Improved estimates for the triangle inequality

Minculete, Nicuşor; Păltănea, Radu

doi:10.1186/s13660-016-1281-z

Cited by 7 publications

(5 citation statements)

References 17 publications

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“…Proof. We use the same idea as in [13]. As we have seen above, the Hessian (F ij ) of F is negative semidefinite, i.e., the function F is concave.…”

Section: Similarly It Holdsmentioning

confidence: 99%

Inequalities from Lorentz-Finsler norms

Minculete,

Pfeifer,

Voicu

2020

Preprint

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We show that Lorentz-Finsler geometry offers a powerful tool in obtaining inequalities. With this aim, we first point out that a series of famous inequalities such as: the (weighted) arithmetic-geometric mean inequality, Aczél's, Popoviciu's and Bellman's inequalities, are all particular cases of a reverse Cauchy-Schwarz, respectively, of a reverse triangle inequality holding in Lorentz-Finsler geometry. Then, we use the same method to prove some completely new inequalities, including two refinements of Aczél's inequality.

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“…Proof. We use the same idea as in [13]. As we have seen above, the Hessian (F ij ) of F is negative semidefinite, i.e., the function F is concave.…”

Section: Similarly It Holdsmentioning

confidence: 99%

Inequalities from Lorentz-Finsler norms

Minculete,

Pfeifer,

Voicu

2020

Preprint

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“…Therefore, we obtain the statement. 7is an improvement of Inequality (5) in an inner product space X over the field of real numbers R with dim R X ≥ n + 1. If dim R X = n, then we have the Parseval identity x 2 = ∑ n k=1 x, e k 2 , for all…”

Section: Some Refinements Of Ostrowski's Inequalitymentioning

confidence: 99%

“…Many refinements and generalizations for the C-B-S inequality can be found in the theory of inequalities (see [2][3][4][5][6]). An improvement of the C-B-S inequality is given by Ostrowski [7], which proved the following: If x = (x 1 , .…”

Section: Introductionmentioning

confidence: 99%

Some Refinements of Ostrowski’s Inequality and an Extension to a 2-Inner Product Space

Minculete

2019

Symmetry

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“…Each part consists of a SN, cluster head, and master node. The network is divided into the diagonal form because the distance travelled by the data packets through the diagonal path is always smaller than the distance travelled by those in which the horizontal path exists, as per the triangular inequality [19].…”

Section: Figure 1 Data Routing Via Cluster Head In Wsnmentioning

confidence: 99%

FEECA: Fuzzy based Energy Efficient Clustering Approach in Wireless Sensor Network

Dwivedi

Sharma

2018

ICST Transactions on Scalable Information Systems

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With huge advancements in Microelectronics, Wireless Sensor Network (WSN) is conquering its domain with low powered tiny sensors. These sensors are commonly used in monitoring and surveillance of remote and urban areas. Since the power supply to these tiny sensors is a battery, for the maximal efficiency of WSN, there arises a need for a maximal lifetime of these tiny sensors wherever they are deployed. Clustering is a promising solution but there is uncertainty in the selection of cluster heads. Fuzzy logic, however, can contribute to the selection of optimal candidates to play the role of such cluster heads. To muddle through the issue, we have proposed a fuzzy-based energy-efficient clustering approach (FEECA) in wireless sensor networks and also designed a fuzzy inference system that defines influential parameters for selecting optimal candidates for cluster head(CH) role. To reduce the expenditure borne by communication, master nodes are selected among the chosen cluster heads so that their communication distance to the sink can be reduced. In the proposed scenario the area is divided into the diagonal form to reduce the load of the network. Each part consists of a Sensor Node (SN), cluster head, and master node. In the diagonal form, the network is divided so that the distance travelled by the data packet through the diagonal path is always smaller than the distance in which the horizontal path exists, based on the triangular inequality theorem. Simulation experiments were conducted and experimental results unveiled better performance of proposed work in terms of stability period, more packet delivery to sink and extended lifetime.

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Improved estimates for the triangle inequality

Abstract: We obtain refined estimates of the triangle inequality in a normed space using integrals and the Tapia semi-product. The particular case of an inner product space is discussed in more detail.

Cited by 7 publications

References 17 publications

Inequalities from Lorentz-Finsler norms

Inequalities from Lorentz-Finsler norms

Some Refinements of Ostrowski’s Inequality and an Extension to a 2-Inner Product Space

FEECA: Fuzzy based Energy Efficient Clustering Approach in Wireless Sensor Network

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