Homometric sets in trees

Let $G = (V,E)$ denote a simple graph with the vertex set $V$ and the edge set $E$. The profile of a vertex set $V'\subseteq V$ denotes the multiset of pairwise distances between the vertices of $V'$. Two disjoint subsets of $V$ are \emph{homometric}, if their profiles are the same. If $G$ is a tree on $n$ vertices we prove that its vertex sets contains a pair of disjoint homometric subsets of size at least $\sqrt{n/2} - 1$. Previously it was known that such a pair of size at least roughly $n^{1/3}$ exists. We get a better result in case of haircomb trees, in which we are able to find a pair of disjoint homometric sets of size at least $cn^{2/3}$ for a constant $c > 0$

Impurity effects on optical response in a finite band electronic system coupled to phonons

The concepts, which have traditionally been useful in understanding the effects of the electron--phonon interaction in optical spectroscopy, are based on insights obtained within the infinite electronic band approximation and no longer apply in finite band metals. Impurity and phonon contributions to electron scattering are not additive and the apparent strength of the coupling to the phonon degrees of freedom is substantially reduced with increased elastic scattering. The optical mass renormalization changes sign with increasing frequency and the optical scattering rate never reaches its high frequency quasiparticle value which itself is also reduced below its infinite band value

The new path of law : from theory of chaos to theory of law

From chaos to chaos theory, from the primordial perception of the world as disorderly to the scientific research of disorder a long distance has been covered. This path implies openness of mind and scientific boldness which connect mythological perceptions of the world with philosophical and scientific interpretations of phenomena throughout the world in a quite distinctive way resting on the creation of a model and application of computing. Owing to this, for the first time instead of asking What awaits us in the future? we can ask What can be done in the future? and get a reliable scientific answer to the question

Associative nature of event participation dynamics: a network theory approach

The affiliation with various social groups can be a critical factor when it comes to quality of life of each individual, making such groups an essential element of every society. The group dynamics, longevity and effectiveness strongly depend on group's ability to attract new members and keep them engaged in group activities. It was shown that high heterogeneity of scientist's engagement in conference activities of the specific scientific community depends on the balance between the numbers of previous attendances and non-attendances and is directly related to scientist's association with that community. Here we show that the same holds for leisure groups of the Meetup website and further quantify individual members' association with the group. We examine how structure of personal social networks is evolving with the event attendance. Our results show that member's increasing engagement in the group activities is primarily associated with the strengthening of already existing ties and increase in the bonding social capital. We also show that Meetup social networks mostly grow trough big events, while small events contribute to the groups cohesiveness.Comment: 16 pages, 6 figs + Supporting information 7 pages, 8 fig

Effect of disorder on the far-infrared conductivity and on the microwave conductivity of two-band superconductors

We consider the far-infrared and the microwave conductivities of a two-band superconductor with non-magnetic impurities. The strong coupling expressions for the frequency and temperature dependent conductivity of a two-band superconductor are developed assuming isotropic bands and interactions. Our numerical results obtained using realistic interaction parameters for MgB$_{2}$ are compared with experiments on this compound. We find that the available experimental results for the far-infrared conductivity of MgB$_{2}$ are consistent with multi-band superconductivity in the presence of a sufficiently strong interband impurity scattering. On the other hand, our numerical results for the microwave conductivity in the superconducting state indicate that the experimental results obtained on samples with the highest transition temperature $T_{c}$ are consistent with a low interband impurity scattering rate but depend sensitively on the ratio of the total scattering rates in the two bands. For the $\pi$-band scattering rate $\gamma_{\pi}$ not greater than the $\sigma$-band scattering rate $\gamma_{\sigma}$ there is a single, broad, low-temperature (at about 0.5$T_{c}$) coherence peak in the microwave conductivity. For $\gamma_{\pi}/\gamma_{\sigma}$=4--7 a high-temperature (at about 0.9$T_{c}$) coherence peak is dominant, but there is also a low-temperature peak/shoulder resulting from the contribution of the $\pi$-band carriers to the microwave conductivity. For $\gamma_{\pi}/\gamma_{\sigma}\gg$1 only the high-temperature coherence peak should be observable.Comment: 11 pages, 6 figure

Sodium channel inactivation kinetics of rat sensory and motor nerve fibres and their modulation by glutathione

Na+ channel currents of rat motor and sensory nerve fibres were studied with the patch-clamp technique on enzymatically demyelinated axons. Differences between motor and sensory fibres in multi-channel inactivation kinetics and the gating of late single-channel currents were investigated. In the axon-attached mode, inactivation of multi-channel Na+ currents in sensory axons was best fitted with a single time constant while for motor axons two time constants were needed. Late single-channel currents in sensory axons were characterized by short openings whereas motor axons exhibited additional long single-channel openings. In contrast, in excised, inside-out membrane patches, no differences between motor and sensory fibres were found: in both types of fibre inactivation of multi-channel Na+ currents proceeded with two time constants and late single-channel currents showed short and long openings. After application of the reducing agent glutathione to the cytoplasmic side of excised inside-out patches, inactivation of Na+ currents in both motor and sensory fibres proceeded with a single, fast exponential time constant and late currents appeared with short openings only. These data indicate that the axonal metabolism may contribute to the different inactivation kinetics of Na+ currents in motor and sensory nerve fibres