A scaling limit for the length of the longest cycle in a sparse random digraph

We discuss the length of the longest directed cycle in the sparse random digraph , constant. We show that for large there exists a function such that a.s. The function where is a polynomial in . We are only able to explicitly give the values , although we could in principle compute any .     

Fast computation of generic bivariate resultants

We prove that the resultant of two “sufficiently generic” bivariate polynomials over a finite field can be computed in quasi-linear expected time, using a randomized algorithm of Las Vegas type. A similar complexity bound is proved for the computation of the lexicographical Gröbner basis for the ideal generated by the two polynomials.     

Hamiltonian cycles in 7-tough (P3 ∪ 2P1)-free graphs

The toughness of a noncomplete graph G is the maximum real number t such that the ratio of $|S|$ to the number of components of $G?S$ is at least t for every cutset S of G. Determining the toughness for a given graph is NP-hard. Chvátal's toughness conjecture, stating that there exists a constant $t_0$ such that every graph with toughness at least $t_0$ is hamiltonian, is still open for general graphs. A graph is called $(P_3\cup 2P_1)$-free if it does not contain any induced subgraph isomorphic to $P_3\cup 2P_1$, the disjoint union of $P_3$ and two isolated vertices. In this paper, we confirm Chvátal's toughness conjecture for $(P_3\cup 2P_1)$-free graphs by showing that every 7-tough $(P_3\cup 2P_1)$-free graph on at least three vertices is hamiltonian.     

Limit cycle bifurcations near a double homoclinic loop with a nilpotent saddle of order m

In this paper, we study the explicit expansion of the first order Melnikov function near a double homoclinic loop passing through a nilpotent saddle of order m in a near-Hamiltonian system. For any positive integer $m(m\ge 1)$, we derive the formulas of the coefficients in the expansion, which can be used to study the limit cycle bifurcations for near-Hamiltonian systems. In particular, for $m=2$, we use the coefficients to consider the limit cycle bifurcations of general near-Hamiltonian systems and give the existence conditions for 10, 11, 13, 15 and 16 (11, 13 and 16, respectively) limit cycles in the case that the homoclinic loop is of cuspidal type (smooth type, respectively) and their distributions. As an application, we consider a near-Hamiltonian system with a nilpotent saddle of order 2 and obtain the lower bounds of the maximal number of limit cycles.     

In vitro biomolecular interaction studies and cytotoxic activities of copper(II) and zinc(II) complexes bearing ONS donor thiosemicarbazones

Among all the bio‐metals, zinc and copper derivatives of ONS donor thiosemicarbazone have aroused great interest because of their potential biological applications. Multisubstituted thiosemicarbazone ligand H₂dspt (3,5‐dichlorosalicylaldehyde‐N⁴‐phenylthiosemicarbazone) derived new ternary complexes like [Zn(dspt)(phen)]?DMF ( 1 ) and [Cu(dspt)(phen)]?DMF ( 2 ), and another thiosemicarbazone, H₂dsct (3,5‐dichlorosalicylaldehyde‐N⁴‐cyclohexylthiosemicarbazone), derived [Cu(dsct)(bipy)]?DMF ( 3 ). These complexes have been characterized by elemental analysis (CHNS), Fourier transform infrared (FT‐IR), ultraviolet–visible (UV–Vis) and proton nuclear magnetic resonance (¹H‐NMR) spectra. The structures of the complexes were obtained by single‐crystal X‐ray diffraction analysis. Compounds 1 and 2 got crystallized in the monoclinic P2₁/c space group. The complexes showed interesting supramolecular interaction, which in turn stabilizes the complexes. The ground state electronic configurations of the complexes were studied using the B3LYP/LANL2DZ basis set, and ESP plots of complexes were investigated. The interaction of the complexes with calf thymus DNA (CT‐DNA) was studied using absorption and fluorescence spectroscopic methods. A UV study of the interaction of the complexes with calf thymus DNA (CT‐DNA) has shown that the complexes can effectively bind to CT‐DNA, and [Cu(dspt)(phen)]·DMF ( 2 ) exhibited the highest binding constant to CT‐DNA (K_b = 3.7 × 10⁴). Fluorescence spectral studies also indicated that Complex 2 binds relatively stronger with CT DNA through intercalative mode, exhibiting higher binding constant (K_q = 4.7 × 10⁵). The DNA cleavage result showed that the complexes are capable of cleaving the DNA without the help of any external agent. Molecular docking studies were carried out to understand the binding of complexes with the molecular target DNA. Complex 2 exhibited the highest cytotoxicity against human breast cancer cell line MD‐MBA‐231 (IC₅₀ = 23.93 μg/mL) as compared to Complex 1 (IC₅₀ = 44.40 μg/mL) .     

On tight 6-cycle decompositions of complete 3-uniform hypergraphs

The complete 3-uniform hypergraph of order v has a vertex set V of size v and the set of all 3-element subsets of V as its edge set. A tight 6-cycle is a hypergraph with vertex set $\{a,b,c,d,e,f\}$ and edge set $\{\{a,b,c\},\{b,c,d\},\{c,d,e\},\{d,e,f\},\{e,f,a\},\{f,a,b\}\}$. We show that there exists a decomposition of the complete 3-uniform hypergraph of order v into isomorphic copies of a tight 6-cycle if and only if $v\equiv 1$, 2, 10, 20, 28, or $29(\mathrm{mod}36)$.     

Synthesis,cytotoxicity, apoptosis and cell cycle arrest of a monoruthenium(II)-substituted Dawson polyoxotungstate

By 5-h reaction of cis-[Ru^IICl₂(DMSO)₄] (M2) with K₁₀[α₂-P₂W₁₇O₆₁] (M3) in ice-cooled, HCl-acidic aqueous solution, a water-soluble 1:2-type diamagnetic ruthenium(II) complex of formula K₁₈[Ru^II(DMSO)₂(P₂W₁₇O₆₁)₂]·35H₂O (M1) was unexpectedly obtained as an analytically pure, homogeneous tan-colored solid, in which two DMSO ligands are coordinated to the ruthenium(II) atom. The cytotoxic potential of the complex was tested on C33A, DLD-1, and HepG-2 cancer cells and human normal embryonic lung fibroblasts cell MRC-5; the viability of the treated cells was evaluated by MTT assay. The mode of cell death was assessed by morphological study of DNA damage and apoptosis assays. Compound M1 induced cell death in a dose-dependent manner, and the mode of cell death was essentially apoptosis though necrosis was also noticed. Cell cycle analysis by flow cytometry indicated that M1 caused cell cycle arrest and accumulated cells in S phase.