Tensors

Deep learning has emerged in the last few years as a premier technology for building intelligent systems that learn from data. Deep neural networks, originally roughly inspired by how the human brain learns, are trained with large amounts of data to solve complex tasks with unprecedented accuracy. With open source frameworks making this technology widely available, it is becoming a must-know for anybody involved with big data and machine learning.

These notes are the second part of the tensor calculus documents which started with the previous set of introductory notes (see T. Sochi, Introduction to Tensor Calculus, arXiv:1603.01660, 2016). In the present text, we continue the discussion of selected topics of the subject at a higher level expanding, when necessary, some topics and developing further concepts and techniques. Unlike the previous notes which are largely based on a Cartesian approach, the present notes are essentially based on assuming an underlying general curvilinear coordinate system.

An introductory overview of vector spaces, algebras, and linear geometries over an arbitrary commutative field is given. Quotient spaces are emphasized and used in constructing the exterior and the symmetric algebras of a vector space. Affine geometries are introduced and generalized by projective completion. General projective geometries are briefly introduced. Tensor products and multilinear functions are treated. The exterior algebra of a vector space and that of its dual are used in treating linear geometry and Grassmann's regressive product is treated. Scalar product spaces, orthogonality, and the Hodge star based on a general basis are covered.

A description of Orthogonal Tensor Hermite Polynomials in 3-D is presented. These polynomials, as introduced by Grad in 1949 [1], can be used to obtain a series solution to the Boltzmann Transport Equation. The properties that are explored are scaling, translation and rotation. Order 6 Hermite Tensors are studied while obtaining the rotation relations. From the scaling of the independent variables of particle velocities, a criterion on temperature is obtained which implies that the equation can be applied to binary gas mixtures only if the temperature of the hotter constituent is less than four times that of the cooler one. This criterion and other properties of the tensor hermite polynomials obtained in this paper can be used to study gas dynamics in the thermosphere.

These notes are the second part of the tensor calculus documents which started with the previous set of introductory notes (see T. Sochi, Introduction to Tensor Calculus, arXiv:1603.01660, 2016). In the present text, we continue the discussion of selected topics of the subject at a higher level expanding, when necessary, some topics and developing further concepts and techniques. Unlike the previous notes which are largely based on a Cartesian approach, the present notes are essentially based on assuming an underlying general curvilinear coordinate system.

Esta apostila de Espaço Tensorial e Funcional Generalizado é resultado da digitação das aulas do curso de Pós-Graduação em Ciência e Engenharia de Matériais da Universidade Estadual de Ponta Grossa- UEPG, na disciplina de Mecânica do Contínuo, ministrado pelo professor Dr. Lucas Máximo Alves e de seus estudos pessoais anteriores, obtidos durante o seu doutorado no Programa de Pós-Graduação de Métodos Numéricos para a Engenharia-PPGMNE da Universidade Federal do Paraná.

In this paper we will see the model of Universe according to Dynamic Universe Model of Cosmology by visualizing various processes that are happening in the Universe as per experimental evidences. For simplifying the matter here we will see in part1: about the Galaxy life cycle, where the birth and death of Galaxies discussed. Probably Universe gives guidance for the movement of Galaxies. We call this Part 1: Thinking and Reproducing Universe or Mindless Universe? (Galaxy life cycle). We see every day Sun, Stars, Galaxies etc., dissipating enormous energy in the form of radiation by the way of fusion of Hydrogen to helium. So after sometime all the Hydrogen is spent and Universe will die, is it not? … Dynamic Universe Model says that the energy in the form of electromagnetic radiation passing grazingly near any gravitating mass changes its in frequency and finally will convert into neutrinos (mass). Hence Dynamic Universe Model proposes another process where energy will be converted back into matter and the cycle energy to mass to energy continues, sustaining the Universe to maintain this present status for ever in this form something like a Steady state model without any expansion. This we will see in Part2: Energy – Mass-Energy Cycle. After converting energy into mass " how various elements are formed and where they are formed? " will be next logical question. Dynamic Universe Model says that these various particles change into higher massive particles or may get bombarded into stars or planets and various elements are formed. Here we bifurcate the formation of elements into 6 processes. They are for Elementary particles and elements generated in frequency changing process, By Cosmic rays, By Small stars, By Large Stars, By Super Novae and Manmade elements By Neutron Stars. This we will discuss in Part3: Nucleosynthesis.

Metamaterials are synthetic materials that exhibit properties not usually found in natural materials. One such property that has been of great importance to the study of optics is a negative refractive index within the metamaterial. This property implies negative permeability and permittivity for the metamaterial. Using and deriving the electromagnetic stress energy tensor in Minkowski space as an approximation in the weak field limit, the gravitational curvature effects of such meta-materials will be investigated. The basic example of a capacitor and inductor will be used to illustrate the mathematical results.

Page 1. AD-A256 672 :. / " PL-TR-92-2108 PHYSICAL CONSTRAINTS ON SEISMIC WAVES FROM CHEMICAL AND NUCLEAR EXPLOSIONS Brian Stump Lane Johnson Robert Reinke Sharon Reamer Ken Olsen Klaus-G. Hlnzen Southern Methodist University ._ ...

These are general notes on tensor calculus which can be used as a reference for an introductory course on tensor algebra and calculus. A basic knowledge of calculus and linear algebra with some commonly used mathematical terminology is presumed.

This guide assumes you know nothing about TensorFlow and takes you from the beginning until understanding the basics of a TensorFlow program including Variables, Placeholders, dataflow graphs, TensorFlow Core API, and TensorBoard for visualization. Because artificial neural networks (ANNs) are in the heart of deep learning models, it is recommended to start learning how they work and implemented programmatically. This guide covers in details all steps required for creating your first ANN using TensorFlow starting by reading input data then building neural networks layers (input, hidden, output) and finally making predictions. This guide helps you how neural networks parameters (e.g. weights) are updated by tracing the dataflow graph. Because the traditional way of reading data using placeholders is not appropriate for large datasets, pandas Series and DataFrame are discussed by understanding the different ways they are created. This is in addition to indexing, updating, deleting items. In addition to TensorFlow Core API, some higher level APIs are discussed including TensorFlow Estimators and train for saving time wasted by implementing some of the frequently used operations.

The paper presents a comprehensive analysis, together with the derivation of a closed-form solution to the two-body problem in rotating non-inertial reference frames. By using an efficient tensor mathematical instrument, which is closely related to the attitude kinematics methods, the motion in the rotating non-inertial reference frame is completely solved. The closed-form solutions for the motion in the non-inertial frame, the motion of the mass center, and the relative motion are presented. Dynamical characteristics similar to linear momentum, angular momentum, and total energy are introduced.

In this paper, we present a moving object detection system named Flux Tensor with Split Gaussian models (FTSG) that exploits the benefits of fusing a motion computation method based on spatio-temporal tensor formulation, a novel foreground and background modeling scheme, and a multi-cue appearance comparison. This hybrid system can handle challenges such as shadows, illumination changes, dynamic background, stopped and removed objects. Extensive testing performed on the CVPR 2014 Change Detection benchmark dataset shows that FTSG outperforms state-of the- art methods.

The present article offers an exposition of the axiomatic definition of tensors and their further developments from this very standpoint. Various types of tensor sand their examples have been included.

1. Introduction

The classical definition of a tensor was introduced by Italian geometer Giovanni Ricci in 1987. Thereafter, tensors have been extensively used as tools in the study of Differential Geometry, Theory of Relativity, Physics, Mechanics, Engineering and many other allied fields. However, the contemporary treatment of tensors is relatively new and its necessity arose mainly due to the following drawbacks in the classical definition:

(i) The components of tensors are not well defined;
(ii) The transformation rules for the tensors relate the components of tensors but not the tensors themselves;
(iii) In the transformation rules there occur the derivatives  which become meaningless if the space under consideration is non-differentiable;
(iv) The sum of two tensors of different types appears to exist but, indeed, it does not.

Owing to these drawbacks the celebrated French geometer Elié Cartan introduced a new definition of a tensor in the early years of the last century. The new definition of tensors is free from above drawbacks. Elié Cartan’s approach to tensors, which forms the subject matter of the present article, not only gives a rigorous foundation of tensors but also stimulates a feeling for the beauty and interest of rigor.

—Many critical EDA problems suffer from the curse of dimensionality, i.e. the very fast-scaling computational burden produced by large number of parameters and/or unknown variables. This phenomenon may be caused by multiple spatial or temporal factors (e.g. 3-D field solvers discretizations and multi-rate circuit simulation), nonlinearity of devices and circuits, large number of design or optimization parameters (e.g. full-chip routing/placement and circuit sizing), or extensive process variations (e.g. variability/reliability analysis and design for manufacturability). The computational challenges generated by such high dimensional problems are generally hard to handle efficiently with traditional EDA core algorithms that are based on matrix and vector computation. This paper presents " tensor computation " as an alternative general framework for the development of efficient EDA algorithms and tools. A tensor is a high-dimensional generalization of a matrix and a vector, and is a natural choice for both storing and solving efficiently high-dimensional EDA problems. This paper gives a basic tutorial on tensors, demonstrates some recent examples of EDA applications (e.g., nonlinear circuit modeling and high-dimensional uncertainty quantification), and suggests further open EDA problems where the use of tensor computation could be of advantage.

Let (,,) be a complete probability measure space and X1 and X2 be separable Banach spaces. For Banach spaces Y1 and Y2, let F1: X1Y1 and F2: X2Y2 be random operators. Using F1 and F2, we construct a random operator F: (X1X2)Y1Y2, which is continuous if F1 and F2 are continuous. We prove that if F1 and F2 are stochastically continuous, then F is also stochastically continuous. Similar result is also established in case of separability of random operators. The fixed points of such random operators on the tensor product of Banach spaces is also studied here.

Most of the available models for reference evapo-transpiration (ET 0) estimation are based upon only an empirical equation for ET 0. Thus, one of the main issues in ET 0 estimation is the appropriate integration of time information and different empirical ET 0 equations to determine ET 0 and boost the precision. The FAO-56 Penman–Monteith, adjusted Hargreaves, Blaney–Criddle, Priestley–Taylor, and Jensen– Haise equations were utilized in this study for estimating ET 0 for two stations of Belgrade and Nis in Serbia using collected data for the period of 1980 to 2010. Three-order tensor is used to capture three-way correlations among months, years, and ET 0 information. Afterward, the latent correlations among ET 0 parameters were found by the multiway analysis to enhance the quality of the prediction. The suggested method is valuable as it takes into account simultaneous relations between elements, boosts the prediction precision , and determines latent associations. Models are compared with respect to coefficient of determination (R 2), mean absolute error (MAE), and root-mean-square error (RMSE). The proposed tensor approach has a R 2 value of greater than 0.9 for all selected ET 0 methods at both selected stations, which is acceptable for the ET 0 prediction. RMSE is ranged between 0.247 and 0.485 mm day −1 at Nis station and between 0.277 and 0.451 mm day −1 at Belgrade station, while MAE is between 0.140 and 0.337 mm day −1 at Nis and between 0.208 and 0.360 mm day −1 at Belgrade station. The best performances are achieved by Priestley–Taylor model at Nis s t a t i o n (R 2 = 0. 9 8 5 , M A E = 0. 1 4 0 m m d a y − 1 , RMSE = 0.247 mm day −1) and FAO-56 Penman–Monteith model at Belgrade station (MAE = 0.208 mm day −1 , RMSE = 0.277 mm day −1 , R 2 = 0.975).

Form-invariance (covariance) and frame-indifference (objectivity) are two notions in classical continuum mechanics which have attracted much attention and controversy over the past decades. Particularly in turbulence modelling it seems that there still is a need for clarification. The aim and purpose of this study is fourfold: (i) To achieve consensus in general on definitions and principles when trying to establish an invariant theory for modelling con-stitutive structures and dynamic processes in mechanics, where special focus is put on the principle of Material Frame-Indifference (MFI), a general principle to model the response of a material in solid and fluid mechanics, stating that only frame-indifferent (objective) terms should enter the constitutive equations. (ii) To show that in constitutive modelling MFI can only be regarded as an approximation that needs to be reduced to a weaker statement when trying to advance it to an axiom of nature. (iii) To convince that in dynamical modelling, as in turbulence, MFI may not be utilized as a modelling guideline, not even in an approx-imative sense. Instead, its reduced form has to be supplemented by a second, independent axiom that includes the microscopic (fluctuating) description of the dynamical processes. Concerning Navier-Stokes turbulence, the axiom of Turbulent Frame-Indifference (TFI) is stated in which turbulence has to be modelled consistently with the invariant properties of the deterministic Navier-Stokes equations, and finally (iv) to propose a novel invariant modelling ansatz both for constitutive and dynamical modelling that allows to include the (mean) velocity field as an own independent modelling variable, however, not in an absolute but only in the relative sense as a velocity difference; a result that would systematically improve current modelling procedures in extended thermodynamics and turbulence theory to be more consistent with physical observations.