Abstract:
In our recent work we described conditions under which a multi-parameter random simplicial complex is connected and simply connected. We showed that the Betti numbers of multi-parameter random simplicial complexes in one specific dimension dominate significantly the Betti numbers in all other dimensions. In this paper we focus mainly on the properties of fundamental groups of multi-parameter random simplicial complexes, which can be viewed as a new class of random groups. We describe thresholds for nontrivially and hyperbolicity (in the sense of Gromov) for these groups. Besides, we find domains in the multi-parameter space where these groups have 2-torsion. We also prove that these groups have never odd-prime torsion and their geometric and cohomological dimensions are either 0,1, 2 or infinity. Another result presented in this paper states that aspherical 2-dimensional subcomplexes of random complexes satisfy the Whitehead Conjecture, i.e. all their subcomplexes are also aspherical (with probability tending to one).

Abstract:
In this paper we introduce a new model of random simplicial complexes depending on multiple probability parameters. This model includes the well-known Linial - Meshulam random simplicial complexes and random clique complexes as special cases. Topological and geometric properties of a multi-parameter random simplicial complex depend on the whole combination of the probability parameters and the thresholds for topological properties are convex sets rather than numbers (as in all previously known models). We discuss the containment properties, density domains and dimension of the random simplicial complexes.

Abstract:
In this paper we develop further the multi-parameter model of random simplicial complexes. Firstly, we give an intrinsic characterisation of the multi-parameter probability measure. Secondly, we show that in multi-parameter random simplicial complexes the links of simplexes and their intersections are also multi-parameter random simplicial complexes. Thirdly, we find conditions under which a multi-parameter random simplicial complex is connected and simply connected.

Abstract:
In this paper we state the homological domination principle for random multi-parameter simplicial complexes, claiming that the Betti number in one specific dimension (which is explicitly determined by the probability multi-parameter) significantly dominates the Betti numbers in all other dimensions. We also state and discuss evidence for two interesting conjectures which would imply a stronger version of the homological domination principle, namely that generically homology of a random simplicial complex coincides with that of a wedges of k-dimensional spheres. These two conjectures imply that under an additional assumption (specified in the paper) a random simplicial complex collapses to a k-dimensional complex homotopy equivalent to a wedge of spheres of dimension k.

Abstract:
Photon counts about 15 cm from the left and right sides of the head while subjects sat quietly during baseline conditions within a hyper-dark chamber were measured by photomultiplier units. Lag/lead analyses for photon emissions between the two hemispheres indicated a weak but statistically significant correlation between the amplitude fluctuations that were separated by about 800 to 900 ms. Analyses of the spectral power densities of photon amplitude variations from the left and right hemispheres revealed peak values between 2 and 3 Hz which were equivalent to a difference of about 900 ms. The radiant flux densities were estimated to be in the order of 10^{?12}？W?m^{?2}？and to include the equivalence of about 10^{7}？neurons. Our calculations, which accounted for the small magnitude of the strength of the interhemispheric coefficients, suggest that the coherence could be strongly correlated with processes associated with the unmyelinated axons with diameters between 400 to 800 nm, the visible wavelengths, within the corpus callosum. When the ratio of the phase shift was applied to the Aharanov-Bohm equation, the time required for a photon-related electron to be within a cerebral magnetic field was the same duration as a single orbit of an electron and a photon’s traversal latency across a neuronal plasma membrane. We suggest that the peak photon decoherence between the two cerebral hemispheres may reveal a neuronal-quanta substrate to the conditions associated with consciousness.

Abstract:
We study random 2-dimensional complexes in the Linial - Meshulam model and prove that for the probability parameter satisfying $$p\ll n^{-46/47}$$ a random 2-complex $Y$ contains several pairwise disjoint tetrahedra such that the 2-complex $Z$ obtained by removing any face from each of these tetrahedra is aspherical. Moreover, we prove that the obtained complex $Z$ satisfies the Whitehead conjecture, i.e. any subcomplex $Z'\subset Z$ is aspherical. This implies that $Y$ is homotopy equivalent to a wedge $Z\vee S^2\vee...\vee S^2$ where $Z$ is a 2-dimensional aspherical simplicial complex. We also show that under the assumptions $$c/n

3$ and $0<\epsilon<1/47$, the complex $Z$ is genuinely 2-dimensional and in particular, it has sizable 2-dimensional homology; it follows that in the indicated range of the probability parameter $p$ the cohomological dimension of the fundamental group $\pi_1(Y)$ of a random 2-complex equals 2.

Abstract:
We study random 2-dimensional complexes in the Linial - Meshulam model and find torsion in their fundamental groups at various regimes. We find a simple algorithmically testable criterion for a subcomplex of a random 2-complex to be aspherical; this implies that any aspherical subcomplex of a random 2-complex satisfies the Whitehead conjecture. We use inequalities for Cheeger constants and systoles of simplicial surfaces to analyse spheres and projective planes lying in random 2-complexes. Our proofs exploit the strong hyperbolicity property of random 2-complexes.

Abstract:
the two-dimensional spin-1 baxter-wu model is studied by using monte carlo simulations. the standard single-spin-flip metropolis algorithm is used to generate the configurations from which the order parameter, specific heat and magnetic susceptibility are measured. the finite-size scaling procedure is employed in order to get the critical behavior. extensive simulations show that the critical exponents are different from those of the spin-1/2 model suggesting that the spin-1 model is in a different universality class.

Abstract:
The two-dimensional spin-1 Baxter-Wu model is studied by using Monte Carlo simulations. The standard single-spin-flip Metropolis algorithm is used to generate the configurations from which the order parameter, specific heat and magnetic susceptibility are measured. The finite-size scaling procedure is employed in order to get the critical behavior. The extensive simulations shown that the critical exponents are different from those of the spin-1/2 model suggesting that the spin-1 model is in a different universality class.

Abstract:
the product development process usually encompasses a very complex and interdisciplinary environment in which product is seen by different views related with the life-cycle functions. an approach based on information models can provide an integrated view of the product, supporting also product information and knowledge (i&k) reuse acquired in previous development processes. this paper discusses the use of additional information and knowledge models to support the capture and reuse of i&k within a brake system friction material development environment. two information models are proposed: the brake friction material product model, which captures information about a specific product and the friction material design knowledge model, which captures design and manufacturing information and knowledge history generated throughout the time. for the representation of the information models object oriented technology is used and case based reasoning is proposed for supporting the i&k retrieving. at the present, work is being performed on the structure definition of the information and knowledge models. a real case in a brazilian brake lining manufacturer is being used.