scholarly journals Algorithm 1 and Algorithm 2 for Determining the Number pi (π)

2019 ◽  
pp. 1-7
Author(s):  
Mieczysław Szyszkowicz
Keyword(s):  
Lower Bounds ◽  
Recurrence Formula ◽  
High Accuracy ◽  
Upper Bounds ◽  
Regular Polygons ◽  
Lower And Upper Bounds ◽  
New Methods ◽  
Richardson’S Extrapolation ◽  
New Algorithms ◽  
Modified Algorithm

Archimedes used the perimeter of inscribed and circumscribed regular polygons to obtain lower and upper bounds of π. Starting with two regular hexagons he doubled their sides from 6 to 12, 24, 48, and 96. Using the perimeters of 96 side regular polygons, Archimedes showed that 3+10/71<π<3+1/7 and his method can be realized as a recurrence formula called the Borchardt-Pfaff-Schwab algorithm. Heinrich Dörrie modified this algorithm to produce better approximations to π than these based on Archimedes’ scheme. Lower bounds generated by his modified algorithm are the same as from the method discovered earlier by cardinal Nicolaus Cusanus (XV century), and again re-discovered two hundred years later by Willebrord Snell (XVII century). Knowledge of Taylor series of the functions used in these methods allows to develop new algorithms. Realizing Richardson’s extrapolation, it is possible to increase the accuracy of the constructed methods by eliminating some terms in their series. Two new methods are presented. An approximation of squaring the circle with high accuracy is proposed.

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) = &Oacute;n 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.

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.

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.

scholarly journals New bounds and algorithms for on-line scheduling: two identical processors, known sum and upper bound on the tasks

2006 ◽  
Vol Vol. 8 ◽  
Author(s):  
Enrico Angelelli ◽  
Maria Grazia Speranza ◽  
Zsolt Tuza
Keyword(s):  
Upper Bound ◽  
Completion Time ◽  
Optimal Algorithm ◽  
Multiprocessor Scheduling ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
International Audience ◽  
On Line ◽  
Maximum Completion Time ◽  
New Algorithms

International audience In this paper we study a semi on-line version of the classical multiprocessor scheduling problem on two identical processors. We assume that the sum of the tasks and an upper bound gamma on the size of each task are known. Each task has to be assigned upon arrival and the assignment cannot be changed later. The objective is the minimization of the maximum completion time on the processors. In this paper we propose new algorithms and improve known lower and upper bounds on the competitive ratio. Algorithms and bounds depend on the value of gamma. An optimal algorithm is obtained for gamma in the interval [ 1/n,2(n+1)/n(2n+1) ] and gamma = (2n-1)/2n(n-1), where n is any integer value larger or equal 2.

scholarly journals Spaces with high topological complexity

2014 ◽  
Vol 144 (4) ◽  
pp. 761-773
Author(s):  
Aleksandra Franc ◽  
Petar Pavešić
Keyword(s):  
Lower Bounds ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Topological Complexity ◽  
Lower And Upper Bounds ◽  
Cw Complex ◽  
The Difference

By a formula of Farber, the topological complexity TC(X) of a (p − 1)-connected m-dimensional CW-complex X is bounded above by (2m + 1)/p + 1. We show that the same result holds for the monoidal topological complexity TCM(X). In a previous paper we introduced various lower bounds for TCM(X), such as the nilpotency of the ring H*(X × X, Δ(X)), and the weak and stable (monoidal) topological complexity wTCM(X) and σTCM(X). In general, the difference between these upper and lower bounds can be arbitrarily large. In this paper we investigate spaces with topological complexity close to the maximal value given by Farber's formula. We show that in these cases the gap between the lower and upper bounds is narrow and TC(X) often coincides with the lower bounds.

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