Some properties of projectors associated with the WLSE under a general linear model

Projectors associated with a particular estimator in a general linear model play an important role in characterizing statistical properties of the estimator. A variety of new properties were derived on projectors associated with the weighted least-squares estimator (WLSE). These properties include maximal and minimal possible ranks, rank invariance, uniqueness, idempotency, and other equalities involving the projectors. Applications of these properties were also suggested. Proofs of the main theorems demonstrate how to use the matrix rank method for deriving various equalities involving the projectors under the general linear model.     

Weighted least-squares estimators of parametric functions of the regression coefficients under a general linear model

The weighted least-squares estimator of parametric functions K β under a general linear regression model { y, X b, s²S }{\{ {\bf y},\,{\bf X \beta}, \sigma^2{\bf \Sigma} \}} is defined to be K[^(b)]{{\bf K}{\hat{\bf {\beta}}}}, where [^(b)]{\hat{{\bf \beta}}} is a vector that minimizes (y − X β)′V(y − X β) for a given nonnegative definite weight matrix V. In this paper, we study some algebraic and statistical properties of K[^(b)]{{\bf K}\hat{{\bf \beta}}} and the projection matrix associated with the estimator, such as, their ranks, unbiasedness, uniqueness, as well as equalities satisfied by the projection matrices.     

Equalities and inequalities for inertias of hermitian matrices with applications

The inertia of a Hermitian matrix is defined to be a triplet composed of the numbers of the positive, negative and zero eigenvalues of the matrix counted with multiplicities, respectively. In this paper, we show some basic formulas for inertias of 2×2 block Hermitian matrices. From these formulas, we derive various equalities and inequalities for inertias of sums, parallel sums, products of Hermitian matrices, submatrices in block Hermitian matrices, differences of outer inverses of Hermitian matrices. As applications, we derive the extremal inertias of the linear matrix expression A-BXB^∗ with respect to a variable Hermitian matrix X. In addition, we give some results on the extremal inertias of Hermitian solutions to the matrix equation AX=B, as well as the extremal inertias of a partial block Hermitian matrix.     

The Minimum Rank of a 3 × 3 Partial Block Matrix

After recalling the definition and some basic properties of finite hypergroups—a notion introduced in a recent paper by one of the authors—several non-trivial examples of such hypergroups are constructed. The examples typically consist of n n×n matrices, each of which is an appropriate polynomial in a certain tri-diagonal matrix. The crucial result required in the construction is the following: ‘let A be the matrix with ones on the super-and sub-diagonals, and with main diagonal given by a ₁…a _n which are non-negative integers that form either a non-decreasing or a symmetric unimodal sequence; then A_k =P_k (A) is a non-negative matrix, where p_k denotes the characteristic polynomial of the top k× k principal submatrix of A, for k=1,…,n. The matrices A_k as well as the eigenvalues of A, are explicitly described in some special cases, such as (i) a_i =0 for all ior (ii) a_i =0 for i<n and a_n =1. Characters ot finite abelian hypergroups are defined, and that naturally leads to harmonic analysis on such hypergroups.     

Consistency check on the fundamental and alternative flux operators in loop quantum gravity

There are different constructions of the flux of triad in loop quantum gravity, namely the fundamental and alternative flux operators. In parallel to the consistency check on the two versions of operator by the algebraic calculus in the literature, we check their consistency by the graphical calculus. Our calculation based on the original Brink graphical method is obviously simpler than the algebraic calculation. It turns out that our consistency check fixes the regulating factor κreg of the Ashtekar-Lewandowski volume operator as1/2, which corrects its previous value in the literature.     

Regioselective addition of Ti enolates to conjugated enones: New insights into the mechanism based on experimental and DFT investigation

The regioselective addition mechanism of the Ti(IV) enolates derived from α-diazo-β-keto carbonyl compounds and α-diazo-β-keto phosphonates to conjugated enones has been studied on the basis of a hypothetical bridging chloride-controlled theory, by density functional theory (DFT), and experimentally. The DFT results indicate that, for the Ti(IV) enolate 3 derived from α-diazo-β-keto carbonyl compounds, the free energy of the bridging chloride-controlled 1,2-addition transition state is 2.4 kcal/mol higher than that of 1,4-addition, and the calculated enthalpies of 1,2-addition is 4.36 kcal/mol more than that of 1,4-addition. For the Ti(IV) enolate 4 derived from α-diazo-β-keto phosphonates, in contrary, the free energy of the bridging chloride-controlled 1,2-addition transition state is 1.1 kcal/mol lower than that of 1,4-addition, and the calculated enthalpy of 1,2-addition is 3.46 kcal/mol less than that of 1,4-addition. Our findings demonstrate that the nucleophilic addition of these Ti(IV) enolates to conjugated enones was carried out not only kinetically but also irreversibly for the first time.     

A new class of functionalized polyoxometalates: synthesis, structure and preliminary antitumor activity studies of three arylimido substituted hexamolybdates bearing a strong electron-withdrawing nitro group, (Bu4N)2[Mo6O18([triple bond]NAr)] (Ar = 3-NO2-C6H4, 2-CH3-4-NO2-C6H3, 2-CH3-5-NO2-C6H3)

Three novel mono-functionalized arylimido derivatives of hexamolybdate bearing the strongest electron-withdrawing nitro group, (Bu(4)N)(2)[Mo(6)O(18)([triple bond, length as m-dash]NAr)] (, and ), have been synthesized for the first time by an improved reaction of octamolybdate ion and 3-nitroaniline hydrochloride, 2-methyl-4-nitroaniline hydrochloride and 2-methyl-5-nitroaniline hydrochloride respectively with DCC (N,N'-dicyclohexylcarbodiimide) as a dehydrating agent. Complete assignments were achieved for the title compounds by elemental analysis, IR, (1)H NMR, UV/visible and single-crystal X-ray diffraction analyses. The preliminary antitumor activity test indicated that the title compounds have some effects on the cellular growth inhibition of K562 cells.