k圈图的最大Laplace分离度

设$ G $是一个$ n $阶$ k $圈图, $ k $圈图为边数等于顶点数加$ k-1 $的简单连通图。$ \mu_{1}(G) $、$ \mu_{2}(G) $分别记为图$ G $的Laplace矩阵的最大特征值和次大特征值, 图$ G $的Laplace分离度定义为$ S_{L}(G)=\mu_{1}(G)-\mu_{2}(G) $。本文研究了给定阶数的$ k $圈图的最大Laplace分离度, 并刻画了相应的极图, 其结果推广了已有当$ k=1, 2, 3 $时的结论。     

Regularity of Viscosity Solutions of the Biased Infinity Laplacian Equation

In this paper, we are interested in the regularity estimates of the nonnegative viscosity super solution of the $β$−biased infinity Laplacian equation $$∆^β_∞u = 0,$$ where $β ∈ \mathbb{R}$ is a fixed constant and $∆^β_∞u := ∆^N_∞u + β|Du|,$ which arises from the random game named biased tug-of-war. By studying directly the $β$−biased infinity Laplacian equation, we construct the appropriate exponential cones as barrier functions to establish a key estimate. Based on this estimate, we obtain the Harnack inequality, Hopf boundary point lemma, Lipschitz estimate and the Liouville property etc.     

Covering with blocks in the non-symmetric case

Considering an infinite string of i.i.d. random letters drawn from a finite alphabet we define the cover timeW _n as the number of random letters needed until each pattern of lenghtn appears at least once as a substring. Sharp weak and a.s. limit results onW _n are known in the symmetric case, i.e., when the random letters are uniformly distributed over the alphabet. In this paper we determine the limit distribution ofW _n in the nonsymmetric case asn. Generalizations in terms of point processes are also proved.Dedicated to Endre Csáki on his 60th birthday.     

Curvature invariants, differential operators and local homogeneity

We first prove that a Riemannian manifold with globally constant additive Weyl invariants is locally homogeneous. Then we use this result to show that a manifold whose Laplacian commutes with all invariant differential operators is a locally homogeneous space.

     

Modèles entiers des courbes hyperelliptiques sur un corps de valuation discrète

Let be a hyperelliptic curve of genus over a discrete valuation field . In this article we study the models of over the ring of integers of . To each Weierstrass model (that is a projective model arising from a hyperelliptic equation of with integral coefficients), one can associate a (valuation of) discriminant. Then we give a criterion for a Weierstrass model to have minimal discriminant. We show also that in the most cases, the minimal regular model of over dominates every minimal Weierstrass model. Some classical facts concerning Weierstrass models over of elliptic curves are generalized to hyperelliptic curves, and some others are proved in this new setting.

     

The regioselectivity of the formation of 2-pyrazolylthiazoles and their precursors from the reaction of 2-hydrazinothiazoles with 4,4,4-trifluoro-1-hetaryl-1,3-butanediones

Reaction of 2-hydrazinothiazoles 1 with 1-thienyl- and 1-furyl-1,3-butanediones 2a,b in methanol in the presence of hydrochloric acid mainly leads to a mixture of pyrazoles 3 and pyrazolines 4 or pyrazoles 3 and 5 in strong acidic conditions. Isomeric hydrazones 6 and pyrazolines 4 were formed and isolated in these reactions in the absence of hydrochloric acid. It has been shown that the regioselectivity in the reaction of diketones 2 with hydrazine 1 is governed by both the concentration of acid and the nature of substituents in the 1,3-diketones 2. Cyclization of hydrazones 6 is shown to occur under milder conditions than dehydration for pyrazolines 4. The new heterocyclic compounds were prepared and fully characterized by NMR spectra and by X-ray analysis for 3c.     

钴配合物的合成晶体结构及活性研究

The complex [Co₂Cl₂(EGTB)]·(ClO₄)₂·5H₂O was synthesized in methanol-aqueous solution and its crystal structure was determined by X-ray diffraction method. The crystal belongs to monoclinic with space group of P2₁/c. The cell parameters are a=1.42231(2)nm,b=2.53181(3)nm,c=1.43392(2)nm,β=102.0518(5)°, Z=4, F(000)=2360,D_c=1.508g·cm^-3, The final R=0.0788,wR₂=0.232, The catalytic rate constant of the dismutation of superoxide radicals has been obtained by means of photo reduction of nitroblue tetrazolium(NBT),and it is 1.875×10⁶mol^-1·L·s^-1.     

Selective formation of AAB- and ABC-type heterotrimeric alpha-helical coiled coils

The alpha-helical coiled coils have a representative amino acid sequence of (abcdefg)(n) heptad repeats. We previously reported that two peptides named IZ-2A and IZ-2W formed an (IZ-2A)(2)/IZ-2W heterotrimer with an Ala-Ala-Trp interaction in the hydrophobic core. In this paper, we describe the selective formation of AAB- and ABC-type heterotrimers. To increase the selectivity of the AAB-type heterotrimeric formation, Lys residues at the f position were mutated to either an Ala or a Gln residue to form IZ-2A(fA) or IZ-2W(fQ). Separately, both IZ-2A(fA) and IZ-2W(fQ) have a random structure at pH 7 and 20 degrees C. However, together IZ-2A(fA) and IZ-2W(fQ) form a 2:1 complex with a thermal transition midpoint (Tm) of 48 degrees C. This procedure was applied to prepare the ABC-type heterotrimer, in which two sets of Ala-Ala-Trp interactions were designed in the hydrophobic core. Interhelical interaction between the e and g positions and the alpha-helical propensity of the amino acid at the f position were also considered in the design. The resultant three peptides selectively formed the ABC-type heterotrimer with a Tm of 51 degrees C. Other peptide combinations had random coil properties.     

Synthese de derives mono- ou disilicies au niveau d'un carbone fonctionnel

Aliphatic saturated amides treated with Me₃SiCl/Li/THF were found to react in two ways. Either alkoxysilanes, mono- or di-silylated at the functional carbon, or C-silylated amines were obtained dependent of the original amide structure and the experimental conditions. (Me₃Si)₂CHN(SiMe₃)₂ exhibited Physicochemical properties that are particular for hindered rotation about the CN bond. A general mechanism is proposed to explain these results.