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We study the Cauchy problem for $p$-adic nonlinear evolutionary pseudo-differential equations for complex-valued functions of a real positive time variable and p-adic spatial variables. Among the equations under consideration there is the p-adic analog of the porous medium equation (or more generally, the nonlinear filtration equation) which arise in numerous application in mathematical physics and mathematical biology. Our approach is based on the construction of a linear Markov semigroup on a p-adic ball and the proof of m-accretivity of the appropriate nonlinear operator. The latter result is equivalent to the existence and uniqueness of a mild solution of the Cauchy problem of a nonlinear equation of the porous medium type.

We prove the local solvability of the p-adic analog of the Navier-Stokes equation. This equation describes, within the p-adic model of porous medium, the flow of a fluid in capillaries., Comment: arXiv admin note: text overlap with arXiv:1708.03261, arXiv:1611.08863

$p$-Adic mathematical physics is a branch of modern mathematical physics based on the application of $p$-adic mathematical methods in modeling physical and related phenomena. It emerged in 1987 as a result of efforts to find a non-Archimedean approach to the spacetime and string dynamics at the Planck scale, but then was extended to many other areas including biology. This paper contains a brief review of main achievements in some selected topics of $p$-adic mathematical physics and its applications, especially in the last decade. Attention is mainly paid to developments with promising future prospects., Comment: 50 pages

Ultrametric approach to the genetic code and the genome is considered and developed. $p$-Adic degeneracy of the genetic code is pointed out. Ultrametric tree of the codon space is presented. It is shown that codons and amino acids can be treated as $p$-adic ultrametric networks. Ultrametric modification of the Hamming distance is defined and noted how it can be useful. Ultrametric approach with $p$-adic distance is an attractive and promising trend towards investigation of bioinformation., Comment: 20 pages. Accepted for publication in Applied Mathematics and Computation

We consider a nonlinear evolution equation for complex-valued functions of a real positive time variable and a p-adic spatial variable. This equation is a non-Archimedean counterpart of the fractional porous medium equation. Developing, as a tool, an $L^1$-theory of Vladimirov's p-adic fractional differentiation operator, we prove m-accretivity of the appropriate nonlinear operator, thus obtaining the existence and uniqueness of a mild solution. We give also an example of an explicit solution of the p-adic porous medium equation., Comment: v2: some typos fixed. To appear in J. Fourier Anal. Appl

We analyze the data from the loophole-free CHSH experiment performed by Hensen et al., and show that it is actually not exempt of an important loophole. By increasing the size of the sample of event-ready detections, one can exhibit in the experimental data a violation of the no-signaling principle with a statistical significance at least similar to that of the reported violation of the CHSH inequality, if not stronger., Comment: This version was updated on the basis of our talk at the conference "Quantum and Beyond", V\"axj\"o, Sweden, June 2016, and a few emails received from the readers of the first version of this arXiv, we improved statistical analysis and clarified essentially our point

Summation of a large class of the functional series, which terms contain factorials, is considered. We first investigated finite partial sums for integer arguments. These sums have the same values in real and all p-adic cases. The corresponding infinite functional series are divergent in the real case, but they are convergent and have p-adic invariant sums in p-adic cases. We found polynomials which generate all significant ingredients of these series and make connection between their real and p-adic properties. In particular, we found connection of one of our integer sequences with the Bell numbers., Comment: 9 pages, accepted for publication in Filomat. arXiv admin note: text overlap with arXiv:1411.4195

We discuss the approach to investigation of molecular machines using systems of integro--differential ultrametric (p-adic) reaction--diffusion equations with drift. This approach combines the features of continuous and discrete dynamic models. We apply this model to investigation of actomyosin molecular motor. The introduced system of equations is solved analytically using p-adic wavelet theory. We find explicit stationary solutions and behavior in the relaxation regime., Comment: 15 pages

Time series defined by a p-adic pseudo-differential equation is investigated using the expansion of the time series over p-adic wavelets. Quadratic correlation function is computed. This correlation function shows a degree--like behavior and is locally constant for some time periods. It is natural to apply this kind of models for the investigation of avalanche processes and punctuated equilibrium as well as fractal-like analysis of time series generated by measurement of pressure in oil wells., Comment: 18 pages

This paper is devoted to wavelet analysis on adele ring $\bA$ and the theory of pseudo-differential operators. We develop the technique which gives the possibility to generalize finite-dimensional results of wavelet analysis to the case of adeles $\bA$ by using infinite tensor products of Hilbert spaces. The adele ring is roughly speaking a subring of the direct product of all possible ($p$-adic and Archimedean) completions $\bQ_p$ of the field of rational numbers $\bQ$ with some conditions at infinity. Using our technique, we prove that $L^2(\bA)=\otimes_{e,p\in\{\infty,2,3,5,...}}L^2({\bQ}_{p})$ is the infinite tensor product of the spaces $L^2({\bQ}_{p})$ with a stabilization $e=(e_p)_p$, where $e_p(x)=\Omega(|x|_p)\in L^2({\bQ}_{p})$, and $\Omega$ is a characteristic function of the unit interval $[0,\,1]$, $\bQ_p$ is the field of $p$-adic numbers, $p=2,3,5,...$; $\bQ_\infty=\bR$. This description allows us to construct an infinite family of Haar wavelet bases on $L^2(\bA)$ which can be obtained by shifts and multi-delations. The adelic multiresolution analysis (MRA) in $L^2(\bA)$ is also constructed. In the framework of this MRA another infinite family of Haar wavelet bases is constructed. We introduce the adelic Lizorkin spaces of test functions and distributions and give the characterization of these spaces in terms of wavelet functions. One class of pseudo-differential operators (including the fractional operator) is studied on the Lizorkin spaces. A criterion for an adelic wavelet function to be an eigenfunction for a pseudo-differential operator is derived. We prove that any wavelet function is an eigenfunction of the fractional operator. These results allow one to create the necessary prerequisites for intensive using of adelic wavelet bases and pseudo-differential operators in applications., Comment: 45 pages

We propose a double blinding-attack on entangled-based quantum key distribution protocols. The principle of the attack is the same as in existing blinding attack except that instead of blinding the detectors on one side only, Eve is blinding the detectors of both Alice and Bob. In the BBM92 protocol, the attack allows Eve to get a full knowledge of the key and remain undetected even if Alice and Bob are using 100% efficient detectors. The attack can be easily extended to Ekert protocol, with an efficiency as high as 85.3%., Comment: 4 pages

In the present paper we propose to describe gene networks in biological systems using probabilistic algorithms. We describe gene duplication in the process of biological evolution using introduction of the replica procedure for probabilistic algorithms. We construct the examples of such a replica procedure for hidden Markov models. We introduce the family of hidden Markov models where the set of hidden states is a finite additive group with a p-adic metric and build the replica procedure for this family of markovian models.

In elaborating on the multiple-photon absorption attack on Ekert protocol proposed in arXiv:1011.4740, we show that it can be used in other entanglement-based protocols, in particular the BBM92 protocol. In this attack, the eavesdropper (Eve) is assumed to be in control of the source, and she sends pulses correlated in polarization (but not entangled) containing several photons at frequencies for which only multiple-photon absorptions are possible in Alice's and Bob's detectors. Whenever the photons stemming from one pulse are dispatched in such a way that the number of photons is insufficient to trigger a multiple-photon absorption in either channel, the pulse remains undetected. We show that this simple feature is enough to reproduce the type of statistics on the detected pulses that are considered as indicating a secure quantum key distribution, even though the source is actually a mixture of separable states. The violation of Bell inequalities measured by Alice and Bob increases with the order of the multiple-photon absorption that Eve can drive into their detectors, while the measured quantum bit error rate decreases as a function of the same variable. We show that the attack can be successful even in the simplest case of a two-photon absorption or three-photon absorption attack, and we discuss possible countermeasures, in particular the use of a fair sampling test., Comment: 11 pages, 7 figures, 1 table

We propose a local scheme to enhance the security of quantum key distribution in Ekert protocol (E91). Our proposal is a fair sampling test meant to detect an eavesdropping attempt that would use a biased sample to mimic an apparent violation of Bell inequalities. The test is local and non disruptive: it can be unilaterally performed at any time by either Alice or Bob during the production of the key, and together with the Bell inequality test., Comment: 4 pages, 3 figures, submitted to Phys. Rev. Lett.

A brief review of some selected topics in p-adic mathematical physics is presented., Comment: 36 pages

In this paper we denonstrate that the use of the system of 2-adic numbers provides a new insight to some problems of genetics, in particular, generacy of the genetic code and the structure of the PAM matrix in bioinformatics. The 2-adic distance is an ultrametric and applications of ultrametrics in bioinformatics are not surprising. However, by using the 2-adic numbers we match ultrametric with a number theoretic structure. In this way we find new applications of an ultrametric which differ from known up to now in bioinformatics. We obtain the following results. We show that the PAM matrix A allows the expansion into the sum of the two matrices A=A^{(2)}+A^{(\infty)}, where the matrix A^{(2)} is 2-adically regular (i.e. matrix elements of this matrix are close to locally constant with respect to the discussed earlier by the authors 2-adic parametrization of the genetic code), and the matrix A^{(\infty)} is sparse. We discuss the structure of the matrix A^{(\infty)} in relation to the side chain properties of the corresponding amino acids., Comment: 11 pages, discussion and references added

In this paper a countable family of new compactly supported {\em non-Haar} $p$-adic wavelet bases in ${\cL}^2(\bQ_p^n)$ is constructed. We use the wavelet bases in the following applications: in the theory of $p$-adic pseudo-differential operators and equations. Namely, we study the connections between wavelet analysis and spectral analysis of $p$-adic pseudo-differential operators. A criterion for a multidimensional $p$-adic wavelet to be an eigenfunction for a pseudo-differential operator is derived. We prove that these wavelets are eigenfunctions of the fractional operator. In addition, $p$-adic wavelets are used to construct solutions of linear and semi-linear pseudo-differential equations. Since many $p$-adic models use pseudo-differential operators (fractional operator), these results can be intensively used in these models.

We study the asymptotical behavior of the $p$-adic singular Fourier integrals $$ J_{\pi_{\alpha},m;\phi}(t) =\bigl< f_{\pi_{\alpha};m}(x)\chi_p(xt), \phi(x)\bigr> =F\big[f_{\pi_{\alpha};m}\phi\big](t), \quad |t|_p \to \infty, \quad t\in \bQ_p, $$ where $f_{\pi_{\alpha};m}\in {\cD}'(\bQ_p)$ is a {\em quasi associated homogeneous} distribution (generalized function) of degree $\pi_{\alpha}(x)=|x|_p^{\alpha-1}\pi_1(x)$ and order $m$, $\pi_{\alpha}(x)$, $\pi_1(x)$, and $\chi_p(x)$ are a multiplicative, a normed multiplicative, and an additive characters of the field $\bQ_p$ of $p$-adic numbers, respectively, $\phi \in {\cD}(\bQ_p)$ is a test function, $m=0,1,2...$, $\alpha\in \bC$. If $Re\alpha>0$ the constructed asymptotics constitute a $p$-adic version of the well known Erd\'elyi lemma. Theorems which give asymptotic expansions of singular Fourier integrals are the Abelian type theorems. In contrast to the real case, all constructed asymptotics have the {\it stabilization} property.

We described a wide class of $p$-adic refinable equations generating $p$-adic multiresolution analysis. A method for the construction of $p$-adic orthogonal wavelet bases within the framework of the MRA theory is suggested. A realization of this method is illustrated by an example, which gives a new 3-adic wavelet basis. Another realization leads to the $p$-adic Haar bases which were known before., Comment: 10 pages

In the present paper we consider countable state $p$-adic Potts model on the Cayley tree. A construction of $p$-adic Gibbs measures which depends on weights $\l$ is given, and an investigation of such measures is reduced to examination of an infinite-dimensional recursion equation. Studying of the derived equation under some condition on weights, we prove absence of the phase transition. Note that the condition does not depend on values of the prime $p$, and an analogues fact is not true when the number of spins is finite. For homogeneous model it is shown that the recursive equation has only one solution under that condition on weights. This means that there is only one $p$-adic Gibbs measure $\m_\l$. The boundedness of the measure is also established. Moreover, continuous dependence the measure $\m_\l$ on $\l$ is proved. At the end we formulate one limit theorem for $\m_\l$., Comment: 13 pages. submitted

For the first time we apply the methodologies of nonlinear analysis to investigate atomic matter. We use these methods in the analysis of Atomic Weights and of Mass Number of atomic nuclei. Using the AutoCorrelation Function and Mutual Information we establish the presence of nonlinear effects in the mechanism of increasing mass of atomic nuclei considered as a function of the atomic number. We find that increasing mass is divergent, possibly chaotic. We also investigate the possible existence of a Power Law for atomic nuclei and, using also the technique of the variogram, we conclude that a fractal regime could superintend to the mechanism of increasing mass for nuclei. Finally, using the Hurst exponent, evidence is obtained that the mechanism of increasing mass in atomic nuclei is in the fractional Brownian regime. The most interesting results are obtained by using Recurrence Quantification Analysis (RQA). New recurrences, psudoperiodicities, self-resemblance and class of self-similarities are identified with values of determinism showing oscillating values indicating the presence of more or less stability during the process of increasing mass of atomic nuclei. In brief, new regimes of regularities are identified for atomic nuclei that deserve to be studied by future researches. In particular an accurate analysis of binding energy values by nonlinear methods is further required., Comment: 42 pages including figures

We propose a model of processing of information in the brain which has the following distinguishing features: a). It is quantum-like (QL). The brain uses the quantum rule (given by von Neumann trace formula) for calculation of averages for psychological functions. b). Those functions are considered as self-observations of the brain. c). The QL-representation has the temporal basis. The brain is a machine transferring time into cognition. d). Any cognitive process is based on (at least) two time scales: precognitive time scale (which is very fine) and cognitive time scale (which is essentially coarser). To couple our model to physiology, behavioral science, and psychology, we consider a number of known fundamental time scales in the brain. Although the elaboration of those scales was based on advanced experimental research, there are still many controversial approaches and results. The temporal structure of the brain functioning is very complex.

We introduce the simple parametrization for the space of codons (triples of nucleotides) by 8\times 8 table. This table (which we call the dyadic plane) possesses the natural 2-adic ultrametric. We show that after this parametrization the genetic code will be a locally constant map of the simple form. The local constancy of this map will describe degeneracy of the genetic code. The map of the genetic code defines 2-adic ultrametric on the space of amino acids. We show that hydrophobic amino acids will be clustered in two balls with respect to this ultrametric. Therefore the introduced parametrization of space of codons exhibits the hidden regularity of the genetic code., Comment: Some gap in the construction was fixed

In this paper we study some problems related with the theory of multidimensional $p$-adic wavelets in connection with the theory of multidimensional $p$-adic pseudo-differential operators (in the $p$-adic Lizorkin space). We introduce a new class of $n$-dimensional $p$-adic compactly supported wavelets. In one-dimensional case this class includes the Kozyrev $p$-adic wavelets. These wavelets (and their Fourier transforms) form an orthonormal complete basis in ${\cL}^2(\bQ_p^n)$. A criterion for a multidimensional $p$-adic wavelet to be an eigenfunction for a pseudo-differential operator is derived. We prove that these wavelets are eigenfunctions of the Taibleson fractional operator. Since many $p$-adic models use pseudo-differential operators (fractional operator), these results can be intensively used in applications. Moreover, $p$-adic wavelets are used to construct solutions of linear and {\it semi-linear} pseudo-differential equations.

In the first part of the present paper we give an analysis of the ontic nature of quantum states to be intended as potentialities and of the central role of spin to be considered as the basic essence of quantum mechanical reality: using an algebraic quantum like structure we give mathematical proof on the transition from potentiality to actualization : we recall here what was recently given by us in arXiv quant-ph/0607196. However, as may be expected, it is not so easy to introduce examples containing an adequate description of ontic potentialities through detailed models of systems. The central aim of this paper is to attempt to reach this objective giving direct cases of systems in which ontic potentialities act jointly to actualization. Our aim is to provide evidence for the possible importance of potential states in the sphere of the biological dynamics giving detailed examples of interest for biological studies. We outline the possible implications of potentialities at the level of linear and non linear biological dynamics.

We analyze optical EPR experimental data performed by Weihs et al in Innsbruck 1997-1998. We show that for some linear combinations of the raw coincidence rates, the experimental results display some anomalous behavior that a more general source state (like non-maximally entangled state) cannot straightforwardly account for. We attempt to explain these anomalies by taking account of the relative efficiencies of the four channels. For this purpose, we use the fair sampling assumption, and assume explicitly that the detection efficiencies for the pairs of entangled photons can be written as a product of the two corresponding detection efficiencies for the single photons. We show that this explicit use of fair sampling cannot be maintained to be a reasonable assumption as it leads to an apparent violation of the no-signalling principle., Comment: 13 pages, 8 figures

Family of replica matrices, related to general ultrametric spaces with general measures, is introduced. These matrices generalize the known Parisi matrices. Some functionals of replica approach are computed. Replica symmetry breaking solution is found., Comment: 21 pages

Replica symmetry breaking solutions for the new replica anzats, related to general ultrametric spaces, are investigated. A variant of analysis on trees is developed and applied to the computation of the n\to0 limit in the new replica anzats., Comment: 22 pages

Family of replica matrices, related to general ultrametric spaces, is introduced. These matrices generalize the known Parisi matrices. Some functionals of replica approach are computed., Comment: 19 pages

Gaussian random field on general ultrametric space is introduced as a solution of pseudodifferential stochastic equation. Covariation of the introduced random field is computed with the help of wavelet analysis on ultrametric spaces. Notion of ultrametric Markovianity, which describes independence of contributions to random field from different ultrametric balls is introduced. We show that the random field under investigation satisfies this property., Comment: 15 pages

In this paper the p -adic Lizorkin spaces of test functions and distributions are introduced, and multidimensional Vladimirov's and Taibleson's fractional operators are studied on these spaces. Since the p -adic Lizorkin spaces are invariant under the Vladimirov and Taibleson operators, they can play a key role in considerations related to fractional operator problems. A class of p -adic pseudo-differential operators in the Lizorkin spaces is also introduced and solutions of pseudo-differential equations are constructed. p -Adic multidimensional Tauberian theorems connected with fractional operators and pseudo-differential operators for the Lizorkin distributions are also proved., Comment: 32 pages

Free evolution for quantum particle in general ultrametric space is considered. We find that if mean zero wave packet is localized in some ball in the ultrametric space then its evolution remains localized in the same ball., Comment: 6 pages, LaTeX

Examples of games between two partners with mixed strategies, calculated by the use of the probability amplitude as some vector in Hilbert space are given. The games are macroscopic, no microscopic quantum agent is supposed. The reason for the use of the quantum formalism is in breaking of the distributivity property for the lattice of yes-no questions arising due to the special rules of games. The rules of the games suppose two parts: the preparation and measurement. In the first part due to use of the quantum logical orthocomplemented non-distributive lattice the partners freely choose the wave functions as descriptions of their strategies. The second part consists of classical games described by Boolean sublattices of the initial non-Boolean lattice with same strategies which were chosen in the first part. Examples of games for spin one half are given. New Nash equilibria are found for some cases. Heisenberg uncertainty relations without the Planck constant are written for the "spin one half game".

A family of orthonormal bases, the ultrametric wavelet bases, is introduced in quadratically integrable complex valued functions spaces for a wide family of ultrametric spaces. A general family of pseudodifferential operators, acting on complex valued functions on these ultrametric spaces is introduced. We show that these operators are diagonal in the introduced ultrametric wavelet bases, and compute the corresponding eigenvalues. We introduce the ultrametric change of variable, which maps the ultrametric spaces under consideration onto positive half-line, and use this map to construct non-homogeneous generalizations of wavelet bases., Comment: 19 pages, LaTeX

We constructed the representation of contextual probabilistic dynamics in the complex Hilbert space. Thus dynamics of the wave function can be considered as Hilbert space projections of realistic dynamics in a ``prespace''. The basic condition for representing of the prespace-dynamics is the law of statistical conservation of energy -- conservation of probabilities. Construction of the dynamical representation is an important step in the development of contextual statistical viewpoint to quantum processes. But the contextual statistical model is essentially more general than the quantum one. Therefore in general the Hilbert space projection of the ``prespace'' dynamics can be nonlinear and even irreversible (but it is always unitary). There were found conditions of linearity and reversibility of the Hilbert space dynamical projection. We also found conditions for the conventional Schroedinger dynamics (including time-dependent Hamiltonians). We remark that in general even the Schroedinger dynamics is based just on the statistical conservation of energy; for individual systems the law of conservation of energy can be violated (at least in our theoretical model).

We continue the development of a so called contextual statistical model (here context has the meaning of a complex of physical conditions). It is shown that, besides contexts producing the conventional trigonometric $\cos$-interference, there exist contexts producing the hyperbolic $\cos$-interference. Starting with the corresponding interference formula of total probability we represent such contexts by hyperbolic probabilistic amplitudes or in the abstract formalism by normalized vectors of a hyperbolic analogue of the Hilbert space. There is obtained a hyperbolic Born's rule. Incompatible observables are represented by noncommutative operators., Comment: We discuss possible applications of hyperbolic quantum mechnaics (in particular, in string theory and cosmology as well as cognitive sciences); there is also given a general scheme of experimental verification of hyperbolic quantum mechanics through statistical verification of hyperbolic interference