Spatial parity violation and the turbulent magnetic Prandtl number

Using the field theory renormalization-group technique in the two-loop approximation, we study the influence of helicity (spatial parity violation) on the turbulent magnetic Prandtl number in the model of kinematic magnetohydrodynamic turbulence, where the magnetic field behaves as a passive vector quantity advected by the helical turbulent environment given by the stochastic Navier-Stokes equation. We show that the presence of helicity decreases the value of the turbulent magnetic Prandtl number and that the two-loop helical contribution to the turbulent magnetic Prandtl number is up to 4.2% of its nonhelical value. This result demonstrates the strong stability of the properties of diffusion processes of the magnetic field in turbulent environments with spatial parity violation compared with the corresponding systems without the helicity.     

The principle of maximum randomness in the theory of fully developed turbulence. II. Isotropic decaying turbulence

A statistical model for describing the decay of developed isotropic turbulence of an incompressible fluid is proposed. The model uses the distribution function of the velocity pulsations introduced earlier by the authors on the basis of the principle of maximum randomness of the velocity field for a given spectral energy flux. The renormalization-group technique and expansion are used to calculate the correlation functions of the velocity that occur in the equation of spectral energy balance. This leads to a closed equation for the dependence of the energy spectrum on the integral turbulence scaler _c(t). In the inertial interval, this equation gives the Kolmogorov asymptotic spectrum, while for the time dependence ofr _c(t) and the pulsation energye(t) it predicts the power lawsr _c(t)t^2/5 andr(t)t ^–6/5.Physics Research Institute of the St Petersburg University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 96, No. 1, pp. 150–159, July, 1993.     

A new type of gravitational interaction with parity violation and its role in gravidynamics and electrodynamics

Lagrangian densities for spinor particles and photons in a gravitational field, including terms with C- and P-parity violations that remove degeneracy in spin states, are postulated. The modified Dirac and Maxwell equations are obtained. Hamiltonians for particles (bodies) possessing quantum (classical in the case of bodies) angular momentum are constructed in the semiclassical approximation. It is shown that the new interactions can produce the following effects: variation of body weight because of the partial spin ordering of electrons and nucleons, angular splitting and the relative retardation of left- and right-handed polarized waves passing near the Sun, modification of atomic spectra and of the atomic shell structure, and generation of high-power X-rays in the vicinity of neutron stars and of isotropically distributed high-energy cosmic rays in the early Universe.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 105, No. 2, pp. 324–340, November, 1995.     

Non-cupping and randomness

Let be Martin-Löf-random. Then there is a promptly simple set such that for each Martin-Löf-random set , . When , one obtains a c.e. non-computable set which is not weakly Martin-Löf cuppable. That is, for any Martin-Löf-random set , if , then .

     

Lowness properties and randomness

The set A is low for (Martin-Löf) randomness if each random set is already random relative to A. A is K-trivial if the prefix complexity K of each initial segment of A is minimal, namely . We show that these classes coincide. This answers a question of Ambos-Spies and Ku?era in: P. Cholak, S. Lempp, M. Lerman, R. Shore, (Eds.), Computability Theory and Its Applications: Current Trends and Open Problems, American Mathematical Society, Providence, RI, 2000: each low for Martin-Löf random set is . Our class induces a natural intermediate ideal in the r.e. Turing degrees, which generates the whole class under downward closure.Answering a further question in P. Cholak, S. Lempp, M. Lerman, R. Shore, (Eds.), Computability Theory and Its Applications: Current Trends and Open Problems, American Mathematical Society, Providence, RI, 2000, we prove that each low for computably random set is computable.     

Truth‐table Schnorr randomness and truth‐table reducible randomness

Schnorr randomness and computable randomness are natural concepts of random sequences. However van Lambalgen’s Theorem fails for both randomnesses. In this paper we define truth‐table Schnorr randomness (defined in 6 too only by martingales) and truth‐table reducible randomness, for which we prove that van Lambalgen's Theorem holds. We also show that the classes of truth‐table Schnorr random reals relative to a high set contain reals Turing equivalent to the high set. It follows that each high Schnorr random real is half of a real for which van Lambalgen's Theorem fails. Moreover we establish the coincidence between triviality and lowness notions for truth‐table Schnorr randomness. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Expected rank and randomness in rooted graphs

For a rooted graph G, let EV_b(G;p) be the expected number of vertices reachable from the root when each edge has an independent probability p of operating successfully. We determine the expected value of EV_b(G;p) for random trees, and include a connection to unrooted trees. We also consider rooted digraphs, computing the expected value of a random orientation of a rooted graph G in terms of EV_b(G;p). We consider optimal location of the root vertex for the class of grid graphs, and we also briefly discuss a polynomial that incorporates vertex failure.     

The perception of randomness

Psychologists have studied people's intuitive notions of randomness by two kinds of tasks: judgment tasks (e.g., “is this series like a coin?” or “which of these series is most like a coin?”), and production tasks (e.g., “produce a series like a coin”). People's notion of randomness is biased in that they see clumps or streaks in truly random series and expect more alternation, or shorter runs, than are there. Similarly, they produce series with higher than expected alternation rates. Production tasks are subject to other biases as well, resulting from various functional limitations. The subjectively ideal random sequence obeys “local representativeness”; namely, in short segments of it, it represents both the relative frequencies (e.g., for a coin, 50%–50%) and the irregularity (avoidance of runs and other patterns). The extent to which this bias is a handicap in the real world is addressed.     

Lowness of higher randomness notions

We study randomness notions given by higher recursion theory, establishing the relationships Π ₁ ¹ -randomness ? Π ₁ ¹ -Martin-Löf randomness ? Δ ₁ ¹ -randomness = Δ ₁ ¹ -Martin-Löf randomness. We characterize the set of reals that are low for Δ ₁ ¹ randomness as precisely those that are Δ ₁ ¹ -traceable. We prove that there is a perfect set of such reals.     

Basic randomness of nature and ether-drift experiments

We re-consider the idea that quantum fluctuations might reflect the existence of an ‘objective randomness’, i.e. a basic property of the vacuum state which is independent of any experimental accuracy of the observations or limited knowledge of initial conditions. Besides being responsible for the observed quantum behavior, this might introduce a weak, residual form of ‘noise’ which is intrinsic to natural phenomena and could be important for the emergence of complexity at higher physical levels. By adopting Stochastic Electro Dynamics as a heuristic model, we are driven to a picture of the vacuum as a form of highly turbulent ether, which is deep-rooted into the basic foundational aspects of both quantum physics and relativity, and to search for experimental tests of this scenario. An analysis of the most precise ether-drift experiments, operating both at room temperature and in the cryogenic regime, shows that, at present, there is some ambiguity in the interpretation of the data. In fact the average amplitude of the signal has precisely the magnitude expected, in a ‘Lorentzian’ form of relativity, from an underlying stochastic ether and, as such, might not be a spurious instrumental effect. This puzzle, however, should be solved in a next future with the use of new cryogenically cooled optical resonators whose stability should improve by about two orders of magnitude. In these new experimental conditions, the persistence of the present amplitude would represent a clean evidence for the type of random vacuum we are envisaging.     

Algorithmic randomness of continuous functions

We investigate notions of randomness in the space ${{\mathcal C}(2^{\mathbb N})}We investigate notions of randomness in the space of continuous functions on . A probability measure is given and a version of the Martin-L?f test for randomness is defined. Random continuous functions exist, but no computable function can be random and no random function can map a computable real to a computable real. The image of a random continuous function is always a perfect set and hence uncountable. For any , there exists a random continuous function F with y in the image of F. Thus the image of a random continuous function need not be a random closed set. The set of zeroes of a random continuous function is always a random closed set. Research partially supported by the National Science Foundation grants DMS 0532644 and 0554841 and 00652732. Thanks also to the American Institute of Mathematics for support during 2006 Effective Randomness Workshop; Remmel partially supported by NSF grant 0400307; Weber partially supported by NSF grant 0652326. Preliminary version published in the Third International Conference on Computability and Complexity in Analysis, Springer Electronic Notes in Computer Science, 2006.