Upper Bound of the Number of Zeros for Abelian Integrals in a Kind of Quadratic Reversible Centers of Genus One

By using the methods of Picard-Fuchs equation and Riccati equation, we study the upper bound of the number of zeros for Abelian integrals in a kind of quadratic reversible centers of genus one under polynomial perturbations of degree $n$. We obtain that the upper bound is $7[(n-3)/2]+5$ when $n\ge 5$, $8$ when $n=4$, $5$ when $n=3$, $4$ when $n=2$, and $0$ when $n=1$ or $n=0$, which linearly depends on $n$.     

Cyclical scheduling and multi-shift scheduling: Complexity and approximation algorithms

We consider the multiple shift scheduling problem modelled as a covering problem. Such problems are characterized by a constraint matrix that has, in every column, blocks of consecutive 1s, each corresponding to a shift. We focus on three types of shift scheduling problems classified according to the column structure in the constraint matrix: columns of consecutive 1s, columns of cyclical 1s, and columns of k consecutive blocks. In particular, the complexity of the cyclical scheduling problem, where the matrix satisfies the property of cyclical 1s in each column, was noted recently by Hochbaum and Tucker to be open. They further showed that the unit demand case is polynomially solvable. Here we extend this result to the uniform requirements case, and provide a 2-approximation algorithm for the non-uniform case. We also establish that the cyclical scheduling problem’s complexity is equivalent to that of the exact matching problem—a problem the complexity status of which is known to be randomized polynomial (RP). We then investigate the three types of shift scheduling problems and show that, while the consecutive ones version is polynomial and the k-block columns version is NP-hard (for k≥2), for the k-blocks problem we give a simple k-approximation algorithm, which is the first approximation algorithm determined for the problem.     

Center, limit cycles and isochronous center of a Z 4-equivariant quintic system

In this paper, we study the limit cycles bifurcations of four fine focuses in Z4-equivariant vector fields and the problems that its four singular points can be centers and isochronous centers at the same time. By computing the Liapunov constants and periodic constants carefully, we show that for a certain Z4-equivariant quintic systems, there are four fine focuses of five order and five limit cycles can bifurcate from each, we also find conditions of center and isochronous center for this system. The process of proof is algebraic and symbolic by using common computer algebra soft such as Mathematica, the expressions after being simplified in this paper are simple relatively. Moreover, what is worth mentioning is that the result of 20 small limit cycles bifurcating from several fine focuses is good for Z4-equivariant quintic system and the results where multiple singular points become isochronous centers at the same time are less in published references.     

Center cyclicity of a family of quartic polynomial differential system

In this paper we study the cyclicity of the centers of the quartic polynomial family written in complex notation as

$$\dot{z} = i z + z \bar{z}\big(A z^2 + B z \bar{z} + C \bar{z}^2 \big),$$

where \({A,B,C \in \mathbb{C}}\). We give an upper bound for the cyclicity of any nonlinear center at the origin when we perturb it inside this family. Moreover we prove that this upper bound is sharp.

     

Commutative Power-Associative Nilalgebras of Nilindex 5

In this paper we study the structure of commutative power-associative nilalgebras of dimension 8 and nilindex ≤ 5 over a field of characteristic different from 2, 3 and 5. We prove that every algebra in this class verifies the identities x⁴y = 0 and x(x(x(x(xy)))) = 0. In particular, we finish the proof of the Albert’s problem [0] in the following case: every commutative power-associative nilalgebra of dimension ≤ 8 over a field of characteristic ≠ 2, 3 and 5 is solvable. The solvability of these algebras for dimension 4, 5 and 6 were proved in [0], [0] and [0] respectively.     

Locally compact covergece groups ad $$$$-trasitive actios

All \(\sigma \)-compact, locally compact groups acting sharply \(n\)-transitively and continuously on compact spaces \(M\) have been classified, except for \(n=2,3\) when \(M\) is infinite and disconnected. We show that no such actions exist for \(n=2\) and that these actions for \(n=3\) coincide with the action of a hyperbolic group on a space equivariantly homeomorphic to its hyperbolic boundary. We further characterize non-compact groups acting 3-properly and transitively on infinite compact sets as non-elementary boundary-transitive hyperbolic groups, which in turn were recently studied by Caprace, de Cornulier, Monod and Tessera. As an important tool, we generalize Bowditch’s topological characterization of discrete hyperbolic groups to locally compact hyperbolic groups. Finally, we show that if a locally compact group acts continuously, 4-properly and 4-cocompactly on a locally connected metrizable compactum M, then M has a global cut point, which is in sharp contrast to the \(3\)-proper, \(3\)-cocompact case due to the solution of Bowditch’s cut-point conjecture.     

-contingent epiderivatives in reflexive spaces

In this paper we introduce a notion of the τ^w-contingent epiderivative of a set-valued map by considering the weak topology in the image space. We establish a chain rule formula for families of τ^w-contingent and contingent epiderivatives. When the image space is reflexive and the ordering cone a half-space, we give conditions for the existence of the family of τ^w-contingent epiderivatives of a stable set-valued map; moreover we obtain an interesting formula that connects this family with the contingent epiderivative of an associated scalar set-valued map. Finally we apply the previous results in order to establish optimality conditions for a set-valued optimization problem via epiderivatives with respect to half-spaces associated with the ordering cone. In particular we provide a scalarization method for computing these conditions in reflexive spaces.     

Centers of quasi-homogeneous polynomial planar systems

In this paper we determine the centers of quasi-homogeneous polynomial planar vector fields of degree 0, 1, 2, 3 and 4. In addition, in every case we make a study of the reversibility and the analytical integrability of each one of the above centers. We find polynomial centers which are neither orbitally reversible nor analytically integrable, this is a new scenario in respect to the one of non-degenerate and nilpotent centers.     

Some permutation groups of degreep=6q+1

Transitive permutation groups of degrees 43, 67, 79, 103 and 139 are classified.In this note we consider insoluble transitive permutation groups of degreeq = 6q+1 wherep andq are primes and summarise the computations whereby these groups have been classified for some small values ofq. The result which allows progress on this problem is due to McDonough [1]; he showed that if such a group has a Sylowp-normaliser of order 3p then it is isomorphic either toPSL(3, 3) orPAL(3, 5) (of degrees 13, 31 respectively). Using this theorem machine computations along the lines of those done by Parker, Nikolai and Appel [3, 2] for degreesp=2q+1 andp=4q+1 give the following     

有时滞的系統的无条件稳定性

<正> §1.問題的提出及解法 錢学森在[1]中提出了有时滞的系統的无条件稳定性的問題,并叙述了Satche的作图法.对于有时滞的系統的稳定性問題,一般化为超越方程的根的实部的符号的判定問題,这方面有及,Hayes及Bellman等人的工     

S~4到CP~n的常曲率弱Lagrangian极小浸入

研究球面S~4到CP~n的常曲率弱Lagrangian极小浸入■:S~4→CP~n.如果■的诱导度量ds~2具有常(截面)曲率c,则存在一个整数s≥1,使得c=4/[s(s+3)].浸入■被两个四元齐次多项式f_s和f_(s-1)唯一确定.当s=1,即c=1,或s=2,即c=2/5时,浸入■是绝对实的.     

与位数码列有关的函数的一个性质

对于正整数ｎ＝ａ１０~ｋｎ＋…ａ,称ａ,ａ是ｎ的位数码,记ｓ（ｎ）＝ａ＋　＋ａ_０,ｓｍ（ｎ）＝（ａ）ｍ＋　＋（ａ_０）~ｍ,称１~ｍ２~ｍ３~ｍ４~ｍ５~ｍ６~ｍ７~ｍ８~ｍ９~ｍ１~ｍ０~ｍ１~ｍ１~ｍ１~ｍ２~ｍ１~ｍ３~ｍ１~ｍ４~ｍ是ｍ阶位数码列,定义函数ｆｍ（ｘ）＝１~ｍ＋２~ｍｘ~１＋ ＋９~ｍｘ~８＋１~ｍｘ~９＋０ｘ~１０＋１~ｍｘ~１１＋１~ｍｘ１２＋…．     

Nonexistence of a code for

In this paper, we shall prove that there is no [3q⁴-q³-q²-3q-1,5,3q⁴-4q³-2q+1]_q code over the finite field for q11. Thus, we conclude the nonexistence of a [g_q(5,d),5,d]_q code for 3q⁴-4q³-2q+1d3q⁴-4q³-q.     

Dedekind zeta-functions and Dedekind sums

In this paper we use Dedekind zeta functions of two real quadratic number fields at -1 to denote Dedekind sums of high rank. Our formula is different from that of Siegel’s. As an application, we get a polynomial representation of ζK(-1): ζK(-1) = 1/45(26n³ -41n± 9),n = ±2(mod 5), where K = Q(√5q), prime q = 4n² + 1, and the class number of quadratic number field K2 = Q(vq) is 1.     

On subgroups of finite <Emphasis Type="Italic">p</Emphasis>-groups

In §2, we prove that if a 2-group G and all its nonabelian maximal sub-groups are two-generator, then G is either metacyclic or minimal non-abelian. In §3, we consider a similar question for p > 2. In §4 the 2-groups all of whose minimal nonabelian subgroups have order 16 and a cyclic subgroup of index 2, are classified. It is proved, in §5, that if G is a nonmetacyclic two-generator 2-group and A, B, C are all its maximal subgroups with d(A) ≤ d(B) ≤ d(C), then d(C) = 3 and either d(A) = d(B) = 3 (this occurs if and only if G/G′ has no cyclic subgroup of index 2) or else d(A) = d(B) = 2. Some information on the last case is obtained in Theorem 5.3.     

Functions for parametrization of solutions of an equation in a free monoid

In this paper we introduce recursive functions

of the word variables , natural number variables and variables whose values are finite sequences of natural number variables. By means of these functions we give finite expressions for the family of solutions of the equation

where is an arbitrary word in the alphabet , in a free monoid.

     

Sign balance for finite groups of Lie type

A product formula for the parity generating function of the number of 1’s in invertible matrices over is given. The computation is based on algebraic tools such as the Bruhat decomposition. It is somewhat surprising that the number of such matrices with odd number of 1’s is greater than the number of those with even number of 1’s. The same technique can be used to obtain a parity generating function also for symplectic matrices over . We present also a generating function for the sum of entries of matrices over an arbitrary finite field calculated in . The Mahonian distribution appears in these formulas.     

完全3-一致超图K_n~((3))的哈密顿圈分解

基于王建方和李东给出的超图哈密顿圈的定义和Katona-Kierstead给出的超图哈密顿链的定义,近年来,国内外学者对一致超图的哈密顿圈分解的研究有一系列结果.特别是Bailey-Stevens和Meszka-Rosa研究了完全3-一致超图K_n~((3))的哈密顿圈分解,得到了n=6k+1,6k+2(k=1,2,3,4,5)的哈密顿圈分解.本文在吉日木图提出的边划分方法的基础上继续研究,得到了完全3-一致超图K_n~((3))的哈密顿圈分解的算法,由此得到了n=6k+2,6k+4(k=1,2,3,4,5,6,7),n=6k+5(k=1,2,3,4,5,6)时的圈分解.这一结果将Meszka-Rosa关于K_n~((3))的哈密顿圈分解结果从n≤32提高到了n≤46(n≠43).     

On the problem op n-stationary centers

Among the various problems of celestial mechanics related to the n-body problem, a certain amount of interest attaches to the specific situation wherein a passive gravitational point mass M moves under the assumption that the relative disposition of the other active gravitational masses experiences no large changes.

If the attracting masses are regarded as points and if changes in the relative disposition of the attracting bodies are neglected, one arrives at the problem of the motion of the point M in a field produced by n-stationary attracting centers (the point M here represents the (n+l)-th body).

In addition to the problem of central motion (n = 1), soluble dynamics problems of this category include Euler's case [1] of two (n= 2) stationary Newtonian attracting centers.

This problem, which for a long time was of solely theoretical Interest as an example of an integrable Liouville system [2], has recently been attracting attention in connection with the mechanics of artificial satellites, particularly after it was shown that the potential of an oblate spheroid can be approximated by the potential of two specifically chosen stationary Newtonian attracting centers [3 and 4].

The solution of the problem for n-attracting centers for n ≥ 3 is unknown, except for a single special case of three centers pointed out by Lagrange and considered In greater detail by J.A. Serre [5].

We shall concern ourselves here with problems on the existence of periodic trajectories in the case of n-attracting centers. An arbitrary and not necessarily Newtonian gravitational law will be assumed.

Our analysis is based on the theory of quasiindices of singular force field points as set forth in [60].     

Classification of Some Optimal Ternary Linear Codes of Small Length

A classification is given of some optimal ternary linear codes of small length. Dimension 2 is classified for every minimum distance. Dimension 3, 4 and 5 is classified up to minimum distance 12. For higher dimension a classification is given where possible.