Let B be a bicyclic graph of order n, and m_B(1) be the multiplicity of Laplacian eigenvalue 1 in B. In this paper, we first establish a sharp upper bound for the multiplicity of Laplacian eigenvalue 1 of B, i.e., m_B(1)≤ n-4 , and moreover, we characterize the extremal graphs with m_B(1)=n-4 . Then we present the distribution of Laplacian eigenvalue 1 of bicyclic graphs. At last, we characterize all bicyclic graphs B with the second largest multiplicity and the third largest multiplicity of Laplacian eigenvalue 1, i.e., m_B(1)=n-5 and n-6 , respectively.
Let 𝔻 represent the open unit disc in ℂ . Denote by A(𝔻) the disc algebra, and ℬ(X, A(𝔻)) the Banach space of all bounded linear operators from a Banach space X into A(𝔻) . We prove that, under the assumption of equicontinuity at a point in ∂𝔻 , the Bishop–Phelps–Bollobás property holds for ℬ(X, A(𝔻)) .
In this paper, we introduce a new class of positive linear operators, namely the bivariate generalized Szász–Mirakyan–Durrmeyer operators, for approximating functions of two variables defined on ℝ_+^2 . The basic approximation properties of the proposed family of operators are investigated by establishing weighted pointwise convergence, uniform convergence on compact subsets, and quantitative estimates concerning the rate of convergence. Moreover, certain direct approximation results are obtained using the Peetre K-functional and the second-order modulus of smoothness. To study the asymptotic behaviour of these operators, a Voronovskaya-type theorem has been proved. At the end, numerical simulation is presented to illustrate the approximation ability of the proposed operator using test function.
In this note, we consider first passage percolation (FPP) on the two-dimensional integer lattice with i.i.d. passage times distributed according to the Durrett-Liggett class of measures. For this specific type of FPP model, we show that along any direction within a deterministic angular sector (known as the percolation cone), there exist infinitely many infinite geodesics directed along that direction. Furthermore, we prove that a bi-infinite geodesic exists almost surely. Our proof does not require any additional assumptions on the limiting shape.
We consider predictive inference in Linear Mixed Models (LMMs). Specifically, we study the problem of estimating the predictive density under Kullback–Leibler (KL) loss in LMMs with a large number of units. We consider flexible classes of empirical Bayes (EB) predictive densities and develop a novel risk estimation based methodology for selecting hyper-parameters of EB predictive density estimates. Our risk estimation based hyper-parameter selection methodology uses the decision-theoretic identity in Lemma 2 of George et al. [18] that connects predictive KL risk for density estimation to Stein’s unbiased estimate of the quadratic risk in point estimation. Direct construction of unbiased KL risk estimates is not possible in LMMs. We leverage information in the covariates and exchangeability of the individual effects to construct asymptotically efficient estimates of the KL risks for wide classes of predictive density estimators. We derive the rates of convergences of the proposed KL risk estimates (KLRE) and show that EB predictors calibrated by minimizing KLREs are asymptotically optimal in LMMs.
