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Multilinear commutators with vector symbol defined by \[ T_{\vec{b}}(f)(x)=\int_{{\bb R}^n}\Bigg[\prod\limits^m_{j=1}(b_j(x)-b_j(y))\Bigg]K(x,y)f(y)dy \] are considered, where is a Calderon–Zygmund kernel. The following a priori estimates are proved for . For , there exists a constant such that \[ \|\dot{T}_{{\vec{b}}}(f)\|_{L^P(w)}\le C\|\vec{b}\|\|M_{L(\log\,L)^{1/r}}(f)\|_{L^P(w)} \] and \[ \sup_{t>0}\frac{1}{\Phi(\frac{1}{t})}w(\{y\in{\bb R}^n:|T_{\vec{b}}f(y)|>t\})\le C\sup_{t>0}\frac{1}{\Phi(\frac{1}{t})}w(\{y\in{\bb R}^n:M_{L(\log\,L)^{1/r}}(\|\vec{b}\|f)(y)>t\}), \] where \begin{eqnarray*} &\|\vec{b}\|=\prod\limits^m_{j=1}\|b_j\|_{osc_{\exp L}^r j},\\ &\Phi(t)=t\log^{1/r}(e+t),\quad \frac{1}{r}=\frac{1}{r_1}+\cdots+\frac{1}{r_m}, \end{eqnarray*} and is an Orlicz type maximal operator. This extends, with a different approach, classical results by Coifman.As a corollary, it is deduced that the operators are bounded on when , and that they satisfy corresponding weighted -type estimates with .

© 2014 IEEE. Similarity measures are the foundations of many research areas, e.g. information retrieval, recommender system and machine learning algorithms. Promoted by these application scenarios, a number of similarity measures have been proposed and proposing. In these state-of-the-art measures, vector-based representation is widely accepted based on Vector Space Model (VSM) in which an object is represented as a vector composed of its features. Then, the similarity between two objects is evaluated by the operations on two corresponding vectors, like cosine, extended jaccard, extended dice and so on. However, there is an assumption that the features are independent of each others. This assumption is apparently unrealistic, and normally, there are relations between features, i.e. the co-occurrence relations between keywords in text mining area. In this paper, a space geometry-based method is proposed to extend the VSM from the orthogonal coordinate system (OVSM) to affine coordinate system (AVSM) and OVSM is proved to be a special case of AVSM. Unit coordinate vectors of AVSM are inferred by the relations between features which are considered as angles between these unit coordinate vectors. At last, five different similarity measures are extended from OVSM to AVSM using unit coordinate vectors of AVSM. Within the numerous application fields of similarity measures, the task of text clustering is selected to be the evaluation criterion. Documents are represented as vectors in OVSM and AVSM, respectively. The clustering results show that AVSM outweighs the OVSM.

An analysis method based on vector triangle and vector locus is introduced in this paper. By using vector triangles, a set of quantities generally used in feedback theory can be intuitively visualized, which is impossible by using other theories. The vector locus records the evolvement of the vector triangle versus frequency in polar coordinates, and combines the magnitude and phase plot together in one track. Examples are given to show that the proposed method gives intuitive insights while still retains a general method to characterize feedback systems.

Having schematically run through the transition from single-phase to three-phase relationships, the energy balance is formalised using Park vector notation. The imaginary power notation emerges. Leading back, in the sinusoidal case, to the usual reactive power, it generalises its specific contents in the case of variable states, explaining in particular a typical aspect of three-phase networks: the energy bouncing from one phase to another. This aspect can be seen as an index of power quality. At last, these same considerations are obtained by means of Lagrangian and thermodynamic approaches that lend a deeper meaning to the energy related quantities.

It is well known that vector–tensor notation is a compact and natural language for the mathematical formulation of continuum mechanics problems. Here we describe the application of vector technique to numerical simulation starting with a mathematical formulation. We provide an efficient numerical scheme and furnish an implementation as a computer program. As a result (in comparison with traditional “component” form), the two last steps are significantly simplified, especially for multidimensional problems with various boundary conditions in irregular geo- metries where nonorthogonal meshes are applied. Therefore, more attention can be focused on the physical nature of problems. Apart from the cleaner syntax, vector notation conserves the structure of traditional numerical algorithms and solves multidimensional problems with minimum additional programming effort. Complex practical applications of this technique are described as well.

Equivalent network for H-plane step is derived from modal analysis and expressed by multi-port ideal transformer or vector notation. Vector notation of equivalent network for waveguide was proposed. Then, whole equivalent network for step discontinuity is given. These conclusions are consistent with conventional results and help one to understand mode behavior at step. Finally, an equivalent network for various H-plane waveguide circuits is derived, Article

Integrated systems for reaction and separation offer promising technology for less expensive and more energy-efficient chemical processes. In this paper, we show how to view reactive separation processes as a vector combination of reaction, separation and mixing. In composition space, vector directions are calculated from physical and chemical data only; thus they are independent of design. On the other hand, vector lengths reflect design decisions which are at our disposal, allowing us to manipulate resultant moves in composition space by how we design and operate the equipment. Of particular interest is the discovery of a reaction difference point. We show that this mathematical artifact is found outside the feasible composition space. It determines the underlying structure of linear composition changes by reaction in mole fraction space. Using the proposed vector notation and the reaction difference point, we provide a simple and visual explanation for the term `reactive azeotrope', herein referred to as a reactive fixed point . The potential for such points arises from collinearity among reaction, separation and mixing as individual phenomena and is independent of operating conditions. We discuss the possible impact of such points on limiting behaviour in systems featuring simultaneous reaction and separation.

In this paper, we extend the Kirchhoff approach, which is widely used for near-nadir backscattering calculations, to include the proper polarization sensitivity for general bistatic scattering from gently sloping, perfectly conducting surfaces. Previously, Holliday has shown how the inclusion of terms from the second iteration of the surface-current integral equation is required to obtain agreement with the small perturbation method for backscattering conditions. Here we employ a similar approach by retaining all terms in this iterative expansion through first order in the surface slope to derive expressions for the standard Kirchhoff field as well as for a supplementary field that contains the polarization sensitivity. A polarization vector notation is introduced to simplify the inclusion of tilting effects from larger-scale features on the scattering surface. In connection with this latter development, we provide a clarification of the earlier work by Valenzuela on this topic together with an e...

A method is presented to calibrate transfer function-noise (TFN) models, operating at the same frequency as the input (auxiliary) variables, to sparsely or irregularly observed time series of the output (target) variable. Once calibrated, the TFN models can be used to predict or simulate the output variable at the same frequency as the input variable. Consequently, the method provides a useful tool for filling in gaps of irregularly or sparsely observed hydrological time series. Although generic and suitable for any type of time series, the method is described through the modeling of a time series of groundwater head data with precipitation surplus (precipitation minus potential evapotranspiration) as input variable. First, the TFN model is written in vector notation, yielding the state equation of a linear discrete stochastic system. Subsequently, the state equation is embedded in a Kalman filter algorithm. The Kalman filter is then combined with a maximum likelihood criterion to obtain estimates of the parameters of the TFN model for small time steps (e.g., 1 day) while using sparsely (e.g., two times a month) or even irregularly observed time series of groundwater head data. The method is illustrated using (subsets of) time series of groundwater head data with varying regular and irregular observation intervals.

This book employs homogeneous coordinate notation to compute the first- and second-order derivative matrices of various optical quantities. It will be one of the important mathematical tools for automatic optical design. The traditional geometrical optics is based on raytracing only. It is very difficult, if possible, to compute the first- and second-order derivatives of a ray and optical path length with respect to system variables, since they are recursive functions. Consequently, current commercial software packages use a finite difference approximation methodology to estimate these derivatives for use in optical design and analysis. Furthermore, previous publications of geometrical optics use vector notation, which is comparatively awkward for computations for non-axially symmetrical systems.

The development of a query-independent ranking approach for user-generated content is the main goal of this thesis. The following chapter presents a modeling and a ranking solution based on the insights presented in the previous chapters. Section 5.1 proposes a vector notation for the central characteristics of a user-generated content unit based on the social media document view introduced in Chapter 3.4. The calculation of a ranking that is applicable to all types of user-generated content is proposed in Section 5.2. The example calculation in Chapter 5.3 illustrates the application of the proposed ranking approach by means of user-generated content units from different social media categories.

In previous work the author has presented a methodology for obtaining closed form equations of motion for general non-holonomic hybrid parameter multiple body systems (HPMBS). It is well known that generating closed form symbolic equations of motion for HPMBS of few degrees of freedom rapidly degenerates to an algebraic soup unless computer tools are utilized. Methods dependent on Lagrange multipliers are virtually unable to provide the constant free (minimal) symbolic equations for systems of moderate to high order due to the complication of eliminating the multipliers. In this paper, the intent is to provide the details, and example use of, a tool, based on commercial symbolic manipulation software, developed to assist the symbolic processing of the equations of motion as previously minimally formulated. The tool presented herein is novel in the way the vector notation of the formulation scheme is retained and the way that frame information is carried in the calculations. The symbolic tool provided in this paper allows the modelling methodology to be a practical tool for generating closed form equations of motion for extremely complicated HPMBS.

An image measurement technique using the possibility of particle passing has been extended to obtain the three-dimensional velocity field. The authors have developed a new measurement technique by means of statistical analysis which gives the absolute value of flow speed distribution in three-dimensional flow field or the velocity vector field in two-dimensional field. Present paper describes the extension of the technique which enables to obtain the three-dimensional vector field using the governing equation expressed by vector notation and the Taylor's expansion of three-dimensional luminance function in time and space. The results of numerical simulation show that the three-dimensional flow field illuminated by twin-sheet light is obtained by simple statistical processing.

This paper presents a systolic design for flexible string matching. Initially a sequential algorithm is discussed which consists of two phases, i.e. pre-processing and searching, and a matrix vector notation of the algorithm is proposed. Then, starting from the computational schedule of the searching phase a systolic algorithm is derived which can be realised directly onto a special purpose VLSI processor array architecture for string matching. Further the pre-processing phase is also accommodated onto the same VLSI design. Finally extensions to approximate pattern matching are discussed.

An embedded image coder based on wavelet transform coding and multistage vector quantization (VQ) is proposed in this research. We have examined several critical issues to make the proposed algorithm practically applicable. They include the complexity of embedded VQ, design of the successive vector quantizer, significance evaluation of a vector symbol, and integration of wavelet transform coding and vector quantization. It is shown in experiments that the new method achieves a superior rate-distortion performance.

Homogeneous coordinate notation is one of powerful mathematical tools for engineering field. It was used to study the motion of rigid bodies [1, 2], robotics [3], theory of gearing [4], and computer graphics [5]. Previous publications of geometrical optics use vector notation, which is comparatively awkward for computations for non-axially symmetrical systems. In order to circumvent these limitations, this book employs homogeneous coordinate notation to compute various optical quantities. The discussion in this chapter is designed mainly for those readers who have not so far encountered this mathematical tool. The treatment is elementary and covers only what will needed to understand the rest of the book.

In nearly all undergraduate textbooks on applied analysis, the vector product is introduced only in R3, giving many students the impression, that this is a special feature of this particular vector space. The author suggests avoiding this conclusion by introducing the complex numbers as an example of a vector product in R2. The purpose of this paper is twofold: first, to show that this approach is a good opportunity to practice vector calculus and algebra, and secondly, to exercise the handling of different notations for vectors and products leading to the definition of, for example, imaginary unit i.

In this paper we redevelop the scalar and dyadic Green functions of electromagnetic theory using differential forms. The Green dyadic becomes a double form, which is a differential form in one space with coefficients that are forms in another space, or a differential form-valued form. The results presented here correspond closely with the usual dyadic treatment, but are clearer and more intuitive. Many of the usual expressions using green functions in vector notation require a surface normal; with the Green forms the surface normal is unnecessary. We illustrate the formalism by computing scattering from a randomly rough conducting surface and deriving the Green form for a dielectric half-space. We also define the interior derivative, which is equivalent to the coderivative but for a constant metric has a computational rule dual to that of the exterior derivative and simplifies calculation in coordinates. This work makes available some of the tools that have not yet been presented in the language of differ...

The paper is devoted to the algorithm for generation of PDF with vector symbols from scanned documents. The complex multi-stage technique includes segmentation of the document to text/drawing areas and background, conversion of symbols to lines and Bezier curves, storing compressed background and foreground. In the paper we concentrate on symbol conversion that comprises segmentation of symbol bodies with resolution enhancement, contour tracing and approximation. Presented method outperforms competitive solutions and secures the best compression rate/quality ratio. Scaling of initial document to other sizes as well as several printing/scanning-to-PDF iterations expose advantages of proposed way for handling with document images. Numerical vectorization quality metric was elaborated. The outcomes of OCR software and user opinion survey confirm high quality of proposed method.

Lagrangian dynamics and the various variants in use (Hamilton’s equations, Kane’s equations, the method of quasicoordinates and the Kane-Hamilton synthesis introduced in this text) use abstract vector spaces (for example configuration space and state space). These are K dimensional vector spaces with length defined as the square root of the dot product of a vector with itself. Two vectors are perpendicular (or orthogonal) if their dot product is zero. The dot product is defined as one would expect by analogy to the dot product for ordinary vectors: the product of the first pair of components, plus the product of the second pair of components and so on out until all K pairs of components have been multiplied and added. Ordinary vector notation, either classical or in the context of linear algebra does not suffice for everything one wants to do, so I will introduce an indicial notation closely related to that of tensor analysis. It will not be tensor analysis, but the reader familiar with tensors will find much that is familiar in this appendix.

In nearly all of the discussion to follow we shall deal with the set of real numbers. Occasionally, however, we shall deal with complex numbers as well. In order to avoid cumbersome repetition we shall denote the set we are dealing with by F and let the context elucidate whether we are speaking of real or complex numbers or both.

Many mathematical relationships are most efficiently expressed in vector form. Many of the operations of algebra and calculus have their analogs in vector analysis. Three important differential vector quantities are the gradient, divergence and curl. The equations of electrodynamics, beginning with Maxwell’s equations, make extensive use vector notation.

International audience; New expressions for the stress field of straight dislocation segments are derived from the de Wit's expressions for an infinite straight dislocation. Restricted to linear isotropic elasticity, these compact formulae are given in tensor and vector notation expressed in an arbitrary Cartesian reference frame. This solution, allows to compute the internal stress fields or the dislocation-dislocation interactions involved in three-dimensional simulations of plastic strain at spatial scales involving large numbers of dislocations.

An image measurement technique using the possibility of particle passing has been extended to obtain the three-dimensional velocity field. The authors have developed a new measurement technique by means of statistical analysis which gives the absolute value of flow speed distribution in three-dimensional flow field or the velocity vector field in two-dimensional field. Present paper describes the extension of the technique which enables to obtain the three-dimensional vector field using the governing equation expressed by vector notation and the Taylor's expansion of three-dimensional luminance function in time and space. The results of numerical simulation show that the three-dimensional flow field illuminated by twin-sheet light is obtained by simple statistical processing.

The objective of this paper is to express a matrix of any dimension in unit vector notation. This is accomplished by first solving the two and three dimensional cases before solving the general n-dimensional case. The fact that matrices can be represented as a (non-linear) combination of standard basis unit vectors shows that matrices are not simply abstract entities used just for representing data, but also have a geometric interpretation. This paper then defines the natural product and explains its applications in matrix calculus, before relating it to the outer product of real vectors, a special case of the tensor product. We then conclude that the space is the intersection of the exterior space and the newly defined \textit{natural space}.

We begin with the vector notation for the most important functions and q-Taylor formulas for power series and functions of inverse q-shifted factorials. We continue with a historical introduction to the rest of this long and interesting chapter and to the next chapter as well. We will also define q-Appell functions together with the normal form. Then follows the two definitions of q-Kampe de Feriet functions due to Karlsson and Srivastava. The q-analogue of Appell and Kampe de Feriet’s transformation formulas require the Watson q-shifted factorial in the definition. We continue with Carlitz’ Saalschutzian formulas, Andrews’s formal transformations and Carlson’s transformations.

Kochen-Specker (KS) vector systems are sets of vectors in ℝ3 with the property that it is impossible to assign 0s and 1s to the vectors in such a way that no two orthogonal vectors are assigned 0 and no three mutually orthogonal vectors are assigned 1. The existence of such sets forms the basis of the Kochen-Specker and Free Will theorems. Currently, the smallest known KS vector system contains 31 vectors. In this paper, we establish a lower bound of 18 on the size of any KS vector system. This requires us to consider a mix of graph-theoretic and topological embedding problems, which we investigate both from theoretical and practical angles. We propose several algorithms to tackle these problems and report on extensive experiments. At the time of writing, a large gap remains between the best lower and upper bounds for the minimum size of KS vector systems.

Introductory textbooks on linear algebra present a great deal of information about vector spaces. This chapter includes the information that pertains directly to linear programming. It omits the rest. You may find that what remains here is coherent and, as a consequence, particularly accessible.

Many authors, who adopt the vector space model, take the view that documents, terms, queries, etc., are all elements within the same (conceptual) space. This view seems to be a natural one, given that documents and queries have the same vector notation. We show, however, that the structure of the query space can be very different from that of the document space. To this end, concepts like preference, similarity, term independence, and linearity, both in the document space and in the query space, are discussed. Our conclusion is that a more realistic and complete view of IR is obtained if we do not consider documents and queries to be elements of the same space

It has been shown that a vector transform is a good tool for digital filtering and image coding. This paper addresses some of the issues related to vector transforms. First, a fundamental difference between vector transforms and conventional transforms is examined. Second, two approaches to constructing vector transforms are discussed. Finally, an application of vector transformation to an efficient implementation of 2-D nonseparable digital filters is proposed. >

1. Let R be the additive group of all real numbers and t → Tt a representation of this group on an n-dimensional complex vector space X. This means that to each real number t a linear operator Tt : X → X is assigned in such a way that $${\text{T}}_{{\text{t + s}}} = {\text{T}}_{\text{t}} {\text{T}}_{\text{s}} , {\text{T}}_0 = {\text{I}}$$ (1) Since the set {Tt : t ∈ R} consists of operators which commute one to the other the irreducible representations are one dimensional. Hence we are lead to find all functions f: R → C (C is the set of all complex numbers) such that $${\text{f}}\left( {{\text{t}}\,{\text{ + }}\,{\text{s}}} \right) = {\text{f}}\left( {\text{t}} \right)\,\,{\text{f}}\,\,{\text{(s),}}\,\,\,\,\,{\text{f}}\,\,{\text{(0)}}\,{\text{ = }}\,\,{\text{1}}{\text{.}}$$ (2)

Lifted system description in Fourier basis is discussed for analysis and design of linear, cycle invariant Iterative Learning (ILC) and Repetitive Control (RC). ILC and RC can be bridged within an algebraic vector notation utilizing an infinite time Fourier basis model. This model allows a very illustrative, decoupled controller design for ILC and RC. To complete the bridging, it is shown that an internal model description for RC can be derived from the vector notation. However, some approximations in modeling are required. Therefore, the validity of the Fourier basis model is discussed and a q-filter design is recommended. The study closes with simulation and experimental results.

The Rodrigues method, formulated first as a geometrical construction and later in vector notation, provides a simple and direct means of visualizing and calculating the total rotation resulting from two sequential rotations. In this paper the case of the five degrees of freedom for a grain boundary are considered. The Rodrigues method is applicable since the total misorientation is a tilt followed by a twist rotation, and calculations are much simplified since vectors rather than matrices are involved.

Representations are created and given meaning in a shared perceptual space, where they are spoken, written, and drawn in the context of social activity. Consequently, problems in science education cross the boundaries of traditional modularization of the mind into separate perceptual, representation, and communication components that act at distinct times, in distinct domains. We illustrate these issues with a case study of physics learning using a simulation program. The learners are initially uncertain about what aspects of motion to see, where the representations are on the screen, and how to express the relationship between vector notation and motion. At a local level, the students jointly coordinate conversational and perception-action processes to maintain a mutually intelligible stream of activity. At a slightly broader level, they use perception, language, and gesture to construct a shared understanding of what the notation on the computer screen means. Even more broadly, the students use their understanding of the notation to relate their activity to ways in which scientists address similar situations. We conclude that learning to make sharp distinctions from a repertoire of fuzzy, everyday descriptions requires simultaneous, coordination of perception, gesture, and language; one cannot assume competence in two areas and analyze only the third.

The primitive logic functions NOT, AND, OR, NAND, NOR, XOR, and XNOR can be connected together to build slightly more complex functions. A single signal carrying one bit of binary data is known as a scalar entity. A set of signals carrying similar data can be gathered together into a group known as a vector. A key advantage of vector notation is that it allows all of the signals comprising the vector to be easily referenced in a single statement. A multiplexer uses a binary value, or address, to select between a number of inputs and to convey the data from the selected input to the output. A decoder uses a binary value, or address, to select between a number of outputs and to assert the selected output by placing it in its active state. There is a special category of gates called tri-state functions whose outputs can adopt three states: 0, 1, and Z. The tri-state buffer’s symbol is based on a standard buffer with an additional control input known as the enable. A more sophisticated function called a D-type latch can be constructed by attaching two AND gates and a NOT gate to the front of an RS latch. In the case of a D-type flip-flop, the data appears to be loaded when a transition, or edge, occurs on the clock input, which is therefore said to be edge-sensitive. Counter functions are also commonly used in digital systems. The number of states that the counter will sequence through before returning to its original value is called the modulus of the counter.

This chapter contains sections titled: Network Structure Feature Selection Types of Learning Interpretation of Output Summary This chapter contains sections titled: Exercises References