On Instability Analysis of Linear Feedback Systems

In this article we consider the numerical approximation of lower bounds of Structured Singular Values, SSV. The SSV is a wellknown mathematical quantity which is widely used to analyse and syntesize the robust stability and instability analysis of linear feedback systems in control theory. It links a bridge between numerical linear algebra and system theory. The computation of lower bounds of SSV by means of ordinary diﬀerential equations based technique is presented. The obtained numerical results for the lower bounds of SSV are compared with the well-known MATLAB function mussv available in MATLAB control toolbox.

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

UF Journal of Undergraduate Research ◽

10.32473/ufjur.v22i0.121801 ◽

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.

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Improve Some of Bounds for the Randic Estrada Index of Graphs

10.20944/preprints202012.0818.v1 ◽

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.

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Variational Quantum Fidelity Estimation

Quantum ◽

10.22331/q-2020-03-26-248 ◽

2020 ◽

Vol 4 ◽

pp. 248 ◽

Cited By ~ 3

Author(s):

Marco Cerezo ◽

Alexander Poremba ◽

Lukasz Cincio ◽

Patrick J. Coles

Keyword(s):

Phase Transitions ◽

Quantum Computer ◽

Quantum Phase Transitions ◽

Upper Bounds ◽

Low Rank ◽

Matrix Elements ◽

Lower And Upper Bounds ◽

Classical Algorithm ◽

Largest Eigenvalues ◽

Quantum Fidelity

Computing quantum state fidelity will be important to verify and characterize states prepared on a quantum computer. In this work, we propose novel lower and upper bounds for the fidelity F(ρ,σ) based on the ``truncated fidelity'' F(ρm,σ), which is evaluated for a state ρm obtained by projecting ρ onto its m-largest eigenvalues. Our bounds can be refined, i.e., they tighten monotonically with m. To compute our bounds, we introduce a hybrid quantum-classical algorithm, called Variational Quantum Fidelity Estimation, that involves three steps: (1) variationally diagonalize ρ, (2) compute matrix elements of σ in the eigenbasis of ρ, and (3) combine these matrix elements to compute our bounds. Our algorithm is aimed at the case where σ is arbitrary and ρ is low rank, which we call low-rank fidelity estimation, and we prove that no classical algorithm can efficiently solve this problem under reasonable assumptions. Finally, we demonstrate that our bounds can detect quantum phase transitions and are often tighter than previously known computable bounds for realistic situations.

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Stability Analysis of Linear Feedback Systems in Control

Symmetry ◽

10.3390/sym12091518 ◽

2020 ◽

Vol 12 (9) ◽

pp. 1518

Author(s):

Mutti-Ur Rehman ◽

Jehad Alzabut ◽

Muhammad Fazeel Anwar

Keyword(s):

System Theory ◽

Stability Analysis ◽

Linear Time ◽

Singular Values ◽

Low Rank ◽

Gradient System ◽

Feedback Systems ◽

Linear Time Invariant ◽

Linear Time Invariant Systems ◽

Time Invariant

This article presents a stability analysis of linear time invariant systems arising in system theory. The computation of upper bounds of structured singular values confer the stability analysis, robustness and performance of feedback systems in system theory. The computation of the bounds of structured singular values of Toeplitz and symmetric Toeplitz matrices for linear time invariant systems is presented by means of low rank ordinary differential equations (ODE’s) based methodology. The proposed methodology is based upon the inner-outer algorithm. The inner algorithm constructs and solves a gradient system of ODE’s while the outer algorithm adjusts the perturbation level with fast Newton’s iteration. The comparison of bounds of structured singular values approximated by low rank ODE’s based methodology results tighter bounds when compared with well-known MATLAB routine mussv, available in MATLAB control toolbox.

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

Combinatorics Probability Computing ◽

10.1017/s096354839800371x ◽

1998 ◽

Vol 7 (4) ◽

pp. 353-364 ◽

Cited By ~ 1

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.

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Round weighting problem and gathering in radio networks with symmetrical interference

Discrete Mathematics Algorithms and Applications ◽

10.1142/s179383091650035x ◽

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.

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

Advances in Applied Probability ◽

10.2307/1427350 ◽

1984 ◽

Vol 16 (4) ◽

pp. 929-932 ◽

Cited By ~ 16

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.

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

Advances in Applied Probability ◽

10.1017/s000186780002303x ◽

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.

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Norm of nonnegative and positive matrices

International Journal of Applied Mathematical Research ◽

10.14419/ijamr.v6i3.7637 ◽

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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