On lower bounds of various dominating codes for locating vertices in cubic graphs

Self-identifying codes, self-locating dominating codes and solid-locating dominating codes are three subsets of vertices of a graph G to locate vertices. The optimal size of them is denoted by γSID (G),γSLD (G) and γDLD (G). In the master thesis, we mainly discuss their lower bound problem in families of graphs. In the first section, we briefly describe the background of the study and some related questions. In the second, third and fourth section, we show some basic definitions, concepts and examples related to self-identifying codes (SID), self-locating dominating codes (SLD) and solid-locating dominating codes (DLD) in rook’s graphs. In the fifth section, we first introduce some known results of lower bounds of open-locating dominating codes in cubic graphs and then in the sixth section we present some new results about the lower bounds of self-identifying codes, self-locating dominating codes and solid-locating dominating codes in cubic graphs

On Maslov's Quantization Condition for Mechanics in a Magnetic Field

This paper considers Maslov's quantization condition for the dynamical system in a magnetic field on the basis of the theory of Fourier integral operators. As a result, it is clarified that a quasi-classical eigenvalue (energy level) according to the quantization rule provides an approximation of order h^2 to the true eigenvalue of the Schrodinger operator

A Note on the Deformations of Hamiltonian Systems on Nilmanifolds

This note deals with the isospectral deformations of metrics on nilmanifolds from the view point of Hamiltonian dynamical systems. It is shown in some examples that the associated Hamiltonian system (the system of geodesic flow) is left invariant under such deformations without a nowhere dense subset of the phase space

Numbers associated to Symmetric Differential Operators and the Bernoulli Numbers

By considering a certain symmetric differential operator we introduce a sequence of numbers {Ck}∞k=0, and clarify their properties, which are similar to those of the Bernoulli numbers. It is shown that the generating function of {Ck} is the hyperbolic tangent function, and some (maybe known) properties of the Bernoulli numbers are derived through those of Ck

A note on a heat invariant and the Ricci flow on surfaces

In this short note，we consider the monotonicity of the heat invariant a2 (g) for a Riemannian metric g under the normalized Ricci flow on a closed surface. We show that a2(g(t)) is decreasing under the normalized Ricci flow g(t) in the space of metrics of non-positive curvature

A Note on Symmetric Differential Operators and Binomial Coefficients

In this note we derive some identities concerning the binomial coefficients by considering a certain n-th order symmetric differential operator on Rm associated to the function p(x，ξ)(x ∈ Rm) which is a homogeneous polynomial in ξ

A Geometrical Formulation for Classical Mechanics in Gauge Fields

We give a geometrical formulation for the classical mechanics in a non-abelian gauge field on a Riemannian manifold. The formulation is based on the reduction procedure associated to the non-abelian symmetry in the principal bundle which describes the gauge field. In the formulation we present explicitly the equation of the motion (called Wong's equation) of a charged particle by using a local coordinate system

On the Classical and the Quantum Mechanics in a Magnetic Field

This paper studies the relationship between the classical mechanics in a magnetic field and its quantized system. The phase space for classical motion in a magnetic field is derived through the Marsden-Weinstein reduction starting with a principal U(1)-bundle. The quantized system associated to the classical system is defined as the Bochner-Laplacian on a line bundle associated to the principal bundle. In this context we obtain a generalization of Helton's theorem which gives a characterization of the periodicity of the classical trajectories by the spectrum of the associated quantum Hamiltonian (the Bochner-Laplacian). 1991 Mathematics Subject Classification. Primary 58G25: Secondary 58F0