High strong order methods for non-commutative stochastic ordinary differential equation systems and the Magnus formula

In recent years considerable attention has been paid to the numerical solution of stochastic ordinary differential equations (SODEs), as SODEs are often more appropriate than their deterministic counterparts in many modelling situations. However, unlike the deterministic case numerical methods for SODEs are considerably less sophisticated due to the difficulty in representing the (possibly large number of) random variable approximations to the stochastic integrals. Although Burrage and Burrage [High strong order explicit Runge-Kutta methods for stochastic ordinary differential equations, Applied Numerical Mathematics 22 (1996) 81-101] were able to construct strong local order 1.5 stochastic Runge-Kutta methods for certain cases, it is known that all extant stochastic Runge-Kutta methods suffer an order reduction down to strong order 0.5 if there is non-commutativity between the functions associated with the multiple Wiener processes. This order reduction down to that of the Euler-Maruyama method imposes severe difficulties in obtaining meaningful solutions in a reasonable time frame and this paper attempts to circumvent these difficulties by some new techniques. An additional difficulty in solving SODEs arises even in the Linear case since it is not possible to write the solution analytically in terms of matrix exponentials unless there is a commutativity property between the functions associated with the multiple Wiener processes. Thus in this present paper first the work of Magnus [On the exponential solution of differential equations for a linear operator, Communications on Pure and Applied Mathematics 7 (1954) 649-673] (applied to deterministic non-commutative Linear problems) will be applied to non-commutative linear SODEs and methods of strong order 1.5 for arbitrary, linear, non-commutative SODE systems will be constructed - hence giving an accurate approximation to the general linear problem. Secondly, for general nonlinear non-commutative systems with an arbitrary number (d) of Wiener processes it is shown that strong local order I Runge-Kutta methods with d + 1 stages can be constructed by evaluated a set of Lie brackets as well as the standard function evaluations. A method is then constructed which can be efficiently implemented in a parallel environment for this arbitrary number of Wiener processes. Finally some numerical results are presented which illustrate the efficacy of these approaches. (C) 1999 Elsevier Science B.V. All rights reserved.

Halun häilyvät rajat : Magnus Enckellin maalausten maskuliinisuuksien ja feminiinisyyksien representaatioista ja itsen luomisesta

My PhD-thesis The uneasy borders of desire Magnus Enckell's representations of masculinities and femininities and the question how to create the self concentrates on the works of Finnish fin-de-siècle artist Magnus Enckell (1870-1925). My thesis deals with representations of masculinities, femininities, sexualities and different identity-positions. My research is about questions concerning representational ways of melancholy, androgyny, narcissism, themes of Golden Age and Double in Enckell s ouvre. These themes are analyzed by contextualizing them with different, but intersecting, discourses of varied scientific, artistic and occult ideas in the fin-de-siècle. The main point is analyze how the subject is constructed in both Foucauldian and Freudian sense and what one has to know about oneself. My approaches are based on ideas expressed in different discourses as queer-theory, Michel Foucault s genealogical epistemology and knowledge-power theory, psychoanalysis, art history and visual culture studies. My starting point lays is Foucault s idea expressed in his The History of Sexuality that the constitution of homosexual or as well as heterosexual subject inaugurates possibilities for transgressive activities e.g. by giving own voice to the sexualized subject. My main thesis is to suggest that Enckell s works in their multiple and ambiguous ways construct a phantasmatic position for viewer who may identify oneself to different desires, may construct or deconstruct a sexual identity for oneself or try to define the truth about oneself. Enckell s works should be considered as a contradictory processes which both seduce person to construct an identity and as well as lure person to pursue for the deconstruction of specific and permanent identity by celebrating the ambiguousness and discontinuity in one s identity. I m suggesting that the gazing subject feels pleasure in finding one s identity but the one must face the exposure of the melancholic structure which forms the basis of sexual desire. The subject may try to resolve one s melancholy by creating a phantasy about the original and unisexual being where desires, sexualities, phantasies and identities haven t been diverged. This can be fantasized in terms of art which forms a double for the melancholic subject who is in this limited and imaginary way able to forget for a while one s existential solitude.

Painting of Felix Mendelssohn-Bartholdy by Eduard Magnus Portraits Men

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Painting of Freiherr Martin von Magnus Family Portraits Family

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Reproduction of painted portrait by Eduard Magnus of Mrs. Karl Steffeck Portraits Women

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A multistep transversal linearization (MTL) method in non-linear dynamics through a Magnus characterization

A new form of a multi-step transversal linearization (MTL) method is developed and numerically explored in this study for a numeric-analytical integration of non-linear dynamical systems under deterministic excitations. As with other transversal linearization methods, the present version also requires that the linearized solution manifold transversally intersects the non-linear solution manifold at a chosen set of points or cross-section in the state space. However, a major point of departure of the present method is that it has the flexibility of treating non-linear damping and stiffness terms of the original system as damping and stiffness terms in the transversally linearized system, even though these linearized terms become explicit functions of time. From this perspective, the present development is closely related to the popular practice of tangent-space linearization adopted in finite element (FE) based solutions of non-linear problems in structural dynamics. The only difference is that the MTL method would require construction of transversal system matrices in lieu of the tangent system matrices needed within an FE framework. The resulting time-varying linearized system matrix is then treated as a Lie element using Magnus’ characterization [W. Magnus, On the exponential solution of differential equations for a linear operator, Commun. Pure Appl. Math., VII (1954) 649–673] and the associated fundamental solution matrix (FSM) is obtained through repeated Lie-bracket operations (or nested commutators). An advantage of this approach is that the underlying exponential transformation could preserve certain intrinsic structural properties of the solution of the non-linear problem. Yet another advantage of the transversal linearization lies in the non-unique representation of the linearized vector field – an aspect that has been specifically exploited in this study to enhance the spectral stability of the proposed family of methods and thus contain the temporal propagation of local errors. A simple analysis of the formal orders of accuracy is provided within a finite dimensional framework. Only a limited numerical exploration of the method is presently provided for a couple of popularly known non-linear oscillators, viz. a hardening Duffing oscillator, which has a non-linear stiffness term, and the van der Pol oscillator, which is self-excited and has a non-linear damping term.

An analogue of the Magnus problem for associative algebras

We prove an analogue of Magnus theorem for associative algebras without unity over arbitrary fields. Namely, if an algebra is given by $n+k$ generators and $k$ relations and has an $n$-element system of generators, then this algebra is a free algebra of rank $n$.

Magnus Ducatus Finlandiae Nova et accurata delineatio

Mittakaavaa ei määritelty.

S. Gregorius Magnus, Moralia in Job, libri 1-16.

Claude Joly, chantre de Notre-Dame, a inscrit sa signature au f. 2v (XVIIe s.). Ce ms. a appartenu à la cathédrale Notre-Dame de Paris, dont il porte l'ex-libris "A la bibliotheque de l'Eglise de Paris" (XVIIe s., f. 2v). Notre-Dame.

1.° Vita sancti Martini, Confessoris : authore Sulpitio Severo. — 2.° Vita sancti Gregorii, Papae. — 3.° Passio sanctorum Tiburtii et Valeriani. — 4.° Vita sanctae Waltburgae, Virginis. — 5.° Gesta Domini nostri Jesu Christi quae invenit Theodosius Magnus, Imperator, in Hierusalem, in Praetorio Pontii Pilati : sive evangelium Nicodemi. — 6.° Anonymi sermo de vindicta Domini. — 7.° Sermo de sancto Joachim. — 8.° Sermo de assumptione beatae Mariae Virginis. — 9.° Fragmentum de inventione sanctae Crucis. — 10.° Vita Adae et Evae. — 11.° Vita sancti Basilii Magni : authore Amphilochio, Iconiensi Episcopo. — 12.° Vita sancti Eligii : authore Dadone, Rothomagensi Episcopo. — 13.° Vita sancti Joannis Evangelistae : authore Miletone, Laodicensi. — 14.° Vita sancti Arnulfi, Episcopi. — 15.° Anonymus de palatio, de quo narratur in passione sancti Thomae, Apostoli. — 16.° Passio sancti Andreae, Apostoli.

Tellerianus

Album amicorum du professeur Magnus GESNER, de Bruxelles.

Figures et blasons dessinés et peints.