The paper relates to possible connection between spectrum power-law decay and correlation dimension estimation for electroencephalogram (EEG). EEG signals recorded during relaxed wakefulness were analysed. Power-law decay of about 2.28 prevailing over the exponential falling off was established when exponent from the whole EEG spectrum was taken. The correlation dimension was also estimated. Th...

We propose a new method of the analytical computation of the spectral dimension which is based on the equivalence of the random walk and the q-state Potts model with non-zero magnetic field in the limit q → 0. Calculating the critical exponent of the magnetization δ of this model on the dynamically triangulated random surface by means of a matrix model technique we obtain that the spectral dime...

In real systems like EEG the chaos is very difficult to prove or exclude. The scepticism against finite dimension estimates is understandable. It is hard to believe that a complicated system as the brain, which is continually interacting with many other complex systems, should manifest as deterministic low-dimensional dynamics. Predominantly, it is a manifestation of a mixture of noise, some cy...

There were great expectations in the 1980s in connection with the practical applications of mathematical processes which were built mainly upon Fractal Dimension (FD) mathematical basis. Significant results were achieved in the 1990s in practical applications in the fields of information technology, certain image processing areas, data compression, and computer classification. In the present pu...

We calculate the eigenspectrum of random walks on the Eden tree in two and three dimensions. From this, we calculate the spectral dimension ds and the walk dimension dw and test the scaling relation ds = 2df/dw (= 2d/dw for an Eden tree). Finite-size induced crossovers are observed, whereby the system crosses over from a short-time regime where this relation is violated (particularly in two dim...

We show that the spectral dimension ds of two-dimensional quantum gravity coupled to Gaussian fields is two for all values of the central charge c ≤ 1. The same arguments provide a simple proof of the known result ds = 4/3 for branched polymers. email [email protected] email [email protected] email [email protected] email [email protected] email [email protected] 1

We study the maximum possible sign rank of N×N sign matrices with a given VC dimension d. For d = 1, this maximum is 3. For d = 2, this maximum is Θ̃(N1/2). Similar (slightly less accurate) statements hold for d > 2 as well. We discuss the tightness of our methods, and describe connections to combinatorics, communication complexity and learning theory. We also provide explicit examples of matric...

We extend the definition of "spectral dimension" d_{s} (usually defined for fractal and lattice geometries) to theories in spacetimes with anisotropic scaling. We show that in gravity with dynamical critical exponent z in D+1 dimensions, the spectral dimension of spacetime is d_{s}=1+D/z. In the case of gravity in 3+1 dimensions with z=3 in the UV which flows to z=1 in the IR, the spectral dime...

The fractal degree of adsorption on the multi-walled carbon nanotube has been investigated. The fractal-like Langmuir kinetics model has been used to obtain the fractal degree of ion adsorption on multi-walled carbon nanotube. The behavior of the fractal-like kinetics equation was compared with some famous rate equations like Langmuir, pseudo-first-order and pseudo-second-order equations. It is...

where f is a positive even function of the real variable and the parameter Λ fixes the mass scale. The dimension of a noncommutative geometry is not a number but a spectrum, the dimension spectrum (cf. [6]) which is the subset Π of the complex plane C at which the spectral functions have singularities. Under the hypothesis that the dimension spectrum is simple i.e. that the spectral functions h...