K-Plane Clustering

Summary: Similarity and categorization. The present paper is a review of the literature dealing with current models of categorization and similarity estimation. The first part reviews models of concept representations based on notions such as features and dimensions. In the case of dimensional representations, each stimulus is defined as a point in a multidimensional space and the similarity between two stimuli is a function of the distance between the corresponding points in the space. In the case of feature representations, a stimulus is defined in terms of features and the similarity between two stimuli is a function of their common and distinctive features. In the second part, we analyze the construction of the feature space that subjects refer to when they have to categorize a stimulus or estimate the similarity between two stimuli. We examine the influence of context and of naive theories on the estimation of similarity. The process of alignment is analyzed. We present a unified view of similarity and categorization based on the notion of a feature space while explaining the dissociations obtained between similarity and categorization. Key words : similarity, categorization, concepts, theories.

Clearly, therefore, it is important to group diamond processing conditions according to one of the key properties of diamond, viz., abrasivity. This can be done by cluster analysis and by the principal components method. These methods make it possible to combine a rigorous formal approach eliminating subjective errors in the model construction stage (classification) with the experience and intuition of the investigator in the interpretation of results. The authors carried out such an analysis on 108 batches of micropowders ranging in grain size from 60/40 to 5/3 ~m (eight size groups). In each size group determinations were made of abrasivity and it was this characteristic that was employed as an attribute. Thus, the physical object under examination was described by eight attributes. Any multimensional observation (in our case eight-dimensional) can be geometrically interpreted as a point in multidimensional space. Geometric proximity of two or more points in this space indicates proximity of the physical states of the corresponding objects, i.e., their uniformity.

The local Hopf bundles at the points and lines of a compact projective plane with manifold lines are all weakly equivalent. In particular, the point space and the line space of such a projective plane are homeomorphic. This is a consequence of the following topological result. Theorem. Let $\xi,\xi'$ be orientable topological $\mathbb{R}^n$ -bundles over an n-dimensional CW-complex. If $\xi$ and $\xi'$ are fibre homotopy equivalent and stably equivalent, then $\xi$ and $\xi'$ are equivalent.

The union bound is known as the most useful tool for the computation of error probability in digital communication schemes (both coded and non-coded). Given a signal set or codebook C={x/sub 0/,x/sub 1/,...,x/sub M-1/}, the conditional error probability P(e|x/sub 0/) is upper bounded by a sum of pairwise error probabilities (PEP). Here we consider the additive white Gaussian channel, where C is a discrete and finite set of points in the N-dimensional real Euclidean space. In this case, the PEPs are the probabilities of the noise taking a signal on the other side of the hyperplane bounding the decision regions of x/sub 0/ and x/sub i/. An important step toward the reduction of the union bound to its minimal form was made by Verdu, who derived a theorem, commonly referred to as the Verdu-Shields theorem showing how terms can be removed from the upper bound to error probability in the case of binary antipodal transmission over the Gaussian channel with intersymbol interference. In this paper we derive a generalization of the Verdu-Shields theorem which provides a sufficient condition for expurgating a given error sequence x/sub k/ from the union bound.

The following article has been included in a multiple retraction: Canonical Duality Theory for Solving Nonconvex/Discrete Constrained Global Optimization Problems; Ning Ruan, David Y Gao. http://mms.sagepub.com/content/early/2015/07/08/1081286515591087.abstract In 2015 SAGE were made aware of concerns regarding the Special Issue of Mathematics & Mechanics of Solids on Advances in Canonical Duality Theory, guest-edited by Professor David Gao. At the request of the Guest Editor, the Special Issue has been retracted, due to conflict of interest regarding Professor Gao’s role as Guest Editor and co-author on a number of submitted papers. In addition the peer review process was less rigorous than the journal requires. The Guest Editor takes full responsibility for the retraction. The following articles that were due to appear in the Special Issue have therefore been retracted: Canonical Duality-Triality: Bridge between Nonconvex Analysis/Mechanics and Global Optimization in complex systems; David Y Gao, Ning Ruan, and Vittorio Latorre. http://mms.sagepub.com/content/early/2015/02/24/1081286514566533.abstract Canonical Dual Approach for Contact Mechanics Problems with Friction; Vittorio Latorre, Simone Sagratella, David Y Gao. http://mms.sagepub.com/content/early/2015/01/20/1081286514566534.abstract Canonical Duality Theory for Solving Non-Monotone Variational Inequality Problems; Guoshan Liu, David Y Gao, Shouyang Wang. http://mms.sagepub.com/content/early/2015/02/04/1081286514566535.abstract Double Well Potential Function and Its Optimization in The n-dimensional Real Space. Part I; Shu-Cherng Fang, David Y Gao, Gang-Xuan Lin, Ruey-Lin Sheu, Wen-Xun Xing. http://mms.sagepub.com/content/early/2015/02/24/1081286514566704.abstract Double Well Potential Function and Its Optimization in The n-dimensional Real Space. Part II; Yong Xia, Ruey-Lin Sheu, Shu-Cherng Fang, Wenxun Xing. http://mms.sagepub.com/content/early/2015/02/09/1081286514566723.abstract Analytic Solutions to 3-D Finite Deformation Problems Governed by St Venant-Kirchhoff Material; David Y Gao and E. Hajilarov. http://mms.sagepub.com/content/early/2015/07/06/1081286515591084.abstract Triality Theory and Complete Post-buckling Solutions of Large Deformed Beam by Canonical Dual Finite Element Method; Kun Cai, David Y Gao, Qinghua Qin. http://mms.sagepub.com/content/early/2015/06/28/1081286515591085.abstract Global Solutions to Spherically Constrained Quadratic Minimization via Canonical Duality Theory; Yi Chen, David Y Gao. http://mms.sagepub.com/content/early/2015/04/08/1081286515577122.abstract Unified Canonical Duality Methodology for Global Optimization; Vittorio Latorre, David Y Gao and N. Ruan. http://mms.sagepub.com/content/early/2015/07/06/1081286515591305.abstract A Framework of Canonical Dual Algorithms for Global Optimization; Xiaojun Zhou, David Y Gao, Chunhua Yang. http://mms.sagepub.com/content/early/2015/07/22/1081286515592190.abstract Canonical Duality Theory for Solving Nonconvex/Discrete Constrained Global Optimization Problems; Ning Ruan, David Y Gao. http://mms.sagepub.com/content/early/2015/07/08/1081286515591087.abstract Global Optimization Solutions to a Class of Non-convex Quadratic Minimization Problems with Quadratic Constraints; Yubo Yuan. http://mms.sagepub.com/content/early/2015/07/06/1081286515591086.abstract On Minimal Distance between Two Non-Convex Surfaces; Daniel Morales-Silva, David Y Gao. http://mms.sagepub.com/content/early/2015/07/27/1081286515592949.abstract The Editor-in-Chief and SAGE strive to uphold the very highest standards of publication ethics and are committed to supporting the high standards of integrity of Mathematics & Mechanics of Solids. Authors, reviewers, editors and interested readers are encouraged to consult SAGE’s ethics statements and the Committee on Publication Ethics (COPE) website for guidelines on publication ethics.

Effects of diet on the childhood gut microbiome and its implications for atopic dermatitis

A general mathematical formulation of the n £ n proper Orthogonal matrix, that corresponds to a rigid rotation in n-dimensional real Euclidean space, is given here. It is shown that a rigid rotation depends on an angle (principal angle) and on a set of (ni2) principal axes. The latter, however, can be more conveniently replaced by only 2 Orthogonal directions that identify the plane of rotation. The inverse problem, that is, how to compute these principal rotation parameters from the rotation matrix, is also treated. In this paper, the Euler Theorem is extended to rotations in n-dimensional spaces by a constructive proof that establishes the relationship between orientation of the displaced Orthogonal axes in n dimensions and a minimum sequence of rigid rotations. This fundamental relationship, which introduces a new decomposition for proper Orthogonal matrices (those identifying an orientation), can be expressed either by a product or a sum of the same rotation matrices. A similar decomposition in terms of the Skew-Symmetric matrices is also given. The extension of the rigid rotation formulation to n-dimensional complex Euclidean spaces, is also provided. Finally, we introduce the Ortho-Skew real matrices, which are simultaneously proper Orthogonal and Skew-Symmetric and which exist in even dimensional spaces only, and the Ortho-Skew-Hermitian complex matrices which are Orthogonal and Skew-Hermitian. The Ortho-Skew and the Ortho-SkewHermitian matrices represent the extension of the scalar imaginary to the matrix fleld. ⁄ Work previously presented in part at the Second International Conference on Non Linear Prob

A set of $n$ points in Euclidean space is partitioned into the $k$ groups that minimize the within groups sum of squares. Under the assumption that the $n$ points come from independent sampling on a fixed distribution, conditions are found to assure asymptotic normality of the vector of means of the $k$ groups. The method of proof makes novel application of a functional central limit theorem for empirical processes--a generalization of Donsker's theorem due to Dudley.

We consider a problem of constructing algebraic surfaces on the basis of Grassmann’s ideas. We consider the surfaces of the second and higher orders with the purpose of obtaining their Grassmann’s extensive equations and their images in Java environment. The works (Banai and Manevich in Proceedings of the 6th International Conference on Engineering Computer Graphics and Descriptive Geometry, Tokyo, Japan, 1994; Manevich and Sluzkin in Linear Constructions of Grassmann, Kwant, 9 (in Russian). Moscow, pp. 19–22, 1977) [1, 2] were devoted to construction of flat curves of high order on the basis of Grassmann’s ideas. It should be noted that in the catalog of mathematical models of surfaces (Schilling in Catalog Mathematischer Modelle Mit 106 Abbildungen. Siebente Auflage. Leipzig 1911) [3], and in Pabler and Lordick (J Geom Graph 21:263–271, 2017), Velichova (J Geom Graph 19:13–29, 2015) [4, 5] one can see the physical models and the computer images of high degree surfaces. For the construction of the algebraic surfaces of high degrees according to Grassmann points, planes and lines are given as a geometric basis. Linear operations such as the joining of two points by a straight line and the construction of intersection point of straight line with a plane are conducted sequentially. The last operation in this sequence—the intersection of a straight line with a plane generally gives the point of the desired surface. It is shown that on certain straight lines in the process of constructing a surface, projective rows of points are formed. The surface equation is seen from the record of Grassmann extensive equations. The proof that the resulting set of points in space belongs to a high degree surfaces for a number of cases is given by the method of projective geometry. Most of the constructions are based on the G. Grassmann statement: Let there be a certain number of fixed points, straight lines and planes in space. An arbitrary point X is associated with these elements by means of connections and intersections. If as a result of such constructions it turns out that certain four points lie in the same plane or certain two straight lines are intersected, then the point X belongs to an algebraic surface whose order is equal to the number indicating how many times the point X participated in the constructions. (Klein in Linear Constructions of Grassmann, pp. 220–224) [6] On the basis of this theorem we write a computation algorithm for finding images of surfaces of high degrees. It should be emphasized that all the extensive equations remain invariant when the coordinate system is replaced. Usage of the proposed constructions makes it possible to understand more deeply the various methods for the formation of algebraic surfaces, which can be useful in the study of various sections of analytic and projective geometry.

How similar are two data-cubes? In other words, the question under consideration is: given two sets of points in a multidimensional hierarchical space, what is the distance value between them? In this paper we explore various distance functions that can be used over multidimensional hierarchical spaces. We organize the discussed functions with respect to the properties of the dimension hierarchies, levels and values. In order to discover which distance functions are more suitable and meaningful to the users, we conducted two user study analysis. The first user study analysis concerns the most preferred distance function between two values of a dimension. The findings of this user study indicate that the functions that seem to fit better the user needs are characterized by the tendency to consider as closest to a point in a multidimensional space, points with the smallest shortest path with respect to the same dimension hierarchy. The second user study aimed in discovering which distance function between two data cubes, is mostly preferred by users. The two functions that drew the attention of users where (a) the summation of distances between every cell of a cube with the most similar cell of another cube and (b) the Hausdorff distance function. Overall, the former function was preferred by users than the latter; however the individual scores of the tests indicate that this advantage is rather narrow.

A (finite or infinite) word is said to be k-th power-free if it does not contain k consecutive equal blocks. A colouring of the integer lattice points in the n-dimensional Euclidean space is power-free if there exists a positive integer k such that the sequence of colours of consecutive points on any straight line is a k-th power-free word. The Thue threshold of mathbb Z^n is the least number of colours t(n) allowing a power-free colouring of the integer lattice points in the n-dimensional Euclidean space.Answering a question of Grytczuk (2008), we prove that t(2)=t(3)=2. Moreover, we show the existence of a 2-colouring of the integer lattice points in the Euclidean plane such that the sequence of colours of consecutive points on any straight line does not contain squares of length larger than 26.In order to obtain these results, we study repetitions in Toeplitz words. We show that the Toeplitz word generated by any sequence of primitive partial words of maximal length k is k-th power-free. Moreover, adding a suitable hypothesis on the positions of the holes in the generating sequence, we obtain that also the subwords occurring in the considered Toeplitz word according to an arithmetic progression of suitable difference, are k-th power-free words.

DNA microarray experiments generate a substantial amount of information about global gene expression. Gene expression profiles can be represented as points in multi-dimensional space. It is essential to identify relevant groups of genes in biomedical research. Clustering is helpful in pattern recognition in gene expression profiles. Some clustering techniques have been introduced. However, these traditional methods mainly utilize shape-based assumption or distance metric to cluster the points in multi-dimension linear Euclidean space. Poor consistence with the functional annotation of genes is shown in their validation study. A fractal clustering method to cluster genes using intrinsic (fractal) dimension from modern geometry is proposed. Fractal dimension is used to characterize the degree of self similarity among the points in the clusters. The main idea of fractal clustering is to group points in a cluster in such a way that none of the points in the cluster changes the cluster's intrinsic dimension radically. Hausdorff fractal dimension is computed through the means of the box-counting plot algorithm, since it is the fastest and also robust enough. This method is assessed using validation assessment using public microarray dataset. It shows that this method is superior in identifying functional related gene groups than other traditional methods.

Let C be an open convex cone in ${\bf R}^n $, n-dimensional real space, such that $\bar C$ does not contain any entire straight line. We consider holomorphic functions in tubes $T^C = {\bf R}^n + iC \subset {\bf C}^n $, n-dimensional complex space, which are known to have $\mathfrak{S}'$ distributional boundary values. We prove that if the $\mathfrak{S}'$ boundary value of such a holomorphic function is an $L^p $ function, $1 \leq p \leq \infty $, then the holomorphic function is in the Hardy space $H^p (T^C )$, $1 \leq p \leq \infty $, corresponding to the tube $T^C $ and can be recovered as the Poisson integral of the distributional boundary value. A similar result is proved wih respect to the holomorphic functions in tubes $T^C $ which are known to have boundary values in the distribution spaces $\mathfrak{Z}',K'_r ,(\mathfrak{S}^\alpha )'$, and $(W^\Omega )'$ and which do not necessarily have $S'$ distributional boundary values. In all cases we prove converse theorems. Our basic results are motivated ...

The overall comfort experienced by an individual in an n-dimensional space can be theoretically constructed using a mathematical equation, that relates the comfort experienced by an individual in a particular point in space with spatial points associated with higher comfort attributes and the desirable factors affecting the current environmental space.

A continuous predator-prey model in which both species diffuse along a spatial gradient is shown to exhibit temporal chaos at a fixed point in space. The model incorporates a nonlinear functional response of the predator and a logistic growth of the prey; the intrinsic growth rate of the prey varies linearly with space. Numerical results demonstrate that low diffusion values drive an otherwise periodic system into aperiodic behaviour with sensitivity to initial conditions. Evidence is provided for a quasiperiodic route to chaos as the diffusion rate decreases. These results suggest that complex temporal dynamics in natural communities may arise through the spatial dimension. Spatially induced chaos may have an important role in spatial pattern generation in heterogeneous environments.