Classification of Plane Congruences in 4 C4 (I)

Nondegenerate plane congruences in the four-dimensional complex projective space with degenerate general focal conic are classified by using the focal method due to Corrado Segre.     

Modular quintics in ℙ4

We will examine the arithmetic of some of the members of a pencil of symmetric quintics in projective 4‐space. We will give evidence for the modularity of some of the exceptional members (even the non‐rigid ones) and give a proof in one rigid case. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Non-special, non-canal isothermic tori with spherical lines of curvature

This article examines isothermic surfaces smoothly immersed in Möbius space. It finds explicit examples of non-special, non-canal isothermic tori with spherical lines of curvature in two systems by analyzing Darboux transforms of Dupin tori. In addition, it characterizes the property of spherical lines of curvature in terms of differential equations on the Calapso potential of the isothermic immersion, and investigates the effect of classical transformations on this property.

     

On the cotangent cohomology of rational surface singularities with almost reduced fundamental cycle

We prove dimension formulas for the cotangent spaces T ¹ and T ² for a class of rational surface singularities by calculating a correction term in the general dimension formulas. We get that it is zero if the dual graph of the rational surface singularity X does not contain a particular type of configurations, and this generalizes a result of Theo de Jong stating that the correction term c (X ) is zero for rational determinantal surface singularities. In particular our result implies that c (X ) is zero for Riemenschneiders quasi‐determinantal rational surface singularities, and this also generalizes results for quotient singularities. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Exact controllability of nodal profile for 1‐D first order quasilinear hyperbolic systems with internal controls

For 1‐D first order quasilinear hyperbolic systems without zero eigenvalues, based on the theory of exact boundary controllability of nodal profile, using an extension method, the exact controllability of nodal profile can be realized in a shorter time by means of additional internal controls acting on suitably small space‐time domains. On the other hand, using a perturbation method, the exact controllability of nodal profile for 1‐D first order quasilinear hyperbolic systems with zero eigenvalues can be realized by additional internal controls to the part of equations corresponding to zero eigenvalues. Furthermore, by adding suitable internal controls to all the equations on suitable domains, the exact controllability of nodal profile for systems with zero eigenvalues can be realized in a shorter time.     

On the oscillation of solutions of first order differential equations with deviating arguments

1.IntroductionConsiderthefirstorderdifferentialequationwithdeviatingargUmelltTheoscillationofEq.(1)wasstudiedextensivelyinthelastthreedecades.See,forexamDleif--101andthereferencescitedtherein.In1972Ladas.LakshlnhanthamanddeceivedApril1,1997.Re~AugUBt...     

Right order spectral gap estimates for generating sets of ℤ4

Using coupling arguments, a distance method and Zeifman's method we give sharp estimates on the spectral gap for a special case of the class of Markov chains on generating n‐tuples of Abelian groups. In our case the group is ?₄. © 2002 Wiley Periodicals, Inc. Random Struct. Alg. 20: 220–238, 2002     

On the existence of resolvable K4 − e designs

In this article, we settle a problem which originated in 4 regarding the existence of resolvable (K₄ ? e)‐design. We solve the problem with two possible exceptions. © 2007 Wiley Periodicals, Inc. J Combin Designs 15: 502–510, 2007     

On Darboux helices in Euclidean 4‐space

The notion of Darboux helix in Euclidean 3‐space was introduced and studied by Yayl? et al. 2012. They show that the class of Darboux helices coincide with the class of slant helices. In a special case, if the curvature functions satisfy the equality κ² + τ² = constant, then these curves are curve of the constant precession. In this paper, we study Darboux helices in Euclidean 4‐space, and we give a characterization for a curve to be a Darboux helix. We also prove that Darboux helices coincide with the general helices. In a special case, if the first and third curvatures of the curve are equal, then Darboux helix, general helix, and V₄‐slant helix are the same concepts.     

On the comparison principle for unbounded solutions of elliptic equations with first order terms

We prove a comparison principle for unbounded weak sub/super solutions of the equation

$\lambda u?\text{div}(A(x)Du)=H(x,Du)\text{ in }\mathrm{\Omega }$

where $A(x)$ is a bounded coercive matrix with measurable ingredients, $\lambda \ge 0$ and $\xi ?H(x,\xi )$ has a super linear growth and is convex at infinity. We improve earlier results where the convexity of $H(x,?)$ was required to hold globally.     

On an asymptotic behavior of elements of order in irreducible representations of the classical algebraic groups with large enough highest weights

The behavior of the images of a fixed element of order in irreducible representations of a classical algebraic group in characteristic with highest weights large enough with respect to and this element is investigated. More precisely, let be a classical algebraic group of rank over an algebraically closed field of characteristic 2$">. Assume that an element of order is conjugate to that of an algebraic group of the same type and rank naturally embedded into . Next, an integer function on the set of dominant weights of and a constant that depend only upon , and a polynomial of degree one are defined. It is proved that the image of in the irreducible representation of with highest weight contains more than Jordan blocks of size if and are not too small and .

     

Nowhere‐Zero 5‐Flows On Cubic Graphs with Oddness 4

Tutte's 5‐flow conjecture from 1954 states that every bridgeless graph has a nowhere‐zero 5‐flow. It suffices to prove the conjecture for cyclically 6‐edge‐connected cubic graphs. We prove that every cyclically 6‐edge‐connected cubic graph with oddness at most 4 has a nowhere‐zero 5‐flow. This implies that every minimum counterexample to the 5‐flow conjecture has oddness at least 6.     

关于二阶线性时变系统在第一临界情况下的稳定性

在A（t）为2×2矩阵，而其特征根一个为零，另一个为负数的情况下讨论了系统x=A（t）x零解的稳定性，给出了保证该系统零解稳定的若干充分条件．     

Acyclic 4‐Choosability of Planar Graphs with No 4‐ and 5‐Cycles

Every planar graph is known to be acyclically 7‐choosable and is conjectured to be acyclically 5‐choosable (O. V. Borodin, D. G. Fon‐Der‐Flaass, A. V. Kostochka, E. Sopena, J Graph Theory 40 (2002), 83–90). This conjecture if proved would imply both Borodin's (Discrete Math 25 (1979), 211–236) acyclic 5‐color theorem and Thomassen's (J Combin Theory Ser B 62 (1994), 180–181) 5‐choosability theorem. However, as yet it has been verified only for several restricted classes of graphs. Some sufficient conditions are also obtained for a planar graph to be acyclically 4‐ and 3‐choosable. In particular, the acyclic 4‐choosability was proved for the following planar graphs: without 3‐, 4‐, and 5‐cycles (M. Montassier, P. Ochem, and A. Raspaud, J Graph Theory 51 (2006), 281–300), without 4‐, 5‐, and 6‐cycles, or without 4‐, 5‐, and 7‐cycles, or without 4‐, 5‐, and intersecting 3‐cycles (M. Montassier, A. Raspaud, W. Wang, Topics Discrete Math (2006), 473–491), and neither 4‐ and 5‐cycles nor 8‐cycles having a triangular chord (M. Chen and A. Raspaud, Discrete Math. 310(15–16) (2010), 2113–2118). The purpose of this paper is to strengthen these results by proving that each planar graph without 4‐ and 5‐cycles is acyclically 4‐choosable.