Nonlinear vibration of a nonlocal functionally graded beam on fractional visco-Pasternak foundation

Nonlinear Dynamics - A (3+1)-dimensional coupled nonautonomous NLS model with partially nonlocal coupled nonlinearities under the linear and harmonic potentials becomes the center of attention. Two...     

Laplacian coefficients of trees with given number of leaves or vertices of degree two

Let G be a simple undirected n-vertex graph with the characteristic polynomial of its Laplacian matrix . It is well known that for trees the Laplacian coefficient c_n-2 is equal to the Wiener index of G, while c_n-3 is equal to the modified hyper-Wiener index of graph. Using a result of Zhou and Gutman on the relation between the Laplacian coefficients and the matching numbers in subdivided bipartite graphs, we characterize the trees with k leaves (pendent vertices) which simultaneously minimize all Laplacian coefficients. In particular, this extremal balanced starlike tree S(n,k) minimizes the Wiener index, the modified hyper-Wiener index and recently introduced Laplacian-like energy. We prove that graph S(n,n-1-p) has minimal Laplacian coefficients among n-vertex trees with p vertices of degree two. In conclusion, we illustrate on examples of these spectrum-based invariants that the opposite problem of simultaneously maximizing all Laplacian coefficients has no solution, and pose a conjecture on extremal unicyclic graphs with k leaves.     

Synthesis,characterization and biological activities of <Emphasis Type="Italic">N</Emphasis>-heteroaromatic hydrazones and their complexes with Pd(II), Pt(II) and Cd(II)

Condensation derivatives of ethyl hydrazinoacetate with 2-formylpyridine and quinoline-2-carboxaldehyde were synthesized. Pd(II), Pt(II) and Cd(II) complexes with the 2-formylpyridine derivative and a Cd(II) complex with the quinoline-2-carboxaldehyde derivative were synthesized and characterized by spectroscopic techniques. In the complexes, both ligands are coordinated in neutral NN bidentate modes, while the remaining two coordination sites are occupied by chloride. All compounds showed biological activity when tested against Escherichia coli, Bacillus subtilis and Staphylococcus aureus.     

The Voronoi Identity via the Laplace Transform

The classical Voronoi identity $$\Delta (x) = - \frac{2}{\pi }\sum\limits_{n = 1}^\infty {d(n)} \left( {\frac{x}{n}} \right)^{1/2} \left( {K_1 (4\pi \sqrt {xn} ) + \frac{\pi }{2}Y_1 (4\pi \sqrt {xn} )} \right)$$ is proved in a relatively simple way by the use of the Laplace transform. Here Δ(x) denotes the error term in the Dirichlet divisor problem, d(n) is the number of divisors of n and K_1, Y_1 are the Bessel functions. The method of proof may be used to yield other identities similar to Voronoi's.     

Size of supramolecular SNARE complex: membrane-directed self-assembly

Full length v-SNARE protein in lipid vesicles when exposed to t-SNARE-reconstituted lipid membrane results in the self-assembly of a t-/v-SNARE complex in a ring pattern, forming pores, and establishing continuity between the opposing bilayers. It is known that smaller vesicles fuse more efficiently than larger ones, and hence the curvature of secretory vesicles may dictate the potency and efficacy of their fusion at the cell plasma membrane. The diameter of t- and v-SNARE vesicles may, therefore, reflect the size of the t-/v-SNARE complex formed. In the present study, this hypothesis was tested, and results from the study demonstrate that the size of the t-/v-SNARE complex is directly proportional to the vesicle diameter (R2 = 0.9725).