This document contains proofs, supplementary details, and supplementary experiments for the paper " Graph Sparsification Approaches for Laplacian Smoothing ". All section numbers, equation numbers, and figure numbers in this supplementary document are preceded by the letter A, to distinguish them from those from the main paper. Part (a). By optimality ofˆθ for problem (5), y − ˆ θ 2 2 + λ ˆ θ T...

Let β > 1, we show that if 1 has a finite greedy expansion in base β, 1=b1/β + b2/β + . . . bn/β with b1 ≥ bi ≥ 1 for i = 1, 2, . . . , n, then given an infinite coin toss, one can associate a new β-expansion of a point x in [0, ⌊β⌋ β−1 ]. We show that all such expansions can be seen as realizations of an appropriate Markov Chain. We also discuss the uniqueness of the β-expansion of 1.

Epithelial junctions depend on intercellular interactions between β(1) subunits of the Na(+)/K(+)-ATPase molecules of neighboring cells. The interaction between dog and rat subunits is less effective than the interaction between two dog β(1) subunits, indicating the importance of species-specific regions for β(1)-β(1) binding. To identify these regions, the species-specific amino acid residues ...

Denote by xn,k(α, β), k = 1, . . . , n, the zeros of the Jacobi polynomial P (α,β) n (x). It is well known that xn,k(α, β) are increasing functions of β and decreasing functions of α. In this paper we investigate the question of how fast the functions 1 − xn,k(α, β) decrease as β increases. We prove that the products tn,k(α, β) := fn(α, β) (1− xn,k(α, β)), where fn(α, β) = 2n2 + 2n(α + β + 1) +...

thermodynamic study on the interaction of β-cd with poly ethylene oxide and poly acrylic acid was performed by isothermal titration calorimetry at 298k. when β-cd is added to the interpolymer complex, competition is created between host-guest and hydrogen bond. enthalpy of interaction between the β-cd and interpolymer complex was calculated using the extended solvation theory. p=1 shows that wi...

Oxygenated polycyclic aromatic hydrocarbons (OPAHs) are toxic and carcinogenic compounds widely present in the natural environment, posing a serious threat to environment human health. However, removal of OPAHs is mainly hindered by their low water solubility. Cyclodextrins (CDs) frequently used form inclusion complexes (ICs) with hydrophobic molecules improve In this study, we investigated sol...

Ficus natalensis subsp. leprieurii also known as the natal fig is a fruit-producing tree belonging to family Moraceae and widely distributed in African countries cultivated Egypt. F. common with myriad of traditional medicinal importance. Owing increased demand for natural products new structural compositions, current study aimed separate elucidate structure triterpenoid saponins from leaves. O...