scholarly journals Generating All 36,864 Four-Color Adinkras via Signed Permutations and Organizing into ℓ- and ℓ˜-Equivalence Classes

Symmetry ◽  
2019 ◽  
Vol 11 (1) ◽  
pp. 120 ◽  
Author(s):  
S. Gates ◽  
Kevin Iga ◽  
Lucas Kang ◽  
Vadim Korotkikh ◽  
Kory Stiffler
Keyword(s):  
Permutation Group ◽  
Quaternion Algebra ◽  
Equivalence Classes ◽  
Inner Product ◽  
Complete Analysis ◽  
Signed Permutation ◽  
Signed Permutations ◽  
Non Linear

Recently, all 1,358,954,496 values of the gadget between the 36,864 adinkras with four colors, four bosons, and four fermions have been computed. In this paper, we further analyze these results in terms of B C 3 , the signed permutation group of three elements, and B C 4 , the signed permutation group of four elements. It is shown how all 36,864 adinkras can be generated via B C 4 boson × B C 3 color transformations of two quaternion adinkras that satisfy the quaternion algebra. An adinkra inner product has been used for some time, known as the gadget, which is used to distinguish adinkras. We show how 96 equivalence classes of adinkras that are based on the gadget emerge in terms of B C 3 and B C 4 . We also comment on the importance of the gadget as it relates to separating out dynamics in terms of Kähler-like potentials. Thus, on the basis of the complete analysis of the supersymmetrical representations achieved in the preparatory first four sections, the final comprehensive achievement of this work is the construction of the universal B C 4 non-linear σ -model.

Novel method for bearing performance degradation assessment – A kernel locality preserving projection-based approach

2013 ◽  
Vol 228 (3) ◽  
pp. 548-560 ◽  
Author(s):  
Chuang Sun ◽  
Zhousuo Zhang ◽  
Zhengjia He ◽  
Zhongjie Shen ◽  
Binqiang Chen ◽  
...  
Keyword(s):  
Test Data ◽  
Performance Degradation ◽  
Inner Product ◽  
Data Sets ◽  
Degradation Index ◽  
Locality Preserving Projection ◽  
Locality Preserving ◽  
Non Linear ◽  
Linear Information ◽  
Novel Method

Bearing performance degradation assessment is meaningful for keeping mechanical reliability and safety. For this purpose, a novel method based on kernel locality preserving projection is proposed in this article. Kernel locality preserving projection extends the traditional locality preserving projection into the non-linear form by using a kernel function and it is more appropriate to explore the non-linear information hidden in the data sets. Considering this point, the kernel locality preserving projection is used to generate a non-linear subspace from the normal bearing data. The test data are then projected onto the subspace to obtain an index for assessing bearing degradation degrees. The degradation index that is expressed in the form of inner product indicates similarity of the normal data and the test data. Validations by using monitoring data from two experiments show the effectiveness of the proposed method.

scholarly journals Diagrams and Essential Sets for Signed Permutations

10.37236/8106 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
David Anderson
Keyword(s):  
Schubert Variety ◽  
Signed Permutation ◽  
Signed Permutations ◽  
Degeneracy Locus ◽  
Essential Set ◽  
Diagrammatic Method ◽  
Essential Sets ◽  
Rank Conditions ◽  
Theoretic Version

We introduce diagrams and essential sets for signed permutations, extending the analogous notions for ordinary permutations.  In particular, we show that the essential set provides a minimal list of rank conditions defining the Schubert variety or degeneracy locus corresponding to a signed permutation.  Our essential set is in bijection with the poset-theoretic version defined by Reiner, Woo, and Yong, and thus gives an explicit, diagrammatic method for computing the latter.

Advantages of Spectrally Analyzed Data: Stochastic Models for Automotive Measured Parameters

2014 ◽  
Vol 1036 ◽  
pp. 493-498 ◽  
Author(s):  
Radu Vilău ◽  
Marin Marinescu ◽  
Octavian Alexa ◽  
Florin Oloeriu ◽  
Marian Truta
Keyword(s):  
Dynamic Models ◽  
Real Life ◽  
Stochastic Approach ◽  
Angular Speed ◽  
Point Of View ◽  
Mechanical Parameter ◽  
Complete Analysis ◽  
Measured Parameter ◽  
Useful Signal ◽  
Non Linear

The paper deals with a new approach in data analysis of a measured mechanical parameter. The classic approach is mainly based on the deterministic statistics that cant cover the whole field of a complete analysis. The stochastic approach, to be used in this paper, offers far more information about the mechanical parameter and can take into account the non-linearity of the signal, eventually, the mechanical parameter itself. Starting from the point of view that, in real life, there is no steady evolution of any parameter, we decide to take into account the importance of the non-linear components of any signal. After e thorough investigation, we hope we could make the difference between the noise, as non-linear components of the measured parameter, and the useful non-linear components (e.g. important shocks, typically met within a vehicles transmission). Using the stochastic modeling procedures, we aimed at issuing comprehensive, accurate and valuable dynamic models of the phenomenon. These models cam be used in a large variety of situations, from describing the process, to evaluating the health of a mechanical system and to controlling a real-time process based on the pre-set models (previously drawing a map of the systems normal behavior and permanently assessing the deviation from it and acting accordingly). The data were measured within the transmission system of a military vehicle. Specifically, we have gathered information about torque and angular speed of different shafts of the driveline. As everybody knows, the power flows within any vehicles transmission in transient modes mainly and it is accompanied by plenty of noise. It is rather challenging to separate (filter) the useful signal form the noise but, on the other hand, it is the only way to achieve useful data. Therefore, a spectral analysis is a must, but not the conventional one, which has its drawbacks, but a multi-spectral one, which is able to insulate the noise. Moreover, starting from the analysis developed with this method, mathematical models, both in discrete and continuos time can be achieved. It is easy to notice that the models that we have achieved are featured by a very good accuracy. We could push the data processing even further, getting generalized models that provide the needs we have mentioned before, with respect to the mapping of a normal (averaged) behavior of a system, to be used in controlling procedures.

Complementary bounds for inner products associated with non-linear equations

1984 ◽  
Vol 96 (1-2) ◽  
pp. 135-142 ◽  
Author(s):  
R. J. Cole ◽  
J. Mika ◽  
D. C. Pack
Keyword(s):  
Real Hilbert Space ◽  
Linear Equations ◽  
Linear Operators ◽  
Inner Product ◽  
Upper And Lower Bounds ◽  
Non Linear Equation ◽  
Non Linear ◽  
Non Linear Operator ◽  
Non Linear Equations ◽  
Unknown Solution

SynopsisFunctionals are found that give upper and lower bounds to the inner product 〈g0, f〉 involving the unknown solution f of a non-linear equation T[f] = f0, with f∈H, a real Hilbert space, g0 a given function in H and f0 a given function in the range of the non-linear operator T. The method depends upon a re-ordering of terms in the expansion of T[f] about a trial function so as to transfer the non-linearity to a secondary problem that requires its own particular treatment and to enable earlier results obtained for linear operators to be used for the main part. First, bivariational bounds due to Barnsley and Robinson are re-derived. The new and more accurate bounds are given under relaxed assumptions on the operator T by introducing a third approximating function. The results are obtained from identities, thus avoiding some of the conditions imposed by the use of variational methods. The accuracy of the new method is illustrated by applying it to the problem of the heat contained in a bar.

scholarly journals Quaternion Collaborative Filtering for Recommendation

2019 ◽  
Author(s):  
Shuai Zhang ◽  
Lina Yao ◽  
Lucas Vinh Tran ◽  
Aston Zhang ◽  
Yi Tay
Keyword(s):  
Collaborative Filtering ◽  
Quaternion Algebra ◽  
Wide Spectrum ◽  
Representation Learning ◽  
Real Space ◽  
Inner Product ◽  
Hypercomplex Numbers ◽  
Learning Capability ◽  
Inductive Bias ◽  
Real World Datasets

This paper proposes Quaternion Collaborative Filtering (QCF), a novel representation learning method for recommendation. Our proposed QCF relies on and exploits computation with Quaternion algebra, benefiting from the expressiveness and rich representation learning capability of Hamilton products. Quaternion representations, based on hypercomplex numbers, enable rich inter-latent dependencies between imaginary components. This encourages intricate relations to be captured when learning user-item interactions, serving as a strong inductive bias  as compared with the real-space inner product. All in all, we conduct extensive experiments on six real-world datasets, demonstrating the effectiveness of Quaternion algebra in recommender systems. The results exhibit that QCF outperforms a wide spectrum of strong neural baselines on all datasets. Ablative experiments confirm the effectiveness of Hamilton-based composition over multi-embedding composition in real space. 

Self-Complementary Generalized Orbits of a Permutation Group

1974 ◽  
Vol 17 (2) ◽  
pp. 203-208 ◽  
Author(s):  
Roberto Frucht ◽  
Frank Harary
Keyword(s):  
Permutation Group ◽  
Equivalence Relation ◽  
Equivalence Classes ◽  
Cycle Index ◽  
Arbitrary Permutation ◽  
Group A ◽  
Special Classes

AbstractA permutation group A of degree n acting on a set X has a certain number of orbits, each a subset of X. More generally, A also induces an equivalence relation on X(k) the set of all k subsets of X, and the resulting equivalence classes are called k orbits of A, or generalized orbits. A self-complementary k-orbit is one in which for every k-subset S in it, X—S is also in it. Our main results are two formulas for the number s(A) of self-complementary generalized orbits of an arbitrary permutation group A in terms of its cycle index. We show that self-complementary graphs, digraphs, and relations provide special classes of self-complementary generalized orbits.

The impact of external debt on total factor productivity and growth in HIPCs: non-linear regression approaches

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Sisay Demissew Beyene ◽  
Balázs Kotosz
Keyword(s):  
Total Factor Productivity ◽  
Threshold Model ◽  
External Debt ◽  
Seemingly Unrelated Regression ◽  
Complete Analysis ◽  
Threshold Values ◽  
Factor Productivity ◽  
Content Type ◽  
Non Linear ◽  
The Impact

Purpose The purpose of this study is to provide an empirical analysis of the impact of external debt on total factor productivity (TFP) and growth along with the TFP channel through which external debt affects the growth of heavily indebted poor countries (HIPCs). Design/methodology/approach This study uses panel data econometrics; basically, the seemingly unrelated regression (SUR) and alternative non-linear (panel threshold) models. For robustness check, it also uses panel-corrected standard errors, feasible generalized least squares and SUR (using alternative variables). Findings External debt significantly reduces both TFP and growth. Besides, it confirms that the relationship between external debt and TFP and gross domestic product growth is non-linear. Further external debt can affect the growth of HIPCs through the TFP channel. However, the threshold model result reveals weak evidence of threshold values although there are some threshold values of 67 and 54 for TFP and growth models, respectively. Originality/value To the best of the authors’ knowledge, this is the first study on most concerned countries (HIPCs) that shows a detailed and complete analysis of the TFP channel and the impact of external debt on growth. Thus, it provides appropriate and sound policies that consider the unique characteristics of the countries. Unlike most previous findings, this study does not support an inverted U-shape relationship between external debt and growth. Further, it provides insights into the relationships among TFP, external debt and growth. Moreover, it considers basic panel econometric tests like cross-sectional dependence, uses a non-linear simultaneous equations model along with the alternative non-linear model and is supported by different robustness checks.

Non-Linear Response Analysis of Jacket Supported Offshore Wind Turbine under Seismic Loads using Hilbert-Huang Transform

2021 ◽  
Author(s):  
Subham Kashyap ◽  
Nilanjan Saha ◽  
K. A. Abhinav
Keyword(s):  
Finite Element ◽  
Wind Turbine ◽  
Offshore Wind ◽  
Resonance Phenomenon ◽  
Complete Analysis ◽  
Offshore Wind Turbine ◽  
Site Class ◽  
Hilbert Huang Transform ◽  
Class D ◽  
Non Linear

Abstract The present work studies the performance of an offshore wind turbine system in an earthquake coupled with wave and wind loading. The NREL 5 MW offshore wind turbine, supported on the OC4 jacket [14], has been analysed within a finite element framework. A coupled model of hydrodynamics and soil-structure interaction has been implemented. The structure-foundation system is analysed under earthquakes recorded close to offshore waters and at sites with shear-wave velocities, classified under Site-Class D or Site-Class E as per API RP: 2EQ [8]. The soil conditions emulate characteristics of a prospective offshore wind turbine site along the west coast of India, which falls within the Site-Class D classification mentioned above. The geotechnical modelling is done as per the soil curves prescribed by the non-linear Winkler springs along the pile’s length. The complete analysis has been processed in a finite-element framework through the commercial program USFOS [16]. The Hilbert-Huang transform [29] of the tower-responses suggests the increased vulnerability to the resonance phenomenon with 1P and 3P loading. It also suggests an involvement of higher modes in the tower-response. The change in the frequency of the structure-foundation system during and post-earthquake has also been studied.

scholarly journals Depth in Coxeter groups of type $B$

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Eli Bagno ◽  
Riccardo Biagioli ◽  
Mordechai Novick
Keyword(s):  
Coxeter Group ◽  
Simple Formula ◽  
Coxeter Groups ◽  
Type B ◽  
Minimal Path ◽  
Signed Permutation ◽  
Signed Permutations ◽  
International Audience ◽  
Path Cost

International audience The depth statistic was defined for every Coxeter group in terms of factorizations of its elements into product of reflections. Essentially, the depth gives the minimal path cost in the Bruaht graph, where the edges have prescribed weights. We present an algorithm for calculating the depth of a signed permutation which yields a simple formula for this statistic. We use our algorithm to characterize signed permutations having depth equal to length. These are the fully commutative top-and-bottom elements defined by Stembridge. We finally give a characterization of the signed permutations in which the reflection length coincides with both the depth and the length. La statistique profondeur a été introduite par Petersen et Tenner pour tout groupe de Coxeter $W$. Elle est définie pour tout $w \in W$ à partir de ses factorisations en produit de réflexions (non nécessairement simples). Pour le type $B$, nous introduisons un algorithme calculant la profondeur, et donnant une formule explicite pour cette statistique. On utilise par ailleurs cet algorithme pour caractériser tous les éléments ayant une profondeur égale à leur longueur. Ces derniers s’avèrent être les éléments pleinement commutatifs “hauts-et-bas” introduits par Stembridge. Nous donnons enfin une caractérisation des éléments dont la longueur absolue, la profondeur et la longueur coïncident.

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