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Register a new user55- and 56-configurations are reducible
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Jin Xu
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Let $G$ be a 4-chromatic maximal planar graph (MPG) with the minimum degree of at least 4 and let $C$ be an even-length cycle of $G$.If $|f(C)|=2$ for every $f$ in some Kempe equivalence class of $G$, then we call $C$ an unchanged bichromatic cycle (UBC) of $G$, and correspondingly $G$ an unchanged bichromatic cycle maximal planar graph (UBCMPG) with respect to $C$, where $f(C)={f(v)| vin V(C)}$. For an UBCMPG $G$ with respect to an UBC $C$, the subgraph of $G$ induced by the set of edges belonging to $C$ and its interior (or exterior), denoted by $G^C$, is called a base-module of $G$; in particular, when the length of $C$ is equal to four, we use $C_4$ instead of $C$ and call $G^{C_4}$ a 4-base-module. In this paper, we first study the properties of UBCMPGs and show that every 4-base-module $G^{C_4}$ contains a 4-coloring under which $C_4$ is bichromatic and there are at least two bichromatic paths with different colors between one pair of diagonal vertices of $C_4$ (these paths are called module-paths). We further prove that every 4-base-module $G^{C_4}$ contains a 4-coloring (called decycle coloring) for which the ends of a module-path are colored by distinct colors. Finally, based on the technique of the contracting and extending operations of MPGs, we prove that 55-configurations and 56-configurations are reducible by converting the reducibility problem of these two classes of configurations into the decycle coloring problem of 4-base-modules.
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We present a high-resolution in-beam $gamma$-ray spectroscopy study of excited states in the mirror nuclei $^{55}$Co and $^{55}$Ni following one-nucleon knockout from a projectile beam of $^{56}$Ni. The newly determined partial cross sections and the $gamma$-decay properties of excited states provide a test of state-of-the-art nuclear structure models and probe mirror symmetry in unique ways. A mirror asymmetry for the partial cross sections leading to the two lowest $3/2^-$ states in the $A = 55$ mirror pair was identified as well as a significant difference in the $E1$ decays from the $1/2^+_1$ state to the same two $3/2^-$ states. The mirror asymmetry in the partial cross sections cannot be reconciled with the present shell-model picture or small mixing introduced in a two-state model. The observed mirror asymmetry in the $E1$ decay pattern, however, points at stronger mixing between the two lowest $3/2^-$ states in $^{55}$Co than in its mirror $^{55}$Ni.
A permutation group is said to be quasiregular if every its transitive constituent is regular, and a quasiregular coherent configuration can be thought as a combinatorial analog of such a group: the transitive constituents are replaced by the homogeneous components. In this paper, we are interested in the question when the configuration is schurian, i.e., formed by the orbitals of a permutation group, or/and separable, i.e., uniquely determined by the intersection numbers. In these terms, an old result of Hanna Neumann is, in a sense, dual to the statement that the quasiregular coherent configurations with cyclic homogeneous components are schurian. In the present paper, we (a) establish the duality in a precise form and (b) generalize the latter result by proving that a quasiregular coherent configuration is schurian and separable if the groups associated with homogeneous components have distributive lattices of normal subgroups.
A matrix is emph{simple} if it is a (0,1)-matrix and there are no repeated columns. Given a (0,1)-matrix $F$, we say a matrix $A$ has $F$ as a emph{configuration}, denoted $Fprec A$, if there is a submatrix of $A$ which is a row and column permutation of $F$. Let $|A|$ denote the number of columns of $A$. Let $mathcal{F}$ be a family of matrices. We define the extremal function $text{forb}(m, mathcal{F}) = max{|A|colon A text{ is an }m-text{rowed simple matrix and has no configuration } Finmathcal{F}}$. We consider pairs $mathcal{F}={F_1,F_2}$ such that $F_1$ and $F_2$ have no common extremal construction and derive that individually each $text{forb}(m, F_i)$ has greater asymptotic growth than $text{forb}(m, mathcal{F})$, extending research started by Anstee and Koch.
A configuration of particles confined to a sphere is balanced if it is in equilibrium under all force laws (that act between pairs of points with strength given by a fixed function of distance). It is straightforward to show that every sufficiently symmetrical configuration is balanced, but the converse is far from obvious. In 1957 Leech completely classified the balanced configurations in R^3, and his classification is equivalent to the converse for R^3. In this paper we disprove the converse in high dimensions. We construct several counterexamples, including one with trivial symmetry group.
Recent works have indicated that the $^{56}$Ni masses estimated for Stripped Envelope SNe (SESNe) are systematically higher than those estimated for SNe II. Although this may suggest a distinct progenitor structure between these types of SNe, the possibility remains that this may be caused by observational bias. One important possible bias is that SESNe with low $^{56}$Ni mass are dim, and therefore they are more likely to escape detection. By investigating the distributions of the $^{56}$Ni mass and distance for the samples collected from the literature, we find that the current literature SESN sample indeed suffers from a significant observational bias, i.e., objects with low $^{56}$Ni mass - if they exist - will be missed, especially at larger distances. Note, however, that those distant objects in our sample are mostly SNe Ic-BL. We also conducted mock observations assuming that the $^{56}$Ni mass distribution for SESNe is intrinsically the same with that for SNe II. We find that the $^{56}$Ni mass distribution of the detected SESNe samples moves toward higher mass than the assumed intrinsic distribution, because of the difficulty in detecting the low-$^{56}$Ni mass SESNe. These results could explain the general trend of the higher $^{56}$Ni mass distribution (than SNe II) of SESNe found thus far in the literature. However, further finding clear examples of low-$^{56}$Ni mass SESNe ($leq 0.01M_{odot}$) is required to add weight to this hypothesis. Also, the objects with high $^{56}$Ni mass ($gtrsim 0.2 M_{odot}$) are not explained by our model, which may require an additional explanation.
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