scholarly journals A Note on the Asymptotic and Threshold Behaviour of Discrete Eigenvalues inside the Spectral Gaps of the Difference Operator with a Periodic Potential

2018 ◽  
Vol 2018 ◽  
pp. 1-5 ◽  
Author(s):  
Gift Muchatibaya ◽  
Josiah Mushanyu
Keyword(s):  
Periodic Potential ◽  
Spectral Gap ◽  
Difference Operator ◽  
Function Theory ◽  
Spectral Gaps ◽  
Threshold Behaviour ◽  
The Difference

The asymptotic and threshold behaviour of the eigenvalues of a perturbed difference operator inside a spectral gap is investigated. In particular, applications of the Titchmarsh-Weyl m-function theory as well as the Birman-Schwinger principle is performed to investigate the existence and behaviour of the eigenvalues of the operator H0+λWn inside the spectral gap of H0 in the limits λ↑∞ and λ↓0.

scholarly journals Extremal spectral gaps for periodic Schrödinger operators

2019 ◽  
Vol 25 ◽  
pp. 40 ◽  
Author(s):  
Chiu-Yen Kao ◽  
Braxton Osting
Keyword(s):  
Periodic Potential ◽  
Spectral Gap ◽  
Step Function ◽  
Two Dimensions ◽  
One Dimension ◽  
Spectral Gaps ◽  
Rearrangement Algorithm ◽  
Almost Everywhere ◽  
Optimal Value ◽  
Geometric Assumption

The spectrum of a Schrödinger operator with periodic potential generally consists of bands and gaps. In this paper, for fixed m, we consider the problem of maximizing the gap-to-midgap ratio for the mth spectral gap over the class of potentials which have fixed periodicity and are pointwise bounded above and below. We prove that the potential maximizing the mth gap-to-midgap ratio exists. In one dimension, we prove that the optimal potential attains the pointwise bounds almost everywhere in the domain and is a step-function attaining the imposed minimum and maximum values on exactly m intervals. Optimal potentials are computed numerically using a rearrangement algorithm and are observed to be periodic. In two dimensions, we develop an efficient rearrangement method for this problem based on a semi-definite formulation and apply it to study properties of extremal potentials. We show that, provided a geometric assumption about the maximizer holds, a lattice of disks maximizes the first gap-to-midgap ratio in the infinite contrast limit. Using an explicit parametrization of two-dimensional Bravais lattices, we also consider how the optimal value varies over all equal-volume lattices.

Periodic Points of the Difference Operator

1997 ◽  
Vol 28 (1) ◽  
pp. 20-26
Author(s):  
Chris Bernhardt ◽  
Thomas Yuster
Keyword(s):  
Difference Operator ◽  
Periodic Points ◽  
The Difference

scholarly journals Design of preview controller for a type of discrete-time interconnected systems

2020 ◽  
Vol 53 (3-4) ◽  
pp. 719-729
Author(s):  
Hao Xie ◽  
Fucheng Liao ◽  
Usman ◽  
Jiamei Deng
Keyword(s):  
Discrete Time ◽  
Difference Operator ◽  
Interconnected Systems ◽  
Preview Control ◽  
Interconnected System ◽  
Lyapunov Function Method ◽  
System Equations ◽  
Error System ◽  
The Difference ◽  
Theoretical Results

This article proposes and studies a problem of preview control for a type of discrete-time interconnected systems. First, adopting the technique of decentralized control, isolated subsystems are constructed by splitting the correlations between the systems. Utilizing the difference operator to the system equations and error vectors, error systems are built. Then, the preview controller is designed for the error system of each isolated subsystem. The controllers of error systems of isolated subsystems are aggregated as a controller of the interconnected system. Finally, by employing Lyapunov function method and the properties of non-singular M-matrix, the guarantee conditions for the existence of preview controllers for interconnected systems are given. The numerical simulation shows that the theoretical results are effective.

scholarly journals Spectral Gaps of Random Graphs and Applications

2019 ◽  
Author(s):  
Christopher Hoffman ◽  
Matthew Kahle ◽  
Elliot Paquette
Keyword(s):  
Fundamental Group ◽  
Free Group ◽  
Large Scale ◽  
Spectral Gap ◽  
Graph Laplacian ◽  
Complex Model ◽  
Constant Factor ◽  
Spectral Geometry ◽  
Giant Component ◽  
Spectral Gaps

Abstract We study the spectral gap of the Erdős–Rényi random graph through the connectivity threshold. In particular, we show that for any fixed $\delta> 0$ if $$\begin{equation*} p \geq \frac{(1/2 + \delta) \log n}{n}, \end{equation*}$$then the normalized graph Laplacian of an Erdős–Rényi graph has all of its nonzero eigenvalues tightly concentrated around $1$. This is a strong expander property. We estimate both the decay rate of the spectral gap to $1$ and the failure probability, up to a constant factor. We also show that the $1/2$ in the above is optimal, and that if $p = \frac{c \log n}{n}$ for $c < 1/2,$ then there are eigenvalues of the Laplacian restricted to the giant component that are separated from $1.$ We then describe several applications of our spectral gap results to stochastic topology and geometric group theory. These all depend on Garland’s method [24], a kind of spectral geometry for simplicial complexes. The following can all be considered to be higher-dimensional expander properties. First, we exhibit a sharp threshold for the fundamental group of the Bernoulli random $2$-complex to have Kazhdan’s property (T). We also obtain slightly more information and can describe the large-scale structure of the group just before the (T) threshold. In this regime, the random fundamental group is with high probability the free product of a (T) group with a free group, where the free group has one generator for every isolated edge. The (T) group plays a role analogous to that of a “giant component” in percolation theory. Next we give a new, short, self-contained proof of the Linial–Meshulam–Wallach theorem [35, 39], identifying the cohomology-vanishing threshold of Bernoulli random $d$-complexes. Since we use spectral techniques, it only holds for $\mathbb{Q}$ or $\mathbb{R}$ coefficients rather than finite field coefficients, as in [35] and [39]. However, it is sharp from a probabilistic point of view, providing for example, hitting time type results and limiting Poisson distributions inside the critical window. It is also a new method of proof, circumventing the combinatorial complications of cocycle counting. Similarly, results in an earlier preprint version of this article were already applied in [33] to obtain sharp cohomology-vanishing thresholds in every dimension for the random flag complex model.

scholarly journals Dichotomy and positivity of neutral equations with nonautonomous past

2014 ◽  
Vol 8 (2) ◽  
pp. 224-242 ◽  
Author(s):  
Nguyen Huy ◽  
Pham Bang
Keyword(s):  
Exponential Dichotomy ◽  
Difference Operator ◽  
Functional Differential Equations ◽  
Sufficient Condition ◽  
Neutral Functional Differential Equations ◽  
Neutral Equations ◽  
Exponentially Stable ◽  
The Difference ◽  
Functional Differential ◽  
Delay Operator

Consider the linear partial neutral functional differential equations with nonautonomous past of the form (?/?t) F(u(t, ?)) = BFu(t, ?) + ?u(t, ?), t ? 0; (? / ?t) u(t, s) = (? / ?s) u(t, s) + A(s)u(t, s), t ? 0 ? s, where the function u(?, ?) takes values in a Banach space X. Under appropriate conditions on the difference operator F and the delay operator ? we prove that the solution semigroup for this system of equations is hyperbolic (or admits an exponential dichotomy) provided that the backward evolution family U = (U(t, s))t?s?0 generated by A(s) is uniformly exponentially stable and the operator B generates a hyperbolic semigroup (etB)t?0 on X. Furthermore, under the positivity conditions on (etB)t?0, U, F and ? we prove that the above-mentioned solution semigroup is positive and then show a sufficient condition for the exponential stability of this solution semigroup.

Difference Operators in Simulation Data Warehouses

2017 ◽  
Vol 12 (2) ◽  
pp. 347-354 ◽  
Author(s):  
Jing Zhao ◽  
◽  
Yoshiharu Ishikawa ◽  
Yukiko Wakita ◽  
Kento Sugiura
Keyword(s):  
Difference Operator ◽  
Difference Operators ◽  
Observation Data ◽  
Temporal Data ◽  
Data Warehouses ◽  
Simulation Data ◽  
Tsunami Disaster ◽  
Spatio Temporal ◽  
The Difference ◽  
Mass Evacuation

In analyzing observation data and simulation results, there are frequent demands for comparing more than one data on the same subject to detect any differences between them. For example, comparison of observation data for an object in a certain spatial domain at different times or comparison of spatial simulation data with different parameters. Therefore, this paper proposes the difference operator in spatio-temporal data warehouses, which store temporal and spatial observation data and simulation data. The requirements for the difference operator are summarized, and the approaches to implement them are presented. In addition, the proposed approach is applied to the mass evacuation of simulation data in a tsunami disaster, and its effectiveness is verified. Extensions of the difference operator and their applications are also discussed.

scholarly journals On the fine spectrum of the generalized difference operatorB(r,s)over the sequence spacesc0andc

2005 ◽  
Vol 2005 (18) ◽  
pp. 3005-3013 ◽  
Author(s):  
Bilâl Altay ◽  
Feyzı Başar
Keyword(s):  
Difference Operator ◽  
Sequence Spaces ◽  
Band Matrix ◽  
Fine Spectrum ◽  
Special Cases ◽  
The Difference ◽  
The Right

We determine the fine spectrum of the generalized difference operatorB(r,s)defined by a band matrix over the sequence spacesc0andc, and derive a Mercerian theorem. This generalizes our earlier work (2004) for the difference operatorΔ, and includes as other special cases the right shift and the Zweier matrices.

scholarly journals Boundary-Value Problems for Differential-Difference Equations with Incommeasurable Shifts of Arguments Reducible to Nonlocal Problems

2019 ◽  
Vol 65 (4) ◽  
pp. 613-622
Author(s):  
E. P. Ivanova
Keyword(s):  
Boundary Value Problems ◽  
Difference Equations ◽  
Original Problem ◽  
Difference Operator ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Nonlocal Problems ◽  
Higher Order Terms ◽  
The Difference

We consider boundary-value problems for differential-difference equations containing incommeasurable shifts of arguments in higher-order terms. We prove that in the case of finite orbits of boundary points generated by the set of shifts of the difference operator, the original problem is reduced to the boundary-value problem for differential equation with nonlocal boundary conditions.

scholarly journals Optimizing Bidding Strategy of the Roadway Projects – An Application of Utility Function Theory

2021 ◽  
Vol 1203 (2) ◽  
pp. 022141
Author(s):  
Wei Tong Chen ◽  
Ferdinan Nikson Liem ◽  
Tai-Jung Chen ◽  
Theresia Avila Bria
Keyword(s):  
Utility Function ◽  
Decision Model ◽  
Function Theory ◽  
Construction Cost ◽  
Bidding Strategy ◽  
Construction Time ◽  
Potential Success ◽  
The Difference ◽  
Set Up

Abstract The "A+B" bid method is determined by the lowest combined bid. Contractors must greatly shorten construction time to be able to win bidding competitions. However, the risk under the shortened duration of construction increased rapidly. For this reason, it is important to propose a bidding strategy to acquire the best composition of construction cost and time. This research targeted contractors with a questionnaire regarding A+B bidding in Taiwan by applied “Utility Function Theory” to construct a decision model of the best construction cost and time. It was found that the optimal profit regressed from the modified utility function differs from contractors’ preliminary bids. In the situation that the difference in basic profits is not large, contractors can set up a decision maker’s utility function based on their ability to take a risk and simulate the major competitors based on experience to judge the potential success and risk from shortening the construction time.

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