Essential Equivariant Maps and Borsuk-Ulam Theorems

A full characterization is given of those compact Lie groupsG with the property that every G-map XX on a finite-dimensionalG-complex X of finite orbit type, X^G = Ø, is (non-equivariantly)essential. For arbitrary G, conditions are given on the G-spaceX which guarantee this property. Finally, conditions are givenfor the non-existence of a G-map XY inducing a homotopy equivalenceX^GY^G on the fixed point sets. These results have applicationsto critical point theory of almost G-invariant functionals.     

Explosive condensation in a mass transport model

We study a far-from-equilibrium system of interacting particles, hopping between sites of a 1D lattice with a rate which increases with the number of particles at interacting sites. We find that clusters of particles, which initially spontaneously form in the system, begin to move at increasing speed as they gain particles. Ultimately, they produce a moving condensate which comprises a finite fraction of the mass in the system. We show that, in contrast with previously studied models of condensation, the relaxation time to steady state decreases as an inverse power of lnL with system size L and that condensation is instantaneous for L→∞.     

Kinetics of H→P cross-polarization in human trabecular bone

Bloch-decay and cross-polarization (CP) ³¹P nuclear magnetic resonance (NMR) spectra of healthy human trabecular bone were acquired under magic-angle spinning (MAS) at 3 kHz. A single peak at 3.1 ppm was detected. Variable-contact time ¹H → ³¹P CP experiments revealed three signal components growing at various rates. The fast, moderate and slow components were assigned and assessed in P atom % to proton-rich (24%), hydroxyapatite (58%) and proton-deficient (18%) phosphate domains, respectively. Examination of CP kinetics is useful for the chemical characterization of bone tissue.     

Phosphorus-31 spin-lattice NMR relaxation in bone apatite and its mineral standards

Phosphorus-31 spin-lattice relaxation, both in the laboratory (B(0)=4.7 T) and rotating frame (B(1)=2.2 mT), was studied in the following samples: mineral of whole human bone (samples B1-B6), apatite prepared from bone (BHA), natural brushite (BRU), synthetic hydroxyapatite hydrated (HAh) and calcined (HAc), and synthetic carbonatoapatite of type B (CHA-B) with 9 wt% of CO(3)(2-). The T(1)(P) relaxation time was determined directly using the saturation recovery technique, while the T(1 rho)(P) relaxation time was measured via (1)H-->(31)P CP by incrementing the (31)P spin-lock. In order to avoid an effect of magic-angle spinning (MAS) on CP and relaxation, the experiments were carried out on static samples. The (31)P spin-lattice relaxation was discussed for trabecular and cortical bone tissue from adult subjects in comparison to the synthetic mineral standards. None of the reference materials has matched accurately the relaxation behaviour of the bone mineral. The most striking differences between the examined substances were observed for T(1)(P), which for human bone was sample dependent and appeared in the range 55-100 s, while for HAh, HAc, and CHA-B was 7.2, 10.0, and 25.8 s, respectively. Possible reasons of so large relaxation diversity were discussed. It has been suggested that T(1)(P) of apatites is to some extent dependent on the concentration of the structural hydroxyl groups, and this in turn is controlled by the material crystallinity. It was also found that T(1)(P) decreased on hydration by ca. 30%. For T(1rho)(P), both its magnitude and dependence on the CP contact time gave useful structural information. The dehydrated samples (HAc and BHA) had long T(1 rho)(P) over 250 ms. Those, which contained water, either structural (BRU) or adsorbed on the crystal surface (HAh, CHA-B, and B1-B6), had shorter T(1 rho)(P) below 120 ms. It was concluded that the effect of water on T(1 rho)(P) is much more pronounced than on T(1)(P). The interpretation has involved P-OH groups and adsorbed water, which cover the apatite crystal surface.     

Balls-in-boxes condensation on networks

We discuss two different regimes of condensate formation in zero-range processes on networks: on a q-regular network, where the condensate is formed as a result of a spontaneous symmetry breaking, and on an irregular network, where the symmetry of the partition function is explicitly broken. In the latter case we consider a minimal irregularity of the q-regular network introduced by a single Q node with degree Q>q. The statics and dynamics of the condensation depend on the parameter alpha=ln Q/q, which controls the exponential falloff of the distribution of particles on regular nodes and the typical time scale for melting of the condensate on the Q node, which increases exponentially with the system size N. This behavior is different than that on a q-regular network, where alpha=0 and where the condensation results from the spontaneous symmetry breaking of the partition function, which is invariant under a permutation of particle occupation numbers on the q nodes of the network. In this case the typical time scale for condensate melting is known to increase typically as a power of the system size.     

Ionic conductivity of electrolytes based on star-branched poly(ethylene oxide) with high concentration of lithium salts

Electrolytes based on star-branched poly(ethylene oxide) with lithium bis(trifluoromethanesulfone)imide LiTFSI and lithium iodide salts were prepared by casting from solution. The electrical properties of electrolytes subjected to various heating and cooling runs were studied by impedance spectroscopy and impedance spectroscopy simultaneous with optical microscope observation. Differential scanning calorimetry was used for additional characterization. The results indicate that in electrolytes with high content of salt, values of ionic conductivity comparable to that of dilute electrolytes can be achieved. Moreover, electrolytes with high amount of salt seem to show weaker temperature dependence of conductivity. Promising results in terms of ionic conductivity were obtained for mixture of LiTFSI and lithium iodide. A few problems which may decrease the performance of studied system as a solid electrolyte were also identified, from which changes of physical properties of samples subjected to thermal cycles and aging seem to be the most important ones.     

Periodic solutions of symmetric autonomous Newtonian systems

In this paper we show the existence and bifurcation of T-periodic solutions of a special form for an autonomous Newtonian system with symmetry. If the phase-space R2n is equipped with the structure of an orthogonal representation (W,ρW) and the potential is invariant, then for every such a solution the set of indices of nonvanishing Fourier coefficients is finite and depends on W only. If the potential V depends on the squares of complex coordinates, then for every such a solution T is the minimal period.     

Power laws in zero-range processes on random networks

We study statistical properties of a zero-range process (ZRP) on random networks.We derive an analytic expression for the distribution of particles (also called node occupation distribution)in the steady state of the ZRP in the ensemble of uncorrelated random graphs. We analyze the dependence of this distribution on the node-degree distribution.In particular, we show that when the degree distribution is tuned properly, one can obtainscale-free fluctuations in the distribution of particles.Such fluctuations lead to a power law in the distribution of particles, just like in the ZRP with the hopping rate u(m) = 1+b/mon homogeneous graphs.