Families of type III0 ergodic transformations in distinct orbit equivalent classes

A new isomorphism invariant of certain measure preserving flows, using sequences of integers, is introduced. Using this invariant, we are able to construct large families of type III₀ systems which are not orbit equivalent. In particular we construct an uncountable family of nonsingular ergodic transformations, each having an associated flow that is approximately transitive (and therefore of zero entropy), with the property that the transformations are pairwise not orbit equivalent.     

Geometric Construction of Linear Complex of Planes of <Emphasis Type="Italic">B</Emphasis><Subscript>3</Subscript> Type

Using invariant geometric images of a trivector of the type (884; 400), we construct its basic group of automorphisms. We formulate and prove a theorem on necessary and sufficient conditions for determining all planes of a linear complex associated with a trivector of the given type up to linear transformations of its automorphism group. Proving the theorem, we find all kinds of singular lines and construct the polar hyperplanes for nonsingular lines.     

Some designs and codes invariant under the groups <Emphasis Type="Italic">S</Emphasis><Subscript>9</Subscript> and <Emphasis Type="Italic">A</Emphasis><Subscript>8</Subscript>

We examine some designs and binary codes constructed from the primitive permutation representations of the groups PSL ₂(8) and PSL ₂(9). For PSL ₂(8) of degree 36, we construct a design and its code with the automorphism groups PSL ₂(8) and S ₉, respectively. For PSL ₂(8) of degree 36 and PSL ₂(9) of degree 15, we construct some designs and its codes invariant under the groups S ₉ and A ₈, respectively. The weight distribution and the dual of these codes are determined. By considering the action of automorphism groups on some of these codes, we obtain the structure of the stabilizer for every codeword and construct some designs such that S ₉ or A ₈ act primitively on them.      

On the properties of solutions of equations in the <Emphasis Type="Italic">K</Emphasis><Subscript>2</Subscript> hierarchy

For the sixth-order equation of the K ₂ hierarchy, we establish conditions for the existence of families of polar solutions, construct polynomial Hamiltonians and Bäcklund transformations, represent meromorphic solutions via a ratio of special entire functions, and give a criterion for the existence of rational solutions.     

Note on the Casson-Gordon invariant of a satellite knot

In this paper we establish a relation between an appropriate version of the Casson-Gordon invariant of a satellite knot and those of its orbit and companion. We note that in some cases the contribution from, the companion falls. This gives a way to construct algebraically but not smoothly slice knots. This article was processed by the author using theLaT_EX style filecljouri from Springer-Verlag.     

INVARIANTS UNDER STABLE EQUIVALENCES OF MORITA TYPE

The aim of this article is to study some invariants of associative algebras under stable equivalences of Morita type.First of all,we show that,if two finite-dimensional selfinjective k-algebras are sta...     

Integral geometry under G<Subscript>2</Subscript> and Spin(7)

A Hadwiger-type theorem for the exceptional Lie groups G₂ and Spin(7) is proved. The algebras of G₂ or Spin(7) invariant, translation invariant continuous valuations are both of dimension 10. Geometrically meaningful bases are constructed and the algebra structures are computed. Finally, the kinematic formulas for these groups are determined.     

(<Emphasis Type="Italic">q</Emphasis><Subscript>1</Subscript>, <Emphasis Type="Italic">q</Emphasis><Subscript>2</Subscript>)-quasimetrics bi-Lipschitz equivalent to 1-quasimetrics

We prove that the conditions of (q₁, 1)- and (1, q₂)-quasimertricity of a distance function ρ are sufficient for the existence of a quasimetric bi-Lipschitz equivalent to ρ. It follows that the Box-quasimetric defined with the use of basis vector fields of class C₁ whose commutators at most sum their degrees is bi-Lipschitz equivalent to some metric. On the other hand, we show that these conditions are not necessary. We prove the existence of (q₁, q₂)-quasimetrics for which there are no Lipschitz equivalent 1-quasimetrics, which in particular implies another proof of a result by V. Schröder.     

Polyvector representations of GL<Subscript>n</Subscript>

In the present paper, we characterize ⋀ⁿ(GL(n, R)) over any commutative ring R as the connected component of the stabilizer of the Plücker ideal. This folk theorem is classically known for algebraically closed fields and should also be well known in general. However, we are not aware of any obvious reference, so we produce a detailed proof, which follows a general scheme developed by W.C.Waterhouse. The present paper is a technical preliminary to a subsequent paper, where we construct the decomposition of transvections in polyvector representations of GL _n. Bibliography: 50 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 338, 2006, pp. 69–97.     

Examples of algebraic groups of type <Emphasis Type="Italic">G</Emphasis> <Subscript>2</Subscript> having the same maximal tori

Answering a question of A. Rapinchuk, we construct examples of non-isomorphic semisimple algebraic groups H₁ and H₂ of type G₂ having coherently equivalent systems of maximal k-tori.     

Invariants of matrix pairs over discrete valuation rings and Littlewood-Richardson fillings

Let M and N be two r×r matrices of full rank over a discrete valuation ring R with residue field of characteristic zero. Let P,Q and T be invertible r×r matrices over R. It is shown that the orbit of the pair (M,N) under the action (M,N)?(PMQ^-1,QNT^-1) possesses a discrete invariant in the form of Littlewood-Richardson fillings of the skew shape λ/μ with content ν, where μ is the partition of orders of invariant factors of M, ν is the partition associated to N, and λ the partition of the product MN. That is, we may interpret Littlewood-Richardson fillings as a natural invariant of matrix pairs. This result generalizes invariant factors of a single matrix under equivalence, and is a converse of the construction in Appleby (1999) [1], where Littlewood-Richardson fillings were used to construct matrices with prescribed invariants. We also construct an example, however, of two matrix pairs that are not equivalent but still have the same Littlewood-Richardson filling. The filling associated to an orbit is determined by special quotients of determinants of a matrix in the orbit of the pair.     

Maximal Poisson Commutative subalgebras for truncated parabolic subalgebras of maximal index in sl<Subscript>n</Subscript>

We show that several properties of the semisimple algebras carry over to a certain family of parabolic subalgebras of maximal index in sl_n. More precisely we prove an analogue of Kostant's slice theorem [B. Kostant, Amer. J. Math. 85 (1963), 327-404] for these algebras and construct a maximal Poisson commutative subalgebra in the symmetric algebra, following the theory presented in [A.S. Mishchenko and A.T. Fomenko, Math. USSR-Izv. 12 (1978), 371-389]. These results are quite remarkable since these algebras do not admit appropriate sl₂-triples.     

Lie group analysis of two-dimensional variable-coefficient Burgers equation

The modern group analysis of differential equations is used to study a class of two-dimensional variable coefficient Burgers equations. The group classification of this class is performed. Equivalence transformations are also found that allow us to simplify the results of classification and to construct the basis of differential invariants and operators of invariant differentiation. Using equivalence transformations, reductions with respect to Lie symmetry operators and certain non-Lie ansätze, we construct exact analytical solutions for specific forms of the arbitrary elements. Finally, we classify the local conservation laws.     

Author index for volume 75

In this paper we present a method for constructing invariant solutions of partial differential equations. Using a computerprogram we derive a simple class of transformations including similarity transformations which leaves invariant a given hydrodynamical equation. Methods from differential geometry will enable us to construct ordinary differential equations leading to invariant solutions of a given equation.     

Equivariant normal form for nondegenerate singular orbits of integrable Hamiltonian systems

We consider an integrable Hamiltonian system with n degrees of freedom whose first integrals are invariant under the symplectic action of a compact Lie group G. We prove that the singular Lagrangian foliation associated to this Hamiltonian system is symplectically equivalent, in a G-equivariant way, to the linearized foliation in a neighborhood of a compact singular nondegenerate orbit. We also show that the nondegeneracy condition is not equivalent to the nonresonance condition for smooth systems.     

对称连续分布函数的最优不变估计

该文考虑了未知对称连续分布函数的不变估计问题.连续分布函数在单调变换群下是不变的^[1], 但这个变换群不能保证对称分布函数的不变性.于是, 所要研究的判决问题在单调变换群下不再是不变的. 为了保证判决问题不变性, 考虑一个新的变换群—单调奇变换群, 它确保了所研究的判决问题的不变性.注意到对称分布函数零点的特殊性质, 即, 对任一对称分布函数F, 均有F(0)=1/2,通过视零点为一伪观察值, 得到了所有的非随机化不变估计, 并在不变估计中找到了最优不变估计.     

Complex dynamics in physical pendulum equation with suspension axis vibrations

The physical pendulum equation with suspension axis vibrations is investigated. By using Melnikov＇s method, we prove the conditions for the existence of chaos under periodic perturbations. By using second-order averaging method and Melinikov＇s method, we give the conditions for the existence of chaos in an averaged system under quasi-periodic perturbations for Ω = nω ＋ εv, n = 1 - 4, where ν is not rational to ω. We are not able to prove the existence of chaos for n = 5 - 15, but show the chaotic behavior for n = 5 by numerical simulation. By numerical simulation we check on our theoretical analysis and further exhibit the complex dynamical behavior, including the bifurcation and reverse bifurcation from period-one to period-two orbits; the onset of chaos, the entire chaotic region without periodic windows, chaotic regions with complex periodic windows or with complex quasi-periodic windows; chaotic behaviors suddenly disappearing, or converting to period-one orbit which means that the system can be stabilized to periodic motion by adjusting bifurcation parameters α, δ, f0 and Ω; and the onset of invariant torus or quasi-periodic behaviors, the entire invariant torus region or quasi-periodic region without periodic window, quasi-periodic behaviors or invariant torus behaviors suddenly disappearing or converting to periodic orbit; and the jumping behaviors which including from period- one orbit to anther period-one orbit, from quasi-periodic set to another quasi-periodic set; and the interleaving occurrence of chaotic behaviors and invariant torus behaviors or quasi-periodic behaviors; and the interior crisis; and the symmetry breaking of period-one orbit; and the different nice chaotic attractors. However, we haven＇t find the cascades of period-doubling bifurcations under the quasi-periodic perturbations and show the differences of dynamical behaviors and technics of research between the periodic perturbations and quasi-periodic perturbations.     

一般非齐次非线性扩散方程的等价变换和高维不变子空间

结合压力变换和不变子空间方法中的等价变换,给出了一般非齐次非线性扩散方程的等价方程,并给出了等价方程的高维不变子空间.由此构造了一般非齐次非线性扩散方程的广义分离变量解,并给出了几个例子解释这个过程.     

On Four-Weight Spin Models and their Gauge Transformations

We study the four-weight spin models (W₁, W₂, W₃, W₄) introduced by Eiichi and Etsuko Bannai (Pacific J. of Math, to appear). We start with the observation, based on the concept of special link diagram, that two such spin models yield the same link invariant whenever they have the same pair (W₁, W₃), or the same pair (W₂, W₄). As a consequence, we show that the link invariant associated with a four-weight spin model is not sensitive to the full reversal of orientation of a link. We also show in a similar way that such a link invariant is invariant under mutation of links.Next, we give an algebraic characterization of the transformations of four-weight spin models which preserve W₁, W₃ or preserve W₂, W₄. Such gauge transformations correspond to multiplication of W₂, W₄ by permutation matrices representing certain symmetries of the spin model, and to conjugation of W₁, W₃ by diagonal matrices. We show for instance that up to gauge transformations, we can assume that W₁, W₃ are symmetric.Finally we apply these results to two-weight spin models obtained as solutions of the modular invariance equation for a given Bose-Mesner algebra B and a given duality of B. We show that the set of such spin models is invariant under certain gauge transformations associated with the permutation matrices in B. In the case where B is the Bose-Mesner algebra of some Abelian group association scheme, we also show that any two such spin models (which generalize those introduced by Eiichi and Etsuko Bannai in J. Alg. Combin. 3 (1994), 243–259) are related by a gauge transformation. As a consequence, the link invariant associated with such a spin model depends only trivially on the link orientation.