K-dimensional Space Research Articles

Structure of strongly proximinal subspaces

In this paper we extended some results of G. Godefroy, originally proved for c0 using its special geometry, to the class of complex Banach spaces X whose dual is isometric to L1(μ). We exhibit classes of complex Banach spaces in which every strongly proximinal finite codimensional subspace is a finite dimensional perturbation of a M-ideal. We also show for this class, when the unit ball has an extreme point, if every proximinal subspace of codimension one is strongly proximinal, then the space is isometric to the k-dimensional space with the maximum norm, ℓ∞(k).

Eigenvectors of the De-Rham Operator

We aim to examine the influence of the existence of a nonzero eigenvector ζ of the de-Rham operator Γ on a k-dimensional Riemannian manifold (Nk,g). If the vector ζ annihilates the de-Rham operator, such a vector field is called a de-Rham harmonic vector field. It is shown that for each nonzero vector field ζ on (Nk,g), there are two operators Tζ and Ψζ associated with ζ, called the basic operator and the associated operator of ζ, respectively. We show that the existence of an eigenvector ζ of Γ on a compact manifold (Nk,g), such that the integral of Ric(ζ,ζ) admits a certain lower bound, forces (Nk,g) to be isometric to a k-dimensional sphere. Moreover, we prove that the existence of a de-Rham harmonic vector field ζ on a connected and complete Riemannian space (Nk,g), having divζ≠0 and annihilating the associated operator Ψζ, forces (Nk,g) to be isometric to the k-dimensional Euclidean space, provided that the squared length of the covariant derivative of ζ possesses a certain lower bound.

Critical Pattern Selection Method Based on CNN Embeddings for Full-Chip Optimization

Source mask optimization (SMO), a primary resolution enhancement technology, is one of the most pivotal technologies for enhancing lithography imaging quality. Due to the high computation complexity of SMO, patterns should be selected by a selection algorithm before optimization. However, the limitations of existing selection methods are twofold: they are computationally intensive and they produce biased selection results. The representative method having the former limitation is the diffraction signature method. And IBM’s method utilizing the rigid transfer function tends to cause biased selection results. To address this problem, this study proposes a novel pattern cluster and selection algorithm architecture based on a convolutional neural network (CNN). The proposed method provides a paradigm for solving the critical pattern selection problem by CNN to transfer patterns from the source image domain to unified embeddings in a K-dimensional feature space, exhibiting higher efficiency and maintaining high accuracy.

Bisections of Mass Assignments Using Flags of Affine Spaces

We use recent extensions of the Borsuk–Ulam theorem for Stiefel manifolds to generalize the ham sandwich theorem to mass assignments. A k-dimensional mass assignment continuously imposes a measure on each k-dimensional affine subspace of $${\mathbb {R}}^d$$ . Given a finite collection of mass assignments of different dimensions, one may ask if there is some sequence of affine subspaces $$S_{k-1} \subset S_k \subset \ldots \subset S_{d-1} \subset {\mathbb {R}}^d$$ such that $$S_i$$ bisects all the mass assignments on $$S_{i+1}$$ for every i. We show it is possible to do so whenever the number of mass assignments of dimensions $$(k,\ldots ,d)$$ is a permutation of $$(k,\ldots ,d)$$ . We extend previous work on mass assignments and the central transversal theorem. We also study the problem of halving several families of $$(d-k)$$ -dimensional affine spaces of $${\mathbb {R}}^d$$ using a $$(k-1)$$ -dimensional affine subspace contained in some translate of a fixed k-dimensional affine space. For $$k=d-1$$ , there results can be interpreted as dynamic ham sandwich theorems for families of moving points.

VIKOR approach for bi-level multi-criteria nonlinear fractional programming problems: new insights

PurposeThe paper aims to introduce a novel concept to solve the bi-level multi-criteria nonlinear fractional programming (BL-MCNFP) problems. Bi-level programming problem (BLPP) is rigorously flourished and studied by several researchers, which deals with decentralized decisions by comprising a sequence of two optimization problems, namely upper and lower-level problems. However, on the other hand, many real-world decision-making problems involve multiple objectives with fraction aspects, called fractional programming problems that reflect technical and economic performance.Design/methodology/approachThis paper introduces a VIKOR (“VlseKriterijumska Optimizacija I Kompromisno Resenje”) approach to solve the BL-MCNFP problem. In this approach, an aggregating function based on LP metrics is formulated on the basis of the “closeness” scheme from the “ideal” solution. The three steps perform the solution process: First, a new concept is attempted to minimize and maximize of the numerators and denominators from their respective ideal solutions and anti-ideal values simultaneously. Second, for each level, the K-dimensional objective space of each level is converted to a one-dimensional space by an aggregating function. Third, to obtain the final solution, all levels are combined into single-level model where the decision variables of upper levels are interrelated with other levels through fuzzy strategy-based linear and nonlinear membership functions.FindingsThe effectiveness of the proposed VIKOR is demonstrated by numerical examples, where the reported results affirm that the extended VIKOR method provides superior results in comparison with the same methods in the literature, and it is a good alternative to BL-MCNFP problems.Originality/valueIn terms of the assistance-based right decision, a parametric analysis for the weight of the majority is provided to exhibit a wide range of compromise solutions for the decision-maker.

Application of Automatic Analysis of Image Data Based on KD-Tree in Ray Tracing Technology

In the field of computer graphics, ray tracing technology is widely used to generate high-quality "photo-level" images. The automatic analysis of KD-Tree image data is the data structure of dividing K-dimensional database space, and it has important application in the fast search of multi-dimensional space feature vector. Based on the automatic analysis of KD-Tree image data, the reasonable and effective application in ray tracing technology can make the verisimilitude of graphics reach the leap-type change, and realize the ideal application effect.

Extended local convergence for Newton-type solver under weak conditions

We present the local convergence of a Newton-type solver for equations involving Banach space valued operators. The eighth order of convergence was shown earlier in the special case of the k-dimensional Euclidean space, using hypotheses up to the eighth derivative although these derivatives do not appear in the method. We show convergence using only the rst derivative. This way we extend the applicability of the methods. Numerical examples are used to show the convergence conditions. Finally, the basins of attraction of the method, on some test problems are presented.

Modeling sentimental bias and temporal dynamics for adaptive deep recommendation system

Recommendation systems rely on the historic data of users' purchases and their feedbacks to profile their preferences and make future recommendations. Most of these systems usually employ Collaborative Filtering (CF) models to analyze users’ ratings and infer the latent factors which represent the user and item features in k-dimensional latent space. However, the historical rating data used for recommendations are usually sparsed and unbalanced. Various approaches have been used to resolve these issues by combining the user’s ratings and reviews to better capture the user’s sentiments and make accurate recommendations. Other challenges comprise changes in users’ preferences and items’ perceptions over time. Therefore, this paper presents a new Sentiment Scoring Model (SSM) based on Long-/Short-Term Memory and a combination function that catches the sentiment bias between user rating and review to relieve the sparsity and unbalanced dataset. Next, we proposed an Adaptive LSTM (ALSTM) method that can model the drifting of user and item features to improve the recommendation accuracy. We show the performance of our model on the three real-world rating datasets from Amazon reviews, which comprises Fine Food, Baby, and Cell-phone & Accessories categories. The result shows the superiority of our proposed model over the existing static and dynamic models. The statistical test shows that all the performance gains are significant at p < 0.05.

Testing Performance (Time Analysis) of Nearest Neighbour (NN) Search Algorithms on K-d Trees

K-d tree (k-dimensional tree) is a space partitioning data structure for organizing points in a k-dimensional space. K-d tree, or Multidimensional Binary Search Tree is a useful data structure for several applications such as searches involving a multidimensional search key (e.g., Range Search and Nearest Neighbour Search). K-d trees are a special case of binary space partitioning trees.KNN Search is a searching algorithm with complexity O(N log N) {N= no. of data points}. This search algorithm is relatively better than brute force search {Complexity= O(n*k); where k=No. of neighbours searched, N=No. of Data Points in Kd tree} for dimensions N&gt;&gt;2D {N=No. of Points, D=Dimensionality of Tree}.Furthermore, Parallel KNN Search is much more efficient and performs better than KNN Search, as it harnesses parallel processing capabilities of computers and thus, results in better search time.This paper tests the time performance of KNN Search and Parallel KNN Search and compares them by plotting it on a 3D graph. A more comprehensive comparison is done by use of 2D graphs for each dimension(from 2 to 20).

Statistical convergence of sequences of subspaces

In this study, we will give the deinition of statistical convergence of a sequences of subspaces of a k-dimensional inner product space and examine some inclusion relations.

Understanding Consumer Preferences Through Latent Spaces / Entendendo as preferências do consumidor através dos espaços latentes

The Latent Spaces technique has applications in areas such as natural language processing, image recognition and multi-language translation. It permits embedding vectors use. These are vectors in a k-dimensional vector space representing increasingly advanced study objects learning models, forming entirely new basis areas. Those vectors can capture semantic study object features and once trained can be reused in other models, decreasing training time and increasing knowledge transfer. This happens in Google Word2Vec and Facebook FastText pre-trained word vectors set. This work explores Latent Spaces techniques to understand preferences through recommendation mechanism implementation on top of MovieLens dataset from University of Minnesota. From it were extracted a sequence of triples (userId, movieId, rating) , representing ratings given by users to particular films. Two k-dimensional Latent Vector Spaces representing film characteristics and the corresponding user preferences were created using Google Tensorflow, Machine Learning techniques like SGD and Matrix Factorization. The performance benchmark was the mean-square error on the dimensionality k of the Latent Spaces. The capacity of the Latent Spaces was than evaluated to abstract non-trivial information about films and compared with small cosine distance technique. The system was able to find subjective related similarities that would be tough to code in a straightforward manner. Finally, it was explored an alternative to generate user vectors through neural networks. The techniques exposed here support the case that Machine Learning techniques like Latent Spaces can be used in business decision making.

\u03b1-Probabilistic flexible aggregate nearest neighbor search in road networks using landmark multidimensional scaling

In spatial database and road network applications, the search for the nearest neighbor (NN) from a given query object q is the most fundamental and important problem. Aggregate nearest neighbor (ANN) search is an extension of the NN search with a set of query objects Q = { q_0, dots , q_{M-1} } and finds the object p^* that minimizes g { d(p^*, q_i), q_i in Q }, where g (max or sum) is an aggregate function and d() is a distance function between two objects. Flexible aggregate nearest neighbor (FANN) search is an extension of the ANN search with the introduction of a flexibility factor phi , (0 < phi le 1) and finds the object p^* and the set of query objects Q^*_phi that minimize g { d(p^*, q_i), q_i in Q^*_phi }, where Q^*_phi can be any subset of Q of size phi |Q|. This study proposes an efficient alpha -probabilistic FANN search algorithm in road networks. The state-of-the-art FANN search algorithm in road networks, which is known as IER-khbox {NN}, used the Euclidean distance based on the two-dimensional coordinates of objects when choosing an R-tree node that most potentially contains p^*. However, since the Euclidean distance is significantly different from the actual shortest-path distance between objects, IER-khbox {NN} looks up many unnecessary nodes, thereby incurring many calculations of ‘expensive’ shortest-path distances and eventually performance degradation. The proposed algorithm transforms road network objects into k-dimensional Euclidean space objects while preserving the distances between them as much as possible using landmark multidimensional scaling (LMDS). Since the Euclidean distance after LMDS transformation is very close to the shortest-path distance, the lookup of unnecessary R-tree nodes and the calculation of expensive shortest-path distances are reduced significantly, thereby greatly improving the search performance. As a result of performance comparison experiments conducted for various real road networks and parameters, the proposed algorithm always achieved higher performance than IER-khbox {NN}; the performance (execution time) of the proposed algorithm was improved by up to 10.87 times without loss of accuracy.

Component-based 2-/3-dimensional nearest neighbor search based on Elias method to GPU parallel 2D/3D Euclidean Minimum Spanning Tree Problem

We present improved data parallel approaches working on graphics processing unit (GPU) compute unified device architecture (CUDA) platform to build hierarchical Euclidean minimum spanning forest or tree (EMSF/EMST) for applications whose input only contains N points with arbitrary data distribution in 2D/3D Euclidean space. Characteristic of the proposed parallel algorithms follows “data parallelism, decentralized control and O(1) local memory occupied by each GPU thread”. This research has to solve GPU parallelism of component-based nearest neighbor search (component-based NNS), tree traversal, and other graph operations like union-find. For exact NNS, instead of using classical K-d tree search or brute-force computing method, we propose a K-d search method working based on dividing the Euclidean K-dimensional space into congruent and non-overlapping square/cubic cells where size of points in each cell is bounded. For component-based NNS, with the uniqueness property based on 2D/3D square/cubic space partition, we propose dynamic and static pruning techniques to prune unnecessary neighbor cells’ search. For tree traversal, instead of using breadth-first-search, this paper proposes CUDA kernels working with a distributed dynamic link list for selecting a local spanning tree’s shortest outgoing edge since size of local EMSTs in EMSF cannot be predicted. Source code is provided online and experimental comparisons are conducted on both 2D and 3D benchmarks with up to 107 points to build final EMST. Results show that applying K-d search with static pruning technique and the proposed operators totally working in parallel on GPU, our current implementation runs faster than our previous work and current optimal sequential dual-tree mlpack EMST library.

Piecewise Fryze power theory analysis applied to PWM DC–DC converters

This study proposes an extension to the well-known Fryze power theory, which allows the development of a mathematical procedure that defines a global factor for the active and non-active power processing in pulse-width modulated (PWM) dc–dc converters. This global factor is the dc power factor. The proposed extension is a vector representation of periodic currents and voltages mapped into a k-dimensional Euclidean space, which permits that all non-active power of all converter elements to be collected into a single figure of merit. To validate the approaches, a 220 W prototype of an isolated dc–dc Ćuk converter architecture was implemented and evaluated. Experimental results have confirmed that both total non-active power, the proposed dc power factor, and system efficiency are correlated. In the worst case of step-down mode, the converter prototype presented the lowest total non-active power of ∼25 var for the turns ratio of 0.567, resulting in the highest dc power factor of 0.135 and prototype efficiency of 80.6%. In step-up mode, it was obtained the lowest total non-active power of ∼1.14 kvar for the turns ratio of 1.764, resulting in the highest efficiency of 88.3% and dc power factor of 0.145.

Percentile Study of χ Distribution. Application to Response Time Data

As a continuation of our previous work, where a Maxwell–Boltzmann distribution was found to model a collective’s reaction times, in this work we will carry out a percentile study of the χ distribution for some freedom ranging from k = 2 to k = 10. The most commonly used percentiles in the biomedical and behavioral sciences have been included in the analysis. We seek to provide a look-up table with percentile ratios, taken symmetrically about the median, such that this distribution can be identified in practice in an easy way. We have proven that these ratios do not depend upon the variance chosen for the k generating Gaussians. In general, the χ probability density, generalized to take any value of the variance, represents an ideal gas in a k-dimensional space. We also derive an approximate expression for the median of the generalized χ distribution. In the second part of the results, we will focus on the practical case of k = 3, which represents the ideal gas in physics, and models quite well the reaction times of a human collective. Accurately, we will perform a more detailed scrutiny of the percentiles for the reaction time distribution of a sample of 50 school-aged children (7200 reaction times).

Noise fit, estimation error and a Sharpe information criterion

When the in-sample Sharpe ratio is obtained by optimizing over a k-dimensional parameter space, it is a biased estimator for what can be expected on unseen data (out-of-sample). We derive (1) an unbiased estimator adjusting for both sources of bias: noise fit and estimation error. We then show (2) how to use the adjusted Sharpe ratio as model selection criterion analogously to the Akaike Information Criterion (AIC). Selecting a model with the highest adjusted Sharpe ratio selects the model with the highest estimated out-of-sample Sharpe ratio in the same way as selection by AIC does for the log-likelihood as a measure of fit.

Ultra-fast analog ensemble using kd-tree

Analog ensemble (AnEn) is a popular probabilistic weather forecasting method based on similarity search. In that, forecasters are tasked to search for the top-m nearest neighbors (e.g., in terms of Euclidean distance) to a length-k query, from a set of historical data points in k-dimensional space. This is a straightforward yet time-consuming procedure, and few methods seem to be significantly better than a brute-force computation of all distances. To that end, I recommend using a kd-tree to perform AnEn, which appears to be one of (if not) the fastest approaches.

The genus of graphs associated with vector spaces

Let [Formula: see text] be a k-dimensional vector space over a finite field [Formula: see text] with a basis [Formula: see text]. The nonzero component graph of [Formula: see text], denoted by [Formula: see text], is a simple undirected graph with vertex set as nonzero vectors of [Formula: see text] such that there is an edge between two distinct vertices [Formula: see text] if and only if there exists at least one [Formula: see text] along which both [Formula: see text] and [Formula: see text] have nonzero scalars. In this paper, we find the vertex connectivity and girth of [Formula: see text]. We also characterize all vector spaces [Formula: see text] for which [Formula: see text] has genus either 0 or 1 or 2.

Research on Semantic Prediction Analysis of Tibetan Text Based on Word2Vec

This article uses Google’s open source Word2Vec tool to input the corpus of the Tibetan text “Sage wedding” after the word segmentation. The words are mapped to a K-dimensional space in the text and transformed into word vectors by using the context information of the vocabulary. The Word2Vec tool then learns to get a vector model, each of which is represented by a unique word vector. A vocabulary is constructed through training text data and then the vector is represented by learning the words. Word vectors capture the laws of many languages, which results the similarity of the distance between words and words. The experimental results show that the accuracy and recall rate based on the Word2Vec training model are very high.

INNOVATIVE CLUSTERS OF BUSINESS ACCELERATORS IN THE SPHERE OF SCIENCE AND TECHNOLOGY ENTREPRENEURSHIP

In a broad sense, the author’s study was preconditioned by the problem of arranging the efficient system for positioning of the enterprises of venture entrepreneurship acceleration and incubation that today is quite a controversial aspect. At the same time, for today there are no studies and publications related to the identification of approaches to the positioning of clusters of the enterprise innovative activity acceleration that will make it possible to find the most efficient approaches to acceleration. Therefore, this paper is aimed at identifying the basic approaches in the positioning of startup acceleration and incubation institutes. Methodology. The study is based on the algorithms of cluster analysis. Having identified the key parameters of the analysed institutions, using the “Statistics7” program package, we have combined the input data to address the challenge of cluster analysis, consisting in classification breakdown, which meets the criterion of optimality. Proceeding from the research target, the rule “Single Linkage” was chosen as the key function. According to this rule, there are two objects, being the closest to each other. At the next step, the object, which has the maximum degree of similarity with one of the cluster objects, joins them. This method is also called the method of the closest neighbour as the distance between two clusters is defined as the distance between the closest two objects in different clusters. 1 - Pearson is chosen as a degree of distance as far as the given data cannot be presented as points in k-dimensional space. Results. The first part of this research contains a justification of the positioning indicators among the analysed startup accelerators, which were chosen for cluster analysis according to functioning models. In the second part, we consider the distinction of the functioning models of the obtained clusters and the features, typical for every enterprise, which has been involved in the study. Proceeding from the calculation of the configuration of distances of cluster formations using the method of k-means while approaching the process of startup acceleration and incubation, we have identified the aspect, which is of primary importance for the formation of resultative startup-accelerator. Practical implications. The obtained results make it possible to optimize the resources in the process of creation and positioning of startup accelerators and incubators. Value/originality. Based on the developed configuration of the distances of cluster formations using the method of к-means in approached-to-startup acceleration and incubation process, we have found out the best practices in the area of venture business acceleration.