THE PRIME GEODESIC THEOREM AND QUANTUM MECHANICS ON FINITE VOLUME GRAPHS: A REVIEW

2001 ◽  
Vol 13 (12) ◽  
pp. 1459-1503 ◽  
Author(s):  
NORMAN E. HURT
Keyword(s):  
Finite Volume ◽  
Riemannian Manifolds ◽  
Riemann Surfaces ◽  
Three Dimensional ◽  
Hyperbolic Spaces ◽  
Quantum Graphs ◽  
Selberg Zeta Function ◽  
Ramanujan Graphs ◽  
Prime Geodesic Theorem ◽  
Ramanujan Conjecture

The prime geodesic theorem is reviewed for compact and finite volume Riemann surfaces and for finite and finite volume graphs. The methodology of how these results follow from the theory of the Selberg zeta function and the Selberg trace formula is outlined. Relationships to work on quantum graphs are surveyed. Extensions to compact Riemannian manifolds, in particular to three-dimensional hyperbolic spaces, are noted. Interconnections to the Selberg eigenvalue conjecture, the Ramanujan conjecture and Ramanujan graphs are developed.

scholarly journals ON THE DISTRIBUTION OF ZEROS OF THE DERIVATIVE OF SELBERG’S ZETA FUNCTION ASSOCIATED TO FINITE VOLUME RIEMANN SURFACES

2016 ◽  
Vol 228 ◽  
pp. 21-71 ◽  
Author(s):  
JAY JORGENSON ◽  
LEJLA SMAJLOVIĆ
Keyword(s):  
Riemann Surface ◽  
Vertical Distribution ◽  
Zeta Function ◽  
Finite Volume ◽  
Riemann Surfaces ◽  
Fuchsian Groups ◽  
Horizontal Distance ◽  
Distribution Of Zeros ◽  
Selberg Zeta Function ◽  
Hyperbolic Riemann Surface

We study the distribution of zeros of the derivative of the Selberg zeta function associated to a noncompact, finite volume hyperbolic Riemann surface $M$. Actually, we study the zeros of $(Z_{M}H_{M})^{\prime }$, where $Z_{M}$ is the Selberg zeta function and $H_{M}$ is the Dirichlet series component of the scattering matrix, both associated to an arbitrary finite volume hyperbolic Riemann surface $M$. Our main results address finiteness of number of zeros of $(Z_{M}H_{M})^{\prime }$ in the half-plane $\operatorname{Re}(s)<1/2$, an asymptotic count for the vertical distribution of zeros, and an asymptotic count for the horizontal distance of zeros. One realization of the spectral analysis of the Laplacian is the location of the zeros of $Z_{M}$, or, equivalently, the zeros of $Z_{M}H_{M}$. Our analysis yields an invariant $A_{M}$ which appears in the vertical and weighted vertical distribution of zeros of $(Z_{M}H_{M})^{\prime }$, and we show that $A_{M}$ has different values for surfaces associated to two topologically equivalent yet different arithmetically defined Fuchsian groups. We view this aspect of our main theorem as indicating the existence of further spectral phenomena which provides an additional refinement within the set of arithmetically defined Fuchsian groups.

Combined Microstructure and Heat Transfer Modeling of Carbon Nanotube Thermal Interface Materials1

2016 ◽  
Vol 138 (4) ◽  
Author(s):  
Sridhar Sadasivam ◽  
Stephen L. Hodson ◽  
Matthew R. Maschmann ◽  
Timothy S. Fisher
Keyword(s):  
Heat Transfer ◽  
Thermal Resistance ◽  
Finite Volume ◽  
Pressure Dependence ◽  
Matrix Material ◽  
Three Dimensional ◽  
Coarse Grain ◽  
Total Thermal Resistance ◽  
Thermal Interface ◽  
Cnt Arrays

A microstructure-sensitive thermomechanical simulation framework is developed to predict the mechanical and heat transfer properties of vertically aligned CNT (VACNT) arrays used as thermal interface materials (TIMs). The model addresses the gap between atomistic thermal transport simulations of individual CNTs (carbon nanotubes) and experimental measurements of thermal resistance of CNT arrays at mesoscopic length scales. Energy minimization is performed using a bead–spring coarse-grain model to obtain the microstructure of the CNT array as a function of the applied load. The microstructures obtained from the coarse-grain simulations are used as inputs to a finite volume solver that solves one-dimensional and three-dimensional Fourier heat conduction in the CNTs and filler matrix, respectively. Predictions from the finite volume solver are fitted to experimental data on the total thermal resistance of CNT arrays to obtain an individual CNT thermal conductivity of 12 W m−1 K−1 and CNT–substrate contact conductance of 7 × 107 W m−2 K−1. The results also indicate that the thermal resistance of the CNT array shows a weak dependence on the CNT–CNT contact resistance. Embedding the CNT array in wax is found to reduce the total thermal resistance of the array by almost 50%, and the pressure dependence of thermal resistance nearly vanishes when a matrix material is introduced. Detailed microstructural information such as the topology of CNT–substrate contacts and the pressure dependence of CNT–opposing substrate contact area are also reported.

Three-Dimensional Parametric Finite-Volume Homogenization of Periodic Materials with Multi-Scale Structural Applications

2018 ◽  
Vol 10 (04) ◽  
pp. 1850045 ◽  
Author(s):  
Qiang Chen ◽  
Guannan Wang ◽  
Xuefeng Chen
Keyword(s):  
Finite Volume ◽  
Three Dimensional ◽  
Functionally Graded ◽  
Attractive Alternative ◽  
Matrix Assembly ◽  
Functionally Graded Composite ◽  
Multi Scale ◽  
Hexahedral Elements ◽  
Stiffness Matrices ◽  
Structural Applications

In order to satisfy the increasing computational demands of micromechanics, the Finite-Volume Direct Averaging Micromechanics (FVDAM) theory is developed in three-dimensional (3D) domain to simulate the multiphase heterogeneous materials whose microstructures are distributed periodically in the space. Parametric mapping, which endorses arbitrarily shaped and oriented hexahedral elements in the microstructure discretization, is employed in the unit cell solution. Unlike the finite-element (FE) technique, the expressions for local stiffness matrices are derived explicitly, enabling efficient global stiffness matrix assembly using an easily implementable algorithm. To demonstrate the accuracy and efficiency of the proposed theory, the homogenized moduli and localized stress distributions produced by the FE analyses are given for comparisons, where excellent agreement is always obtained for the 3D microstructures with different geometrical and material properties. Finally, a multi-scale stress analysis of functionally graded composite cylinders is conducted. This extension further increases the FVDAM’s range of applicability and opens new opportunities for pursuing other areas, providing an attractive alternative to the FE-based approaches that may be compared.

A Quasi-three-dimensional Finite-volume Shallow Water Model for Green Water on Deck

2013 ◽  
Vol 57 (03) ◽  
pp. 125-140
Author(s):  
Daniel A. Liut ◽  
Kenneth M. Weems ◽  
Tin-Guen Yen
Keyword(s):  
Shallow Water ◽  
Finite Volume ◽  
Time Domain ◽  
Degrees Of Freedom ◽  
Initial Conditions ◽  
Three Dimensional ◽  
Green Water ◽  
Water Model ◽  
Ship Motions ◽  
Shallow Water Model

A quasi-three-dimensional hydrodynamic model is presented to simulate shallow water phenomena. The method is based on a finite-volume approach designed to solve shallow water equations in the time domain. The nonlinearities of the governing equations are considered. The methodology can be used to compute green water effects on a variety of platforms with six-degrees-of-freedom motions. Different boundary and initial conditions can be applied for multiple types of moving platforms, like a ship's deck, tanks, etc. Comparisons with experimental data are discussed. The shallow water model has been integrated with the Large Amplitude Motions Program to compute the effects of green water flow over decks within a time-domain simulation of ship motions in waves. Results associated to this implementation are presented.

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