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ВВ криптографических системах шифрования информации используются гиперкомплексные числа: кватернионы и октонионы. В качестве ключа применяется кватернион, который производит вращения группы выборок информации. Кватернионы и бикватернионы являются частными случаями геометрической алгебры Клиффорда. Использование векторов и мультивекторов геометрической алгебры для шифрования информации позволяет расширить разнообразие этих векторов. Для шифрования информации, представленной совокупностью векторов геометрической алгебры, эти векторы умножаются на мультивектора, которые осуществляют операцию ротор (rotor). В качестве ключа используется мультивектор (ротор). Для дешифрования информации применяется операция, которая соответствует обратному ротору. Алгоритмы геометрической алгебры повышают безопасность шифрования информации за счет повышения размерности алгебры. Для повышения производительности шифрования предлагается коэффициенты информационного вектора и мультивектора вращения выбирать из поля Z256. Предлагается вектор информации с коэффициентами из Z256 складывать со случайным вектором с коэффициентами из Z256 и считать эти коэффициенты ключами шифрования. Приведены базисные векторы применяемых геометрических алгебр и таблицы геометрических произведений базисных векторов.

The authors solve the problem of estimating the best rotation aligning two sets of corresponding vectors (also known as Wahba's problem or point cloud registration). The proposed method is among the fastest methods reported in recent literatures, moreover it is robust to noise, accurate and simpler than most other methods. It is based on solving the linear equations derived from the formulation of the problem in Euclidean Geometric Algebra. The authors show its efficiency in two applications: the as-rigid-as-possible (ARAP) surface modelling and the more smooth rotation enhanced ARAP mesh animation which is the only method capable of deforming surface modes with quality of tetrahedral models. Mesh deformation is a key technique in games, automated construction and robotics. The ARAP technique along with its improved variants, although have been extensively studied, can still not be achieved efficiently. Linear geometric algebra based rotor solution proposed in this study gives another perspective of the kernel problem. This, however, not only improves the real performance of the three-dimensional mesh deformation, but also provides a brand new computationally efficient solution to the Wahba's problem and point cloud registration, which has been closely related to the automation science and engineering.

We study the breaking of time-reversal invariance (TRI) by the application of a magnetic field in the quantum kicked rotor (QKR), using Izrailev's finite-dimensional model. There is a continuous crossover from TRI to time-reversal non-invariance (TRNI) in the spectral and eigenvector fluctuations of the QKR. We show that the properties of this TRI $\rightarrow$ TRNI transition depend on $\alpha^2/N$, where $\alpha$ is the chaos parameter of the QKR and $N$ is the dimensionality of the evolution operator matrix. For $\alpha^2/N \gtrsim N$, the transition coincides with that in random matrix theory. For $\alpha^2/N < N$, the transition shows a marked deviation from random matrix theory. Further, the speed of this transition as a function of the magnetic field is significantly enhanced as $\alpha^2/N$ decreases.

In complex-valued Hopfield neural networks (CHNNs), the neuron states are complex numbers whose amplitudes are: 1) they can also be described in special orthogonal matrices of order and 2) here, we propose a new Hopfield model, the $O(2)$ -valued Hopfield neural network [ $O(2)$ -HNN], whose neuron states are extended to orthogonal matrices. Its neuron states are embedded in 4-D space, while those of CHNNs are embedded in 2-D space. Computer simulations were conducted to compare the noise tolerance (NT) and storage capacity (SC) of CHNNs, $O(2)$ -HNNs, and rotor Hopfield neural networks. In terms of SC, $O(2)$ -HNNs outperformed the others, while in NT, they outdid CHNNs.

This work is interested in studying the motion of a rigid body carrying a rotor that rotates with a constant angular velocity about an axis parallel to the axis of dynamical symmetry. This motion is assumed to take place due to the effect of a combination of both uniform fields of gravity and magnetism that do not possess an axis of common symmetry. The equations of motion are constructed, and they are rewritten by means of the Hamiltonian function in the framework of the Lie–Poisson system. The equilibrium positions are inserted. The necessary conditions for the stability are introduced by applying the linear approximation method, while the sufficient conditions for stability are determined by utilizing the energy-Casimir method.

The quantum kicked rotor is well-known to display dynamical localization in the non-interacting limit. In the interacting case, while the mean-field (Gross-Pitaevskii) approximation displays a destruction of dynamical localization, its fate remains debated beyond mean-field. Here we study the kicked Lieb-Liniger model in the few-body limit. We show that for any interaction strength, two kicked interacting bosons always dynamically localize, in the sense that the energy of the system saturates at long time. However, contrary to the non-interacting limit, the momentum distribution $\Pi(k)$ of the bosons is not exponentially localized, but decays as $\mathcal C/k^4$, as expected for interacting quantum particles, with Tan's contact $\mathcal C$ which remains finite at long time. We discuss how our results will impact the experimental study of kicked interacting bosons., Comment: v1) 10 pages, 10 figures; v2) minor typos corrected, published version

We provide a pair of ribbon graphs that have the same rotor routing and Bernardi sandpile torsors, but different topological genus. This resolves a question posed by M. Chan [Cha]. We also show that if we are given a graph, but not its ribbon structure, along with the rotor routing sandpile torsors, we are able to determine the ribbon graph's genus., Reformatted for DMTCS

The paper focuses on the loads applied to the helicopter blade cross-sections in the multibody dynamics system. The main objective is to simplify the blade aerodynamics calculation and avoid time-consuming CFD methods. For this reason, the way of computing blade aerodynamics is proposed by using multibody dynamics methods with a linear-elastic blade model. As the primary tool for further research, the MCS Adams software package is selected. Splitting the main rotor blade into a finite number of sections, each having its own average value of installation and coning angles, simplifies the calculation. Afterward, expressions for the total flow velocity around the blade section and its angle of attack are obtained through vector operations. This provides a measure of aerodynamic forces acting on each section in its cross-sectional coordinate system. In conclusion, the article provides the formalized method of aerodynamic force distribution between blade sections in the multibody model as well as the correlation between the flow coordinate system and the blade chord coordinate system.

We apply the semi-classical limit of the generalized $SO(3)$ map for representation of variable-spin systems in a four-dimensional symplectic manifold and approximate their evolution terms of effective classical dynamics on $T^{\ast }\mathcal{S}_{2}$. Using the asymptotic form of the star-product, we manage to "quantize" one of the classical dynamic variables and introduce a discretized version of the Truncated Wigner Approximation (TWA). Two emblematic examples of quantum dynamics (rotor in an external field and two coupled spins) are analyzed, and the results of exact, continuous, and discretized versions of TWA are compared., Comment: 19 pages, 3 figures. https://doi.org/10.3390/e23060684

Let $G$ be a ribbon graph. Matthew Baker and Yao Wang proved that the rotor-routing torsor and the Bernardi torsor for $G$, which are two torsor structures on the set of spanning trees for the Picard group of $G$, coincide when $G$ is planar. We prove the conjecture raised by them that the two torsors disagree when $G$ is non-planar.

The wobbling spectrum of $^{163}$Lu is described through a novel approach, starting from a triaxial rotor model within a semi-classical picture, and obtaining a new set of equations for all four rotational bands that have wobbling character. Redefining the band structure in the present model is done by adopting the concepts of Signature Partner Bands and Parity Partner Bands. Indeed, describing a wobbling spectrum in an even-odd nucleus through signature and parity quantum numbers is an inedited interpretation of the triaxial super-deformed bands., Comment: 8 pages, 4 figures

The coupled aerodynamic performance influence of multiple deviations in the blade profile is a complex phenomena. In this paper, a high-pressure compressor rotor blade mid-span profile is studied with the design of experiments (DOE), numerical simulation and surrogate model to analyze the influence of the deviations on the blade aerodynamic performance. The purpose of this work is to provide a new rapid evaluation approach for the blade aerodynamic performance under multiple geometric deviations influence. First, the Hicks-Henne function was used to model the local geometric deviations of the blade profile, and the blade aerodynamic performance was calculated by using the computational fluid dynamics tools. By analyzing the calculation results, the momentum thickness of the boundary layer, the deviations height and the distance between the deviations are combined into a coupled effect model. Then, the coupled effect model was used to rapidly evaluate the blade aerodynamic performance when the two-position local geometric deviations exist on the blade surface. Finally, the evaluated performance were compared with the results predicted by a high-precision surrogate model, which verifies the high accuracy of the coupled effect model in evaluating the positive incidence range of the blade profile.

In this paper we will address the problem of recovering covariant transformations between objects—specifically; lines, planes, circles, spheres and point pairs. Using the covariant language of conformal geometric algebra (CGA), we will derive such transformations in a very simple manner. In CGA, rotations, translations, dilations and inversions can be written as a single rotor, which is itself an element of the algebra. We will show that the rotor which takes a line to a line (or plane to a plane etc) can easily be formed and we will investigate the nature of the rotors formed in this way. If we can recover the rotor between one object and another of the same type, a useable metric which tells us how close one line (plane etc) is to another, can be a function of how close this rotor is to the identity. Using these ideas, we find that we can define metrics for a number of common problems, specifically recovering the transformation between sets of noisy objects.

We analyze the out-of-time-order correlation functions of a solvable model of a large number, $N$, of $M$-component quantum rotors coupled by Gaussian-distributed random, infinite-range exchange interactions. We focus on the growth of commutators of operators at a temperature $T$ above the zero temperature quantum critical point separating the spin-glass and paramagnetic phases. In the large $N,~M$ limit, the squared commutators of the rotor fields do not display any exponential growth of commutators, in spite of the absence of any sharp quasiparticle-like excitations in the disorder-averaged theory. We show that in this limit, the problem is integrable and point out interesting connections to random-matrix theory. At leading order in $1/M$, there are no modifications to the critical behavior but an irrelevant term in the fixed-point action leads to a small exponential growth of the squared commutator. We also introduce and comment on a generalized model involving $p$-pair rotor interactions., Comment: 5+6 pages, 3+2 figures

A reflection-asymmetric triaxial particle rotor model with three quasiparticles and a reflection-asymmetric triaxial rotor is developed, and applied to investigate the observed multiple chiral doublets (M\c{hi}D) candidates with octupole correlations in 131Ba, i.e., two pairs of positive-parity bands D3-D4 and D5-D6, as well as one pair of negative-parity bands D7-D8. The energy spectra, the energy staggering parameters, the B(M1)/B(E2) ratios, and the B(E1)/B(E2) ratios are reproduced well. The chiral geometries for these M\c{hi}D candidates are examined by the azimuthal plots, and the evolution of chiral geometry with spin is clearly demonstrated. The intrinsic structure for the positive-parity bands is analyzed and the possible pseudospin-chiral quartet bands are suggested., 16 pages, 8 figures, PHYSICAL REVIEW C (in press)

A rotor Hopfield neural network (RHNN) is an extension of a complex-valued Hopfield neural network (CHNN) and has excellent noise tolerance. The RHNN decomposition theorem says that an RHNN decomposes into a CHNN and a symmetric CHNN. For a large number of training patterns, the projection rule for RHNNs generates large self-feedbacks, which deteriorates the noise tolerance. To remove self-feedbacks, we propose a projection rule using quaternions based on the decomposition theorem. Using computer simulations, we show that the quaternion projection rule improves noise tolerance.

Classical rotations of asymmetric rigid bodies are unstable around the axis of intermediate momentof inertia, causing a flipping of rotor orientation. This effect, known as the tennis racket effect,quickly averages to zero in classical ensembles since the flipping period varies significantly uponapproaching the separatrix. Here, we explore the quantum rotations of rapidly spinning thermalasymmetric nanorotors and show that classically forbidden tunnelling gives rise to persistent tennisracket dynamics, in stark contrast to the classical expectation. We characterise this effect, demon-strating that quantum coherent flipping dynamics can persist even in the regime where millions ofangular momentum states are occupied. This persistent flipping offers a promising route for observ-ing and exploiting quantum effects in rotational degrees of freedom for molecules and nanoparticles., Comment: 12 pages, 8 figures

In this paper, an analytical method for predicting the rotor overhang (OH) effect is presented using two types of conformal mappings (CMs), which are the ${r}$ – ${\theta }$ and zr plane CMs. First, the no-load air-gap flux density (AFD) of a slotted air gap without OH is determined using the ${r} {\theta }$ plane CM. Second, the complex relative air-gap permeance (CRAP) in the zr plane is obtained by the zr plane CM. This helps in taking the OH effect into account. Finally, the no-load AFD of the slotted and OH air gaps in cylindrical coordinates is derived from the no-load AFD in the ${r} {\theta }$ plane with CRAP along the ${z}$ -axis. The accuracy of the presented method is verified by comparing the results of no-load analysis with those of 3-D finite-element analysis.

In order to overcome the problem of “singular points”, a method has been developed for the kinematically accurate separation of a large rotation into an axial Euler vector and a transverse Euler vector. The proposal is based on the fact that in the problems of the rotor dynamics of machines consisting of shafts, gears, bearings, etc., the transverse rotation never reaches a value of 2π (a critical value for the Euler vector). The axial rotation is not limited in any way. A numerical dynamics example illustrating the method is presented. The result of the dynamics problem is checked by observing the law of conservation of total energy.

We show that a heavy, magnetized and electrically charged asymmetric rigid body moving about a fixed point while carrying a rotor and acted upon by three uniform fields can perform a regular precession about a nonvertical axis, of the type described for the case of a single field by Grioli in 1947. This is the first, and the only known by now, non-equilibrium solution of the problem of motion of a body in the presence of three classical fields, which are irreducible to a less number of fields.

The chirality suggested for the doublet bands B2 and B2a in $^{195}$Tl in $A\sim190$ region is reexamined. The potential-energy curves and the configurations together with the deformation parameters are obtained by the constrained covariant density functional theory. The corresponding experimental energy spectra, energy differences between doublet bands, and the available $B(M1)/B(E2)$ values are investigated by the fully quantal particle rotor model. Analysis on the basis of the angular momentum components, the $K$-plots, and the azimuthal plots suggest a planar rotation interpretation for the bands B2 and B2a. Hence, it coexists with the aplanar rotation in the other doublet bands B4 and B4a in $^{195}$Tl., Comment: 7 pages, 5 figures. arXiv admin note: text overlap with arXiv:1904.12100

In this paper, the dynamics of a tailored composite rotating blade with curved tips are investigated, with a view to improving the dynamic behaviour of the blade in flight. The blade tip is curved either in the out-of-plane or in the in-plane directions. The composite blade is modelled by using the exact beam formulation, and the cross-sectional properties of the blade are obtained using the variational asymptotic method. The resulting nonlinear partial differential equations are discretised using a time-space scheme, and the stationary and rotating frequencies of the blade are obtained from the eigenvalues of the linearised system. Three case studies are considered here each of them representing one of the main elastic couplings that might happen in a composite blade. These three elastic couplings are the flap-twist, lag-twist, and extension-twist couplings. All these couplings are very important in the blade design as they can affect the twist and hence the dynamics of the blade. The blade tip length and curvature value are two main parameters that this paper is focused on. It is shown that the curved tip of the blade affects the blade frequencies by adding extra couplings, and therefore could be used as a potential morphing concept for tuning the frequencies, enhancing the aeroelastic stability or performance of the blade in flight.

We consider Villarceau circles as the orbits of specific composite rotors in 3D conformal geometric algebra that generate knots on nested tori. We compute the conformal parametrization of these circular orbits by giving an equivalent, position-dependent simple rotor that generates the same parametric track for a given point. This allows compact derivation of the quantitative symmetry properties of the Villarceau circles. We briefly derive their role in the Hopf fibration and as stereographic images of isoclinic rotations on a 3-sphere of the 4D Clifford torus. We use the CGA description to generate 3D images of our results, by means of GAviewer. This paper was motivated by the hope that the compact coordinate-free CGA representations can aid in the analysis of Villarceau circles (and torus knots) as occurring in the Maxwell and Dirac equations.

Quantum-chemical INDO/S calculations of Thiofl avin T have been carried out taking account of variation of the angle φ between the planes of the benzothiazole (BTZ) and dimethylaniline (DMA) rings. It was found that when the angle φ changes from 40° to 90° the energy of the triplet state increases by 4000 cm−1, whereas the energy of the singlet state S 1 decreases by 1900 cm−1 and reaches a minimum. The function $$ {E}_{T_1}\left(\upvarphi \right) $$ has a minimum at φ = 30°, which is 300 cm−1 less than at φ = 40°. The calculated T 1 S 0 interval at φ = 30–40° amounts to 15,600–16,000 cm−1, which agrees well with the phosphorescence data (17,100–17,400 cm−1). For φ = 80–90° the T 1, T 2 , and T 3 levels (T 1 and T 2 are lower than S 1 ) are close to the S 1 level. The S 1 and T 3 levels intersect at φ ~ 85°, and at φ = 90° the interval $$ \Delta {E}_{S_1{T}_3} $$ amounts to only 100 cm−1 due to the small value of the exchange integral corresponding to electron transfer DMA → BTZ.

We show that a new attempt by Corda (2016), just like his previous attempt (Corda, 2015) that we had answered before (A.L. Kholmetskii et al., 2015), to reinterpret Mossbauer experiments in a rotating system as a “new, strong and independent proof of the correctness of Einstein’s vision of gravity” is erroneous. In addition, we demonstrate that Corda’s criticism of Yarman–Arik–Kholmetskii gravitation theory (in short YARK), is based on the application of ill-posed logic; thus rendering his claims against YARK as unfounded.

In this paper, the effect of various parameters of a specific rotor blade cross-section on the effectiveness of a twist morphing concept is investigated. Then, by considering different constraints, a cross-section consistent with this morphing concept with high lag-twist coupling and low extension-twist, is developed. This lag bending-torsion coupling is used to change the twist of the blade during the flight, while the high values of extension-twist coupling is avoided. To this end, a concentrated mass is added to the blade, where its chordwise location varies in flight. When the mass moves in the chordwise direction, a local lag bending is introduced into the blade. This in-plane bending moment then changes the blade twist distribution through lag-twist coupling induced through stiffness tailoring in the blade cross-section. Therefore, this coupling plays an important role in this morphing concept. The one-dimensional dynamics of the blade is modelled by using the geometrically exact fully intrinsic bean equations while the 2D cross-sectional stiffness values are determined by using the VABS software. First, a blade which resembles the BO-105 main rotor blade in the fundamental frequencies is designed. Then, the effect of various parameters of the cross-section on the fundamental frequencies, the lag-twist coupling, and the extension-twist coupling are determined. It is found that the skin of the spar has the highest contribution to both the extension-twist and the lag-twist coupling. Finally, a cross-section compatible with the proposed morphing concept is designed and it is demonstrated that the twist of the blade may be changed significantly.

Dynamical kicking systems possess rich topological structures. In this work, we study Floquet states of matter in a non-Hermitian extension of double kicked rotor model. Under the on-resonance condition, we find various non-Hermitian Floquet topological phases, with each being characterized by a pair of topological winding numbers. A generalized mean chiral displacement is introduced to detect these winding numbers dynamically in two symmetric time frames. Furthermore, by mapping the system to a periodically quenched lattice model, we obtain the topological edge states and unravel the bulk-edge correspondence of the non-Hermitian double kicked rotor. These results uncover the richness of Floquet topological states in non-Hermitian dynamical kicking systems., Comment: 13 pages, 9 figures

The behavior of the collective rotor in the chiral motion of triaxially deformed nuclei is investigated using the particle rotor model by transforming the wave functions from the $K$-representation to the $R$-representation. After examining the energy spectra of the doublet bands and their energy differences as functions of the triaxial deformation, the angular momentum components of the rotor, proton, neutron, and the total system are investigated. Moreover, the probability distributions of the rotor angular momentum ($R$-plots) and their projections onto the three principal axes ($K_R$-plots) are analyzed. The evolution of the chiral mode from a chiral vibration at the low spins to a chiral rotation at high spins is illustrated at triaxial deformations $\gamma=20^\circ$ and $30^\circ$., Comment: 21 pages, 6 figures

We report the derivation and experimental evaluation of a stable adaptive identifier to estimate rigid body rotations using rotors in Geometric Algebra GA. This work is motivated by the need for in situ estimation of the alignment between sensors commonly used in underwater vehicle navigation. Here we derive an adaptive identifier using a geometric interpretation of the error to drive first-order rotor kinematics. We prove that it is Lyapunov stable, and we show that it is asymptotically stable in the presence of persistent excitation. We use the identifier to estimate the alignment between the Doppler velocity log sonar and the fiber optic gyrocompass used by underwater vehicles for dead reckoning DR. We evaluate this method in the laboratory with a remotely operated vehicle ROV, and then with an autonomous underwater vehicle AUV operating in the field at 1,200i¾?m depth. Our results show that this technique reduces dead reckoning navigation errors on these platforms and provides comparable performance to previously reported SO3 constrained Linear Algebra LA approaches. The rotor identifier has a number of advantages over these previously reported methods, including a more straightforward derivation, simpler gain tuning, increased computational efficiency, and reduced data manipulation.

An asymmetric mechanism for correlated motion occurring in noninteracting pairs of adjacent orthogonal 1,4-bis(carboxyethynyl)bicyclo[1.1.1]pentane (BCP) rotators 1 in the solid state is unraveled and shown to play an important role in understanding the dynamics in the crystalline rotor, Bu4N+[1–]·H2O. Single crystal X-ray diffraction and calculation of rotor–rotor interaction energies combined with variable-temperature, variable-field 1H spin–lattice relaxation experiments led to the identification and microscopic rationalization of two distinct relaxation processes., Work in Bellaterra and Tarragona was supported by the Spanish Ministerio de Economía y Competitividad (projects FIS2015-64886-C5-4-P and CTQ2017-87269-P) and Generalitat de Catalunya (2014GR301 and 2014GR199). E.C. acknowledges the support of the Spanish MINECO through the Severo Ochoa Centers of Excellence Program under grant SEV-2015-0496. J.K. and J.M. gratefully acknowledge financial support from the European Research Council under the European Community’s Framework Programme (FP7/2007-2013) ERC grant agreement no. 227756 and the Institute of Organic Chemistry and Biochemistry, Academy of Sciences of the Czech Republic (RVO: 61388963). This material is based upon work supported by the National Science Foundation under grant no. CHE-1566435. Work at Orsay was supported by the CNRS. Work at Angers was supported by the CNRS and the Région des Pays de la Loire grant MOVAMOL. Joint work at Angers, Orsay, and Prague was supported by the Ministère de l’Europe et des Affaires Etrangères under the grant PHC Barrande no. 864563J.

In this work we present the extended two-dimensional torsion (E2DT) method and use it to analyze the performance of several methods that incorporate torsional anharmonicity more approximately for calculating rotational–vibrational partition functions. Twenty molecules having two hindered rotors were studied for temperatures between 100 and 2500 K. These molecules present several kinds of situations; they include molecules with nearly separable rotors, molecules in which the reduced moments of inertia change substantially with the internal rotation, and molecules presenting compound rotation. Partition functions obtained by the rigid-rotor harmonic oscillator approximation, a method involving global separability of torsions and the multistructural methods without explicit potential coupling [MS-T(U)] and with explicit potential coupling [MS-T(C)] of torsions, are compared to those obtained with a quantized version – called the extended two-dimensional torsion (E2DT) method – of the extended hindered rotor approximation of Vansteenkiste et al. (Vansteenkiste et al. J. Chem. Phys. 2006, 124, 044314). In the E2DT method, quantum effects due to the torsional modes were incorporated by the two-dimensional nonseparable method, which is a method that is based on the solution of the torsional Schrödinger equation and that includes full coupling in both the kinetic and potential energy. By comparing other methods to the E2DT method and to experimental thermochemical data, this study concludes that the harmonic approximation yields very poor results at high temperatures; the global separation of torsions from the rest of the degrees of freedom is not justified even when an accurate method to treat the torsions is employed; it is confirmed that methods based on less complete potential energy coupling of torsions, such as MS-T(U), are not accurate when dealing with rotors with different barrier heights, and more complete inclusion of torsional coupling to the method in MS-T(C) improves substantially the results in such a way that it could be used in cases where the E2DT method is unaffordable Financial support from the Ministerio de Economia y Competitividad of Spain (Research Grant No CTQ2014-58617-R), the Conselleria de Cultura, Educación e Ordenación Universitaria (Centro singular de investigación de Galicia acreditación 2016-2019, ED431G/09), and the European Regional Development Fund (ERDF) is gratefully acknowledged SI

In spaces Rn, n ≥ 2, it has been proved that a solenoidal vector field and its rotor satisfy the series of new integral identities which have covariant form. The interest in them is explained by hydrodynamics problems for an ideal fluid.

Commonly, in the flutter analysis of a turbomachine blade row the blade row studied for stability is treated as isolated from its neighbouring blade rows. This approach neglects the effects caused by the reflection of upstream and downstream propagating waves at the neighbouring blade rows. These reflected waves are the source of additional unsteady forces at the blade surface which may have significant effects on the aeroelastic stability of the blade row. To take this influence in the flutter analysis of a blade row into account a coupled unsteady simulation of all rows considered has to be performed. The usual approach to solve for the unsteady flow with a non-linear time-domain solver is expen-sive in terms of computational resources. Especially if in a multi-row configuration the blade counts of the blade rows considered do not allow restricting the simulation to a few passages in each row. To solve for the unsteady flow field, in this paper an alternative ap-proach based on the nonlinear frequency domain Harmonic Balance method is presented. In the frequency-domain the unsteady flow field can be approximated with a small number of relevant frequencies and the circumferential and temporal periodicity of the problem reduces the computational domain to a single passage per blade row only. The efficiency of this approach for flutter analysis in a multi-row configuration is demonstrated by applying it to an embedded rotor blade row.

The aim of this paper is to consider from a utopian point of view the activities and theoretical positions of Rotor, a collective of architects and designers founded in 2005 in Brussels, which, for the moment, has never explicitly written about utopia. First, we underline how this group is deeply engaged with a question that in the last decades has become more and more crucial for utopian theories and projects: the question of sustainability. We will analyse one of its most recent exhibitions devoted to this topic, “Behind the Green Door”, as well a series of examples that show how their work can change our relationship primarily with construction materials, but not only with them. Then, we consider the relationship between this approach and the work of Charles Fourier. We suggest that despite their apparent distance, their proximity becomes manifest when, instead of the stereotypical notion of phalanstère, we look closer into Fourier's texts and into the crucial role he gives to details. In this perspective, we show how Rotor is valuable to rethink the relationship between architecture and utopia, in coherence with the most recent critical reflections on both of them.

This paper gives the flutter analysis for a model of aerodynamically coupled five blade rows in which three rotor cascades are simultaneously vibrating. The displacement of a blade vibrating under aerodynamic force is expanded in a modal series with the natural mode shape functions, and the modal amplitudes are treated as the generalized coordinates. The inertial forces and elastic forces are formulated on the basis of the finite element method. The aerodynamic forces are evaluated by applying the unsteady lifting-surface theory for multistage blade rows. The numerical results are presented. It is shown that the risk of flutter occurrence for the model of five blade rows is higher than that for the model of three blade rows previously studied.

Based on the Riemann and Caputo definition of the fractional derivative we use the fractional extensions of the standard rotation group S O ( 3 ) to construct a higher dimensional representation of a fractional rotation group with mixed derivative types. An analytic extended symmetric rotor model is derived, which correctly predicts the sequence of magic numbers in metal clusters. It is demonstrated, that experimental data may be described by assuming a sudden change in the fractional derivative parameter α which is interpreted as a second order phase transition in the region of cluster size with 200 ≤ N ≤ 300 .

This article presents a design study into the redesign of a wind turbine blade tip seeking to increase the energy production subject to the loads constraints of the existing blade. The blade shape is parameterized to allow for planform changes in the tip region with respect to chord, twist and blade length extension, and additionally three parameters that allow to explore winglet-like shapes. The design strategy uses 3D computational fluid dynamics computations of the geometrically resolved rotor to create a surrogate model, after which the tip shape is numerically optimized based on the surrogate model, subject to a number of geometric and loads-based constraints. The study shows that it is possible to increase power production by 2.6% for a blade extension with a winglet, without increasing the flapwise bending moment at 90% radius, whereas for a straight blade extension it was only possible to achieve an increase of 0.76%.

This paper examines numerically the complex classical trajectories of the kicked rotor and the double pendulum. Both of these systems exhibit a transition to chaos, and this feature is studied in complex phase space. Additionally, it is shown that the short-time and long-time behaviors of these two PT-symmetric dynamical models in complex phase space exhibit strong qualitative similarities., 22 page, 16 figures

We discuss a periodic variant of the Heisenberg-Weyl algebra, associated with the group of translations and modulations on the circle. Our study of uncertainty minimizers leads to a periodic version of canonical coherent states. Unlike the canonical, Cartesian case, there are states for which the uncertainty product associated with the generators of the algebra vanishes. Next, we explore the supersymmetric (SUSY) quantum mechanical setting for the uncertainty-minimizing states and interpret them as leading to a family of "hindered rotors". Finally, we present a standard quantum mechanical treatment of one of these hindered rotor systems, including numerically generated eigenstates and energies.

This paper presents a novel viscous inverse method for blade design. In this method, viscosity is directly taken into account using a flow analysis code based on the Navier-Stokes equations. In this approach, the blade pressure loading and the blade thickness are prescribed and the corresponding blade profile is sought. By specifying the target pressure loading and the blade thickness, the camber line instead of the blade surface deforms. The prescription of the blade thickness ensures that a realistic blade shape with closed leading and trailing edges can be obtained. The redesign of an axial rotor and a stator blade, starting from an initial arbitrary profile in subsonic flow regimes, demonstrates the merits and robustness of this approach.

The rotor-router model is a deterministic analogue of random walk invented by Jim Propp. It can be used to define a deterministic aggregation model analogous to internal diffusion limited aggregation. We prove an isoperimetric inequality for the exit time of simple random walk from a finite region in Z d , and use this to prove that the shape of the rotor-router aggregation model in Z d , suitably rescaled, converges to a Euclidean ball in R d .

We study the rotor-router walk with the clockwise ordering of outgoing edges on the semi-infinite cylinder. Imposing uniform conditions on the boundary of the cylinder, we consider growth of the cluster of visited sites and its internal structure. The average width of the surface region of the cluster evolves with time to the stationary value by a scaling law whose parameters are close to the standard KPZ exponents. We introduce characteristic labels corresponding to closed clockwise contours formed by rotors and show that the sequence of labels has in average an ordered helix structure., Comment: 17 pages, 6 figures

Multiple chiral doublet bands (M$\chi$D) in the $80$, 130 and $190$ mass regions are studied by the model of $\gamma$=90$^{\circ}$ triaxial rotor coupled with identical symmetric proton-neutron configurations. By selecting the suitable basis, the calculated wave functions are explicitly exhibited to be symmetric under the operator $\hat{A}$, which is defined as rotation by $90^{\circ}$ about 3-axis with the exchange of valance proton and neutron. We found that both $M1$ and $E2$ transitions are allowed between the levels with different values of $A$, while are forbidden between the levels with same values of $A$. Such a selection rule holds true for M$\chi$D in different mass regions., Comment: 16 pages, 3 figures in Chinese Physics C 2016

The periodically $\ensuremath{\delta}$-kicked quantum linear rotor is known to experience nonclassical bounded energy growth due to quantum dynamical localization in angular momentum space. We study the effect of random deviations of the kick period in simulations and experiments. This breaks the energy and angular momentum localization and increases the rotational alignment, which is the analog of the onset of Anderson localization in one-dimensional chains.

Mathematical physics problems are often formulated by means of the vector analysis difierential operators: divergence, gradient and rotor. For approximate solutions of such problems it is natural to use the corresponding operator statements for the grid problems, i.e., to use the so-called VAGO (Vector Analys Grid Operators) method. In this paper, we discuss the possibilities of such an approach in using gen- eral irregular grids. The vector analysis difierence operators are constructed using the Delaunay triangulation and the Voronoi diagrams. The truncation error and the consistency property of the difierence operators constructed on two types of grids are investigated. Construction and analysis of the difierence schemes of the VAGO method for applied problems are illustrated by the examples of stationary and non-stationary convection-difiusion problems. The other examples concerned the solution of the non- stationary vector problems described by the second-order equations or the systems of flrst-order equations. 2000 Mathematics Subject Classiflcation: 65N06; 65M06.

An analysis of simulated framework structures based on the rigid-unit model quantifies the distortion of polyhedra from their ideal geometric forms and decomposes the motions of the structure into components of rigid-unit displacement, rigid-unit rotation and distortion. Case studies analysing the behaviour of quartz and of other silicates demonstrate that the method provides a novel way of extracting information from reverse Monte Carlo simulations. A program calledGASPhas been developed to perform this analysis and is freely available to researchers. Rotations are handled using the rotor method of geometric algebra.

This paper develops a stability theorem and response bounds for non-conservative systems of the form Mx+(D+G)x+(K+N)x=f(t), with hermitian positive-definite matrices M,D and K, and skew-hermitian matrices G and N. To this end, we first find a Lyapunov function by solving the Lyapunov matrix equation. Then, if a system satisfies the condition of the stability theorem, the associated Lyapunov function can be used to obtain response bounds for the norms as well as for the individual coordinates of the solution. Examples from rotor dynamics illustrate the results.

The triaxial dynamics of the quadrupole-deformed rotor model of both the rigid and the irrotational type have been investigated in detail. The results indicate that level patterns and E2 transitional characters of the two types of the model can be matched with each other to the leading order of the deformation parameter $\beta$. Especially, it is found that the dynamical structure of the irrotational type with most triaxial deformation ($\gamma=30^\circ$) is equivalent to that of the rigid type with oblate deformation ($\gamma=60^\circ$), and the associated spectrum can be classified into the standard rotational bands obeying the rotational $L(L+1)$-law or regrouped into a new ground- and $\gamma$-band with odd-even staggering in the new $\gamma$-band commonly recognized as a signature of the triaxiality. The differences between the two types of the model in this case are emphasized especially on the E2 transitional characters., Comment: 8 pages, 5 figures

In this work we present a method to build in a systematic way a many-body quon basis state. In particular, we show a closed expression for a given number N of quons, restricted to the permutational symmetric subspace, which belongs to the whole quonic space. The method is applied to two simple problems: the three-dimensional harmonic oscillator and the rotor model and compared to previous quantum algebra results. The differences obtained and possible future applications are also discussed., Comment: 13 pages, 2 figures