In this work, a co-variational inequality problem and a co-resolvent equation problem are introduced and investigated. It is shown that the fixed point problem is equivalent to the problem of co-variational inequality. The co-variational inequality problem and the co-resolvent equation problem are equivalent due to this resemblance. The coresolvent equation problem is solved using an iterative approach that we provide at the final stage. Our findings may be seen as an enhancement of several established findings.
Complete non-ambiguous trees (CNATs) are combinatorial objects which appear in various contexts.Recently, Chen and Ohlig studied the notion of permutations associated to these objects, and proposed a series of nice conjectures.Most of them were proved by Selig and Zhu, through a connection with the abelian sandpile model.But one conjecture remained open, about the distribution of a natural statistic named determinant.We prove this conjecture, in a bijective way.
Typhoid fever and cholera remain a huge public health problem on the African continent due to deteriorating infrastructure and declining funding for infrastructure development. The diseases are both caused by bacteria, and they are associated with poor hygiene and waste disposal systems. In this paper, we consider a nonlinear system of ordinary differential equations for the co-infection of typhoid and cholera in a homogeneously mixing population. The model's steady states are determined and analyzed in terms of the model's reproduction number. Impact analysis—how the diseases impact on each other—is carried out. Numerical simulations and sensitivity analysis are also given. The results show that the control of the diseases should be carried out in tandem for the greatest impact of disease control. The results have important implications in the management of the two diseases.
Kenyon and Pemantle (2014) gave a formula for the entries of a square matrix in terms of connected principal and almost-principal minors. Each entry is an explicit Laurent polynomial whose terms are the weights of domino tilings of a half Aztec diamond. They conjectured an analogue of this parametrization for symmetric matrices, where the Laurent monomials are indexed by Catalan paths. In this paper we prove the Kenyon-Pemantle conjecture, and apply this to a statistics problem pioneered by Joe (2006). Correlation matrices are represented by an explicit bijection from the cube to the elliptope.
Enlighted by the recent work of Hao Huang on sensitivity conjecture [arXiv:1907.00847], we propose a new definition of the dimension of graphs and establish a relationship between the chromatic number and the dimension.
Various authors have studied a natural operation (under various names) on the order ideals (equivalently antichains) of a finite poset, here called \emphrowmotion. For certain posets of interest, the order of this map is much smaller than one would naively expect, and the orbits exhibit unexpected properties. In very recent work (inspired by discussions with Berenstein) Einstein and Propp describe how rowmotion can be generalized: first to the piecewise-linear setting of order polytopes, then via detropicalization to the birational setting. In the latter setting, it is no longer \empha priori clear even that birational rowmotion has finite order, and for many posets the order is infinite. However, we are able to show that birational rowmotion has the same order, p+q, for the poset P=[p]×[q] (product of two chains), as ordinary rowmotion. We also show that birational (hence ordinary) rowmotion has finite order for some other classes of posets, e.g., the upper, lower, right and left halves of the poset above, and trees having all leaves on the same level. Our methods are based on those used by Volkov to resolve the type AA (rectangular) Zamolodchikov Periodicity Conjecture.
Using a recursive approach, we obtain a simple exact expression for the $L^2$-distance from the limit in the classical limit theorem of Régnier (1989) for the number of key comparisons required by $\texttt{QuickSort}$. A previous study by Fill and Janson (2002) using a similar approach found that the $d_2$-distance is of order between $n^{-1} \log{n}$ and $n^{-1/2}$, and another by Neininger and Ruschendorf (2002) found that the Zolotarev $\zeta _3$-distance is of exact order $n^{-1} \log{n}$. Our expression reveals that the $L^2$-distance is asymptotically equivalent to $(2 n^{-1} \ln{n})^{1/2}$.