Scalar Products and Orthogonality

1996 ◽  
pp. 65-84
Author(s):  
Rami Shakarchi
Keyword(s):  
Scalar Products

Features of determining the deformed state of a particle material during hot stamping of porous moldings

2021 ◽  
pp. 21-30
Author(s):  
B. G. Gasanov ◽  
A. A. Aganov ◽  
P. V. Sirotin
Keyword(s):  
Displacement Vector ◽  
Hot Stamping ◽  
Strain Tensor ◽  
Particle Material ◽  
Software Algorithms ◽  
Deformed State ◽  
Metal Strain ◽  
Scalar Products ◽  
Analytical Expressions ◽  
Kinetics Of

The paper describes main methods for assessing the deformed state of porous body metal frames developed by different authors based on the analysis of yield conditions and governing equations, using the principle of equivalent strains and stresses, and studying the kinetics of metal strain during pressing. Formulas were derived to determine the components of the powder particle material strain tensor through dyads, as scalar products of the basis vectors of the convected coordinate system at each moment of porous molding strain. The expediency of using the analytical expressions developed to determine the deformed state of the particle material was experimentally substantiated subject to the known displacement vector parameters of representative elements (macrostrains) of porous billets. The applications of well-known analytical expressions were established, and the proposed formulas proved applicable for the deformed state assessment of particle metal during the pressure processing of powder products of different configurations and designing billets with a defined porosity and geometric parameters as a basis for compiling software algorithms for the computer simulation of porous molding hot stamping.

Sunbeam vectors and scalar products

1997 ◽  
Vol 35 (5) ◽  
pp. 310-311
Author(s):  
Ernest Zebrowski
Keyword(s):  
Scalar Products

A WAVE APPROACH TO PATTERN RECOGNITION (WITH APPLICATION TO OPTICAL CHARACTER RECOGNITION)

1994 ◽  
Vol 04 (01) ◽  
pp. 193-207 ◽  
Author(s):  
VADIM BIKTASHEV ◽  
VALENTIN KRINSKY ◽  
HERMANN HAKEN
Keyword(s):  
Pattern Recognition ◽  
Character Recognition ◽  
Optical Character Recognition ◽  
Dissipative Structures ◽  
Binary Images ◽  
Optical Character ◽  
Nonlinear Version ◽  
Medium Quality ◽  
Scalar Products ◽  
Machine Reading

The possibility of using nonlinear media as a highly parallel computation tool is discussed, specifically for image classification and recognition. Some approaches of this type are known, that are based on stationary dissipative structures which can “measure” scalar products of images. In this paper, we exploit the analogy between binary images and point sets, and use the Hausdorff metrics for comparing the images. It does not require the measure at all, and is based only on the metrics of the space whose subsets we consider. In addition to Hausdorff distance, we suggest a new “nonlinear” version of this distance for comparison of images, called “autowave” distance. This distance can be calculated very easily and yields some additional advantages for pattern recognition (e.g. noise tolerance). The method was illustrated for the problem of machine reading (Optical Character Recognition). It was compared with some famous OCR programs for PC. On a medium quality xerocopy of a journal page, in the same conditions of learning and recognition, the autowave approach resulted in much fewer mistakes. The method can be realized using only one chip with simple uniform connection of the elements. In this case, it yields an increase in computation speed of several orders of magnitude.

scholarly journals Regular Bohr-Sommerfeld quantization rules for a h-pseudo-differential operator. The method of positive commutators

2016 ◽  
Vol Volume 23 - 2016 - Special... ◽  
Author(s):  
Abdelwaheb Ifa ◽  
Michel Rouleux
Keyword(s):  
Differential Operator ◽  
Gram Matrix ◽  
Pseudo Differential Operator ◽  
Quantization Rule ◽  
International Audience ◽  
Scalar Products ◽  
Quantization Rules

International audience We revisit in this Note the well known Bohr-Sommerfeld quantization rule (BS) for a 1-D Pseudo-differential self-adjoint Hamiltonian within the algebraic and microlocal framework of Helffer and Sjöstrand; BS holds precisely when the Gram matrix consisting of scalar products of some WKB solutions with respect to the " flux norm " is not invertible. Dans le cadre algébrique et microlocal élaboré par Helffer et Sjöstrand, on propose une ré-écriture de la règle de quantification de Bohr-Sommerfeld pour un opérateur auto-adjoint h-Pseudo-différentiel 1-D; elle s'exprime par la non-inversibilité de la matrice de Gram d'un couple de solutions WKB dans une base convenable, pour le produit scalaire associé à la " norme de flux " .

Solving Systems of Linear Algebraic Equations with the Use of an Selective Regularization

2021 ◽  
pp. 11-15
Author(s):  
Vladimir N. Lutay
Keyword(s):  
Diagonal Element ◽  
Decomposition Process ◽  
Algebraic Equations ◽  
Original Matrix ◽  
Triangular Matrices ◽  
First Case ◽  
Linear Algebraic Equations ◽  
The Matrix ◽  
Gauss Method ◽  
Scalar Products

The solution of systems of linear algebraic equations, which matrices can be poorly conditioned or singular is considered. As a solution method, the original matrix is decomposed into triangular components by Gauss or Chole-sky with an additional operation, which consists in increasing the small or zero diagonal terms of triangular matrices during the decomposition process. In the first case, the scalar products calculated during decomposition are divided into two positive numbers such that the first is greater than the second, and their sum is equal to the original one. In further operations, the first number replaces the scalar product, as a result of which the value of the diagonal term increases, and the second number is stored and used after the decomposition process is completed to correct the result of calculations. This operation increases the diagonal elements of triangular matrices and prevents the appearance of very small numbers in the Gauss method and a negative root expression in the Cholesky method. If the matrix is singular, then the calculated diagonal element is zero, and an arbitrary positive number is added to it. This allows you to complete the decomposition process and calculate the pseudo-inverse matrix using the Greville method. The results of computational experiments are presented.

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