Sharp Minimum Degree Conditions for the Existence of Disjoint Theta Graphs

In 1963, Corrádi and Hajnal showed that if $G$ is an $n$-vertex graph with $n \ge 3k$ and $\delta(G) \ge 2k$, then $G$ will contain $k$ disjoint cycles; furthermore, this result is best possible, both in terms of the number of vertices as well as the minimum degree. In this paper we focus on an analogue of this result for theta graphs. Results from Kawarabayashi and Chiba et al. showed that if $n = 4k$ and $\delta(G) \ge \lceil \frac{5}{2}k \rceil$, or if $n$ is large with respect to $k$ and $\delta(G) \ge 2k+1$, respectively, then $G$ contains $k$ disjoint theta graphs. While the minimum degree condition in both results are sharp for the number of vertices considered, this leaves a gap in which no sufficient minimum degree condition is known. Our main result in this paper resolves this by showing if $n \ge 4k$ and $\delta(G) \ge \lceil \frac{5}{2}k\rceil$, then $G$ contains $k$ disjoint theta graphs. Furthermore, we show this minimum degree condition is sharp for more than just $n = 4k$, and we discuss how and when the sharp minimum degree condition may transition from $\lceil \frac{5}{2}k\rceil$ to $2k+1$.

W(5): Wobbling Mode in the Framework of the X(5) Model

Using in the Bohr Hamiltonian the approximations leading to the Bohr and Mot- telson description of wobbling motion in even nuclei, a W(5) model for wobbling bands, coexisting with a X(5) ground state band, is obtained. Separation of vari­ ables is achieved by assuming that the relevant potential has a sharp minimum at 70, which is the only parameter entering in the spectra and B(E2) transition rates (up to overall scale factors). B(E2) transition rates exhibit the features expected in the wobbling case.

Dwarf Galaxies as Cosmological Probes

The Lambda Cold Dark Matter (LCDM) paradigm makes specific predictions for the abundance, structure, substructure and clustering of dark matter halos, the sites of galaxy formation. These predictions can be directly tested, in the low-mass halo regime, by dark matter-dominated dwarf galaxies. A number of potential challenges to LCDM have been identified when confronting the expected properties of dwarfs with observation. I review our understanding of a few of these issues, including the “missing satellites” and the “too-big-to-fail” problems, and argue that neither poses an insurmountable challenge to LCDM. Solving these problems requires that most dwarf galaxies inhabit halos of similar mass, and that there is a relatively sharp minimum halo mass threshold to form luminous galaxies. These predictions are eminently falsifiable. In particular, LCDM predicts a large number of “dark” low-mass halos, some of which should have retained enough primordial gas to be detectable in deep 21 cm or Hα surveys. Detecting this predicted population of “mini-halos” would be a major discovery and a resounding success for LCDM on small scales.

Giant Magnetothermopower in Sm0.55Sr0.45MnO3 Manganite

Thermopower α and magnetothermopower ∆α/α were studied in the Sm0.55Sr0.45MnO3 samples, containing clusters of three types: ferromagnetic clusters with the Curie temperature TC = 126 K, A-type antiferromagnetic clusters with the Neel temperature TNA ≥ TC and CE-type antiferromagnetic clusters with the TNCE = 240 K. The curves of temperature dependence of α (T) have a large maximum including TC and TNCE and the sharp minimum on the {∆α/α}(T) curves in the TC-region. Negative magnetothermopower in minimum achieves the giant value ~ 85% in magnetic field 14.17 kOe. It is shown that thermopower is largely caused by the presence of ferromagnetic nanoclusters of ferron-type and to a lesser degree of CE-type antiferromagnetic clusters, in which there is a charge ordering, displacing oxygen ions.

The Influence of Magnetic Inhomogeneous State on Thermopower and Magnetothermopower in Sm0.55Sr0.45MnO3 Manganites

Thermopower α and magnetothermopower α/α were studied in the single-crystal Sm0.55Sr0.45MnO3 samples, containing clusters of three types: ferromagnetic clusters with the Curie temperature TC = 134 K, A-type antiferromagnetic clusters with the Neel temperature TNATC and CE-type antiferromagnetic clusters with the TNCE = 240 K. The curves of temperature dependence of α (T) and {α/α}(T) have extrema in the TNCE-region: large maximum on the first and sharp minimum on the second. Negative magnetothermopower in minimum achieves the giant value 50% in magnetic field 13,2 kOe. It is shown that thermopower is essentially caused by the presence of CE-type antiferromagnetic clusters, in which exists charge order, displacing oxygen ions

Degree Conditions for H-Linked Digraphs

Given a (multi)digraph H, a digraph D is H-linked if every injective function ι:V(H) → V(D) can be extended to an H-subdivision. In this paper, we give sharp degree conditions that ensure a sufficiently large digraph D is H-linked for arbitrary H. The notion of an H-linked digraph extends the classes of m-linked, m-ordered and strongly m-connected digraphs.First, we give sharp minimum semi-degree conditions for H-linkedness, extending results of Kühn and Osthus on m-linked and m-ordered digraphs. It is known that the minimum degree threshold for an undirected graph to be H-linked depends on a partition of the (undirected) graph H into three parts. Here, we show that the corresponding semi-degree threshold for H-linked digraphs depends on a partition of H into as many as nine parts.We also determine sharp Ore–Woodall-type degree-sum conditions ensuring that a digraph D is H-linked for general H. As a corollary, we obtain (previously undetermined) sharp degree-sum conditions for m-linked and m-ordered digraphs.

Mechanical behaviour of polymer mixtures in the phase separation region

The liquid–liquid phase transition triggered by changes in the composition of polymer mixtures in solution or melt is often accompanied by "critical" opalescence, which signals the appearance of a microemulsion in the mixture. The viscosity of the polymer mixture in this region is characterized by a sharp minimum, observed, as a rule, over an extremely narrow range of concentration. Depending on the concentration of the solution, the type of polymer, and the solvent, the viscosity may decrease by a factor of 8–10. On transition from micro- to macro-separation, viscosity rapidly increases back to the original level. Changes in the composition of the mixture can alter the concentration at which phase separation occurs, but the minimum in viscosity invariably corresponds to the moment of phase separation. The critical opalescence region represents the formation of phase particles up to 80–100 nm in size, and this corresponds to the point of viscosity drop. This effect is due to the appearance of thermodynamically stable microemulsions in the polymer mixture, in the region between the binodal and the spinodal in the phase diagram. These emulsions are characterized by lower molecular interaction of incompatible polymers in the highly developed interfacial layer. Extremal changes at the point of phase separation are also observed for other mechanical characteristics of polymer mixtures in solutions or melts, for example, G′ and G″ dynamic moduli or complex viscosity η*. Keywords: polyblends, critical phenomena, viscosity, emulsions, phase separation.