scholarly journals On a Simple Relation between Duffin-Kemmer-Petiau and Tzou Algebras

2018 ◽  
Vol 2018 ◽  
pp. 1-4
Author(s):  
Andrzej Okniński
Keyword(s):  
Unitary Transformation ◽  
Dimensional Subspace ◽  
Simple Relation ◽  
Simple Link ◽  
Lower Dimensional

A simple link between βμ matrices of the Duffin-Kemmer-Petiau theory and ρμ matrices of Tzou representations is constructed. The link consists of a constant unitary transformation of the βμ matrices and a projection onto a lower-dimensional subspace.

scholarly journals Eigencages: Learning a Latent Space of Porous Cage Molecules

2018 ◽  
Author(s):  
Arni Sturluson ◽  
Melanie T. Huynh ◽  
Arthur H. P. York ◽  
Cory Simon
Keyword(s):  
Dimensional Subspace ◽  
Singular Value ◽  
Adsorption Properties ◽  
3D Image ◽  
Latent Space ◽  
Cage Molecules ◽  
Gas Molecules ◽  
Low Dimensional ◽  
Value Decomposition ◽  
Lower Dimensional

<div>Porous organic cage molecules harbor nano-sized cavities that can selectively adsorb gas molecules, lending them applications in separations and sensing. The geometry of</div><div>the cavity strongly influences their adsorptive selectivity. </div><div><br></div><div>For comparing cages and predicting their adsorption properties, we embed/encode a set of 74 porous organic</div><div>cage molecules into a low-dimensional, latent “cage space” on the basis of their intrinsic porosity. </div><div><br></div><div>We first computationally scan each cage to generate a 3D image of its porosity. Leveraging the singular value decomposition, in an unsupervised manner, we then learn across all cages an approximate, lower-dimensional subspace in which the 3D porosity images lay. The “eigencages” are the set of orthogonal characteristic 3D porosity images that span this lower-dimensional subspace, ordered in terms of importance. A latent representation/encoding of each cage follows from expressing it as a combination of the eigencages. </div><div><br></div><div>We show that the learned encoding captures salient features of the cavities of porous cages and is predictive of properties of the cages that arise from cavity shape.</div>

scholarly journals Design and Implementation of Ethical Values Created Curriculum Development to Improve Mobile Security

2019 ◽  
Vol 9 (2S4) ◽  
pp. 625-628
Keyword(s):  
Curriculum Development ◽  
Dimensional Subspace ◽  
Mobile Security ◽  
Ethical Values ◽  
3D Perception ◽  
Design And Implementation ◽  
3D Surface ◽  
Relevant Calculation ◽  
Lower Dimensional

Medicinal imaging has assumed a key job in the direction of MIS strategies to expand the specialists' spatial introduction and help with the distinguishing proof of basic life systems and pathology. Current intransigent perception frameworks are promising. Be that as it may, they can barely meet the necessities of high goals and continuous 3D perception of the careful scene to help the acknowledgment of anatomic structures for safe MIS techniques. In this exploration we present a by and large relevant calculation which plans to furnish specialists with constant 3D perception of complete organ misshapen utilizing 3D optical fix pictures with constrained perspectives and a solitary preoperative MRI or CT filter. The proposed calculation is stretched out to remake the inside structures of an organ by just testing on the outside surface. Reconstructing is persuaded by our exact perception that the round consonant coefficients comparing to mutilated surfaces of a given organs lie in lower dimensional subspace in an organized lexicon that can be gained from a lot of agents preparing surfaces. The proposed methodology recognizes a structured scanty portrayal of every 3D surface. This enables the method to recreate discretionary organ misshapen utilizing exceptionally restricted watched information with high exactness.

Fractal properties of products and projections of measures in ℝd

1994 ◽  
Vol 115 (3) ◽  
pp. 527-544 ◽  
Author(s):  
Xiaoyu Hu ◽  
S. James Taylor
Keyword(s):  
Dimensional Subspace ◽  
Fractal Measure ◽  
Borel Measures ◽  
Fractal Properties ◽  
Packing Dimensions ◽  
Fractal Measures ◽  
Almost All ◽  
Lower Dimensional

AbstractBorel measures in ℝd are called fractal if locally at a.e. point their Hausdorff and packing dimensions are identical. It is shown that the product of two fractal measures is fractal and almost all projections of a fractal measure into a lower dimensional subspace are fractal. The results rely on corresponding properties of Borel subsets of ℝd which we summarize and develop.

scholarly journals Eigencages: Learning a Latent Space of Porous Cage Molecules

2018 ◽  
Author(s):  
Arni Sturluson ◽  
Melanie T. Huynh ◽  
Arthur H. P. York ◽  
Cory Simon
Keyword(s):  
Dimensional Subspace ◽  
Singular Value ◽  
Adsorption Properties ◽  
3D Image ◽  
Latent Space ◽  
Cage Molecules ◽  
Gas Molecules ◽  
Low Dimensional ◽  
Value Decomposition ◽  
Lower Dimensional

<div>Porous organic cage molecules harbor nano-sized cavities that can selectively adsorb gas molecules, lending them applications in separations and sensing. The geometry of</div><div>the cavity strongly influences their adsorptive selectivity. </div><div><br></div><div>For comparing cages and predicting their adsorption properties, we embed/encode a set of 74 porous organic</div><div>cage molecules into a low-dimensional, latent “cage space” on the basis of their intrinsic porosity. </div><div><br></div><div>We first computationally scan each cage to generate a 3D image of its porosity. Leveraging the singular value decomposition, in an unsupervised manner, we then learn across all cages an approximate, lower-dimensional subspace in which the 3D porosity images lay. The “eigencages” are the set of orthogonal characteristic 3D porosity images that span this lower-dimensional subspace, ordered in terms of importance. A latent representation/encoding of each cage follows from expressing it as a combination of the eigencages. </div><div><br></div><div>We show that the learned encoding captures salient features of the cavities of porous cages and is predictive of properties of the cages that arise from cavity shape.</div>

scholarly journals Eigencages: Learning a Latent Space of Porous Cage Molecules

2018 ◽  
Author(s):  
Arni Sturluson ◽  
Melanie T. Huynh ◽  
Arthur H. P. York ◽  
Cory Simon
Keyword(s):  
Dimensional Subspace ◽  
Singular Value ◽  
Adsorption Properties ◽  
3D Image ◽  
Latent Space ◽  
Cage Molecules ◽  
Gas Molecules ◽  
Low Dimensional ◽  
Value Decomposition ◽  
Lower Dimensional

<div>Porous organic cage molecules harbor nano-sized cavities that can selectively adsorb gas molecules, lending them applications in separations and sensing. The geometry of the cavity strongly influences adsorptive selectivity.</div><div><br></div><div>For comparing cages and predicting their adsorption properties, we embed/encode the cavities of a set of 74 porous organic cage molecules into a low-dimensional, latent "cage space".</div><div><br></div><div>We first scan the cavity of each cage to generate a 3D image of its porosity. Leveraging the singular value decomposition, in an unsupervised manner, we then learn across all cages an approximate, lower-dimensional subspace in which the 3D cage cavity images lay. The "eigencages" are the set of characteristic 3D cage cavity images that span this lower-dimensional subspace. A latent representation/encoding of each cage then follows from expressing it as a combination of the eigencages.</div><div><br></div><div>We show that the learned encoding captures salient features of the cavities of porous cages and is predictive of properties of the cages that arise from cavity shape.</div>

scholarly journals High-dimensional Bayesian optimization using low-dimensional feature spaces

2020 ◽  
Vol 109 (9-10) ◽  
pp. 1925-1943 ◽  
Author(s):  
Riccardo Moriconi ◽  
Marc Peter Deisenroth ◽  
K. S. Sesh Kumar
Keyword(s):  
Optimization Problem ◽  
Feature Space ◽  
Dimensional Subspace ◽  
Global Optimum ◽  
Black Box ◽  
Fine Tuning ◽  
Bayesian Optimization ◽  
Feature Spaces ◽  
Low Dimensional ◽  
Lower Dimensional

Abstract Bayesian optimization (BO) is a powerful approach for seeking the global optimum of expensive black-box functions and has proven successful for fine tuning hyper-parameters of machine learning models. However, BO is practically limited to optimizing 10–20 parameters. To scale BO to high dimensions, we usually make structural assumptions on the decomposition of the objective and/or exploit the intrinsic lower dimensionality of the problem, e.g. by using linear projections. We could achieve a higher compression rate with nonlinear projections, but learning these nonlinear embeddings typically requires much data. This contradicts the BO objective of a relatively small evaluation budget. To address this challenge, we propose to learn a low-dimensional feature space jointly with (a) the response surface and (b) a reconstruction mapping. Our approach allows for optimization of BO’s acquisition function in the lower-dimensional subspace, which significantly simplifies the optimization problem. We reconstruct the original parameter space from the lower-dimensional subspace for evaluating the black-box function. For meaningful exploration, we solve a constrained optimization problem.

scholarly journals Behavior of the iterative ensemble-based variational method in nonlinear problems

2021 ◽  
Vol 28 (1) ◽  
pp. 93-109
Author(s):  
Shin'ya Nakano
Keyword(s):  
Data Assimilation ◽  
Variational Method ◽  
Local Maximum ◽  
Nonlinear Problems ◽  
Dimensional Subspace ◽  
High Dimensional ◽  
Dimensional Approximation ◽  
Monotonic Convergence ◽  
System Nonlinearity ◽  
Lower Dimensional

Abstract. The behavior of the iterative ensemble-based data assimilation algorithm is discussed. The ensemble-based method for variational data assimilation problems, referred to as the 4D ensemble variational method (4DEnVar), is a useful tool for data assimilation problems. Although the 4DEnVar is derived based on a linear approximation, highly uncertain problems, in which system nonlinearity is significant, are solved by applying this method iteratively. However, the ensemble-based methods basically seek the solution within a lower-dimensional subspace spanned by the ensemble members. It is not necessarily trivial how high-dimensional problems can be solved with the ensemble-based algorithm which employs the lower-dimensional approximation based on the ensemble. In the present study, an ensemble-based iterative algorithm is reformulated to allow us to analyze its behavior in high-dimensional nonlinear problems. The conditions for monotonic convergence to a local maximum of the objective function are discussed in a high-dimensional context. It is shown that the ensemble-based algorithm can solve high-dimensional problems by distributing the ensemble in different subspace at each iteration. The findings as the results of the present study were also experimentally supported.

scholarly journals Eigencages: Learning a Latent Space of Porous Cage Molecules

2018 ◽  
Author(s):  
Arni Sturluson ◽  
Melanie T. Huynh ◽  
Arthur H. P. York ◽  
Cory Simon
Keyword(s):  
Dimensional Subspace ◽  
Singular Value ◽  
Adsorption Properties ◽  
3D Image ◽  
Latent Space ◽  
Cage Molecules ◽  
Gas Molecules ◽  
Low Dimensional ◽  
Value Decomposition ◽  
Lower Dimensional

<div>Porous organic cage molecules harbor nano-sized cavities that can selectively adsorb gas molecules, lending them applications in separations and sensing. The geometry of the cavity strongly influences adsorptive selectivity.</div><div><br></div><div>For comparing cages and predicting their adsorption properties, we embed/encode the cavities of a set of 74 porous organic cage molecules into a low-dimensional, latent "cage space".</div><div><br></div><div>We first scan the cavity of each cage to generate a 3D image of its porosity. Leveraging the singular value decomposition, in an unsupervised manner, we then learn across all cages an approximate, lower-dimensional subspace in which the 3D cage cavity images lay. The "eigencages" are the set of characteristic 3D cage cavity images that span this lower-dimensional subspace. A latent representation/encoding of each cage then follows from expressing it as a combination of the eigencages.</div><div><br></div><div>We show that the learned encoding captures salient features of the cavities of porous cages and is predictive of properties of the cages that arise from cavity shape.</div>

New Bounds of Weyl Sums

2019 ◽  
Author(s):  
Changhao Chen ◽  
Igor E Shparlinski
Keyword(s):  
Hausdorff Dimension ◽  
Infinite Series ◽  
Dimensional Subspace ◽  
Point Of View ◽  
Orthogonal Projections ◽  
Additional Advantage ◽  
Real Polynomials ◽  
New Ideas ◽  
Technical Innovations ◽  
Lower Dimensional

Abstract We augment the method of Wooley (2016) by some new ideas and in a series of results, improve his metric bounds on the Weyl sums and the discrepancy of fractional parts of real polynomials with partially prescribed coefficients. We also extend these results and ideas to principally new and very general settings of arbitrary orthogonal projections of the vectors of the coefficients $(u_1, \ldots , u_d)$ onto a lower-dimensional subspace. This new point of view has an additional advantage of yielding an upper bound on the Hausdorff dimension of sets of large Weyl sums. Among other technical innovations, we also introduce a “self-improving” approach, which leads to an infinite series of monotonically decreasing bounds, converging to our final result.

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