scholarly journals A concise proof to the spectral and nuclear norm bounds through tensor partitions

2019 ◽  
Vol 17 (1) ◽  
pp. 365-373 ◽  
Author(s):  
Xu Kong
Keyword(s):  
Upper Bounds ◽  
Spectral Norm ◽  
Nuclear Norm ◽  
Short Paper ◽  
Lower And Upper Bounds ◽  
Dual Norm ◽  
Numerical Examples ◽  
Matrix Anal

Abstract On estimations of the lower and upper bounds for the spectral and nuclear norm of a tensor, Li established neat bounds for the two norms based on regular tensor partitions, and proposed a conjecture for the same bounds to be hold based on general tensor partitions [Z. Li, Bounds on the spectral norm and the nuclear norm of a tensor based on tensor partition, SIAM J. Matrix Anal. Appl., 37 (2016), pp. 1440-1452]. Later, Chen and Li provided a solution to the conjecture [Chen B., Li Z., On the tensor spectral p-norm and its dual norm via partitions]. In this short paper, we present a concise and different proof for the validity of the conjecture, which also offers a new and simpler proof to the bounds of the spectral and nuclear norms established by Li for regular tensor partitions. Two numerical examples are provided to illustrate tightness of these bounds.

Reliability Analysis of Truss Structures by Using Matrix Method

1980 ◽  
Vol 102 (4) ◽  
pp. 749-756 ◽  
Author(s):  
Y. Murotsu ◽  
H. Okada ◽  
K. Niwa ◽  
S. Miwa
Keyword(s):  
Lower Bound ◽  
Failure Probability ◽  
Matrix Method ◽  
Failure Criteria ◽  
Upper Bounds ◽  
Truss Structures ◽  
Lower And Upper Bounds ◽  
Numerical Examples ◽  
Modes Of Failure ◽  
Complete Failure

This paper proposes a method of systematically generating the failure criteria of truss structures by using Matrix Method. The resulting criterion for a statically determinate truss is simple and its failure probability is easily evaluated. In case of a statically indeterminate truss, however, there are many possible modes or paths to complete failure of the structure and it is impossible in practice to generate all of them. Hence, the failure probability is estimated by evaluating its lower and upper bounds. The lower bound is evaluated by selecting the dominant modes of failure and calculating their probabilities. The upper bound is evaluated by assuming that the redundant truss behaves itself like a statically determinate truss, i.e., the structure fails if any one member is subject to failure. Numerical examples are provided to demonstrate the applicability of the propsed methods.

scholarly journals Convergence ofp-series revisited with applications

2006 ◽  
Vol 2006 ◽  
pp. 1-8
Author(s):  
Elom K. Abalo ◽  
Kokou Y. Abalo
Keyword(s):  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Numerical Examples ◽  
Partial Sum

We construct two adjacent sequences that converge to the sum of a given convergentp-series. In case of a divergentp-series, lower and upper bounds of the(kn)th partial sum are constructed. In either case, we extend the results obtained by Hansheng and Lu (2005) to any integerk≥2. Some numerical examples are given.

Harmonic Waves in Layered Composites: Comparison Among Several Schemes

1975 ◽  
Vol 42 (3) ◽  
pp. 699-704 ◽  
Author(s):  
S. Nemat-Nasser ◽  
S. Minagawa
Keyword(s):  
Rayleigh Quotient ◽  
Upper Bounds ◽  
Harmonic Waves ◽  
Layered Composites ◽  
Test Functions ◽  
Lower And Upper Bounds ◽  
Numerical Examples ◽  
Elastic Composites

For harmonic waves in layered elastic composites the results obtained by means of the Rayleigh quotient based on displacement and the Rayleigh quotient based on stress, are compared with those obtained by a new quotient recently proposed by one of the authors, in an effort to examine reasons for the astonishing accuracy of the new quotient for this class of problems. This comparison leads to a scheme for obtaining improved test functions which then give very accurate lower and upper bounds for the wave frequencies. Results are illustrated by numerical examples.

scholarly journals New bounds for the minimum eigenvalue of 𝓜-tensors

2017 ◽  
Vol 15 (1) ◽  
pp. 296-303 ◽  
Author(s):  
Jianxing Zhao ◽  
Caili Sang
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Minimum Eigenvalue ◽  
Numerical Examples ◽  
Theoretical Results

Abstract A new lower bound and a new upper bound for the minimum eigenvalue of an 𝓜-tensor are obtained. It is proved that the new lower and upper bounds improve the corresponding bounds provided by He and Huang (J. Inequal. Appl., 2014, 2014, 114) and Zhao and Sang (J. Inequal. Appl., 2016, 2016, 268). Finally, two numerical examples are given to verify the theoretical results.

Extension of the Spatial PDI Model to Three Dimensions

1997 ◽  
Vol 84 (1) ◽  
pp. 176-178
Author(s):  
Frank O'Brien
Keyword(s):  
Population Density ◽  
Dimensional Space ◽  
Finite Lattice ◽  
Three Dimensional ◽  
Upper Bounds ◽  
Three Dimensions ◽  
Lower And Upper Bounds ◽  
Density Index ◽  
Three Dimensional Space

The author's population density index ( PDI) model is extended to three-dimensional distributions. A derived formula is presented that allows for the calculation of the lower and upper bounds of density in three-dimensional space for any finite lattice.

Energy of spherical fuzzy graphs

2020 ◽  
pp. 321-332
Author(s):  
S. Yahya Mohamed ◽  
A. Mohamed Ali
Keyword(s):  
Adjacency Matrix ◽  
Upper Bounds ◽  
Fuzzy Graph ◽  
Lower And Upper Bounds ◽  
Fuzzy Graphs

In this paper, the notion of energy extended to spherical fuzzy graph. The adjacency matrix of a spherical fuzzy graph is defined and we compute the energy of a spherical fuzzy graph as the sum of absolute values of eigenvalues of the adjacency matrix of the spherical fuzzy graph. Also, the lower and upper bounds for the energy of spherical fuzzy graphs are obtained.

scholarly journals Relationship between Age and Value of Information for a Noisy Ornstein–Uhlenbeck Process

Entropy ◽  
2021 ◽  
Vol 23 (8) ◽  
pp. 940
Author(s):  
Zijing Wang ◽  
Mihai-Alin Badiu ◽  
Justin P. Coon
Keyword(s):  
Value Of Information ◽  
Markov Models ◽  
Upper Bounds ◽  
Distribution Functions ◽  
Lower And Upper Bounds ◽  
Uhlenbeck Process ◽  
Source Data ◽  
Non Linear ◽  
Queue Model ◽  
Selection Of

The age of information (AoI) has been widely used to quantify the information freshness in real-time status update systems. As the AoI is independent of the inherent property of the source data and the context, we introduce a mutual information-based value of information (VoI) framework for hidden Markov models. In this paper, we investigate the VoI and its relationship to the AoI for a noisy Ornstein–Uhlenbeck (OU) process. We explore the effects of correlation and noise on their relationship, and find logarithmic, exponential and linear dependencies between the two in three different regimes. This gives the formal justification for the selection of non-linear AoI functions previously reported in other works. Moreover, we study the statistical properties of the VoI in the example of a queue model, deriving its distribution functions and moments. The lower and upper bounds of the average VoI are also analysed, which can be used for the design and optimisation of freshness-aware networks. Numerical results are presented and further show that, compared with the traditional linear age and some basic non-linear age functions, the proposed VoI framework is more general and suitable for various contexts.

The Lower and Upper Bounds of Turán Number for Odd Wheels

2021 ◽  
Vol 37 (3) ◽  
pp. 919-932
Author(s):  
Byeong Moon Kim ◽  
Byung Chul Song ◽  
Woonjae Hwang
Keyword(s):  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Turán Number
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