Characterization of ergodicity of p adic
ISSN 1064-5624, Doklady Mathematics, 2011, Vol. 83, No. 3, pp. 306–308. © Pleiades Publishing, Ltd., 2011.
Original Russian Text © V.S. Anashin, A.Yu. Khrennikov, E.I. Yurova, 2011, published in Doklady Akademii Nauk, 2011, Vol. 438, No. 2, pp. 151–153.
MATHEMATICS
Characterization of Ergodicity of p-Adic Dynamical Systems
by Using the van der Put Basis
V. S. Anashina, A. Yu. Khrennikovb, and E. I. Yurovab
Presented by Academician V.S. Vladimirov December 23, 2010
Received December 24, 2010
DOI: 10.1134/S1064562411030100
The study of p-adic dynamical systems [1–3] is
motivated by their applications in various areas of
mathematics. First, we note applications in mathematical physics [2], with their foundations described
in [4] (see also [5]), in psychology [6], in various
domains of computer science [7, 1, 8–10], and in
genetics [1, 11, 12]. The results presented in this paper
can find wide applications in cryptography, for
example, in the construction of special maps (known
as T-functions), which are used in modern cryptographic algorithms (see, e.g., [13]). It is well known
that the ergodicity of p-adic dynamical systems [7, 9,
10, 2, 3] can be used to construct pseudorandom number generators (see [14, 7, 8, 13, 10]). We obtained several results for p-adic maps that provide the opportunity to characterize their important properties, including ergodicity and the preservation of the Haar
measure, in terms of coefficients with respect to the
van der Put basis. To the best of our knowledge, this is
the first attempt to use the van der Put basis to examine
the properties of (discrete) dynamical systems in fields
of p-adic numbers. Note that the van der Put basis differs fundamentally from previously used ones, for
example, the monomial and Mahler bases, which are
related to the algebraic structure of p-adic fields. The
van der Put basis is related to the zero-dimensional
topology of these fields (ultrametric structure), since it
consists of characteristic functions of p-adic balls; i.e.,
the basic point in the construction of this basis is the
continuity of the characteristic function of a p-adic
ball.
Let f: ⺪p → ⺪p be a function defined (and valued)
on the set ⺪p of all p-adic integers (p is a prime), and
let f satisfy the Lipschitz condition with a constant of 1
a
Institute of Information Security, Moscow State University,
Moscow, 119991Russia
e-mail: Vladimir.anashin@u-picardie.fr
b International Center for Mathematical Modeling,
Linnaeus University, SE-351 95 Växjö, Sweden
e-mail: khrennikov@lnu.se, ekaterina.yurova@lnu.se
with respect to the p-adic metric | · |p : | f (x) – f (y)|p ≤
|x – y|p for all x, y ∈ ⺪p. Recall that a mapping of an
algebraic system A to itself is called compatible if it
preserves all the congruences of A. Since ⺪p is a commutative ring with respect to addition and multiplication, a function f: ⺪p → ⺪p satisfies the Lipschitz condition with a constant of 1 if and only if it is compatible; i.e., f(x) ≡ f(y) (modpk) whenever x ≡ y (mod pk);
x, y ∈ ⺪p (in this case, a reduction modulo pk is an epimorphism modpk: ⺪p → ⺪/pk⺪ (with the kernel pk⺪p)
of the ring ⺪p to the ring ⺪/pk⺪ of residues modulo pk).
Along with compatible functions, we also consider
locally compatible functions of p-adic integer argument taking values in ⺪p: locally compatible functions
are ones satisfying the p-adic Lipschitz condition with
a constant of 1 locally, i.e., in a suitable neighborhood
of each point from ⺪p. Since ⺪p is compact, a function
f : ⺪p → ⺪p is locally compatible if and only if there
exists N ∈ ⺞ = {1, 2, 3, …} such that a ≡ b (modpk)
implies f(a) ≡ f(b) (modpk) for k ≥ N. Note that, given
a (locally) compatible function f, the mapping
fmodpk: a 哫 f(a)modpk of the ring ⺪/pk⺪ to itself is
well defined for all k = 1, 2, 3, … (for all k ≥ N) in the
sense that the image f(a)modpk of an element a ∈
⺪/pk⺪ is independent of the choice of a representative
in the coset a + pk⺪p (the elements of ⺪/pk⺪ are identified with the numbers 0, 1, 2, …, pk – 1).
The space ⺪p is equipped by the natural probability
measure, namely, the Haar measure μp normalized
so that μp(⺪p) = 1. Elementary μp-measurable sets are
p-adic balls. Recall that the p-adic ball of radius p–k is
the set a + pk⺪p of all p-adic integers congruent to a
modulo pk. The volume of this ball is defined as μp (a +
pk⺪p) = p–k.
Recall that a mapping f: ⺣ → ⺣ of a measurable
space S with a probability measure μ is said to preserve
the measure μ (in what follows, measure-preserving) if
μ(f–1(S)) = μ(S) for each measurable subset S ⊂ ⺣.
A measure-preserving mapping f: ⺣ → ⺣ is
called ergodic if it has no proper invariant subsets, i.e.,
if f –1(S) = S for a measurable subset S ⊂ ⺣ implies
μ(S) = 0 or μ(S) = 1.
306
CHARACTERIZATION OF ERGODICITY OF p-ADIC DYNAMICAL SYSTEMS
Since we study the conditions under which a locally
compatible function f : ⺪p → ⺪p preserves the normalized Haar measure μp or is ergodic with respect to μp,
in what follows, we use the term “measure-preserving” or “ergodic,” respectively.
We say that a (locally) compatible function f : ⺪p →
⺪p is bijective (transitive) modulo pk if the induced
mapping x 哫 f(x)mod pk is a permutation (a permutation of a single cycle, respectively) on Z/pk⺪. In the
case of a locally compatible function f, we assume, of
course, that k ≥ N. It was shown in [9] (see also [1, Section 4.4]) that a compatible function f : ⺪p → ⺪p is
measure-preserving (ergodic) if and only if it is bijective (respectively, transitive) modulo pk for all k = 1,
2, …. As was noted in [1, Section 4.4], the corresponding proof implies that this criterion (with “for all
k” replaced by “for all sufficiently large k”) holds for
locally compatible functions.1
It is well known (see, e.g., [15]) that any mapping g:
⺞0 → ⺪p of the set ⺞0 = {0, 1, 2, …} to the space ⺪p
can be defined using a van der Put series. Namely, for
g there is a unique sequence {B0, B1, B2, …} of p-adic
numbers such that
g(x) =
∑B
m
⋅ χ ( m, x )
(1)
– log m
p
f is (locally) compatible if and only if |Bm|p ≤ p
for
all (sufficiently large) m ∈ ⺞0.
In other words, f is (locally) compatible if and only
Bm
- = bm are p-adic integers for all (sufficiently
if -----------log m
p p
large) m = 1, 2, 3, …. Namely, a compatible function f
is compatible if and only if it can be represented as the
series
if
x–mp≤p
–n
otherwise,
where n = 1 if m = 0 and n is uniquely determined by
the inequality pn – 1 ≤ m ≤ pn – 1 if m ≥ 1. The number
n in the definition of χ(m, x) has a clear interpretation:
it is equal to the number of digits in the representation
of m ∈ ⺞0 in the base-p numeral system. Namely,
given m ∈ ⺞, if log pm | denotes the largest rational
integer not greater than log pm , then log pm is the
length of the representation of m in the p-ary numeral
system minus 1; i.e., n = log pm + 1 for m ∈ ⺞0 (by
definition, we set log p0 = 0). The coefficients Bm
are related to the values of g as follows. Let m = m0 +
m1 + … + mn – 2pn – 2 + mn – 1pn – 1 be the representation of
m in the p-ary number system, i.e., mj ∈ {0, 1, …, p – 1,
j = 0, 1, …, n – 1 and mn – 1 ≠ 0. Then
g ( m ) – g ( m – mn – 1 pn – 1 )
Bm = ⎨
⎩ g ( m ) otherwise.
if
m≥p
Note also that χ(m, x) is the characteristic function
– log pm – 1
centered at the point
1 Locally
compatible functions are called in [1] asymptotically
compatible. We use the former term as more appropriate.
DOKLADY MATHEMATICS
f(x) =
∑p
log pm
b m χ ( m, x )
(2)
for suitable p-adic integers bm ∈ ⺪p, m = 0, 1, 2, ….
Moreover, for the given f, the numbers bm ∈ ⺪p (m = 0,
1, 2, …) are uniquely defined; i.e., there exists a
unique sequence b0, b1, b2, … p-adic integers such that
equality (2) holds for all x ∈ ⺪p.
for all x ∈ ⺞0; here,
of the ball of radius p
m ∈ ⺞0.
CHARACTERIZATION OF LOCALLY
COMPATIBLE p-ADIC FUNCTIONS USING
VAN DER PUT SERIES
Theorem 1. Let f : ⺪p → ⺪p, and let the restriction of
f to ⺞0 be represented as the van der Put series (1). Then
m=0
m=0
1
χ ( m, x ) = ⎨
⎩0
Since every locally compatible function f : ⺪p → ⺪p
is uniformly continuous on ⺪p and ⺞0 is dense in ⺪p,
then function f is uniquely defined by its values at the
points 0, 1, 2, … . Since f is continuous, the coefficients Bm of its van der Put series tend p-adically to 0
with increasing m (see, e.g., [15]).
∞
∞
307
Vol. 83
No. 3
2011
Measure-Preserving Compatible p-Adic Functions
Using the above p-adic integers b0, b1, b2, …, we
can characterize measure-preserving functions in the
class of all compatible functions.
Theorem 2. Let f : ⺪p → ⺪p be a compatible function
represented in the form of series (2). Then f is measurepreserving if and only if
(i) f is bijective modulo pN for some (equivalently,
any) N ∈ ⺞ and
(ii) for all k > N and all m = 0, 1, 2, …, pk – 1, the
numbers
b m + pk modp, b m + 2pk modp, …, b m + ( p – 1 )pk modp
form the complete system of all positive residues modulo p.
Measure Preservation and Ergodicity
of Compatible 2-Adic Functions
If p = 2 (which is the most important case for applications in computer science and cryptography), Theorem 2 can be strengthened and, additionally, an
ergodicity criterion can be obtained for compatible
functions.
Theorem 3. A compatible function f : ⺪2 → ⺪2 is
measure-preserving if and only if f can be represented as
308
ANASHIN et al.
series (2), where the 2-adic integers bm ∈ ⺪2 satisfy the
following conditions:
(i) b0 + b1 ≡ 1 (mod2) and
(ii) |bm|2 = 1 for all m ≥ 2.
Theorem 4. A compatible function f : ⺪2 → ⺪2 is
ergodic if and only if f can be represented as series (2),
where the 2-adic integers bm ∈ ⺪2 satisfy the following
conditions:
(i) b0 ≡ 1 (mod2), b0 + b1 ≡ 3 (mod 4), b2 + b3 ≡ 2
(mod4);
(ii) |bm|2 = 1 for m ≥ 2;
n
2 –1
(iii)
b m ≡ 0 (mod4) for n ≥ 3.
∑
m=2
n–1
ACKNOWLEDGMENTS
The authors are grateful to V.M. Maksimov for
helpful discussions.
This work was supported by the Russian Foundation for Basic Research (project no. 09-01-00653-a,
V. S. Anashin) and by the Research Council of Sweden
(project “Non-Archimedean Analysis: From Foundations to Applications”).
REFERENCES
1. V. S. Anashin and A. Yu. Khrennikov, Applied Algebraic
Dynamics (Walter de Gruyter, Berlin, 2009).
2. A. Yu. Khrennikov, Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems, and Biological
Models (Kluwer, Dordrecht, 1997).
3. A. Yu. Khrennikov and M. Nilsson, p-Adic Deterministic and Random Dynamical Systems (Kluwer, Dordrecht, 2004).
4. V. S. Vladimirov, I. V. Volovich, and E. I. Zelenov, pAdic Analysis and Mathematical Physics (Fizmatlit,
Moscow, 1994) [in Russian].
5. A. Yu. Khrennikov, Non-Archimedean Analysis and
Applications (Fizmatlit, Moscow, 2003) [in Russian].
6. A. Yu. Khrennikov, Modeling of Processes of Thinking in
the p-Adic Coordinate Systems (Fizmatlit, Moscow,
2004) [in Russian].
7. V. S. Anashin, Diskret. Mat. 14 (4), 3–64 (2002).
8. V. S. Anashin, J. Math. Sci. 89, 1355–1390 (1998).
9. V. Anashin, Proceedings of 2nd International Conference on
p-Adic Mathematical Physics, Belgrade, September 15–21,
2005, AIP Conference Proc. Melville (Am. Inst. Phys,
New York, 2006), Vol. 826, pp. 3–24.
10. V. Anashin, Comput. J. 53, 370–392 (2010).
11. A. Yu. Khrennikov, “p-Adic Information Space and Gene
Expression,” International Conference on Integrative
Approaches to Brain Complexity (Wellcome Trust, Cambridge, 2006), p. 14.
12. A. Yu. Khrennikov and S. V. Kozyrev, Phys. A. Stat.
Mech. Appl. 381, 265–272 (2007).
13. V. Anashin, NATO Sci. Peace Secur. Ser. D Inf. Commun. Secur. 18, 33–57 (2008).
14. V. S. Anashin, Mat. Zametki 55 (2), 3–46.
15. K. Mahler, p-Acid Numbers and Their Functions (Cambridge Univ. Press, Cambridge, 1981).
DOKLADY MATHEMATICS
Vol. 83
No. 3
2011
Original Russian Text © V.S. Anashin, A.Yu. Khrennikov, E.I. Yurova, 2011, published in Doklady Akademii Nauk, 2011, Vol. 438, No. 2, pp. 151–153.
MATHEMATICS
Characterization of Ergodicity of p-Adic Dynamical Systems
by Using the van der Put Basis
V. S. Anashina, A. Yu. Khrennikovb, and E. I. Yurovab
Presented by Academician V.S. Vladimirov December 23, 2010
Received December 24, 2010
DOI: 10.1134/S1064562411030100
The study of p-adic dynamical systems [1–3] is
motivated by their applications in various areas of
mathematics. First, we note applications in mathematical physics [2], with their foundations described
in [4] (see also [5]), in psychology [6], in various
domains of computer science [7, 1, 8–10], and in
genetics [1, 11, 12]. The results presented in this paper
can find wide applications in cryptography, for
example, in the construction of special maps (known
as T-functions), which are used in modern cryptographic algorithms (see, e.g., [13]). It is well known
that the ergodicity of p-adic dynamical systems [7, 9,
10, 2, 3] can be used to construct pseudorandom number generators (see [14, 7, 8, 13, 10]). We obtained several results for p-adic maps that provide the opportunity to characterize their important properties, including ergodicity and the preservation of the Haar
measure, in terms of coefficients with respect to the
van der Put basis. To the best of our knowledge, this is
the first attempt to use the van der Put basis to examine
the properties of (discrete) dynamical systems in fields
of p-adic numbers. Note that the van der Put basis differs fundamentally from previously used ones, for
example, the monomial and Mahler bases, which are
related to the algebraic structure of p-adic fields. The
van der Put basis is related to the zero-dimensional
topology of these fields (ultrametric structure), since it
consists of characteristic functions of p-adic balls; i.e.,
the basic point in the construction of this basis is the
continuity of the characteristic function of a p-adic
ball.
Let f: ⺪p → ⺪p be a function defined (and valued)
on the set ⺪p of all p-adic integers (p is a prime), and
let f satisfy the Lipschitz condition with a constant of 1
a
Institute of Information Security, Moscow State University,
Moscow, 119991Russia
e-mail: Vladimir.anashin@u-picardie.fr
b International Center for Mathematical Modeling,
Linnaeus University, SE-351 95 Växjö, Sweden
e-mail: khrennikov@lnu.se, ekaterina.yurova@lnu.se
with respect to the p-adic metric | · |p : | f (x) – f (y)|p ≤
|x – y|p for all x, y ∈ ⺪p. Recall that a mapping of an
algebraic system A to itself is called compatible if it
preserves all the congruences of A. Since ⺪p is a commutative ring with respect to addition and multiplication, a function f: ⺪p → ⺪p satisfies the Lipschitz condition with a constant of 1 if and only if it is compatible; i.e., f(x) ≡ f(y) (modpk) whenever x ≡ y (mod pk);
x, y ∈ ⺪p (in this case, a reduction modulo pk is an epimorphism modpk: ⺪p → ⺪/pk⺪ (with the kernel pk⺪p)
of the ring ⺪p to the ring ⺪/pk⺪ of residues modulo pk).
Along with compatible functions, we also consider
locally compatible functions of p-adic integer argument taking values in ⺪p: locally compatible functions
are ones satisfying the p-adic Lipschitz condition with
a constant of 1 locally, i.e., in a suitable neighborhood
of each point from ⺪p. Since ⺪p is compact, a function
f : ⺪p → ⺪p is locally compatible if and only if there
exists N ∈ ⺞ = {1, 2, 3, …} such that a ≡ b (modpk)
implies f(a) ≡ f(b) (modpk) for k ≥ N. Note that, given
a (locally) compatible function f, the mapping
fmodpk: a 哫 f(a)modpk of the ring ⺪/pk⺪ to itself is
well defined for all k = 1, 2, 3, … (for all k ≥ N) in the
sense that the image f(a)modpk of an element a ∈
⺪/pk⺪ is independent of the choice of a representative
in the coset a + pk⺪p (the elements of ⺪/pk⺪ are identified with the numbers 0, 1, 2, …, pk – 1).
The space ⺪p is equipped by the natural probability
measure, namely, the Haar measure μp normalized
so that μp(⺪p) = 1. Elementary μp-measurable sets are
p-adic balls. Recall that the p-adic ball of radius p–k is
the set a + pk⺪p of all p-adic integers congruent to a
modulo pk. The volume of this ball is defined as μp (a +
pk⺪p) = p–k.
Recall that a mapping f: ⺣ → ⺣ of a measurable
space S with a probability measure μ is said to preserve
the measure μ (in what follows, measure-preserving) if
μ(f–1(S)) = μ(S) for each measurable subset S ⊂ ⺣.
A measure-preserving mapping f: ⺣ → ⺣ is
called ergodic if it has no proper invariant subsets, i.e.,
if f –1(S) = S for a measurable subset S ⊂ ⺣ implies
μ(S) = 0 or μ(S) = 1.
306
CHARACTERIZATION OF ERGODICITY OF p-ADIC DYNAMICAL SYSTEMS
Since we study the conditions under which a locally
compatible function f : ⺪p → ⺪p preserves the normalized Haar measure μp or is ergodic with respect to μp,
in what follows, we use the term “measure-preserving” or “ergodic,” respectively.
We say that a (locally) compatible function f : ⺪p →
⺪p is bijective (transitive) modulo pk if the induced
mapping x 哫 f(x)mod pk is a permutation (a permutation of a single cycle, respectively) on Z/pk⺪. In the
case of a locally compatible function f, we assume, of
course, that k ≥ N. It was shown in [9] (see also [1, Section 4.4]) that a compatible function f : ⺪p → ⺪p is
measure-preserving (ergodic) if and only if it is bijective (respectively, transitive) modulo pk for all k = 1,
2, …. As was noted in [1, Section 4.4], the corresponding proof implies that this criterion (with “for all
k” replaced by “for all sufficiently large k”) holds for
locally compatible functions.1
It is well known (see, e.g., [15]) that any mapping g:
⺞0 → ⺪p of the set ⺞0 = {0, 1, 2, …} to the space ⺪p
can be defined using a van der Put series. Namely, for
g there is a unique sequence {B0, B1, B2, …} of p-adic
numbers such that
g(x) =
∑B
m
⋅ χ ( m, x )
(1)
– log m
p
f is (locally) compatible if and only if |Bm|p ≤ p
for
all (sufficiently large) m ∈ ⺞0.
In other words, f is (locally) compatible if and only
Bm
- = bm are p-adic integers for all (sufficiently
if -----------log m
p p
large) m = 1, 2, 3, …. Namely, a compatible function f
is compatible if and only if it can be represented as the
series
if
x–mp≤p
–n
otherwise,
where n = 1 if m = 0 and n is uniquely determined by
the inequality pn – 1 ≤ m ≤ pn – 1 if m ≥ 1. The number
n in the definition of χ(m, x) has a clear interpretation:
it is equal to the number of digits in the representation
of m ∈ ⺞0 in the base-p numeral system. Namely,
given m ∈ ⺞, if log pm | denotes the largest rational
integer not greater than log pm , then log pm is the
length of the representation of m in the p-ary numeral
system minus 1; i.e., n = log pm + 1 for m ∈ ⺞0 (by
definition, we set log p0 = 0). The coefficients Bm
are related to the values of g as follows. Let m = m0 +
m1 + … + mn – 2pn – 2 + mn – 1pn – 1 be the representation of
m in the p-ary number system, i.e., mj ∈ {0, 1, …, p – 1,
j = 0, 1, …, n – 1 and mn – 1 ≠ 0. Then
g ( m ) – g ( m – mn – 1 pn – 1 )
Bm = ⎨
⎩ g ( m ) otherwise.
if
m≥p
Note also that χ(m, x) is the characteristic function
– log pm – 1
centered at the point
1 Locally
compatible functions are called in [1] asymptotically
compatible. We use the former term as more appropriate.
DOKLADY MATHEMATICS
f(x) =
∑p
log pm
b m χ ( m, x )
(2)
for suitable p-adic integers bm ∈ ⺪p, m = 0, 1, 2, ….
Moreover, for the given f, the numbers bm ∈ ⺪p (m = 0,
1, 2, …) are uniquely defined; i.e., there exists a
unique sequence b0, b1, b2, … p-adic integers such that
equality (2) holds for all x ∈ ⺪p.
for all x ∈ ⺞0; here,
of the ball of radius p
m ∈ ⺞0.
CHARACTERIZATION OF LOCALLY
COMPATIBLE p-ADIC FUNCTIONS USING
VAN DER PUT SERIES
Theorem 1. Let f : ⺪p → ⺪p, and let the restriction of
f to ⺞0 be represented as the van der Put series (1). Then
m=0
m=0
1
χ ( m, x ) = ⎨
⎩0
Since every locally compatible function f : ⺪p → ⺪p
is uniformly continuous on ⺪p and ⺞0 is dense in ⺪p,
then function f is uniquely defined by its values at the
points 0, 1, 2, … . Since f is continuous, the coefficients Bm of its van der Put series tend p-adically to 0
with increasing m (see, e.g., [15]).
∞
∞
307
Vol. 83
No. 3
2011
Measure-Preserving Compatible p-Adic Functions
Using the above p-adic integers b0, b1, b2, …, we
can characterize measure-preserving functions in the
class of all compatible functions.
Theorem 2. Let f : ⺪p → ⺪p be a compatible function
represented in the form of series (2). Then f is measurepreserving if and only if
(i) f is bijective modulo pN for some (equivalently,
any) N ∈ ⺞ and
(ii) for all k > N and all m = 0, 1, 2, …, pk – 1, the
numbers
b m + pk modp, b m + 2pk modp, …, b m + ( p – 1 )pk modp
form the complete system of all positive residues modulo p.
Measure Preservation and Ergodicity
of Compatible 2-Adic Functions
If p = 2 (which is the most important case for applications in computer science and cryptography), Theorem 2 can be strengthened and, additionally, an
ergodicity criterion can be obtained for compatible
functions.
Theorem 3. A compatible function f : ⺪2 → ⺪2 is
measure-preserving if and only if f can be represented as
308
ANASHIN et al.
series (2), where the 2-adic integers bm ∈ ⺪2 satisfy the
following conditions:
(i) b0 + b1 ≡ 1 (mod2) and
(ii) |bm|2 = 1 for all m ≥ 2.
Theorem 4. A compatible function f : ⺪2 → ⺪2 is
ergodic if and only if f can be represented as series (2),
where the 2-adic integers bm ∈ ⺪2 satisfy the following
conditions:
(i) b0 ≡ 1 (mod2), b0 + b1 ≡ 3 (mod 4), b2 + b3 ≡ 2
(mod4);
(ii) |bm|2 = 1 for m ≥ 2;
n
2 –1
(iii)
b m ≡ 0 (mod4) for n ≥ 3.
∑
m=2
n–1
ACKNOWLEDGMENTS
The authors are grateful to V.M. Maksimov for
helpful discussions.
This work was supported by the Russian Foundation for Basic Research (project no. 09-01-00653-a,
V. S. Anashin) and by the Research Council of Sweden
(project “Non-Archimedean Analysis: From Foundations to Applications”).
REFERENCES
1. V. S. Anashin and A. Yu. Khrennikov, Applied Algebraic
Dynamics (Walter de Gruyter, Berlin, 2009).
2. A. Yu. Khrennikov, Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems, and Biological
Models (Kluwer, Dordrecht, 1997).
3. A. Yu. Khrennikov and M. Nilsson, p-Adic Deterministic and Random Dynamical Systems (Kluwer, Dordrecht, 2004).
4. V. S. Vladimirov, I. V. Volovich, and E. I. Zelenov, pAdic Analysis and Mathematical Physics (Fizmatlit,
Moscow, 1994) [in Russian].
5. A. Yu. Khrennikov, Non-Archimedean Analysis and
Applications (Fizmatlit, Moscow, 2003) [in Russian].
6. A. Yu. Khrennikov, Modeling of Processes of Thinking in
the p-Adic Coordinate Systems (Fizmatlit, Moscow,
2004) [in Russian].
7. V. S. Anashin, Diskret. Mat. 14 (4), 3–64 (2002).
8. V. S. Anashin, J. Math. Sci. 89, 1355–1390 (1998).
9. V. Anashin, Proceedings of 2nd International Conference on
p-Adic Mathematical Physics, Belgrade, September 15–21,
2005, AIP Conference Proc. Melville (Am. Inst. Phys,
New York, 2006), Vol. 826, pp. 3–24.
10. V. Anashin, Comput. J. 53, 370–392 (2010).
11. A. Yu. Khrennikov, “p-Adic Information Space and Gene
Expression,” International Conference on Integrative
Approaches to Brain Complexity (Wellcome Trust, Cambridge, 2006), p. 14.
12. A. Yu. Khrennikov and S. V. Kozyrev, Phys. A. Stat.
Mech. Appl. 381, 265–272 (2007).
13. V. Anashin, NATO Sci. Peace Secur. Ser. D Inf. Commun. Secur. 18, 33–57 (2008).
14. V. S. Anashin, Mat. Zametki 55 (2), 3–46.
15. K. Mahler, p-Acid Numbers and Their Functions (Cambridge Univ. Press, Cambridge, 1981).
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Vol. 83
No. 3
2011