and g(x) be any continuous and differentiable functions of the real variable x. Then, the following function can be constructed that behaves asymptotically like f(x) as x → ∞ and like g( since, for κ 0, lim x→∞ B κ (0, x) = −lim x→∞ B κ (0, −x) = 1. Furthermore, for κ > 0, the resulting function h(x; κ) will be continuous and differentiable if both f(x) and g(x) are.

Compactification in unified physical theories
We now show how the ideas in the previous section can be applied to model compactification in the passage from supergravity to pure four-dimensional gravity. The main object of general relativity is the metric tensor with elements g μν (μ, ν ∈ {0, 1, 2, 3}) that governs the geometry of spacetime. It is a 4 × 4 symmetric matrix (g μν = g νμ ) with 10 different components. In our notation, the metric tensor can be specified as In 11-dimensional supergravity (10-dimensional supergravity would proceed on analogous lines), the metric tensor has the form The problem that leads to introduce the idea of compactification is that 11 dimensions are necessary in supergravity (seen as the low energy limit of M-theory) to consistently bring together general relativity and quantum mechanics and, hence, it may be described by a tensor of the form of equation (71). However, the observable universe has four dimensions and, hence, it is described by an object of the form of equation (70). If we regard the extra seven dimensions as true, physical dimensions, on a par with the four observed dimensions [5] this suggests to embed both the four-dimensional metric tensor of pure gravity and the 11dimensional tensor in a single nonlinear B κ -embedding. Let be the scale at which spacetime is probed. If we identify (where L p = G c 3 = 1.6 × 10 −33 cm is the Planck length, with G being the gravitational constant, c the speed of light and Planck's constant), we can translate the problem of compactification to the description of a mathematical mechanism involved in the passage from g 11D to g 4D as κ → ∞. This suggests the construction of a nonlinear B κ -embedding that yields a four-dimensional metric tensor g 4D in the limit κ → ∞ (thus being able to reproduce general relativity in this limit) and the 11-dimensional metric tensor g 11D in the limit κ → 0 (thus being potentially useful in unified field theory).
First, we note that we can rewrite equation (71) as We can now apply the replacement Mode I to every B-function involving a dummy index higher or equal than four. In this way, we construct a nonlinear B κ -embedding that includes equation (71) in the limit κ → 0. We obtain We have, by using equations (20) and (23) lim κ→0 If we denote by (dx 0 , dx 1 , . . . , dx 11 ) any 11-dimensional spacetime infinitesimal displacement we can write the nonlinear B κ -embedding of the squared differential of the arc-length ds 2 κ as Again, we note that, by using equations (20) and (23) These equations provide the right differential of the arc-lengths for the respective metric tensors. We thus see that the κ-deformed structure, equation (74) contains the metric tensors of 4-dimensional gravity and 11-dimensional supergravity as specific limiting cases, regardless of the specific form of their elements g μν . Although we have assumed a simple dependence of the parameter κ on the Planck length, equation (72), this dependence can be more involved and might be derived from first-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 that the nonlinear B κ -embeddings given by equations (74) and (77) are smooth, differentiable and well defined for any value of κ.

Connecting cellular automata and (nonlinear) partial differential equations through nonlinear B κ -embeddings
In section 4 we have shown how nonlinear B κ -embeddings can be used to construct generalized structures that connect mathematical objects with different dimensionality. In this and the following section, we show how they can be used indeed to smoothly connect different qualitative dynamical behaviors governed by different dynamical (evolution) rules. An increased value of the continuous parameter κ ∈ R is in all the following associated to the loosening of the rigidity of the dynamical rules (CA-like rules) found in the limit κ → 0. Consequently, an increased value of the reciprocal 1/κ represents the loosening of the dynamical rules found at κ → ∞. Thus, an increased value of κ provides a smooth, continuous and differentiable connection between two limits in which dynamical rules can be either discrete or continuous. If the dynamics is continuous, the embeddings converge uniformly to the limiting cases. If it is discrete, convergence is, necessarily, pointwise. Note that, depending on the physical application, κ can be regarded as a temperature-like parameter (so that when κ is increased a 'thermal motion' is intensified that weakens the dynamical rules) or even as time. Nonlinear B κ -embeddings can therefore be used to construct 'cartographies' of physical theories and bifurcation scenarios where qualitative changes in dynamical behavior are induced by tuning the continuous parameter κ.
In these cartographies, theories (specific models) are encompassed by more advanced theories in a hierarchical manner. CAs [11,12,[14][15][16][17][18][19][20][21][22], CMLs [13,[23][24][25][26][27][28] and (nonlinear) PDEs [40,41] constitute the different mathematical approaches to model spatiotemporal pattern formation outside of equilibrium, as found in experimental physical systems [15]. CAs are fully discrete coupled maps in which space and time are discrete and the local phase space is both discrete and finite. CAs serve as toy models for the overall observed features of complex physical systems [14]. An example of this is spatiotemporal intermittency [14,42,43]. CMLs [23,24] are discrete maps ruling the evolution of a dynamical system on a discrete spacetime but for which the local phase space is continuous. Finally, PDEs constitute continuous models of dynamical system evolving on continuous and differentiable spacetimes and with a continuous local phase space. In this section we show how all CAs can be encompassed by means of B κ -embeddings that connect them to certain CMLs and nonlinear PDEs. In particular, we show how some CAs lead to nonlinear diffusion equations.
We define the alphabet A p ≡ {0, 1, . . . , p − 1}, p 2, p ∈ N, as the set of integers in the interval [0, p − 1]. We write A N p for the Cartesian product of N copies of A p . Let x j t ∈ A p be a dynamical variable at time t ∈ Z and position j ∈ Z on a ring of N s sites, j ∈ [0, N s − 1] and let l, r be non-negative integers. A CA, with rule vector (a 0 , a 1 , . . . , a p l+r+1 −1 ), a n∈Z ∈ A p , ∀n ∈ [0, p l+r+1 − 1], range N = l + r + 1 and Wolfram code R = p l+r+1 −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 N p → A p and synchronously at every t according to the universal map [11] x j t+1 = where 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 of the code l R r p [11]. For example, all 256 Wolfram elementary CA 1 R 1 2 are obtained by taking p = 2, l = r = 1 in equation (80). Thus, Wolfram rule 30 is denoted by 1 30 1 2 and has rule vector (a 0 , a 1 , . . . , a 7 ) = (0, 1, 1, 1, 1, 0, 0, 0). In general, the coefficients a n of any CA can be directly obtained from the Wolfram code R by means of the following expression [21] where . . . denotes the lower closest integer (floor) function. Specially interesting for physical applications are those CAs that are locally isotropic so that the CA output does not depend on the particular arrangement of the dynamical states within a neighborhood, but on the sum of the cell values. These are called totalistic CAs and are a subset of those described by equation (80). Totalistic CAs are given by the map where σ n ∈ A p . The Wolfram code of a totalistic CA is given by RT ≡ This generalization amounts to use in equation (80) the replacement Mode I described in section 2. Therefore, by using equation (22), equation (83) becomes equal to equation (80) in the limit κ → 0. In the limit κ → ∞ one has, from equation (37), x j t+1 ∼ 1 2κ [13] so that, if the limit is strictly taken x j t = 0, ∀j and ∀t > 0. If we use mode II instead, we obtain the limit κ → 0 is as before but now the limit κ → ∞ is because of equation (20). Equations (83) and (84) describe respective CMLs taking place on the real numbers.
We thus see how B κ -embeddings connect CA and certain CMLs. If we consider the replacement Mode II on totalistic CA, we have, from equations (82) and (39) where the last equation is obtained in the asymptotic limit κ large. Let us consider, more specifically, local rules for which l = r = 1. We obtain where we have defined By further introducing we, finally, obtain This is a difference equation involving a discretized Laplacian Δ 2 L x j t and the first-order time difference Δ T x j t . By making the following transformations and by taking the limits N T → ∞, N L → ∞, τ → 0, → 0, so that N T τ = t, N L = x remain constant, we obtain from equation (94) The function P x j t is a quadratic polynomial governing the homogeneous dynamics. The function If 3D < 1 (which can always be the case for κ sufficiently large) the homogeneous dynamics of equation (105) converges to the stable homogeneous fixed point given by f(u * ) = 0, i.e.
We note that the conditions expressed by equations (103) and (104) are curiously the same as those found in the construction of appropriate difference operators for generalized logarithms and group entropies (see equation (2) in [44]).

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 → 0 and κ → ∞ each of the CAs entering in the connection is obtained. We call such nonlinear B κ -embedding, generally behaving as a CML, a CA 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 qualitatively 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 the CA limits and we believe that the general concept of CA connections introduced here may be useful in, e.g. biophysical models of multicellular ensembles [45], where variability and network heterogeneity need to be taken into account and may incorporate several kinds of typical CA dynamics. In these applications κ may be related to biological time/or and to the connectivity of the multicellular ensemble mediated by gap junctions [46]. The strength of the coupling is given by 1/κ so that, when κ → 0 (strong coupling regime), the cells in the ensemble are tightly coupled and one can qualitatively describe the ensemble by means of a CA. However, aging of the network may lead to a lower value of 1/κ leading to conformational changes that, in turn, may lead to dynamical changes so that the network is loosened. Finally, the presence of other agents in the network, facilitated by the decreased network connectivity may induce a different CA dynamics that is qualitatively different than the one obtained in the limit κ → 0. In this article we construct the whole general class of CA connections (CMLs) in which any two arbitrary CAs in rule space can be present in the CA limits.
The question now arises whether we can modify equation (84) so that the limit κ → ∞ is another cellular automaton of the form of equation (80) for all initial conditions x j 0 ∈ A p (∀j) but with a generally different rule vector We construct a κ-deformed formula so that two such CA are connected in the limits κ → 0 and κ → ∞. We note that x j t+1 ∈ A p is any of the integers m in the interval [0, p − 1]. Therefore, we observe that By using the replacement mode II and the κ → 1/κ transformation, we construct from here a nonlinear B κembedding    where a t (κ) ≡ 7 n=0 a n B κ n − 7u t , with (a 0 , a 1 , . . . , a 7 ) = (0, 1, 1, 1, 1, 0, 0, 0) and (b 0 , b 1 , . . . , b 7 ) = (0, 1, 1, 1, 0, 1, 1, 0). The bifurcation diagram can be readily calculated by iterating equation (121) for sufficiently long times, starting from initial conditions u 0 ∈ [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 bifurcation diagram is shown in figure 5 in which u ∞ is plotted vs κ.
Some observations made in figure 4 can be qualitatively understood by means of the bifurcation diagram in figure 5: • In the limits κ → 0 and κ → ∞ we have u ∞ → 0. This corresponds to the CA limits and both rules 1 30 1 2 and 1 110 1 2 in the connection fix the quiescent state and no other homogeneous configuration. The mean-field model is not able to capture the complex dynamics of equation (117) in these CA limits because the dynamical behavior is highly correlated through the CA dynamics and the correlations, that are affected by the neighborhood configurations, are lost in the mean-field model.
• For κ ≈ 1 the coexistence between two alternating period-2 branches at low values of u ∞ and a chaotic stripe at high values of u ∞ is observed. A gap in u ∞ is seen separating both behaviors. This qualitatively captures the observation made in figure 4 for this value of κ. • In the interval 2.75 κ 4, u ∞ displays a wide variety of possible states that are all reached in an aperiodic, chaotic, manner. The branch at low u ∞ is now fused with the chaotic stripe and there is no gap. This matches the observation made on the turbulent behavior described in figure 4 and the mean-field model allows to relate that turbulence in the full model to low-dimensional chaos in the reduced one. • In the interval 5 κ 8 there is a coexistence between stable homogeneous neighborhoods (branch with low u ∞ ) and a chaotic stripe at high values of u ∞ . Again, these behaviors are separated by a wide gap in the possible values of u ∞ .

Conclusions
In this work nonlinear B κ -embeddings have been constructed that are able to yield mathematical objects with different dimensionality (scalars, vectors, matrices) and dynamical classes of models (CAs, CMLs, nonlinear PDEs, CA connections) as a continuous parameter κ ∈ R is varied. We note that κ should have a wide physical significance. If one considers, for example, many particle systems governed by statistical laws, κ can be thought as a coarse-graining parameter, an increased value of it leading to fuzzier descriptions involving a lower number of degrees of freedom. κ can also be considered as a scale parameter in unified field theories, involved in the connection of objects (tensors) of different dimensions at different scales in which spacetime is probed. When connecting two mathematical structures of different dimensions, the technique presented in this manuscript is equivalent to embedding both of them in the space in which the structure of higher dimension lives and smoothly interpolating between them in that space. The major advantage of the embeddings presented is that the embedded structures are fixed points of the embeddings, when looked at dynamically.
Based on appropriate nonlinear B κ -embeddings [2], a new approach to compactification in unified physical theories (e.g. supergravity in 10 or 11-dimensional spacetimes) has been suggested. The method involves no Fourier expansion and no truncation, as is usually performed on extra dimensions to account for the observable four-dimensional universe. The limits κ → 0 and κ → ∞ of the B κ -embeddings are robust and yield the metric tensors of the spacetime with 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 with nonlinear PDEs through appropriate CMLs, all these structures being particular instances of the embedding. In particular, we have shown how (nonlinear) diffusion equations naturally emerge asymptotically from this construction. This mathematical approach sheds light, therefore, on why the Laplacian operator has such a tremendous importance in physical theories, since it already emerges from the most elementary dynamical systems and interactions when the continuum limit is performed.
In this article, we have also introduced the concept of CA connections. These are CMLs obtained from nonlinear B κ -embeddings, that depend on a control parameter κ such that in the limits κ → 0 and κ → ∞ the CML collapses to a CA. We have shown that any two CAs in rule space can be connected in this way. A mean-field, reduced model allows a bifurcation diagram to be calculated that qualitatively captures the features observed in the spatiotemporal evolution of the connection (in those parameter regimes where the neighborhood dynamics is approximately homogeneous). We have illustrated these general results with the specific example of Wolfram elementary Boolean CA rules 30 and 110 [14] constructing a connection between both rules. At intermediate κ values, a wide variety of dynamical behavior has been observed ranging from coherent to seemingly chaotic behavior, as well as the coexistence of coherence and disorder for simple initial conditions. These behaviors have been qualitatively investigated by means of a mean-field model derived from the connection. The results presented in this article can be easily generalized to more dimensions and arbitrary order in time [11].
If 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 biophysical applications to model changes in dynamical behavior induced by fuzziness or the coarsening of network connectivity [46]. If one considers physical models of networks governed by a finite set of strict rules (CA-like), a non-vanishing value for the parameter κ may incorporate the overall effect of the network heterogeneity, as well as the weakening of cooperative phenomena as κ is increased. If κ is made explicitly dependent on time, specific CA connections may also account for the effect of aging in the evolution of a system dynamics. We, therefore, believe that the structures here introduced can be helpful to model the long time evolution of biological organisms [45].

Data availability statement
Data sharing is not applicable to this article as no new data were created or analysed in this study.