New Class Of Transformations Research Articles

Decomposition of transformations

The decomposition theorem for transformations without any additivity assumptions is proved. This result is new even for image transformations. We also introduce a new class of transformations with some additivity assumpt ions which includes image transformations. Using the transformation from this class we generalize the Aarnes factorization theorem to representable deficient topological measures. We also establish the relationship between the decomposition of a representable deficient topological measure and the decomposition of the transformation mentioned above.

Aspects of reduction and transformation of Lagrangian systems with symmetry

This paper contains results on geometric Routh reduction and it is a continuation of a previous paper [7] where a new class of transformations is introduced between Lagrangian systems obtained after Routh reduction. In general, these reduced Lagrangian systems have magnetic force terms and are singular in the sense that the Lagrangian does not depend on some velocity components. The main purpose of this paper is to show that the Routh reduction process itself is entirely captured by the application of such a new transformation on the initial Lagrangian system with symmetry.

Canonical and Anti-Canonical Transformations Preserving Convexity of Potentials

The aim of the paper is to characterize transformations that preserve the potential structure of a relationship between dual variables. The first step consists in deriving a geometric definition of the condition for the existence of a potential. Having at hand this formulation, it becomes clear that the canonical similitudes represents the class of transformations that preserves the potential form of a relationship. Next, we derive the conditions under which canonical similitudes preserve the convexity of the potential or change it into concavity. This new class of transformations can be viewed as a generalization of the Legendre-Fenchel transformation. These concepts are applied to the Hooke constitutive relationship.

Box–Cox transforms for realized volatility

The log transformation of realized volatility is often preferred to the raw version of realized volatility because of its superior finite sample properties. One of the possible explanations for this finding is the fact the skewness of the log transformed statistic is smaller than that of the raw statistic. Simulation evidence presented here shows that this is the case. It also shows that the log transform does not completely eliminate skewness in finite samples. This suggests that there may exist other nonlinear transformations that are more effective at reducing the finite sample skewness. The main goal of this paper is to study the accuracy of a new class of transformations for realized volatility based on the Box–Cox transformation. This transformation is indexed by a parameter β and contains as special cases the log (when β = 0 ) and the raw (when β = 1 ) versions of realized volatility. Based on the theory of Edgeworth expansions, we study the accuracy of the Box–Cox transforms across different values of β . We derive an optimal value of β that approximately eliminates skewness. We then show that the corresponding Box–Cox transformed statistic outperforms other choices of β , including β = 0 (the log transformation). We provide extensive Monte Carlo simulation results to compare the finite sample properties of different Box–Cox transforms. Across the models considered in this paper, one of our conclusions is that β = − 1 (i.e. relying on the inverse of realized volatility also known as realized precision) is the best choice if we want to control the coverage probability of 95% level confidence intervals for integrated volatility.

Transformations of measures by Gauss maps

The goal of this paper is to introduce a new class of transformations of measures on R having the form (heuristically) T = φ · ∇φ/|∇φ|, where φ is some function; our construction exhibits an interesting link between two actively developing areas: optimal transportation and geometrical glows (see [1], [2], and [3] concerning these directions). Let A ⊂ R, d ≥ 2, be a compact convex set and let μ = %0 dx be a probability measure on A equivalent to the restriction of Lebesgue measure. Let ν = %1 dx be a probability measure on the ball Br := {x : |x| ≤ r} equivalent to the restriction of Lebesgue measure. We prove that there exists a mapping T such that ν = μ ◦ T−1 and T = φ · n, where φ : A → [0, r] is a continuous potential with convex sub-level sets and n is the Gauss map of the corresponding level sets of φ. Moreover, the mapping T is invertible and essentially unique. Our proof employs the optimal transportation techniques. In the case of smooth φ the level sets of φ are governed by the Gauss curvature flow ẋ(s) = −sd−1 %1(sn) %0(x) K(x) · n(x), where K is the Gauss curvature. Throughout we assume that d ≥ 2 and denote by H the n-dimensional Hausdorff measure. Let IntA denote the interior of a set A. Given a compact convex set V ⊂ R with boundary M = ∂V , for an arbitrary point x ∈M , let us set

Dynamical Spatial Warping: A Novel Method for the Conformational Sampling of Biophysical Structure

The difficulties encountered in sampling of systems with rough energy landscapes using present methodology significantly limit the impact of simulation on molecular biology, in particular protein folding and design. Here, we present a major methodological development based on a promising new technique, the reference potential spatial warping algorithm (REPSWA) [Z. Zhu et al., Phys. Rev. Lett., 88 (2002), pp. 100201-100204], and present applications to several realistic systems. REPSWA works by introducing a variable transformation in the classical partition function that reduces the volume of phase space associated with a priori known barrier regions while increasing that associated with attractive basins. In this way, the partition function is preserved so that enhanced sampling is achieved without the need for reweighting phase-space averages. Here, a new class of transformations, designed to overcome the barriers induced by intermolecular/nonbonded interactions, whose locations are not known a priori, is introduced. The new transformations are designed to work in synergy with transformations originally introduced for overcoming intramolecular barriers. The new transformation adapts to the fluctuating local environment and is able to handle barriers that arise “on the fly.” Thus, the new method is referred to as dynamic contact REPSWA (DC-REPSWA). In addition, combining hybrid Monte Carlo (HMC) with DC-REPSWA allows more aggressive sampling to take place. The combined DC-REPSWA-HMC method and its variants are shown to substantially enhance conformational sampling in long molecular chains composed of interacting single beads and beads with branches. The latter topologies characterize the united residue and united side chain representation of protein structures.

Preservers on Hilbert space effects

In this paper we study a new class of transformations on the set of all Hilbert space effects. This consists of the bijective maps which preserve the order and zero product in both directions. The main result of the paper gives a complete description of the structure of those transformations. As applications we obtain additional new results and some former ones as easy corollaries. In particular, we obtain the form of the ortho-order automorphisms as well as that of the sequential automorphisms. In the last paragraph of the paper we show that the automorphisms of these two kinds belong to our class of transformations even when their domain is the set of all effects in a general von Neumann algebra.

Hyperbolic quantum mechanics

We start to develop the quantization formalism in a hyperbolic Hilbert space. Generalizing Born’s probability interpretation, we found that unitary transformations in such a Hilbert space represent a new class of transformations of probabilities which describe a kind of hyperbolic interference. The most interesting problem which prompted by our investigation is to find experimental evidence of hyperbolic interference. The hyperbolic quantum formalism can also be interesting as a new theory of probability waves that can be developed in parallel with the standard quantum theory. Comparative analysis of these two wave theories could be useful for understanding of the role of various structures of the standard quantum formalism. In particular, one of distinguishing feature of the hyperbolic quantum formalism is the restricted validity of the superposition principle.

Detection of weak signals based on a new class of transformations of random series

For detecting weak signals (when their amplitude is comparable with the amplitude of the noise track) a new class of transformations of random series to a straight line has been suggested. It has been shown that these transformations are quasi-linear and can be defined as a signal-to-staircase transformation (SST). The height of a step defines an amplitude of the detected signal and the step length its duration. The SST can be applicable for a wide class of random series having different statistical nature. The verification of this new method based on the analysis of the real signal/noise tracks containing registration of different earthquakes with small amplitudes has been realized. It has also been shown that different situations which have been found from the real-data analysis demonstrate the high sensitivity and efficiency of the new method suggested.

Lattice gravity near the continuum limit

We prove that the lattice gravity always approaches the usual continuum limit when the link length ℓ → 0, provided that certain general boundary conditions are satisfied. This result holds for any lattice, regular or irregular. Furthermore, for a given lattice, the deviation from its continuum limit can be expressed as a power series in ℓ 2. General formulas for such a perturbative calculation are given, together with a number of illustrative examples, including the graviton propagator. The lattice gravity satisfies all the invariance properties of Einstein's theory of general relativity. In addition, it is symmetric under a new class of transformations that are absent in the usual continuum theory. The possibility that the lattice theory (with a nonzero ℓ) may be more fundamental is discussed.

A Modern Look at Conformal Mapping Including Multiply Connected Regions

With the advent of large-scale digital computers, numerical solution of the transonic flow equations in two dimensions has proven feasible. In such calculations it is convenient, and often crucial, to map the region of in- terest conformally onto a simple domain so that a simple finite difference system can be employed. This paper updates a well-known airfoil conformal mapping method (Theodorsen and Garrick) with modern techniques, greatly enhancing the mapping speed and accuracy while simplifying the analysis. Also, a powerful new class of conformal transformations is introduced, and applied to a two-element airfoil. N the calculation of transonic flow over airfoils it has proven convenient to map the infinite region of the flow exterior to an airfoil conformally onto a finite region, such as the interior of a circle, as in Sells,l Melnik and Ives,2 Jameson,3 and Bauer, Garabedian, and Korn.4 Such a mapping places a fine grid in the physical plane where needed, while retaining a uniform grid in the (con- formally mapped) computational plane. The gradients of flow properties in the computational plane are smaller than in the physical plane, so that finite difference approximations in the computational plane are more accurate than in the physical plane. This allows the use of a rather coarse grid. It is easier to apply boundary conditions in the mapped plane than it is in the physical plane, as the boundaries are coordinate lines in the mapped plane. The conformal mapping ensures that the nonlinear partial differential equations in the computational plane are only slightly more complicated than those in the physical plane.1'5 To take advantage of the above con- siderations, there is a need for conformal mapping techniques that are simple and rapid. This paper presents a conformal mapping technique which is at least an order of magnitude faster than conventional mapping techniques. In analogy to the Fourier transform techniques utilized here, the present work can be considered a fast conformal mapping. This paper also introduces a powerful new class of con- formal transformations. These new transformations are an extension of the well-known von Karman-Trefftz trans- formation to multiply-conn ected regions, and can be used in the analysis of flow over multielement airfoils. They have the property that any number N of airfoil elements are simultaneously (with one application of the transformation) mapped to TV near circles. One member of this new class of transformations is used in mapping the region exterior to a two-element airfoil onto the region between two concentric circles. By combining the known solution for incompressible flow between two con- centric circles6 with this two-element mapping, we can very simply solve for the inviscid incompressible flow over a two- element air foil.