We characterize the bipartite graphs that minimize the (first degree-based) entropy among all bipartite graphs of given size. For bipartite graphs given size and (upper bound on the) order, we give a ...lower bound for this entropy. The extremal graphs turn out to be complete bipartite graphs, or nearly complete bipartite. Here we make use of an equivalent representation of bipartite graphs by means of Young diagrams, which make it easier to compare the entropy of related graphs. We conclude that the general characterization of the extremal graphs is a difficult problem, due to its connections with number theory. However, it is easier to identify them for particular values of the order n and size m because we have narrowed down the possible extremal graphs. We indicate that some of our ideas extend to other degree-based topological indices as well.
•Characterize the bipartite graphs that minimize the first-degree based entropy given the size or both the size and the order.•Prove that the extremal graphs turn out to be complete bipartite graphs, or nearly complete bipartite.•Make use of an equivalent representation of bipartite graphs by means of Young diagrams.•Extend our ideas to some degree-based topological indices.
We prove that Schubert varieties in potentially different Grassmannians are isomorphic as varieties if and only if their corresponding Young diagrams are identical up to a transposition. We also ...discuss a generalization of this result to Grassmannian Richardson varieties. In particular, we prove that Richardson varieties in potentially different Grassmannians are isomorphic as varieties if their corresponding skew diagrams are semi-isomorphic as posets, and we conjecture the converse. Here, two posets are said to be semi-isomorphic if there is a bijection between their sets of connected components such that the corresponding components are either isomorphic or opposite.
We consider two natural statistics on pairs of histograms, in which the n bins have weights 0,…,n−1. The signed difference (D) between the weighted totals of the histograms is, in a sense, refined by ...the earth mover's distance (EMD), which measures the amount of work required to equalize the histograms. We were recently surprised, however, by how little the EMD actually does refine D in certain real-world applications, which led to the main problem in this paper: what is the probability that EMD=|D|? We derive a formula for this probability, as well as the expected value of |D|, via the combinatorics of Young diagrams and plane partitions. We then generalize our results to an arbitrary number of histograms, where we realize the higher-dimensional analogue |D| as distance on the Type-A root lattice.
Within the framework of the modified potential cluster model with a classification of orbital states according to Young diagrams, the possibility of describing experimental data for total cross ...sections of the neutron radiative capture on 11B to the ground state of 12B at energies of 10 meV (1 meV = 10−3 eV) to 7 MeV was considered. It was shown that, taking into account only the E1 transition from the S state of the n11B scattering to the ground state of 12B, it is quite possible to explain the magnitude of the known experimental cross sections at energies of 25.3 meV to 70 keV. Furthermore, on the basis of the total cross sections of 10 meV to 7 MeV, but excluding resonances above 5 MeV, the reaction rate is calculated in the temperature range of 0.01 to 10.0 T9. It is shown that the inclusion of low-lying resonance states makes a significant contribution to the reaction rate, starting already with temperatures of 0.2–0.3 T9.
In their study of Jack polynomials, Nazarov-Sklyanin introduced a remarkable new graded linear operator ${\mathcal L} \colon Fw \rightarrow Fw$ where $F$ is the ring of symmetric functions and $w$ is ...a variable. In this paper, we (1) establish a cyclic decomposition $Fw \cong \bigoplus_{\lambda} Z(j_{\lambda}, {\mathcal L})$ into finite-dimensional ${\mathcal L}$-cyclic subspaces in which Jack polynomials $j_{\lambda}$ may be taken as cyclic vectors and (2) prove that the restriction of ${\mathcal L}$ to each $Z(j_{\lambda}, {\mathcal L})$ has simple spectrum given by the anisotropic contents $s$ of the addable corners $s$ of the Young diagram of $\lambda$. Our proofs of (1) and (2) rely on the commutativity and spectral theorem for the integrable hierarchy associated to ${\mathcal L}$, both established by Nazarov-Sklyanin. Finally, we conjecture that the ${\mathcal L}$-eigenfunctions $\psi_{\lambda}^s {\in Fw}$ {with eigenvalue $s$ and constant term} $\psi_{\lambda}^s|_{w=0} = j_{\lambda}$ are polynomials in the rescaled power sum basis $V_{\mu} w^l$ of $Fw$ with integer coefficients.
Self-conjugate core partitions, bar-core partitions, core shifted Young diagrams (or CSYDs), and doubled distinct core partitions have been studied independently. In this paper, we show that doubled ...distinct cores and CSYDs are special cases of bar-core partitions. From this fact together with the abacus construction, we find path interpretations of bar-cores, CSYDs, and doubled distinct cores on (s,t)-cores and (s,s+d,s+2d)-cores, respectively. As a result, we get the number of these core partitions.
A family of rational solutions of the Kadomtsev–Petviashvili I equation with
N
distinct peaks as
|
t
|
→
∞
, is characterized in terms of the partitions of a positive integer
N
. This new approach ...leads to a complete classification of these
N
-lump solutions whose properties including the asymptotic location of the peaks are investigated using the Schur function associated with a given partition of
N
. Relationship between the geometric structures of the
N
-lump wave pattern and the Young diagram of the associated integer partition is explored.
Macdonald processes are certain probability measures on two-dimensional arrays of interlacing particles introduced by Borodin and Corwin in 7. They are defined in terms of nonnegative specializations ...of the Macdonald symmetric functions and depend on two parameters q,t∈0;1). Our main result is a classification of continuous time, nearest neighbor Markov dynamics on the space of interlacing arrays that act nicely on Macdonald processes.
The classification unites known examples of such dynamics and also yields many new ones.
When t=0, one dynamics leads to a new integrable interacting particle system on the one-dimensional lattice, which is a q-deformation of the PushTASEP (= long-range TASEP).
When q=t, the Macdonald processes become the Schur processes of Okounkov and Reshetikhin 41. In this degeneration, we discover new Robinson–Schensted-type correspondences between words and pairs of Young tableaux that govern some of our dynamics.
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