Indecomposable Representations with Invariant Inner Product

1985 ◽  
pp. 149-159 ◽  
Author(s):  
Huzihiro Araki
Keyword(s):  
Inner Product ◽  
Indecomposable Representations

scholarly journals On a New Geometric Constant Related to the Euler-Lagrange Type Identity in Banach Spaces

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 116
Author(s):  
Qi Liu ◽  
Yongjin Li
Keyword(s):  
Banach Spaces ◽  
Product Space ◽  
Sufficient Conditions ◽  
Normal Structure ◽  
The Other ◽  
Inner Product ◽  
Equivalent Characterization ◽  
Geometric Constant ◽  
Geometric Constants

In this paper, we will introduce a new geometric constant LYJ(λ,μ,X) based on an equivalent characterization of inner product space, which was proposed by Moslehian and Rassias. We first discuss some equivalent forms of the proposed constant. Next, a characterization of uniformly non-square is given. Moreover, some sufficient conditions which imply weak normal structure are presented. Finally, we obtain some relationship between the other well-known geometric constants and LYJ(λ,μ,X). Also, this new coefficient is computed for X being concrete space.

scholarly journals Fuzzy Inner Product Space: Literature Review and a New Approach

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 765
Author(s):  
Lorena Popa ◽  
Lavinia Sida
Keyword(s):  
Literature Review ◽  
Product Space ◽  
Inner Product Space ◽  
Inner Product ◽  
Product Spaces ◽  
Comprehensive Overview ◽  
New Approach ◽  
Inner Product Spaces ◽  
Product Function ◽  
Suitable Definition

The aim of this paper is to provide a suitable definition for the concept of fuzzy inner product space. In order to achieve this, we firstly focused on various approaches from the already-existent literature. Due to the emergence of various studies on fuzzy inner product spaces, it is necessary to make a comprehensive overview of the published papers on the aforementioned subject in order to facilitate subsequent research. Then we considered another approach to the notion of fuzzy inner product starting from P. Majundar and S.K. Samanta’s definition. In fact, we changed their definition and we proved some new properties of the fuzzy inner product function. We also proved that this fuzzy inner product generates a fuzzy norm of the type Nădăban-Dzitac. Finally, some challenges are given.

scholarly journals Identification of coexisting dynamics in boundary layer flows through proper orthogonal decomposition with weighting matrices

Meccanica ◽  
2021 ◽  
Author(s):  
Matteo Dellacasagrande ◽  
Dario Barsi ◽  
Patrizia Bagnerini ◽  
Davide Lengani ◽  
Daniele Simoni
Keyword(s):  
Experimental Data ◽  
Proper Orthogonal Decomposition ◽  
Fourier Transforms ◽  
Free Stream ◽  
Numerical Test ◽  
Separated Flow ◽  
Orthogonal Decomposition ◽  
Inner Product ◽  
Test Case ◽  
Proper Orthogonal

AbstractA different version of the classic proper orthogonal decomposition (POD) procedure introducing spatial and temporal weighting matrices is proposed. Furthermore, a newly defined non-Euclidean (NE) inner product that retain similarities with the POD is introduced in the paper. The aim is to emphasize fluctuation events localized in spatio-temporal regions with low kinetic energy magnitude, which are not highlighted by the classic POD. The different variants proposed in this work are applied to numerical and experimental data, highlighting analogies and differences with respect to the classic and other normalized variants of POD available in the literature. The numerical test case provides a noise-free environment of the strongly organized vortex shedding behind a cylinder. Conversely, experimental data describing transitional boundary layers are used to test the capability of the procedures in strongly not uniform flows. By-pass and separated flow transition processes developing with high free-stream disturbances have been considered. In both cases streaky structures are expected to interact with other vortical structures (i.e. free-stream vortices in the by-pass case and Kelvin–Helmholtz rolls in the separated type) that carry a significant different amount of energy. Modes obtained by the non-Euclidean POD (NE-POD) procedure (where weighted projections are considered) are shown to better extract low energy events sparse in time and space with respect to modes extracted by other variants. Moreover, NE-POD modes are further decomposed as a combination of Fourier transforms of the related temporal coefficients and the normalized data ensemble to isolate the frequency content of each mode.

Mackey Continuity of Characteristic Functionals

2002 ◽  
Vol 9 (1) ◽  
pp. 83-112
Author(s):  
S. Kwapień ◽  
V. Tarieladze
Keyword(s):  
Product Space ◽  
Locally Convex Space ◽  
Inner Product ◽  
Nuclear Space ◽  
Probability Measures ◽  
Linear Kernel ◽  
Characteristic Functional ◽  
Initial Space ◽  
Radon Probability Measure ◽  
Locally Convex

Abstract Problems of the Mackey-continuity of characteristic functionals and the localization of linear kernels of Radon probability measures in locally convex spaces are investigated. First the class of spaces is described, for which the continuity takes place. Then it is shown that in a non-complete sigmacompact inner product space, as well as in a non-complete sigma-compact metizable nuclear space, there may exist a Radon probability measure having a non-continuous characteristic functional in the Mackey topology and a linear kernel not contained in the initial space. Similar problems for moment forms and higher order kernels are also touched upon. Finally, a new proof of the result due to Chr. Borell is given, which asserts that any Gaussian Radon measure on an arbitrary Hausdorff locally convex space has the Mackey-continuous characteristic functional.

scholarly journals Human-Mimetic Estimation of Food Volume from a Single-View RGB Image Using an AI System

Electronics ◽  
2021 ◽  
Vol 10 (13) ◽  
pp. 1556
Author(s):  
Zhengeng Yang ◽  
Hongshan Yu ◽  
Shunxin Cao ◽  
Qi Xu ◽  
Ding Yuan ◽  
...  
Keyword(s):  
Chronic Diseases ◽  
Volume Estimation ◽  
Portion Size ◽  
Inner Product ◽  
Probability Vector ◽  
Internal Reference ◽  
Single View ◽  
Unhealthy Diet ◽  
Simplified Procedure ◽  
Traditional Image

It is well known that many chronic diseases are associated with unhealthy diet. Although improving diet is critical, adopting a healthy diet is difficult despite its benefits being well understood. Technology is needed to allow an assessment of dietary intake accurately and easily in real-world settings so that effective intervention to manage being overweight, obesity, and related chronic diseases can be developed. In recent years, new wearable imaging and computational technologies have emerged. These technologies are capable of performing objective and passive dietary assessments with a much simplified procedure than traditional questionnaires. However, a critical task is required to estimate the portion size (in this case, the food volume) from a digital image. Currently, this task is very challenging because the volumetric information in the two-dimensional images is incomplete, and the estimation involves a great deal of imagination, beyond the capacity of the traditional image processing algorithms. In this work, we present a novel Artificial Intelligent (AI) system to mimic the thinking of dietitians who use a set of common objects as gauges (e.g., a teaspoon, a golf ball, a cup, and so on) to estimate the portion size. Specifically, our human-mimetic system “mentally” gauges the volume of food using a set of internal reference volumes that have been learned previously. At the output, our system produces a vector of probabilities of the food with respect to the internal reference volumes. The estimation is then completed by an “intelligent guess”, implemented by an inner product between the probability vector and the reference volume vector. Our experiments using both virtual and real food datasets have shown accurate volume estimation results.

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