ISSN 1439-8516 CN 11-2039/O1
Complex periodical sequences with lower autocorrelation values are used in CDMA communication systems and cryptography. In this paper we present new nonexistence results on perfect p-ary sequences and almost p-ary sequences and related difference sets by using some knowledge on cyclotomic fields and their subfields.
Given a domain Ω ⊂ Rn, let λ > 0 be an eigenvalue of the elliptic operator L :=Σi,jn =1 ∂/∂xj(aij ∂/∂xj) on Ω for Dirichlet condition. For a function ƒ ∈ L2(Ω), it is known that the linear resonance equation Lu + λu = ƒ in Ω with Dirichlet boundary condition is not always solvable. We give a new boundary condition Pλ(u|∂Ω) = g, called to be projective Dirichlet condition, such that the linear resonance equation always admits a unique solution u being orthogonal to all of the eigenfunctions corresponding to λ which satisfies ‖u‖2,2 ≤ C(‖ƒ‖2 +‖g‖2,2) under suitable regularity assumptions on ∂Ω and L, where C is a constant depends only on n, Ω, and L. More a priori estimates, such as W2,p-estimates and the C2,α-estimates etc., are given also. This boundary condition can be viewed as a generalization of the Dirichlet condition to resonance equations and shows its advantage when applying to nonlinear resonance equations. In particular, this enables us to find the new indicatrices with vanishing mean (Cartan) torsion in Minkowski geometry. It is known that the geometry of indicatries is the foundation of Finsler geometry.
The Conway potential function (CPF) for colored links is a convenient version of the multivariable Alexander-Conway polynomial. We give a skein characterization of CPF, much simpler than the one by Murakami. In particular, Conway's “smoothing of crossings” is not in the axioms. The proof uses a reduction scheme in a twisted group-algebra PnBn, where Bn is a braid group and Pn is a domain of multi-variable rational fractions. The proof does not use computer algebra tools. An interesting by-product is a characterization of the Alexander-Conway polynomial of knots.
Recently, Cristofaro-Gardiner and Hutchings proved that there exist at least two closed characteristics on every compact star-shaped hypersuface in R4. Then Ginzburg, Hein, Hryniewicz, and Macarini gave this result a second proof. In this paper, we give it a third proof by using index iteration theory, resonance identities of closed characteristics and a remarkable theorem of Ginzburg et al.
We determine the maximum order Eg of finite groups G acting on the closed surface Σg of genus g which extends over (S3,Σg) for all possible embeddings Σg → S3, where g > 1.
We study space-like self-shrinkers of dimension n in pseudo-Euclidean space Rmm+n with index m. We derive drift Laplacian of the basic geometric quantities and obtain their volume estimates in pseudo-distance function. Finally, we prove rigidity results under minor growth conditions in terms of the mean curvature or the image of Gauss maps.
We give a summary on the recent development of chaos theory in topological dynamics, focusing on Li-Yorke chaos, Devaney chaos, distributional chaos, positive topological entropy, weakly mixing sets and so on, and their relationships.
We first determine in this paper the structure of the generalized Fitting subgroup F*(G) of the finite groups G all of whose defect groups (of blocks) are conjugate under the automorphism group Aut(G) to either a Sylow p-subgroup or a fixed p-subgroup of G. Then we prove that if a finite group L acts transitively on the set of its proper Sylow p-intersections, then either L/Op(L) has a T.I. Sylow p-subgroup or p = 2 and the normal closure of a Sylow 2-subgroup of L/O2(L) is 2-nilpotent with completely descripted structure. This solves a long-open problem. We also obtain some generalizations of the classic results by Isaacs and Passman on half-transitivity.
In geometry, there are several challenging problems studying numbers associated to convex bodies. For example, the packing density problem, the kissing number problem, the covering density problem, the packing-covering constant problem, Hadwiger's covering conjecture and Borsuk's partition conjecture. They are fundamental and fascinating problems about the same objects. However, up to now, both the methodology and the technique applied to them are essentially different. Therefore, a common foundation for them has been much expected. By treating problems of these types as functionals defined on the spaces of n-dimensional convex bodies, this paper tries to create such a foundation. In particular, supderivatives for these functionals will be studied.
