scholarly journals Minimization of the lowest eigenvalue for a vibrating beam

2018 ◽  
Vol 38 (4) ◽  
pp. 2079-2092 ◽  
Author(s):  
Quanyi Liang ◽  
◽  
Kairong Liu ◽  
Gang Meng ◽  
Zhikun She ◽  
...  
Keyword(s):  
Lowest Eigenvalue

Transfer Matrix of the Two-dimensional Ising Model

2020 ◽  
pp. 211-234
Author(s):  
Giuseppe Mussardo
Keyword(s):  
Ising Model ◽  
Critical Point ◽  
Transfer Matrix ◽  
Functional Equations ◽  
Baxter Equation ◽  
Matrix Formalism ◽  
Two Dimensional ◽  
R Matrix ◽  
Transfer Matrix Formalism ◽  
Lowest Eigenvalue

This chapter deals with the exact solution of the two-dimensional Ising model as it is achieved through the transfer matrix formalism. It discusses the crucial role played by the commutative properties of the transfer matrices, which lead to a functional equation for their eigenvalues. The exact free energy of the Ising model and its critical point can be identified by means of the lowest eigenvalue. The chapter covers Baxter's approach, the Yang–Baxter equation and its relation to the Boltzmann weights, the R-matrix, and discusses activity away from the critical point, the six-vertex model, as well as functional equations and symmetries.

scholarly journals Counterexample to Strong Diamagnetism for the Magnetic Robin Laplacian

2020 ◽  
Vol 23 (3) ◽  
Author(s):  
Ayman Kachmar ◽  
Mikael P. Sundqvist
Keyword(s):  
Magnetic Field ◽  
Boundary Condition ◽  
Laplace Operator ◽  
Unit Disc ◽  
Uniform Magnetic Field ◽  
Robin Boundary Condition ◽  
Robin Laplacian ◽  
The Laplace Operator ◽  
Robin Boundary ◽  
Lowest Eigenvalue

Abstract We determine a counterexample to strong diamagnetism for the Laplace operator in the unit disc with a uniform magnetic field and Robin boundary condition. The example follows from the accurate asymptotics of the lowest eigenvalue when the Robin parameter tends to $-\infty $ − ∞ .

scholarly journals Spectral analysis of non-commutative harmonic oscillators: The lowest eigenvalue and no crossing

2014 ◽  
Vol 415 (2) ◽  
pp. 595-609 ◽  
Author(s):  
Fumio Hiroshima ◽  
Itaru Sasaki
Keyword(s):  
Spectral Analysis ◽  
Harmonic Oscillators ◽  
Lowest Eigenvalue

Maximizing the lowest eigenvalue of a constrained affine sum with application to the optimal design of structures and vibrating systems

2008 ◽  
Vol 223 (3) ◽  
pp. 583-593 ◽  
Author(s):  
Y M Ram ◽  
S G Krishna
Keyword(s):  
Optimal Design ◽  
Discrete Model ◽  
Natural Frequencies ◽  
Symmetric Matrices ◽  
Vibratory System ◽  
Newton Iterative Method ◽  
Lagrange Problem ◽  
Repeated Use ◽  
Lowest Eigenvalue ◽  
Vibrating Systems

This article deals with the problem of maximizing the lowest eigenvalue of an affine sum of symmetric matrices subject to a constraint. It is shown that by the repeated use of eliminants, the problem may be reduced in a systematic manner to that of finding the roots of certain polynomials. However, the process of finding the analytical solution is tedious. Therefore, a Newton iterative method, which solves the problem numerically, is developed. To demonstrate the results, the Lagrange problem of determining the shape of the strongest column is formulated in the discrete model setting and solved by using the developed method. The design problem of finding the mass distribution in a vibratory system that optimizes its extreme natural frequencies is also given.

Distribution of the lowest eigenvalue for a class of random matrices

1974 ◽  
Vol 15 (6) ◽  
pp. 880-883
Author(s):  
Nazakat Ullah
Keyword(s):  
Random Matrices ◽  
Lowest Eigenvalue

scholarly journals SURVIVAL PROBABILITY (HEAT CONTENT) AND THE LOWEST EIGENVALUE OF DIRICHLET LAPLACIAN

2011 ◽  
Vol 25 (15) ◽  
pp. 1993-2007
Author(s):  
PAVOL KALINAY ◽  
LADISLAV ŠAMAJ ◽  
IGOR TRAVĚNEC
Keyword(s):  
Survival Probability ◽  
Heat Content ◽  
Simple Algorithm ◽  
Time Behavior ◽  
Dirichlet Laplacian ◽  
Geometric Characteristics ◽  
Long Time ◽  
Time Expansion ◽  
Short Time ◽  
Lowest Eigenvalue

We study the survival probability of a particle diffusing in a two-dimensional domain, bounded by a smooth absorbing boundary. The short-time expansion of this quantity depends on the geometric characteristics of the boundary, whilst its long-time asymptotics is governed by the lowest eigenvalue of the Dirichlet Laplacian defined on the domain. We present a simple algorithm for calculation of the short-time expansion for an arbitrary "star-shaped" domain. The coefficients are expressed in terms of powers of boundary curvature, integrated around the circumference of the domain. Based on this expansion, we look for a Padé interpolation between the short-time and the long-time behavior of the survival probability, i.e., between geometric characteristics of the boundary and the lowest eigenvalue of the Dirichlet Laplacian.

A Hybrid Inverse Mode Problem for Fixed-Fixed Mass-Spring Models

1996 ◽  
Vol 118 (4) ◽  
pp. 641-648 ◽  
Author(s):  
Izuru Takewaki ◽  
Tsuneyoshi Nakamura ◽  
Yasumasa Arita
Keyword(s):  
Inverse Problem ◽  
Closed Form ◽  
Sufficient Conditions ◽  
Eigenvalue Analysis ◽  
Spring Model ◽  
Mass Spring Model ◽  
Left And Right ◽  
The Masses ◽  
Mass Spring ◽  
Lowest Eigenvalue

A hybrid inverse mode problem is formulated for a fixed-fixed mass-spring model. A problem of eigenvalue analysis and its inverse problem are combined in this hybrid inverse mode formulation. It is shown if all the masses and the mid-span stiffnesses of the model are prescribed, then the stiffnesses of the left and right spans (side-spans) can be found for a specified lowest eigenvalue and a specified set of lowest-mode drifts in the side-spans. Sufficient conditions are introduced and proved for a specified eigenvalue and a specified set of drifts in the side-spans to provide positive stiffnesses of the side-spans and to be those in the lowest eigenvibration. A set of solution stiffnesses in the side-spans is derived uniquely in closed form.

Sign in / Sign up

Export Citation Format

Share Document