p-adic Cellular Neural Networks

In this article we introduce the p-adic cellular neural networks which are mathematical generalizations of the classical cellular neural networks (CNNs) introduced by Chua and Yang. The new networks have infinitely many cells which are organized hierarchically in rooted trees, and also they have infinitely many hidden layers. Intuitively, the p-adic CNNs occur as limits of large hierarchical discrete CNNs. More precisely, the new networks can be very well approximated by hierarchical discrete CNNs. Mathematically speaking, each of the new networks is modeled by one integro-differential equation depending on several p-adic spatial variables and the time. We study the Cauchy problem associated to these integro-differential equations and also provide numerical methods for solving them.


Introduction
In the late 80s Chua and Yang introduced a new natural computing paradigm called the cellular neural networks (or cellular nonlinear networks) CNN which includes the cellular automata as a particular case [6], [7], [9].From the beginning the CNN paradigm was intended for applications as an integrated circuit.This paradigm has been extremely successful in various applications in vision, robotics and remote sensing, see e.g.[8], [25] and the references therein.
In this article we present a mathematical generalization of the CNNs of Chua and Yang called p-adic cellular neural networks.The p-adic continuous CNNs offer a theoretical framework to study the emergent patterns of hierarchical discrete CNNs having arbitrary many hidden layers.
Nowadays, it is widely accepted that the analysis on ultrametric spaces is the natural tool for formulating models where the hierarchy plays a central role.An ultrametric space (M, d) is a metric space M with a distance satisfying d(A, B) ≤ max {d (A, C) , d (B, C)} for any three points A, B, C in M .Ultrametricity in physics means the emergence of ultrametric spaces in physical models.Ultrametricity was discovered in the 80s by Parisi and others in the theory of spin glasses and by Frauenfelder and others in physics of proteins.In both cases, the space of states of a complex system has a hierarchical structure which play a central role in the physical behavior of the system, see e.g.[10], [11], [15], [20], [21], [24], [27], [28]- [30], and the references therein.
On the other hand, Khrennikov and his collaborators have studied neural network models where p-state neurons take their values in p-adic numbers, see [2], [19].These models are completely different to the ones considered here.In addition, Khrennikov has developed non-Archimedean models of brain activity and mental processes, see e.g.[18] and the references therein.
Among the ultrametric spaces, the field of p-adic numbers Q p plays a central role.A p-adic number is a series of the form (1.1) x = x −k p −k + x −k+1 p −k+1 + . . .+ x 0 + x 1 p + . . ., with x −k = 0, where p is a prime number, the x j s are p-adic digits, i.e. numbers in the set {0, 1, . . ., p − 1}.The set of all the possible series of form (1.1) constitutes the field of p-adic numbers Q p .There are natural field operations, sum and multiplication, on series of form (1.1), see e.g.[16].There is also a natural norm in Q p defined as |x| p = p k , for a nonzero p-adic number x of the form (1.1).The field of p-adic numbers with the distance induced by |•| p is a complete ultrametric space.The ultrametric property refers to the fact that |x − y| p ≤ max |x − z| p , |z − y| p for any x, y, z in Q p .We denote by G M the set all the p-adic numbers of the form i , where the i j s belong to {0, 1, . . ., p − 1}.Then (G M , |•| p ) is a finite ultrametric space.Geometrically speaking, G M is a regular rooted tree with 2M layers, here regular means that exactly p edges emanate from each vertex.A (1-dimensional) p-adic discrete CNN is a dynamical system of the form i ∈ G M , where Y (j, t) = f (X(j, t)), with f (x) = 1 2 (|x + 1| − |x − 1|).Here X(i, t), Y (i, t) ∈ R are the state, respectively the output, of cell i at the time t.The function U (i) ∈ R is the input of the cell i, Z(i) ∈ R is the threshold of cell i, and the matrices A, B : G N M ×G N M → R are the feedback operator and feedforward operator, respectively.Notice that matrices A, B are functions on the Cartesian product of two rooted trees.The Chua-Yang CNNs are a particular case of (1.2).In this article we study N -dimensional, discrete hierarchical CNNs having arbitrary many layers.For the seek of simplicity, we focus on space-invariant networks, i.e. in the case in which In this article we initiate the study of the emergent patterns produced by the p-adic discrete CNNs.Since we are interested in arbitrary large trees, the description of these networks requires literally of millions of integro-differential equations, consequently a numerical approach seems not suitable, instead of this, we construct a p-adic continuous model that can be very well approximated by (1.2).
The study of the qualitative behavior of differential equations on large graphs is a relevant matter due its applications.In [23] Nakao and Mikhailov proposed using continuous models to study reaction-diffusion systems on networks and the corresponding Turing patterns.In [28] the second author showed that p-adic analysis is the natural tool to carry out this program.Models constructed using energy landscapes naturally drive to a large systems of differential equations (the master equation of the system), see e.g.[5], [20], [21].p-Adic continuous versions of some of these systems were constructed by Avetisov, Kozyrev and others in connection with models of protein folding, see e.g.[20], [21] for a general discussion.Another relevant system is the Eigen-Schuster model in biology.In [29] a p-adic continuous version of this model was introduced, this p-adic version allows to explain the Eigen paradox.Recently Hua and Hovestadt pointed out that the p-adic number system offers a natural representation of hierarchical organization of complex networks [15].
Intuitively, in the space-invariant case, the continuous model corresponding to (1.2) is obtained by taking the limit as M tends to infinity: For the sake of simplicity, in the introduction we discuss our results in dimension one.We study the case where A(|x| p ), B(|x| p ) are integrable, and U , Z are continuous functions vanishing at infinity.Under these hypotheses the initial value problem attached to (1.4), with initial datum X 0 (a continuous function vanishing at infinity) has a unique solution X(x, t) which is a continuos function vanishing at infinity in x for any t ≥ 0, satisfying |X(x, t)| ≤ X max , where the constant X max is completely determined by A, B, U , Z and f , see Theorem 2. An analog result is valid for discrete CNNs, see Theorem 3.
The solution X(x, t) can be very well approximated in the In practical applications it is natural to assume that radial functions A, B have compact support or that they are test functions.Under this hypothesis we study the patterns produced by p-adic continuous CNNs when U , Z and X 0 are test functions.The hypothesis that X 0 is a test functions means that at time t = 0 only certain clusters of cells are excited.Each cluster corresponds to a p-adic ball centered at some cell with radius, say p −L .The intensity of the excitation is the same for all cells in a given cluster.The fact that U , Z are test functions can be interpreted in an analogous way.Let B M0 denote the ball centered at the origin with radius p M0 , which the smallest ball containing the supports of A, B, U , Z, X 0 .Then the solution X(x, t) of the initial value problem attached to (1.4) is a test function supported in B M0 of the form j∈G M 0 X(j, t)Ω p M0 |x − j| p for t ≥ 0, with M 0 ≥ L, see Theorem 1.This means that a p-adic continuous CNN produces a pattern which is organized in a finite number of disjoint clusters, each of them supporting a time varying pattern.We also show the existence of two steady state patterns X + (x), X − (x), which are test functions, such that X − (x) ≤ lim t→∞ X(x, t) ≤ X + (x), see Theorem 2. We conjecture that for generic p-adic continuous CNNs, lim t→∞ X(x, t) is a test function, which means that the steady state pattern is organized in a finite number of disjoint clusters, each of them supporting a constant pattern.This is exactly the multistability property reported in [23], see also [28], for reaction-diffusion networks.
We have conducted a large number of numerical simulations.Such simulations require solving integro-differential equations on a tree.The numerical study of p-adic continuous CNNs offers two big challenges.The first, the need of dealing with matrices having millions of entries, the second, the visualization of functions depending on p-adic variables.Due to the first problem, we use small trees with 16 to 64 leaves.The p-adic numbers have a fractal nature, then, it is necessary to visualize real-valuated functions defined on the Cartesian product of a fractal times the real line.To deal with this problem we us systematically heat maps which allow us to get a glimpse of the hierarchical nature of the CNNs.Our numerical simulations show that the solutions of continuous CNNs exhibit a very complex behavior, including self-similarity and multistability, depending on the interaction of all the parameters defining the network and initial datum.
2. p-Adic Analysis: Essential Ideas 2.1.The field of p-adic numbers.Along this article p will denote a prime number.The field of p−adic numbers Q p is defined as the completion of the field of rational numbers Q with respect to the p−adic norm | • | p , which is defined as b , where a and b are integers coprime with p.The integer γ := ord(x), with ord(0) := +∞, is called the p−adic order of x.We extend the p−adic norm to Q N p by taking We define ord(x) = min 1≤i≤N {ord(x i )}, then ||x|| p = p −ord(x) .The metric space Q N p , || • || p is a complete ultrametric space.As a topological space Q p is homeomorphic to a Cantor-like subset of the real line, see e.g.[1], [27].
Any p−adic number x = 0 has a unique expansion of the form x j p j , where x j ∈ {0, 1, . . ., p − 1} and x 0 = 0.

Topology of
0 equals the product of N copies of B 0 = Z p , the ring of p−adic integers.We also denote by S N r (a) = {x ∈ Q N p ; ||x − a|| p = p r } the sphere of radius p r with center at a = (a 1 , . . ., a N ) ∈ Q N p , and take S N r (0) := S N r .We notice that S We will use Ω (p −r ||x − a|| p ) to denote the characteristic function of the ball B N r (a).For more general sets, we will use the notation 1 A for the characteristic function of a set A.

The Bruhat-Schwartz space. A real-valued function ϕ defined on
is locally constant with compact support.Any test function can be represented as a linear combination, with real coefficients, of characteristic functions of balls.The R-vector space of Bruhat-Schwartz functions is denoted by For an in depth discussion about p-adic analysis the reader may consult [1], [17], [26], [27].
| and the bar means the completion with respect the metric induced by • ∞ .Notice that all the functions in X ∞ are continuous and that On the other hand, since for j = 1, . . ., N , with i j k ∈ {0, 1, . . ., p − 1}.From now on, we fix a set of representatives in Q N p for G N M of the form (2.2).Notice that where are orthogonal with respect to the standard L 2 inner product, since We denote by D M Q N p := D M the R-vector space spanned by (2.3).We set Notice that X M is isomorphic as a Banach space to R #G N M , • R , where 2.5.Tree-like structures and p-adic numbers.Take N = 1 and fix where the i j s belong to {0, 1, . . ., p − 1}.Furthermore, the restriction of We endow G M with the metric induced by |•| p , and thus G M becomes a finite ultrametric space.In addition, G M can be identified with the set of branches (vertices at the top level) of a rooted tree with 2M + 1 levels and p 2M branches.Any element i ∈ G M can be uniquely written as p −M i, where with the i j s belonging to {0, 1, . . ., p − 1}.The elements of the Z p /p 2M Z p are in bijection with the vertices at the top level of the above mentioned rooted tree.By definition the root of the tree is the only vertex at level 0. There are exactly p vertices at level 1, which correspond with the possible values of the digit i 0 in the p-adic expansion of i.Each of these vertices is connected to the root by a nondirected edge.At level , with 1 ≤ ≤ 2M , there are exactly p vertices, each vertex corresponds to a truncated expansion of i of the form In conclusion, Z p /p 2M Z p is a rooted tree, and Z p is an infinite rooted tree.Now, the 1-dimensional unit sphere Z × p is the disjoint union of sets of the form j + pZ p , for j ∈ {1, . . ., p − 1}.Each set of the form j + pZ p is an infinite rooted tree.Then, Z × p is a forest formed by the disjoint union of p − 1 infinite rooted trees.On the other hand, The field of p-adic numbers has a fractal structure, see e.g.[1], [27].
Figure 1.The rooted tree associated with the group Z 2 /2 3 Z 2 .We identify the elements of Z 2 /2 3 Z 2 with the set of integers {0, . . ., 7} with binary representation . Two leaves i, j ∈ Z 2 /2 3 Z 2 have a common ancestor at level 2 if and only if i ≡ j mod 2 2 , i.e., i = a 0 + a 1 2 + i 2 2 2 and j = a 0 + a 1 2 + j 2 2 2 with i 2 , j 2 ∈ {0, 1}.Now, for i, j ∈ Z 2 /2 3 Z 2 have a common ancestor at level 1 if and only if i ≡ j mod 2. Notice that that the p-adic distance satisfies − log 2 |i − j| 2 = − (level of the first common ancestor of i, j).

p-Adic CNNs: basic definitions
We say that a function f : R → R is called a Lipschitz function if there exists a real constant L(f The state X i (t) ∈ R of cell i is described by the following differential equations: (i) state equation: where Y (i, t) ∈ R is the output of cell i at the time t, f : R → R is a bounded Lipschitz function satisfying f (0) = 0.The function U (i) ∈ R is the input Not all the cells of G N M are active.A cell i is connected with cell j if A(i, j) = 0 or B(i, j) = 0 for some j ∈ G N M .Then, a p-adic discrete CNN is a dynamical system on C N,M := i ∈ G N M ; A(i, j) = 0 or B(i, j) = 0 for some j ∈ G N M .The topology of a p-adic discrete CNN depends on the functions A, B : For general matrices A, B, it is difficult to give a graph-type description of the topology of the network.Our p-adic CNNs contain as a particular case the CNNs of Chua and Yang, see e.g.[8], [25].In this article we focus on p-adic CNNs satisfying (3.1) A(i, j) = A( i − j p ), B(i, j) = B( i − j p ), which are discrete CNNs having the space-invariant property.The fact that A and B are radial functions of • p implies that the cells are organized in a tree like-structure with many layers.

p-Adic continuous CNNs
, and U , Z ∈ X ∞ , a p-adic continuous CNN, denoted as CNN(A, B, U, Z), is the dynamical system given by the following differential equations: (i) state equation: where x ∈ Q N p , t ≥ 0, and (ii) output equation: Y (x, t) = f (X(x, t)).We say that X(x, t) ∈ R is the state of cell x at the time t, Y (x, t) ∈ R is the output of cell x at the time t.Function A(x, y) is the kernel of the feedback operator, while function B(x, y) is the kernel of the feedforward operator.Function U is the input of the CNN, while function Z is the threshold of the CNN.
We set B = 0 and A(i, j) = [a i,j ], with a i,j = 0 if |i − j| 3 = 1 and i, j ∈ C 1,2 ; a i,j = 0 otherwise.We focus mainly in continuous CNNs having the space invariant property, i.e.A(x, y) = A( x − y p ) and B(x, y) = B( x − y p ) for some A, B ∈ L 1 , however our results are valid for general p-adic continuous CNNs.Along this article the function f (x) = 1 2 (|x + 1| − |x − 1|) will be fixed, for this reason it does not appear in the list of parameters of the CNNs.

Discretization of p-adic continuous CNNs.
A central result of the present work is the fact that p-adic continuous CNNs are 'continuous versions' of p-adic discrete CNNs.More precisely, p-adic discrete CNNs are very good approximations of p-adic continuous CNNs for sufficiently large M .We discuss here the discretization process in an intutive way (a formal theorem will be provided later on).
Intuitively, a discretization of a p-adic continuous CNN(A, B, U, Z) is obtained assuming that X(•, t), A, Y (•, t), B, U and Z belong to D M , i.e. Similarly, and Y (i, t) = f (X(i, t)), for i ∈ G N M .This is exactly a p-adic discrete CNN with A(i, j) = p −M N A(i, j), B(i, j) = p −M N B(i, j).
Proof.By Lemma 1, H : X ∞ → X ∞ is a well-defined operator.Take g, g ∈ X ∞ , then Remark 1. (i) Lemma 1 remains valid if we replace the condition E is radial and integrable by the condition E(x, y) is a continuous function with compact support.
(ii) Under the hypothesis of part (i), Lemma 2 is valid for operators of the form Proposition 1. Assume that A, B, f satisfy hypotheses of Lemma 2 and that U , Z ∈ X ∞ .Let τ be a fixed positive real number.Then for each X 0 ∈ X ∞ there exists a unique X ∈ C([0, τ ], X ∞ ) which satisfies (4.2) X(x, t) = e −t X 0 (x) + t 0 e −(t−s) H(X(x, s))ds where The function X(x, t) is differentiable in t for all x, and it is a solution of equation (3.2) with initial datum X 0 .
Proof.The result follows from Lemma 2, by using standard techniques in PDEs, see e.g.[22,Theorem 5.1.2].To make the treatment comprehensive to a general audience, we provide some details here.First, define and And hence, for M ≥ 1.By the contraction mapping theorem, there is a unique unique X ∈ Y which T (X) = X.Moreover, since the right-hand side of (4.2) is differentiable in t, X is a solution of (3.2) with initial condition X 0 .
Remark 2. The contraction mapping theorem provides an iterative formula for X(x, t).Set X 1 (x, t) = X 0 (x) and are radial functions, for some M 0 ∈ N, and that U , Z, X 0 ∈ X M0 .We also assume that f is a Lipschitz functions with f (0) = 0. Then there is a unique 2), which is a solution of equation (3.2) with initial datum X 0 .
Remark 3.This theorem remains valid if A(x, y), B(x, y) are continuous functions with compact support, see Remark 1.
Proof.Since X M0 is a subspace of X ∞ , by applying Proposition 1, there exists a unique and that for any E ∈ L 1 (p −M0 Z N p ) are radial function, with the convention that the support of E is the ball p −M0 Z N p , as M → ∞, see e.g.[26, Theorem 1.14], (4.4) provides an explicit approximation of the continuous CNN described in Theorem 1.
Lemma 3. Let τ be a fixed positive real number, let X(x, t) be the solution given in Proposition 1, with X(x, 0) = X 0 .Then, for all x, y ∈ Q N p and t ∈ (0, τ ), Moreover, if X 0 is a locally-constant function, i.e.X 0 (x) = X 0 (y) for y ∈ B l (x), with l = l(x) ∈ Z, for any x ∈ Q N p , then X(•, t) is a locally-constant function and X(x, t) = X(y, t) for y ∈ B l (x) for any x ∈ Q N p .
Proof.Fix x, y ∈ Q N p , the by Proposition 1 and Lemma 2, for all t ∈ (0, τ ] Thus, by Gronwall theorem, see [22,Theorem 5.1.1], for all t ∈ (0, τ ). where then the CNN(A, B, U, Z) has a unique stationary state.This follows by the fact that, under this condition, H(X) becomes a contraction map in X ∞ , cf.Lemmas 1, 2.
Theorem 2. All the states X(x, t) of a p-adic continuous CNN(A, B, U, Z) are bounded for all time t ≥ 0.More precisely, if |X(x, t)| ≤ X max for all t ≥ 0 and for all x ∈ Q N p .In addition Remark 6. Condition (4.5) implies that X(x, t) does not blow-up at finite time.
The existence of a stationary state X * (x) means that the state of each cell of a p-adic continuous CNN most settle at stable equilibrium point after the transient has decayed to zero.
Proof.By Proposition 1, see (4.2)-(4.3),by using that |Y (y, This bound is valid for any t ∈ [0, τ ], but τ is an arbitrary, the bound is valid for any t ≥ 0. The bound (4.5) implies existence of the functions: Now assume that lim t→∞ X(x, t) = X * (x) exists.By using that and the dominated convergence and Lemma 2, it follows from (4.2) that

Stability of p-adic discrete CNN and Approximation of Continuous CNNs
5.1.The operators P M , E M .We now define for M ≥ 1, P M : X ∞ → X M as Therefore P M is a linear bounded operator, indeed, P M ≤ 1.We denote by E M , M ≥ 1, the embedding X M → X ∞ .The following result is a consequence of the above observations.If Z, Y are real Banach spaces, we denote by B(Z, Y), the space of all linear bounded operators from Z into Y.Lemma 4. [30, Lemma 2] With the above notation, the following assertions hold true: (i) X ∞ , X M for M ≥ 1, are real Banach spaces, all with the norm We now assume that A, B are radial integrable functions, and that U , Z, X 0 ∈ X ∞ .Based on the continuity of operators A, B : X ∞ → X ∞ and the formula given in Lemma 5, we can approximate the solution X(x, t) of a p-adic continuous CNN(A, B, U, Z) by p 2k ODEs, k ≥ 1, of the form In the simulations the parameters k, k max were chosen by trial and error on a case by case approach.The sum kmax l=k A(p −l )p −l can be approximated by A(p −k )p −k in the cases were Qp Without loss of generality, we may assume that k H ≤ k, and since any two balls are disjoint or one contains the other, then We now assume that U , Z, X 0 ∈ X ∞ and that A, B are test functions of the form Based on the continuity of operators A, B : X ∞ → X ∞ and the formula given in Lemma 6, we can approximate the solution X(x, t) of a p-adic continuous CNN by p 2k ODEs, k ≥ 1, of the form It is possible to combine the approximations given in numeric schemes A, B.

6.3.
A remark on the visualization of finite rooted trees.The discretizations of the kernels A, B are functions on G k ×G k , while the input U and X 0 are functions on G k .We use systematically heat maps to present these functions.We always include a plot of the tree G k .By convention we identify the leaves of the tree G k with the set of rational numbers {0, 1/p k , 2/p k , . . ., (p 2k − 1)/p k }.Furthermore, we label the levels of G k with integers from the set {−k, −k + 1, . . ., 0, 1, . . .k − 1}.
The level l consists of the cells i, j such that − log p (|i − j| p ) = (the level of the first common ancestor of i, j) = l.
6.4.First Simulation.In this example, we take k = 2, p = 2, which means that we use a tree with 2 4 = 16 leaves and 4 levels.A basic application of the classical CNNs is image processing, see e.g.[8].In this example we present a one-dimensional edge detector, which is a p-adic, one-dimensional analog of the examples 3.1 and 3.2 in [8].The input U is a image having three levels: As in [8] we take X 0 (x) = 0, A(x) = 0. To construct template B, we identify a matrix with a test function.We use Finally, we take Z The output Y (x, t) consists of the edges on the input U , see Figure 7.   Step 0.05.6.5.Second Simulation.In this example, we take k = 2, p = 2, which means that we use a tree with 2 4 = 16 leaves and 4 levels.We consider a CNN with the followin parameters: We set X 0 (x) = 0 and f (x) = 1 2 (|x + 1| − |x − 1|).The numerical solutions is given in Figure 11.We now take A(x) = B(x) = Ω(2 2 |x| 2 ).In this case the output Y (x, t) changes completely, see Figure 12.As a consequence of the fractal nature of the p-adic numbers, the p-adic CNNs exhibit self-similarity in several ways.For instance, the graph of the kernel A(x, y) is a self-similar set, this follows by comparing the graphs given in simulations 2 and 3 for this kernel.In addition, the output Y (x, t) = 0 when the norm |x| 2 is sufficiently large.In this simulation the CNN produces a pattern similar to the input.

CONCLUSIONS
In this article, we present a p-adic generalization of Chua-Yang CNNs.In the p-adic framework, a continuous CNN is modeled by just one integro-differential equation depending on several p-adic variables and the time.In contrast, the classical CNNs are described by a discrete system of integro-differential equations.The need of constructing continuous models of discrete CNNs whose modeling requires millions of integro-differential equations is quite natural.
A one-dimensional p-adic continuous CNN has infinitely many cells which are hierarchically organized in rooted trees, also a such network has infinitely many hidden layers.The topology of the network, which lately controls the interaction of the cells, depends on the supports of the kernels of the feedback and feedforward operators.Under mild hypotheses, there is a natural discretization process of p-adic continuous CNNs that produces standard discrete CNNs.The solutions of the continuous CNNs can be very well approximated by the solutions of discrete CNNs.Then, for practical purposes, a p-adic continuous CNN is a hierarchical discrete CNN with many hidden layers.
Our numerical simulations show that the solutions of continuous CNN exhibit a very complex behavior, including self-similarity and multistability, depending on the interaction of the parameters defining the network and the initial datum.
In the p-adic framework, the class of continuous CNNs is huge, for instance, consider equations of type ∂X(x, t) ∂t = −LX(x, t) + where ∂X(x,t) ∂t = −LX(x, t) is a p-adic heat equation, i.e. the fundamental solution of a such equation is the transition probability density of a Markov process on Q N p .The class of p-adic heat equations is extremely large, see e.g.[20], [31].By incorporating a 'diffusion term' is natural to expect that the corresponding network will produce more complex patterns.We plan to study these networks in a forthcoming article.In the classical framework the reaction-diffusion CNNs have been studied intennsively, see e.g.[12,13,14].

Figure 2 .
Figure 2. The heat map associated with the p-adic distance function on Z 2 /2 3 Z 2 .

Proof.
It is sufficient to consider the case whereH(x) = Ω(p k H |x − b H | p ) for some k H ∈ Z and b H ∈ Q p .Since g(x) = lim k→∞ i∈G k g(i)Ω(p k |x − a| p ), we have Qp H(x − y)g(y)dy = lim k→∞ i∈G k g(i) Qp Ω(p k H |x − b H − y| p )Ω(p k |y − i| p
y)Y (y, t)d N y + Q N p B(x, y)U (y)d N y + Z(x), N 0 .The balls and spheres are both open and closed subsets in Q N p .In addition, two balls in Q N p are either disjoint or one is contained in the other.As a topological space Q N p , || • || p is totally disconnected, i.e. the only connected subsets of Q N p are the empty set and the points.A subset of Q N p is compact if and only if it is closed and bounded in Q N p , see e.g.[27, Section 1.3], or [1, Section 1.8].The balls and spheres are compact subsets.Thus Q N p , || • || p is a locally compact topological space.
1 0 = Z × p (the group of units of Z p ), but Z × p N S