Phase Space Transformations of Gaussian Diffusions

It is shown that for Gaussian diffusions, the transformation back to Brownian motion, usually accomplished via the Girsanov (or Feynman–Kac) formula and time-shift, can be obtained by a classical canonical, i.e. symplectic, transformation in phase space. The method is based on constants of motion, in this case the Wronskian. Similar transformations for general diffusions are briefly discussed.     

The Darboux transformation of some Manakov systems

The Darboux transformation is explicitly constructed for a coupled system of an arbitrary number of focusing or defocusing nonlinear Schrödinger fields with cubic nonlinearity (i.e., a Manakov system with an arbitrary number of fields). An immediate consequence is an explicit formula for an auto-Bäcklund transformation for the Manakov system.     

Transformation Kernel Density Estimation With Applications

One of the main objectives of this article is to derive efficient nonparametric estimators for an unknown density fX. It is well known that the ordinary kernel density estimator has, despite several good properties, some serious drawbacks. For example, it suffers from boundary bias and it also exhibits spurious bumps in the tails. We propose a semiparametric transformation kernel density estimator to overcome these defects. It is based on a new semiparametric transformation function that transforms data to normality. A generalized bandwidth adaptation procedure is also developed. It is found that the newly proposed semiparametric transformation kernel density estimator performs well for unimodal, low, and high kurtosis densities. Moreover, it detects and estimates densities with excessive curvature (e.g., modes and valleys) more effectively than existing procedures. In conclusion, practical examples based on real-life data are presented.     

Micromechanical modeling of bainitic phase transformation

We develop a micromechanical material model for phase transformation from austenite to bainite for a polycrystalline low alloys steel. In this material (e.g. 51CrV4) the phase changes from austenite to perlite-ferrite, bainite or martensite, respectively. This work is concerned with phase transformation between austenite and n-bainite variants in differently orientated grains. The characteristic features of bainite formation are the combination of time-dependent transformation kinetics and lattice shearing in the microstructure. These effects are considered on the microscale and transferred to the polycrystalline macroscale by means of homogenisation of stochastically orientated grains. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Well‐posed fractional calculus operator: Obtaining new transformations formulas involving Gauss hypergeometric functions with rational quadratic,cubic, and higher degree arguments

In this paper, we propose a systematic method for discovering new transformation formulas for the Gauss hypergeometric function with quadratic and rational (quadratic, cubic, and of higher degree) arguments. These new transformation formulas are obtained from known transformation formulas given in 1881 by Goursat (E. Goursat, Sur l'Équation différentielle linéaire qui admet pour intégrale la série hypergéométrique, Annales scientifique de l'É. N. S., 2e série tome 10 [1881], 3–142). This method relies on the use of the well‐posed fractional calculus operator introduced by Tremblay (R. Tremblay, Une contribution à la théorie de la dérivée fractionnaire, Doctoral thesis, Université Laval, Québec, Canada [1974]). We illustrate the effectiveness of the method by giving several presumably new transformation formulas for the Gauss hypergeometric function.     

A single-period inventory placement problem for a supply system with the satisficing objective

Consider the inventory placement problem in an N-stage supply system facing a stochastic demand for a single planning period. Each stage is a stocking point holding some form of inventory (e.g., raw materials, subassemblies, product returns or finished products) that after a suitable transformation can satisfy demand. Stocking decisions are made before demand occurs. Unsatisfied demands are lost. The revenue, salvage value, ordering, transformation, and lost sales costs are proportional. There are fixed costs for utilizing stages for stock storage. The objective is to maximize the probability of achieving a given target profit level.     

A computational two scaled model for the simulation of micro-heterogeneous low-alloyed TRIP steels

In transformation induced plasticity (TRIP) steel a diffusionless austenitic-martensitic phase transformation induced by plastic deformation can be observed, resulting in excellent macroscopic properties. In particular low-alloyed TRIP steels, which can be obtained at lower production costs than high-alloyed TRIP steel, combine this mechanism with a heterogeneous arrangement of different phases at the microscale, namely ferrite, bainite, and retained austenite. The macroscopic behavior is governed by a complex interaction of the phases at the micro-level and the inelastic phase transformation from retained austenite to martensite. A reliable model for low-alloyed TRIP steel should therefore account for these microstructural processes to achieve an accurate macroscopic prediction. To enable this, we focus on a multiscale method often referred to as FE² approach, see [6]. In order to obtain a reasonable representative volume element, a three-dimensional statistically similar representative volume element (SSRVE) [1] can be used. Thereby, also computational costs associated with FE² calculations can be significantly reduced at a comparable prediction quality. The material model used here to capture the above mentioned microstructural phase transformation is based on [3] which was proposed for high alloyed TRIP steels, see also e.g. [8]. Computations based on the proposed two-scale approach are presented here for a three dimensional boundary value problem to show the evolution of phase transformation at the microscale and its effects on the macroscopic properties. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Softness of hypercoherences and MALL full completeness

We prove a full completeness theorem for multiplicative–additive linear logic (i.e. MALL) using a double gluing construction applied to Ehrhard’s *-autonomous category of hypercoherences. This is the first non-game-theoretic full completeness theorem for this fragment. Our main result is that every dinatural transformation between definable functors arises from the denotation of a cut-free MALL proof.Our proof consists of three steps. We show:

• •Dinatural transformations on this category satisfy Joyal’s softness property for products and coproducts.
• •Softness, together with multiplicative full completeness, guarantees that every dinatural transformation corresponds to a Girard MALL proof-structure.
• •The proof-structure associated with any dinatural transformation is a MALL proof-net, hence a denotation of a proof. This last step involves a detailed study of cycles in additive proof-structures.

The second step is a completely general result, while the third step relies on the concrete structure of a double gluing construction over hypercoherences.     

Determining conformal transformations in from minimal correspondence data

In this paper, we derive a method to determine a conformal transformation in n‐dimensional Euclidean space in closed form given exact correspondences between data. We show that a minimal data set needed for correspondence is a localized vector frame and an additional point. In order to determine the conformal transformation, we use the representation of the conformal model of geometric algebra by extended Vahlen matrices— 2 ×2 matrices with entries from Euclidean geometric algebra (the Clifford algebra of ). This reduces the problem on the determination of a Euclidean orthogonal transformation from given vector correspondences, for which solutions are known. We give a closed form solution for the general case of conformal (in contrast, anti‐conformal) transformations, which preserve (in contrast, reverse) angles locally, as well as for the important special case when it is known that the conformal transformation is a rigid body motion—also known as a Euclidean transformation—which additionally preserves Euclidean distances. Copyright © 2011 John Wiley & Sons, Ltd.     

A Levin-type algorithm for accelerating the convergence of Fourier series

A nonlinear sequence transformation is presented which is able to accelerate the convergence of Fourier series. It is tailored to be exact for a certain model sequence. As in the case of the Levin transformation and other transformations of Levin-type, in this model sequence the partial sum of the series is written as the sum of the limit (or antilimit) and a certain remainder, i.e., it is of Levin-type. The remainder is assumed to be the product of a remainder estimate and the sum of the first terms oftwo Poincaré-type expansions which are premultiplied by two different phase factors. This occurrence of two phase factors is the essential difference to the Levin transformation. The model sequence for the new transformation may also be regarded as a special case of a model sequence based on several remainder estimates leading to the generalized Richardson extrapolation process introduced by Sidi. An algorithm for the recursive computation of the new transformation is presented. This algorithm can be implemented using only two one-dimensional arrays. It is proved that the sequence transformation is exact for Fourier series of geometric type which have coefficients proportional to the powers of a numberq, |q|<1. It is shown that under certain conditions the algorithm indeed accelerates convergence, and the order of the convergence is estimated. Finally, numerical test data are presented which show that in many cases the new sequence transformation is more powerful than Wynn's epsilon algorithm if the remainder estimates are properly chosen. However, it should be noted that in the vicinity of singularities of the Fourier series the new sequence transformation shows a larger tendency to numerical instability than the epsilon algorithm.     

Use of an orthoexponential transformation in the study of the non-stationary equations of thermoelasticity and thermoviscoelasticity

A system of orthoexponential polynomials (OEP) orthogonal in the interval t ε [0, ∞) representing a special case of the orthoexponential Jacobi polynomials /1/ is studied. It is proposed to use the OEP as the kernels of an integral transformation (the OEP transformation) in time, since, compared with Laplace transformations, its use simplifies the procedure for obtaining the originals of the quantities required. The OEP transformation is used to solve the non-stationary equations of thermoelasticity and thermoviscoelasticity. The initial equations are reduced to the corresponding systems of ordinary triangular differential equations, and their general solutions are constructed.     

Extending a finite strain hyperelastic micro-sphere framework towards phase transformations

A finite strain micro-sphere framework for hyperelastic solids elaborated by Carol et al. is extended towards the modelling of phase transformations in order to simulate polycrystalline solids under large deformations such as, e.g., shape memory alloys and shape memory polymers. The implemented phase transformation mechanism is based on statistical physics and is not restricted in terms of the number of solid material phases that can be considered, though we restrict the provided examples to two phases for the sake of conceptual clarity. The specifically chosen non-quadratic format of the Helmholtz free energy functions considered on the micro-plane level includes Bain-type transformation strains for each of the phases considered. Following the Voigt assumption on the micro-scale, identical total micro-stretches act in each of the material phases, where a multiplicative decomposition into elastic and transformation-related contributions is applied. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Spectral Properties of Interval Exchange Maps

 In [7], Nogueira and Rudolph proved that for irreducible permutations not of rotation class almost every (a.e.) interval exchange transformation (i.e.t.) is topological weak mixing. It is conjectured that the claim holds if topological weak mixing is replaced by weak mixing. Here we study the behaviour of eigenfunctions of i.e.t. Our analysis gives alternative proofs of results due to Katok and Stepin [4] and Veech [10]: for certain permutations a.e. i.e.t. is weak mixing and for irreducible permutations a.e. i.e.t. is totally ergodic.     

Spectral Properties of Interval Exchange Maps

 In [7], Nogueira and Rudolph proved that for irreducible permutations not of rotation class almost every (a.e.) interval exchange transformation (i.e.t.) is topological weak mixing. It is conjectured that the claim holds if topological weak mixing is replaced by weak mixing. Here we study the behaviour of eigenfunctions of i.e.t. Our analysis gives alternative proofs of results due to Katok and Stepin [4] and Veech [10]: for certain permutations a.e. i.e.t. is weak mixing and for irreducible permutations a.e. i.e.t. is totally ergodic. (Received 1 February 2001)     

Asymptotic stability and smooth equivalence op ordinary equations : PMM vol.40, n 6, 1976, pp. 1048–1057

Under certain specified conditions the asymptotic stability is a coarse property [1],(i.e. addition of fairly smooth functions to the right-hand sides of equations, does not disturb the asymptotic stability). It is shown below that in this cage the unperturbed system is coarse in a more general sense, namely, any smooth system acted upon by fairly small smooth perturbations, can be returned to its unperturbed state by a smooth reversible transformation. The value and order of the perturbations and the domain of existence of the transformation are all estimated explicitly. The condition required for the above assertion to hold, is that of the existence of a Liapunov function admitting, together with its derivative, specified estimates. This requirement holds, in particular, in the case when the right-hand sides of the unperturbed system are homogeneous functions, the position of equilibrium is asymptotically stable, and its neighborhood contains no solutions bounded when −∞ <t < ∞ (see [1]). If the system is analytic, the requirement will hold in at least all critical cases investigated in which the asymptotic stability with t → ∞ or t → −∞ is fixed, since in these cases the Liapunov function will be analytic, or simply polynomial. It follows therefore from the theorem which we prove, that in all the cases in question, the system is reduced by a smooth transformation, to the polynomial form. If the unperturbed system is linear, then from the theorem proved follows a theorem on linearization appearing in [2]; if the system is nonlinear but of second order, a theorem from [3] ensues. The results obtained in this paper for the nonlinear autonomous systems are extended to the case when the perturbations are continuous and bounded functions of time. This makes possible the investigation of the dynamics of the process in the neighborhood of asymptotically stable equilibria and of periodic modes, ignoring a wide range of external perturbations.     

Exact solutions of some mixed problems of uncoupled thermoelasticity for a truncated hollow circular cone with a groove along the generatrix

An elastic body of finite dimensions in the form of a truncated hollow circular cone with a groove along the generatrix is considered. The uncoupled problem of thermoelasticity is formulated for this body for different types of boundary conditions on all the surfaces. These are the conditions for specifying the displacements or sliding clamping on surfaces with fixed angular coordinates and the conditions for specifying the stresses on surfaces with a fixed radial coordinate (shear stresses are assumed to be zero). It is assumed that the temperature is a specified function of all the spherical coordinates. Some auxiliary functions, related to the displacements, are introduced first, and equations for these functions are then derived using Lamé's equations. A finite integral Fourier transformation with respect to one of the angular variables is then employed. After this, by solving certain Sturm-Liouville problems, a new integral transformation is constructed and is applied to the equations with respect to the other angular variable. As a result a one-dimensional system of differential equations is obtained, to solve which an integral Mellin transformation is employed in a special way. Finally, exact solutions of some problems of thermoelasticity are constructed in series for this body.     

Oblique magnetohydrodynamic shock wave decay

Summary A modified hodograph transformation is used to obtain an exact solution of the equations governing the one-dimensional unsteady flow of an ideal, inviscid, perfectly conducting compressible fluid, subjected to an oblique magnetic field. This solution is used to obtain an approximate representation of the path of an initially uniform shock wave which intersects a centered simple wave. The solutions for the corresponding problems in the conventional, non-magnetic case and for a transverse orientation of the applied magnetic field are contained as special limiting cases of the solutions of the present paper. This provides a valuable check on the theory.

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Résumé Une transformation hodographe modifiée est employée pour obtenir une solution exacte des équations relatives aux écoulements unidimensionnels non-stationnaires et non-isentropiques d'un fluide non visqueux idéal, parfaitement conducteur d'électricité et compressible, soumis à l'action d'un champ magnétique oblique avec deux composantes différentes de zero. On utilise cette solution pour obtenir une représentation approximative de la trajectoire d'une onde de choc magnétohydrodynamique initialement uniforme, rencontrant une onde simple centrée. Les solutions pour le cas non magnétique et pour un champ transversal apparaissent comme des cas limites particuliers de la solution présentée ici, c'est là une confirmation de la validité de la theorie.

     

极分解与广义*-Aluthge变换

首先给出了Hilbert空间上有界线性算子极分解的的若干性质.其次指出广义的*-Aluthge变换与*-Aluthge变换具有许多相似性质;例如,T_(α,β)^((*))=U|T_(α,β)((*))=U|T_(α,β)^((*))|当且仅当T是双正规的,即[|T|,|T*|]=0,其中对任意两个算子A和B,[A,B]=AB-BA.     

Classes of convolution type operators on unions of bounded intervals

An invertibility theory for classes of convolution type operators on a union of bounded intervals whose kernels have Fourier transforms which are related to solutions of corona problems is established and the corresponding formulas for the inverse operators are given. A generalization of the portuguese transformation for matrix functions is obtained and is used to establish the invertibility theory for one of the above mentioned classes of operators. The same transformation allows, also, to establish the equivalence between convolution type operators on an union of disjoint intervals and convolution type operators on a bounded interval.