Geometric Comments on Music

In fact degrees of the middle chromatic scales are related with the logarithm (base 2). With relying on this fact, In this study by using logarithm and string oscillation properties, we show that music notes (A − B − · · ·) are related with the interval (0, a] (a is constant) and then we define a topology on this interval. Finally Lie algebra for the mentioned interval of music will be characterized.

OSCILLATION PROPERTIES FOR NON-CLASSICAL STURM-LIOUVILLE PROBLEMS WITH ADDITIONAL TRANSMISSION CONDITIONS

This work is aimed at studying some comparison and oscillation properties of boundary value problems (BVP’s) of a new type, which differ from classical problems in that they are defined on two disjoint intervals and include additional transfer conditions that describe the interaction between the left and right intervals. This type of problems we call boundary value-transmission problems (BVTP’s). The main difficulty arises when studying the distribution of zeros of eigenfunctions, since it is unclear how to apply the classical methods of Sturm’s theory to problems of this type. We established new criteria for comparison and oscillation properties and new approaches used to obtain these criteria. The obtained results extend and generalizes the Sturm’s classical theorems on comparison and oscillation.

Spectral properties of a beam equation with eigenvalue parameter occurring linearly in the boundary conditions

In this paper, we consider an eigenvalue problem for ordinary differential equations of fourth order with a spectral parameter in the boundary conditions. The location of eigenvalues on real axis, the structure of root subspaces and the oscillation properties of eigenfunctions of this problem are investigated, and asymptotic formulas for the eigenvalues and eigenfunctions are found. Next, by the use of these properties, we establish sufficient conditions for subsystems of root functions of the considered problem to form a basis in the space $L_p,1 < p < \infty$ .

The Properties of Eigenvalues and Eigenfunctions for Nonlocal Sturm–Liouville Problems

The present paper is concerned with the spectral theory of nonlocal Sturm–Liouville eigenvalue problems on a finite interval. The continuity, differentiability and comparison results of eigenvalues with respect to the nonlocal potentials are studied, and the oscillation properties of eigenfunctions are investigated. The comparison result of eigenvalues and the oscillation properties of eigenfunctions indicate that the spectral properties of nonlocal problems are very different from those of classical Sturm–Liouville problems. Some examples are given to explain this essential difference.

On Oscillation Properties of Self-Adjoint Boundary Value Problems of Fourth Order

The connection between the number of internal zeros of nontrivial solutions to fourth-order self-adjoint boundary value problems and the inertia index of these problems is studied. We specify the types of problems for which such a connection can be established. In addition, we specify the types of problems for which a connection between the inertia index and the number of internal zeros of the derivatives of nontrivial solutions can be established. Examples demonstrating the effectiveness of the proposed new approach to an oscillatory problem are considered.