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An equiangular tight frame (ETF) is a type of optimal packing of lines in Euclidean space. A regular simplex is a special type of ETF in which the number of vectors is one more than the dimension of the space they span. In this paper, we consider ETFs that contain a regular simplex, that is, have the property that a subset of its vectors forms a regular simplex. As we explain, such ETFs are characterized as those that achieve equality in a certain well-known bound from the theory of compressed sensing. We then consider the so-called binder of such an ETF, namely the set of all regular simplices that it contains. We provide a new algorithm for computing this binder in terms of products of entries of the ETF's Gram matrix. In certain circumstances, we show this binder can be used to produce a particularly elegant Naimark complement of the corresponding ETF. Other times, an ETF is a disjoint union of regular simplices, and we show this leads to a certain type of optimal packing of subspaces known as an equichordal tight fusion frame. We conclude by considering the extent to which these ideas can be applied to numerous known constructions of ETFs, including harmonic ETFs. [ABSTRACT FROM AUTHOR]

Two n -dimensional vectors A and B , A , B ∈ R n , are said to be trivially orthogonal if in every coordinate i ∈ [ n ] , at least one of A ( i ) or B ( i ) is zero. Given the n -dimensional Hamming cube { 0 , 1 } n , we study the minimum cardinality of a set V of n -dimensional { − 1 , 0 , 1 } vectors, each containing exactly d non-zero entries, such that every ‘possible’ point A ∈ { 0 , 1 } n in the Hamming cube has some V ∈ V which is orthogonal, but not trivially orthogonal, to A . We give asymptotically tight lower and (constructive) upper bounds for such a set V except for the case where d ∈ Ω ( n 0 . 5 + ϵ ) and d is even, for any ϵ , 0 < ϵ ≤ 0 . 5 . [ABSTRACT FROM AUTHOR]

This paper provides sufficient conditions for the asymptotic normality of quadratic forms of averages of random vectors of increasing dimension and improves on conditions found in the literature. Such results are needed in applications of Owen’s empirical likelihood when the number of constraints is allowed to grow with the sample size. Indeed, the results of this paper are already used in Peng and Schick (2013) for this purpose. We also demonstrate how our results can be used to obtain the asymptotic distribution of the empirical likelihood with an increasing number of constraints under contiguous alternatives. In addition, we discuss potential applications of our result. The first example focuses on a chi-square test with an increasing number of cells. The second example treats testing for the equality of the marginal distributions of a bivariate random vector. The third example generalizes a result of Schott (2005) by showing that a standardized version of his test for diagonality of the dispersion matrix of a normal random vector is asymptotically standard normal even if the dimension increases faster than the sample size. Schott’s result requires the dimension and the sample size to be of the same order. [ABSTRACT FROM AUTHOR]

The resolvent Krylov subspace method builds approximations to operator functions f(A) times a vector v. For the semigroup and related operator functions, this method is proved to possess the favorable property that the convergence is automatically faster when the vector v is smoother. The user of the method does not need to know the presented theory and alterations of the method are not necessary in order to adapt to the (possibly unknown) smoothness of v. The findings are illustrated by numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper we consider the Wafer-to-Wafer Integration problem. A wafer can be seen as a p -dimensional binary vector. The input of this problem is described by m multisets (called “lots”), where each multiset contains n wafers. The output of the problem is a set of n disjoint stacks, where a stack is a set of m wafers (one wafer from each lot). To each stack we associate a p -dimensional binary vector corresponding to the bit-wise AND operation of the wafers of the stack. The objective is to maximize the total number of “1” in the n stacks. We provide m 1 − ϵ and p 1 − ϵ non-approximability results even for n = 2 , f ( n ) non-approximability for any polynomial-time computable function f , as well as a p r -approximation algorithm for any constant r . Finally, we show that the problem is FPT when parameterized by p , and we use this FPT algorithm to improve the running time of the p r -approximation algorithm. [ABSTRACT FROM AUTHOR]

This study presents the calculation of spectral angle beyond two endmember vectors to the n -dimensional solid spectral angle (NSSA). The calculation of the NSSA is used to characterize the local spectral shape difference among a set of endmembers, leading to a methodology for band selection based on spectral shape variations of more than two spectra. Equidistributed sequences used in the quasi-Monte Carlo method (ESMC) for numerical simulations are shown to expedite the calculation of the NSSA. We develop a band selection method using the computation of NSSA( ϑ n ) in the context of a sliding window. By sliding the window over all bands available for varying band intervals, the calculated solid spectral angle values can capture the similarity of the endmembers over all spectral regions available and for spectral features of varying widths. By selecting a subset of spectral bands with largest solid spectral angles, a methodology can be developed to capture the most important spectral information for the separation or mapping of endmembers. We provide an example of the merits of the NSSA-ESMC method for band selection as applied to linear spectral unmixing. Specifically, we examine the endmember abundance errors resulting from the NSSA band selection method as opposed to using the full spectral dimensionality available. [ABSTRACT FROM AUTHOR]

In the paper, we introduce the concepts of $$G$$ -type I and generalized $$G$$ -type I functions for a new class of nonconvex multiobjective variational control problems. For such nonconvex vector optimization problems, we prove sufficient optimality conditions for weakly efficiency, efficiency and properly efficiency under assumptions that the functions constituting them are $$G$$ -type I and/or generalized $$G$$ -type I objective and constraint functions. Further, for the considered multiobjective variational control problem, its dual multiobjective variational control problem is given and several duality results are established under (generalized) $$G$$ -type I objective and constraint functions. [ABSTRACT FROM AUTHOR]

The paper deals with the problem of characterizing Pareto optima (efficient solutions) for the difference of two mappings vector-valued in a finite or infinite-dimensional preordered space. Closely related to the well-known optimality criterion of scalar DC optimization, a mixed vectorial condition is obtained in terms of both strong (Fenchel) and weak (Pareto) $$\epsilon $$ -subdifferentials that completely characterizes the exact or approximate weak efficiency. This condition also allows to deal with some special restricted mappings. Moreover, the condition established in the literature in terms of strong $$\epsilon $$ -subdifferentials for characterizing the strongly efficient solutions (usual optima), is shown here to remain valid without assuming that the objective space is order-complete. [ABSTRACT FROM AUTHOR]

We solve a problem of Littlewood: there exist seven infinite circular cylinders of unit radius which mutually touch each other. In fact, we exhibit two such sets of cylinders. Our approach is algebraic and uses symbolic and numerical computational techniques. We consider a system of polynomial equations describing the position of the axes of the cylinders in the 3 dimensional space. To have the same number of equations (namely 20) as the number of variables, the angle of the first two cylinders is fixed to 90 degrees, and a small family of direction vectors is left out of consideration. Homotopy continuation method has been applied to solve the system. The number of paths is about 121 billion, it is hopeless to follow them all. However, after checking 80 million paths, two solutions are found. Their validity, i.e., the existence of exact real solutions close to the approximate solutions at hand, was verified with the alphaCertified method as well as by the interval Krawczyk method. [ABSTRACT FROM AUTHOR]

One classical result of Freiman gives the optimal lower bound for the cardinality of $$A+A$$ if $$A$$ is a $$d$$ -dimensional finite set in $$\mathbb R^d$$ . Matolcsi and Ruzsa have recently generalized this lower bound to $$|A+kB|$$ if $$B$$ is $$d$$ -dimensional, and $$A$$ is contained in the convex hull of $$B$$ . We characterize the equality case of the Matolcsi-Ruzsa bound. The argument is based partially on understanding triangulations of polytopes. [ABSTRACT FROM AUTHOR]

The goal of this work is to examine the efficacy of interpolatory model order reduction on frequency sweep problems with many forcing vectors. The adaptive method proposed relies on the implicit interpolatory properties of subspace projection with basis vectors spanning the forced response of the system and its derivatives. The algorithm is similar to a recently proposed adaptive scheme in that it determines both interpolation location and order within the frequency domain of interest. The bounds of efficiency of the approach as the number of forcing vectors grows are explored through the use of rough floating operation counts. This, in turn, guides choices made in the adaptive algorithm, including a new technique that prevents excessive subspace growth by capitalizing on subspace direction coupling. In order to demonstrate the algorithm's utility, a series of frequency sweep problems drawn from the field of structural acoustics is analyzed. It is demonstrated that increased order of interpolation can actually degrade the efficiency of the algorithm as the number of forcing vectors grows but that this can be limited by the subspace size throttling procedure developed here. [ABSTRACT FROM AUTHOR]

A consistency criteria is given for a certain class of finite positive measures on the surfaces of the finite dimensional unit balls in a real separable Hilbert space. It is proved, through a Kolmogorov type existence theorem, that the class induces a unique positive measure on the surface of the unit ball in the Hilbert space. As an application, this will naturally accomplish the work of Kanter on the existence and uniqueness of the spectral measures of finite dimensional stable random vectors to the infinite dimensional ones. The approach presented here is direct and different from the functional analysis approach in the work of Kuelbs and Linde and the indirect approach of Tortrat and Dettweiler. [ABSTRACT FROM AUTHOR]

Screening is the problem of finding a superset of the set of non-zero entries in an unknown p-dimensional vector \beta^* given n noisy observations. Naturally, we want this superset to be as small as possible. We propose a novel framework for screening, which we refer to as Multiple Grouping (MuG), that groups variables, performs variable selection over the groups, and repeats this process multiple number of times to estimate a sequence of sets that contains the non-zero entries in \beta^*. Screening is done by taking an intersection of all these estimated sets. The MuG framework can be used in conjunction with any group based variable selection algorithm. In the high-dimensional setting, where p \gg n, we show that when MuG is used with the group Lasso estimator, screening can be consistently performed without using any tuning parameter. Our numerical simulations clearly show the merits of using the MuG framework in practice. [ABSTRACT FROM AUTHOR]

This study introduces a classifier founded on k-nearest neighbours in the complementary subspaces (NCS). The global space, spanned by all training samples, can be decomposed into the direct sum of two subspaces in terms of one class: the projection vectors of this class into one subspace are nonzero, and that into another subspace are zero. A query sample is projected into the two subspaces for each class, respectively. In each subspace, the distance from the projection vector to the mean of its k-nearest neighbours can be calculated, and the final classification rules are designed in terms of the two distances calculated in the two complementary subspaces, respectively. Allowing for the geometric meaning of Gram determinant and kernel trick, the classifier is naturally implemented in the kernel space. The experimental results on 1 synthetic, 13 IDA binary class, and five UCI multi-class data sets show that NCS compares favourably to the comparing classifiers, which is founded on the k-nearest neighbours or the nearest subspace, on almost all the data sets. The classifier can straightforwardly solve multi-classification problems, and the performance is promising. [ABSTRACT FROM AUTHOR]

Abstract: In this study, a 12-dimensional feature vector is constructed to reflect the general contents and spatial arrangements of the secondary structural elements of a given protein sequence. Among the 12 features, 6 novel features are specially designed to improve the prediction accuracies for α/β and α + β classes based on the distributions of α-helices and β-strands and the characteristics of parallel β-sheets and anti-parallel β-sheets. To evaluate our method, the jackknife cross-validating test is employed on two widely-used datasets, 25PDB and 1189 datasets with sequence similarity lower than 40% and 25%, respectively. The performance of our method outperforms the recently reported methods in most cases, and the 6 newly-designed features have significant positive effect to the prediction accuracies, especially for α/β and α + β classes. [Copyright &y& Elsevier]

Sparse linear array has some advantages over uniform linear array in some applications. However, it is well known that sparse array always suffers from manifold ambiguity, which has significant influence on direction-of-arrival (DOA) estimation and resolution. In this study, a new method based on the multiple signal classification (MUSIC) for resolving manifold ambiguities of uncorrelated sources by using semi-circular substrate is proposed for DOA estimation of sparse array. Spurious MUSIC peak spectrums are generated because of a linear combination of the steering vectors of true DOAs. The main idea of the proposed method is to setup one or several semi-circular medium substrates at the front of some elements. The phases of these elements are changed through the refractive index and the radii of those semi-circular substrates. This treatment breaks down the previous linearity of the steering vectors for the array without adding the substrates. Thus, true DOA peaks can be discriminated according to this feature. Simulation results show good performance of the proposed method. Trivial and nontrivial ambiguities are efficiently resolved. [ABSTRACT FROM AUTHOR]

The present study experimentally examines how an electron vortex beam with orbital angular momentum (OAM) undergoes diffraction through a forked grating. The nth-order diffracted electron vortex beam after passing through a forked grating with a Burgers vector of 1 shows an OAM transfer of nħ. Hence, the diffraction patterns become mirror asymmetric owing to the size difference between the electron beams. Such a forked grating, when used in combination with a pinhole located at the diffraction plane, could act as an analyzer to measure the OAM of input electrons. [ABSTRACT FROM AUTHOR]

This paper is mainly a review of the dualities between Vasiliev's higher spin gauge theories in AdS4 and three dimensional large N vector models, with focus on the holographic calculation of correlation functions of higher spin currents. We also present some new results in the computation of parity odd structures in the three point functions in parity violating Vasiliev theories. [ABSTRACT FROM AUTHOR]

The article discusses the building of vector dimensional models for event structure and verbal semantics related to the representation of verb meaning and interpretation of linguistic expressions. Topics discussed include three-dimensional representation for the semantics of verbs having time quality profile, three-way distinction among stative phases, and use of force vectors for the similarity of meanings.

The article discusses the single-domain constraint for the verbs to lexicalize in a pattern, express force dynamics, and the vector based theories on event structure. Topics discussed include force dynamics and verbs involving conventional consequences, innovative aspects of the semantics of verbs, analysis of the verbal meanings.

The article discusses the semantics based on the notions of vectors in property conceptual spaces and lacunas in linguistic semantics and also informs the meaning of the small words, lexemes, and small part of sentences. Topics discussed include symbolic representation of the word meaning in generative semantics, identity vectors denoted by statives, and extension of vectors to temporally situated vectors and application of vector semantics.

The article discusses the study done to examine the two vector model with respect to dynamic models of the lexical semantics of the literature for representation of the meanings of the verbs used. Topics discussed include distinction between cognitive representations and linguistic representations, using a pattern of interacting forces for representing actions including walking and swimming, and interpreting the meanings of the verb by separation of force and result vectors.

The article discusses the concepts of cognitive science approach for conceptual spaces for modeling the lexical semantics particularly in the semantics of the verbs. Topics discussed include establishment of a bridge between cognitive science and linguistics, relationship between path and vectors, casual and logical reasoning.

A correction to the article "A method to determine fault vectors in 4H-SiC from stacking sequences observed on high resolution transmission electron microscopy images" that was published in the October 22, 2014 issue is presented.

Multi body system contains finite number of rigid bodies that are placed on each axis, which do not intersect each other. Derivatives of the linear momentum and angular momentum with respect to time by use mass moment vectors and vector rotators for pole and axis, are determined. A number of theorems are defined. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim) [ABSTRACT FROM AUTHOR]