Lévy processes and quasi-shuffle algebras

We investigate the algebra of repeated integrals of semimartingales. We prove that a minimal family of semimartingales generates a quasi-shuffle algebra. In essence, to fulfil the minimality criterion, first, the family must be a minimal generator of the algebra of repeated integrals generated by its elements and by quadratic covariation processes recursively constructed from the elements of the family. Second, recursively constructed quadratic covariation processes may lie in the linear span of previously constructed quadratic covariation processes and of the family, but may not lie in the linear span of repeated integrals of these. We prove that a finite family of independent Lévy processes that have finite moments generates a minimal family. Key to the proof are the Teugels martingales and a strong orthogonalization of them. We conclude that a finite family of independent Lévy processes forms a quasi-shuffle algebra. We discuss important potential applications to constructing efficient numerical methods for the strong approximation of stochastic differential equations driven by Lévy processes.     

Conservation of integrals and symplectic structure in the integration of differential equations by multistep methods

Summary We consider the question of whether multistep methods inherit in some sense quadratic first integrals possessed by the differential system being integrated. We also investigate whether, in the integration of Hamiltonian systems, multistep methods conserve the symplectic structure of the phase space.     

Spectra and semigroup smoothing for non-elliptic quadratic operators

We study non-elliptic quadratic differential operators. Quadratic differential operators are non-selfadjoint operators defined in the Weyl quantization by complex-valued quadratic symbols. When the real part of their Weyl symbols is a non-positive quadratic form, we point out the existence of a particular linear subspace in the phase space intrinsically associated to their Weyl symbols, called a singular space, such that when the singular space has a symplectic structure, the associated heat semigroup is smoothing in every direction of its symplectic orthogonal space. When the Weyl symbol of such an operator is elliptic on the singular space, this space is always symplectic and we prove that the spectrum of the operator is discrete and can be described as in the case of global ellipticity. We also describe the large time behavior of contraction semigroups generated by these operators.     

La formule de Plancherel pour les groupes de Lie presque algébriques réels

We give a proof of the Plancherel formula for real almost algebraic groups in the philosophy of the orbit method, following the lines of the one given by M. Duflo and M. Vergne for simply connected semisimple Lie groups. Main ingredients are: (1) Harish-Chandra's descent method which, interpreting Plancherel formula as an equality of semi-invariant generalized functions, allows one to reduce it to a neighbourhood of zero in the Lie algebra of the centralizer of any elliptic element; (2) character formula for representations constructed by M. Duflo, we recently proved; (3) Poisson-Plancherel formula near elliptic elements s in good position, a generalization of the classical Poisson summation formula expressing the Fourier transform of the sum of a series of Harish-Chandra type elliptic orbital integrals in the Lie algebra centralizing s as a generalized function supported on a set of admissible regular forms in the dual of this Lie algebra.     

Normal forms and versal deformations of linear Hamiltonian systems

Equivalence classes of time independent, linear, real Hamiltonian systems can be identified, up to canonical transformations, with the orbits of the adjoint action of the real symplectic group on its Lie algebra. A new set of representatives, also called normal forms, for these orbits is given. Versal deformations of systems in normal form are constructed. Applications of versal deformations to the study of bifurcations of linear systems with small codimension are indicated.     

Strongly stable real infinitesimally symplectic mappings

We prove that a mapA εsp(σ,R), the set of infinitesimally symplectic maps, is strongly stable if and only if its centralizerC(A) insp(σ,R) contains only semisimple elements. Using the theorem that everyB insp(σ,R) close toA is conjugate by a real symplectic map to an element ofC(A), we give a new proof of the openness of the set of strongly stable maps. Then we prove that the set of strongly stable maps is the interior of the set of all infinitesimally symplectic maps with purely imaginary or zero eigenvalues, and the connected components of this set are described. Finally, we give a new proof of the analytic conjugacy theorem for an analytic curve through a given strongly stable map.     

哈密尔顿系统的有限元法

利用常微分方程的连续有限元法,结合函数的M-型展开,对非线性哈密尔顿系统证明了连续一、二次有限元分在3阶量、5阶量意义下近似保辛,且保持能量守恒.在数值实验中结合庞加莱截面,哈密尔顿混沌数值试验结果与理论相吻合.     

Metric Lie algebras and quadratic extensions

The present paper contains a systematic study of the structure of metric Lie algebras, i.e., finite-dimensional real Lie algebras equipped with a nondegenerate invariant symmetric bilinear form. We show that any metric Lie algebra g without simple ideals has the structure of a so called balanced quadratic extension of an auxiliary Lie algebra l by an orthogonal l-module a in a canonical way. Identifying equivalence classes of quadratic extensions of l by a with a certain cohomology set H²_Q(l,a), we obtain a classification scheme for general metric Lie algebras and a complete classification of metric Lie algebras of index 3.     

3-李代数的辛结构

研究了3-李代数和度量3-李代数的辛结构.对任意3-李代数L,构造了无限多个度量辛3-李代数.证明了度量3-李代数(A,B)是度量辛3-李代数的充要条件,即存在可逆导子D,使得D∈Der_B(A).同时证明了每一个度量辛3-李代数(A,B,ω)是度量辛3-李代数(A,B,ω)的T_θ~*-扩张.最后,利用度量辛3-李代数经过特殊导子的双扩张得到了新的度量辛3-李代数.     

Caractérisation spectrale des algèbres de Banach de dimension finie

Let f be an analytic function mapping a domain in $\text{C}$ into a complex Banach algebra. Using potential theory and a new result on almost continuity of the spectrum, which extends the theorem of Newburgh, we prove that either the set of λ such that the spectrum of f(λ) is finite is of outer capacity zero, or there exists an integer n such that the spectrum of f(λ) has at most n elements for every λ. From this we get extensions of a theorem given, in the complex case, by Kaplansky in 1954 and Hirschfeld and Johnson in 1972. More precisely we show that, if the spectrum is finite for every element of an open set of a real algebra or of the set of Hermitian elements of an algebra with an involution, then the quotient of this algebra by its radical is finite-dimensional.     

The Power Matrix, coadjoint action and quadratic differentials

The coefficients of a quadratic differential which is changing under the Loewner flow satisfy a well-known differential system studied by Schiffer, Schaeffer and Spencer, and others. By work of Roth, this differential system can be interpreted as Hamilton's equations. We apply the power matrix to interpret this differential system in terms of the coadjoint action of the matrix group on the dual of its Lie algebra. As an application, we derive a set of integral invariants of Hamilton's equations which is in a certain sense complete. In function theoretic terms, these are expressions in the coefficients of the quadratic differential and Loewner map which are independent of the parameter in the Loewner flow.     

A representation of solution of stochastic differential equations

We prove that the logarithm of the formal power series, obtained from a stochastic differential equation, is an element in the closure of the Lie algebra generated by vector fields being coefficients of equations. By using this result, we obtain a representation of the solution of stochastic differential equations in terms of Lie brackets and iterated Stratonovich integrals in the algebra of formal power series.     

A note on ideals in synaptic algebras

The notion of a synaptic algebra was introduced by David Foulis. Synaptic algebras unite the notions of an order-unit normed space, a special Jordan algebra, a convex effect algebra and an orthomodular lattice. In this note we study quadratic ideals in synaptic algebras which reflect its Jordan algebra structure. We show that projections contained in a quadratic ideal from a p-ideal in the orthomodular lattice of projections in the synaptic algebra and we find a characterization of those quadratic ideals which are generated by their projections.     

The reducibility of the equations of free motion of a complex mechanical system

A mechanical system consists of an unchangeable rigid body (a carrier) and a subsystem whose configuration and composition may vary with time (the motion of its elements relative to the carrier is given). The free motion of the system in a uniform gravitational field is investigated, on the assumption that there is no dynamic symmetry. Necessary and sufficient conditions are derived for the existence of two integrals, each quadratic in the components of the absolute angular velocity of the carrier. Lt is shown that the initial dynamical system can be reduced to an autonomous gyrostat system if and only if the motion has these two quadratic integrals; the explicit form of a linear transformation to the autonomous system is indicated. The explicit form of the integrals and conditions for their existence are obtained. Examples of motion with two quadratic integrals are considered.     

Lie algebras admitting a unique quadratic structure

We prove that any Lie algebra g over a field K of characteristic zero admitting a unique up to a constant quadratic structure is necessarily a simple Lie algebra. If the field K is algebraically closed, such condition is also sufficient.

Further, a real Lie algebra g admits a unique quadratic structure if and only if its complexification g^C is a simple Lie algebra over C     

UNITARY GRASSMANNIANS OF DIVISION ALGEBRAS

We consider a central division algebra over a separable quadratic extension of a base field endowed with a unitary involution and prove 2-incompressibility of certain varieties of isotropic right ideals of the algebra. The remaining related projective homogeneous varieties are shown to be 2-compressible in general. Together with [17], where a similar issue for orthogonal and symplectic involutions has been treated, the present paper completes the study of Grassmannians of isotropic right ideals of division algebras.     

Linearization of Hamiltonian and Gradient Systems

Necessary and sufficient conditions are derived in order totransform a nonlinear Hamiltonian or gradient system by a changeof coordinates of its state space into a linear Hamiltonianor gradient system. It is shown that such a transformation necessarilyrespects the symplectic or metrical structure. The conditionsare given in terms of the observation space of the system, whichin the Hamiltonian case is a Lie algebra and in the gradientcase a symmetric algebra.     

CCC forcing and splitting reals

The results of this paper were motivated by a problem of Prikry who asked if it is relatively consistent with the usual axioms of set theory that every nontrivial ccc forcing adds a Cohen or a random real. A natural dividing line is into weakly distributive posets and those which add an unbounded real. In this paper I show that it is relatively consistent that every nonatomic weakly distributive ccc complete Boolean algebra is a Maharam algebra, i.e. carries a continuous strictly positive submeasure. This is deduced from theP-ideal dichotomy, a statement which was first formulated by Abraham and Todorcevic [AT] and later extended by Todorcevic [T]. As an immediate consequence of this and the proof of the consistency of theP-ideal dichotomy we obtain a ZFC result which says that every absolutely ccc weakly distributive complete Boolean algebra is a Maharam algebra. Using a previous theorem of Shelah [Sh1] it also follows that a modified Prikry conjecture holds in the context of Souslin forcing notions, i.e. every nonatomic ccc Souslin forcing either adds a Cohen real or its regular open algebra is a Maharam algebra. Finally, I also show that every nonatomic Maharam algebra adds a splitting real, i.e. a set of integers which neither contains nor is disjoint from an infinite set of integers in the ground model. It follows from the result of [AT] that it is consistent relative to the consistency of ZFC alone that every nonatomic weakly distributive ccc forcing adds a splitting real.     

Stochastic discrete Hamiltonian variational integrators

Variational integrators are derived for structure-preserving simulation of stochastic Hamiltonian systems with a certain type of multiplicative noise arising in geometric mechanics. The derivation is based on a stochastic discrete Hamiltonian which approximates a type-II stochastic generating function for the stochastic flow of the Hamiltonian system. The generating function is obtained by introducing an appropriate stochastic action functional and its corresponding variational principle. Our approach permits to recast in a unified framework a number of integrators previously studied in the literature, and presents a general methodology to derive new structure-preserving numerical schemes. The resulting integrators are symplectic; they preserve integrals of motion related to Lie group symmetries; and they include stochastic symplectic Runge–Kutta methods as a special case. Several new low-stage stochastic symplectic methods of mean-square order 1.0 derived using this approach are presented and tested numerically to demonstrate their superior long-time numerical stability and energy behavior compared to nonsymplectic methods.     

位势问题边界元法中几乎奇异积分的正则化

将一种通用算法应用于平面位势问题边界元法中近边界点几乎奇异积分的正则化。对线性单元,位势问题近边界点的几乎强和超奇异积分可归纳为两种形式。通过分部积分,将引起奇异的积分元素变换到积分号之外,从而对这两种积分分别给出了无奇异的正则化计算公式。除了线性元,二次元也应用于该算法。与近边界点临近的二次单元划分为两段线性单元,该算法仍然适用。算例证明了方法的有效性和精确性。对曲线边界问题,联合二次元和线性元可提高计算结果精确度。