Lower Bounds for the Probability of Intersection of Several Unions of Events

1998 ◽  
Vol 7 (4) ◽  
pp. 353-364 ◽  
Author(s):  
TUHAO CHEN ◽  
E. SENETA
Keyword(s):  
Lower Bounds ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Numerical Example ◽  
Bonferroni Inequalities ◽  
Indicator Functions ◽  
Single Set ◽  
Better Than

To bound the probability of a union of n events from a single set of events, Bonferroni inequalities are sometimes used. There are sharper bounds which are called Sobel–Uppuluri–Galambos inequalities. When two (or more) sets of events are involved, bounds are considered on the probability of intersection of several such unions, one union from each set. We present a method for unified treatment of bivariate lower and upper bounds in this note. The lower bounds obtained are new and at least as good as lower bounds appearing in the literature so far. The upper bounds coincide with existing bivariate Sobel–Uppuluri–Galambos type upper bounds derived by the method of indicator functions. A numerical example is given to illustrate that the new lower bounds can be strictly better than existing ones.

Optimization Design of Electromagnetic Devices Using an Enhanced Salp Swarm Algorithm

2021 ◽  
Vol 35 (12) ◽  
pp. 1471-1476
Author(s):  
Houssem Bouchekara ◽  
Mostafa Smail ◽  
Mohamed Javaid ◽  
Sami Shamsah
Keyword(s):  
Optimization Design ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Design Problems ◽  
Salp Swarm Algorithm ◽  
Electromagnetic Devices ◽  
Swarm Algorithm ◽  
Better Than

An Enhanced version of the Salp Swarm Algorithm (SSA) referred to as (ESSA) is proposed in this paper for the optimization design of electromagnetic devices. The ESSA has the same structure as of the SSA with some modifications in order to enhance its performance for the optimization design of EMDs. In the ESSA, the leader salp does not move around the best position with a fraction of the distance between the lower and upper bounds as in the SAA; rather, a modified mechanism is used. The performance of the proposed algorithm is tested on the widely used Loney’s solenoid and TEAM Workshop Problem 22 design problems. The obtained results show that the proposed algorithm is much better than the initial one. Furthermore, a comparison with other well-known algorithms revealed that the proposed algorithm is very competitive for the optimization design of electromagnetic devices.

scholarly journals Estimating and Approximating the Total π-Electron Energy of Benzenoid Hydrocarbons

2000 ◽  
Vol 55 (5) ◽  
pp. 507-512 ◽  
Author(s):  
I. Gutman ◽  
J. H. Koolen ◽  
V. Moulto ◽  
M. Parac ◽  
T. Soldatović ◽  
...  
Keyword(s):  
Electron Energy ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Benzenoid Hydrocarbons ◽  
Carbon Carbon Bonds ◽  
Carbon Bonds ◽  
Approximate Formulas ◽  
Better Than

Abstract Lower and upper bounds as well as approximate formulas for the total π-electron energy (E) of benzenoid hydrocarbons are deduced, depending only on the number of carbon atoms (n) and number of carbon-carbon bonds (to). These are better than the several previously known (n, m)-type estimates and approximations for E.

scholarly journals A lower bound for general t-stack sortable permutations via pattern avoidance

2020 ◽  
Vol 22 ◽  
Author(s):  
Pranav Chinmay
Keyword(s):  
Lower Bound ◽  
Recurrence Relation ◽  
Lower Bounds ◽  
Upper Bounds ◽  
Pattern Avoidance ◽  
Lower And Upper Bounds

There is no formula for general t-stack sortable permutations. Thus, we attempt to study them by establishing lower and upper bounds. Permutations that avoid certain pattern sets provide natural lower bounds. This paper presents a recurrence relation that counts the number of permutations that avoid the set (23451,24351,32451,34251,42351,43251). This establishes a lower bound on 3-stack sortable permutations. Additionally, the proof generalizes to provide lower bounds for all t-stack sortable permutations.

scholarly journals Improve Some of Bounds for the Randic Estrada Index of Graphs

2020 ◽  
Author(s):  
Akbar Jahanbani
Keyword(s):  
Lower Bounds ◽  
Upper Bounds ◽  
Randić Index ◽  
Lower And Upper Bounds ◽  
Graph Invariants ◽  
Randic Index ◽  
Estrada Index ◽  
Eigenvalues Of Graphs

Let G be a graph with n vertices and let 1; 2; : : : ; n be the eigenvalues of Randic matrix. The Randic Estrada index of G is REE(G) = Ón i=1 ei . In this paper, we establish lower and upper bounds for Randic index in terms of graph invariants such as the number of vertices and eigenvalues of graphs and improve some previously published lower bounds.

On the classical Bonferroni inequalities and the corresponding Galambos inequalities

1981 ◽  
Vol 18 (03) ◽  
pp. 757-763 ◽  
Author(s):  
A. M. Walker
Keyword(s):  
Lower Bounds ◽  
Probability Space ◽  
Upper And Lower Bounds ◽  
Partial Sums ◽  
Bonferroni Inequalities ◽  
Indicator Functions

Let (A 1 A 2, · ··, An ) be a set of n events on a probability space. Let be the sum of the probabilities of all intersections of r events, and Mn the number of events in the set which occur. The classical Bonferroni inequalities provide upper and lower bounds for the probabilities P(Mn = m), and equal to partial sums of series of the form which give the exact probabilities. These inequalities have recently been extended by J. Galambos to give sharper bounds. Here we present straightforward proofs of the Bonferroni inequalities, using indicator functions, and show how they lead naturally to new simple proofs of the Galambos inequalities.

scholarly journals Round weighting problem and gathering in radio networks with symmetrical interference

2016 ◽  
Vol 08 (02) ◽  
pp. 1650035
Author(s):  
Jean-Claude Bermond ◽  
Cristiana Gomes Huiban ◽  
Patricio Reyes
Keyword(s):  
Lower Bounds ◽  
Access Network ◽  
Upper Bounds ◽  
Radio Networks ◽  
Interference Model ◽  
Lower And Upper Bounds ◽  
Single Node ◽  
Model Based ◽  
Weighting Problem ◽  
General Graphs

In this paper, we consider the problem of gathering information in a gateway in a radio mesh access network. Due to interferences, calls (transmissions) cannot be performed simultaneously. This leads us to define a round as a set of non-interfering calls. Following the work of Klasing, Morales and Pérennes, we model the problem as a Round Weighting Problem (RWP) in which the objective is to minimize the overall period of non-interfering calls activations (total number of rounds) providing enough capacity to satisfy the throughput demand of the nodes. We develop tools to obtain lower and upper bounds for general graphs. Then, more precise results are obtained considering a symmetric interference model based on distance of graphs, called the distance-[Formula: see text] interference model (the particular case [Formula: see text] corresponds to the primary node model). We apply the presented tools to get lower bounds for grids with the gateway either in the middle or in the corner. We obtain upper bounds which in most of the cases match the lower bounds, using strategies that either route the demand of a single node or route simultaneously flow from several source nodes. Therefore, we obtain exact and constructive results for grids, in particular for the case of uniform demands answering a problem asked by Klasing, Morales and Pérennes.

scholarly journals Bounds for the variance of the busy period of the M/G/∞ queue

1984 ◽  
Vol 16 (4) ◽  
pp. 929-932 ◽  
Author(s):  
M. F. Ramalhoto
Keyword(s):  
Distribution Function ◽  
Lower Bounds ◽  
Service Time ◽  
Time Distribution ◽  
Busy Period ◽  
Random Variable ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Service Time Distribution ◽  
Lower And Upper Bounds

Some bounds for the variance of the busy period of an M/G/∞ queue are calculated as functions of parameters of the service-time distribution function. For any type of service-time distribution function, upper and lower bounds are evaluated in terms of the intensity of traffic and the coefficient of variation of the service time. Other lower and upper bounds are derived when the service time is a NBUE, DFR or IMRL random variable. The variance of the busy period is also related to the variance of the number of busy periods that are initiated in (0, t] by renewal arguments.

scholarly journals Bounds for the variance of the busy period of the M/G/∞ queue

1984 ◽  
Vol 16 (04) ◽  
pp. 929-932
Author(s):  
M. F. Ramalhoto
Keyword(s):  
Distribution Function ◽  
Lower Bounds ◽  
Service Time ◽  
Time Distribution ◽  
Busy Period ◽  
Random Variable ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Service Time Distribution ◽  
Lower And Upper Bounds

Some bounds for the variance of the busy period of an M/G/∞ queue are calculated as functions of parameters of the service-time distribution function. For any type of service-time distribution function, upper and lower bounds are evaluated in terms of the intensity of traffic and the coefficient of variation of the service time. Other lower and upper bounds are derived when the service time is a NBUE, DFR or IMRL random variable. The variance of the busy period is also related to the variance of the number of busy periods that are initiated in (0, t] by renewal arguments.

scholarly journals On Instability Analysis of Linear Feedback Systems

Computation ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 16
Author(s):  
Mutti-Ur Rehman ◽  
Jehad Alzabut
Keyword(s):  
Differential Equation ◽  
System Theory ◽  
Numerical Simulation ◽  
Lower Bounds ◽  
Numerical Approximation ◽  
Upper Bounds ◽  
Low Rank ◽  
Linear Feedback ◽  
Feedback Systems ◽  
Lower And Upper Bounds

The numerical approximation of the μ -value is key towards the measurement of instability, stability analysis, robustness, and the performance of linear feedback systems in system theory. The MATLAB function mussv available in MATLAB Control Toolbox efficiently computes both lower and upper bounds of the μ -value. This article deals with the numerical approximations of the lower bounds of μ -values by means of low-rank ordinary differential equation (ODE)-based techniques. The numerical simulation shows that approximated lower bounds of μ -values are much tighter when compared to those obtained by the MATLAB function mussv.

scholarly journals Norm of nonnegative and positive matrices

2017 ◽  
Vol 6 (3) ◽  
pp. 98
Author(s):  
Alaa Abu Alroz
Keyword(s):  
Spectral Radius ◽  
Lower Bounds ◽  
Maximum Modulus ◽  
Upper Bounds ◽  
Nonnegative Matrices ◽  
Lower And Upper Bounds ◽  
Positive Matrix ◽  
Eigen Value ◽  
Eigen Values ◽  
Positive Matrices

The spectral radius r(A) of matrix A is the maximum modulus of the Eigen values. In this paper, the studies about the lower and upper bounds for the spectral radius and the lower bounds for the minimum eigen value of appositive and nonnegative matrices are investigate.The matrix norm, the spectral radius norm,and the column (row) sums of nonnegative and positive matrices are widely used to establish some inequalities for matrices. Then several existing results are improved for these inequalities for nonnegative and positive matrix. Furthermore, the lower and upper bounds of the Perron roots for nonnegative matrices are examined, and some upper bounds are computed.

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