Algebraic cycles and very special cubic fourfolds

Informed by the Bloch–Beilinson conjectures, Voisin has made a conjecture about 0-cycles on self-products of Calabi–Yau varieties. In this note, we consider variant versions of Voisin’s conjecture for cubic fourfolds, and for hyperkähler varieties. We present examples for which these conjectures are verified, by considering certain very special cubic fourfolds and their Fano varieties of lines.     

How Can Theory Predict the Selectivity of Palladium‐Catalyzed Cross‐Coupling of Pristine Aromatic Molecules?

The new approach for palladium‐catalyzed cross‐coupling of two non‐activated aromatic compounds (D. R. Stuart, K. Fagnou, Science 2007 , 316, 1172) was studied theoretically. The energetic span model (S. Kozuch, S. Shaik, Acc. Chem. Res. 2011 , 44, 101, and references therein) was employed to analyze the kinetic behavior of the catalytic cycle. The computed energy profile, combined with the energetic span model, accounts for the experimental selectivity, which favors the hetero‐coupling of benzene with indole. This selectivity is driven by a fine balance of the entropic contributions and the high ratio of concentrations used for benzene over indole. This analysis may allow future theoretical predictions of how different aromatic compounds can be effectively coupled.     

具有一类三次曲线解的Kolmogorov三次系统的极限环的存在性

证明了具有三次曲线解y=-x(x-1)2 4/24的Kolmogorov三次系统是有存在极限环可能的.     

Shock wave driven out by a piston in a mixture of a non-ideal gas and small solid particles under the influence of the gravitation field with monochromatic radiation

Similarity solutions for a spherical shock wave in a mixture of small solid particles of micro size and a non-ideal gas are discussed under the influence of the gravitational field with monochromatic radiation. The solid particles are uniformly distributed in the mixture, and the shock wave is assumed to be driven by a piston. It is assumed that the equilibrium flow-conditions are maintained and the moving piston continuously supplies the variable energy input. Due to the central mass ($\overline{m}$) at the origin (Roche model), the medium is considered to be under the influence of the gravitational field. In comparison to the attraction of the central mass at the origin, the gravitational effect of the mixture itself is neglected. The density of the undisturbed medium is assumed to be constant in order to obtain the self-similar solutions. The effect of the parameter of non-idealness of the gas $\overline{b}$, the mass concentration of solid particles in the mixture μ_p, the ratio of the density of solid particles to the initial density of the gas G_a and the gravitational parameter G₀ are obtained. It is shown that due to an increase in the gravitational parameter the compressibility of the medium at any point in the flow field behind the shock front decrease and the flow variables velocity, pressure, radiation flux and shock strength are increased. Also, an increase in the ratio of the density of solid particles to the initial density of the gas G_a and the gravitational parameter G₀ has the same effect on the shock strength and the reverse effect on the compressibility. The non-idealness of the gas causes a decrease in the shock strength and widens the disturbed region between the piston and the shock.     

Euclidean distance degree and limit points in a Morsification

Motivated by finding an effective way to compute the algebraic complexity of the nearest point problem for algebraic models, we introduce an efficient method for detecting the limit points of the stratified Morse trajectories in a small perturbation of any polynomial function on a complex affine variety. We compute the multiplicities of these limit points in terms of vanishing cycles. In the case of functions with only isolated stratified singularities, we express the local multiplicities in terms of polar intersection numbers.     

Thioalkynes in Ring Forming Reactions

Organic cycles play an important role in chemistry, pharmacology and material science for their unique properties. Construction of organic cycles from thioalkynes attracted increasing attention due to the facile access of thioalkynes. 2H-Azirines were synthesized successfully from thioalkynyl oxime ethers. Cyclobutanes were formed through chiral titanium catalyzed cycloaddition of thioalkynes. Cyclopentenes were afforded by annulation of thioalkynes. Thioalkynes could be also applied to synthesize thiophenes, oxazoles, benzo[b]thiophenes, 2H-chromenes, 2-phenylbenzothiazoles, diazacyclobutene, etc. In this review, construction of organic cycles from thioalkynes were highlighted. Firstly, the property and application of organic cyclic compounds were simply introduced. After presenting the general methods to access organic cycles, applications of thioalkynes as synthons to prepare organic cycles were classified and presented in detail. Based on different kinds of organic cycles obtained from thioalkynes, organic reactions for synthesis of three-, four-, five-, six-membered as well as fused cycles would be summarized and the plausible reaction mechanisms could be presented if available.     

Supereulerian regular matroids without small cocircuits

A cycle of a matroid is a disjoint union of circuits. A matroid is supereulerian if it contains a spanning cycle. To answer an open problem of Bauer in 1985, Catlin proved in [J. Graph Theory 12 (1988) 29–44] that for sufficiently large $n$, every 2-edge-connected simple graph $G$ with $\text{◂=▸}n=\mid V\left(G\right)\mid$ and minimum degree $\text{◂≥▸}\delta \left(G\right)\ge \frac{n}{5}$ is supereulerian. In [Eur. J. Combinatorics, 33 (2012), 1765–1776], it is shown that for any connected simple regular matroid $M$, if every cocircuit $D$ of $M$ satisfies $\text{◂≥▸}\mid D\mid \ge \max \left\{\frac{\text{◂−▸}r\left(M\right)-5}{5},6\right\}$, then $M$ is supereulerian. We prove the following. (i) Let $M$ be a connected simple regular matroid. If every cocircuit $D$ of $M$ satisfies $\text{◂≥▸}\mid D\mid \ge \max \left\{\frac{\text{◂+▸}r\left(M\right)+1}{10},9\right\}$, then $M$ is supereulerian. (ii) For any real number $c$ with , there exists an integer $f\left(c\right)$ such that if every cocircuit $D$ of a connected simple cographic matroid $M$ satisfies $\mid D\mid \ge \text{◂lim▸}\max \text{◂{}▸}\{c\text{◂()▸}\left(r\left(M\right)+1\right),f\left(c\right)\}$, then $M$ is supereulerian.