• Bulletin of the Korean Chemical Society
• /
• v.35 no.3
• /
• pp.805-810
• /
• 2014

The shape invariant potentials are proved to be exactly solvable, i.e. the wave functions and energies of a particle moving under the influence of the shape invariant potentials can be algebraically determined without any approximations. It is well known that the SWKB quantization is exact for all shape invariant potentials though the SWKB quantization itself is approximate. This mystery has not been mathematically resolved yet and may not be solved in a concrete fashion even in the future. Therefore, in the present work, to understand (not prove) the mystery an attempt of deriving exactly solvable potentials directly from the SWKB quantization has been made. And it turns out that all the derived potentials are shape invariant. It implicitly explains why the SWKB quantization is exact for all known shape invariant potentials. Though any new potential has not been found in this study, this brute-force derivation of potentials helps one understand the characteristics of shape invariant potentials.

• Journal of the Korean Mathematical Society
• /
• v.54 no.2
• /
• pp.647-662
• /
• 2017

We analyze properties of solvable and nilpotent algebras with bracket. The class of solvability and nilpotency of the tensor square of an algebra with bracket is obtained. Homological characterizations of nilpotent algebras with bracket are presented.

• Kyungpook Mathematical Journal
• /
• v.47 no.2
• /
• pp.203-219
• /
• 2007

We estimate the stable rank and connected stable rank of group $C^*$-algebra of certain disconnected solvable Lie groups such as semi-direct products of connected solvable Lie groups by the integers.

• 제어로봇시스템학회:학술대회논문집
• /
• 1992.10b
• /
• pp.405-412
• /
• 1992

The problem we consider in this paper is more demanding than the problem of input-output linearization with state equivalence recently solved by Cheng, Isidori, Respondek, and Tarn. We request that the MIMO nonlinear system, for which the problem of input-output linearization with state-equivalence is solvable, can be decoupled. In exchange for further requirement like this, our problem produces more usable and informative results than the problem of input-output linearization with state-equivalence. We present the necessary and sufficient conditions for our problem to be solvable. We characterize each of the nonlinear systems satisfying these conditions by a set of parameters which are invariant under the group action of state feedback and transformation. Using this set of parameters, we can determine directly the unique one, among the canonical forms of decouplable and controllable linear systems, to which a nonlinear system can be transformed via appropriate state feedback and transformation. Finally, we present the necessary and sufficient conditions for our problem to be solvable with internal stability, that is, for stable decoupling.

• Bulletin of the Korean Chemical Society
• /
• v.34 no.1
• /
• pp.197-200
• /
• 2013

The bound state wave functions for all the known exactly solvable potentials can be expressed in terms of orthogonal polynomials because the polynomials always satisfy the boundary conditions with a proper weight function. The orthogonality of polynomials is of great importance because the orthogonality characterizes the wave functions and consequently the quantum system. Though the orthogonality of orthogonal polynomials has been known for hundred years, the known orthogonality is found to be inadequate for polynomials appearing in some exactly solvable potentials, for example, Ginocchio potential. For those potentials a more general orthogonality is defined and algebraically derived. It is found that the general orthogonality is valid with a certain constraint and the constraint is very useful in understanding the system.

• Proceedings of the Korean Operations and Management Science Society Conference
• /
• 2007.11a
• /
• pp.309-312
• /
• 2007

We consider a proportionate flow shop problem with controllable processing times. The objective is to minimize the sum of total completion time and total compression cost, in which the cost function of compressing the processing times is non-decreasing concave. We show that the problem can be solved in polynomial time by reducing it to a polynomially solvable 0-1 unconstrained quadratic program.

• Journal of applied mathematics & informatics
• /
• v.4 no.1
• /
• pp.285-297
• /
• 1997

We will study the generalization of theorems on the pe-riodic locally - solvable FC-groups to the theorems on the periodic locally-solvable CC-groups. The main theorem is the Theorem A. For the proof the inverse limit of inverse system and topological ap-proch developed by Dixon is useful.

• Bulletin of the Korean Chemical Society
• /
• v.32 no.12
• /
• pp.4233-4238
• /
• 2011

Based on the exact quantization rule for the nonrelativistic Schrodinger equation, the exact quantization rule for the relativistic one-dimensional Klein-Gordon equation is suggested. Using the new quantization rule, the exact relativistic energies for exactly solvable potentials, e.g. harmonic oscillator, Morse, and Rosen-Morse II type potentials, are obtained. Consequently the new quantization rule is found to be exact for one-dimensional spinless relativistic quantum systems. Though the physical meanings of the new quantization rule have not been fully understood yet, the present formal derivation scheme may shed light on understanding relativistic quantum systems more deeply.

• Communications of the Korean Mathematical Society
• /
• v.15 no.2
• /
• pp.349-355
• /
• 2000

In this paper we generalize the Whitehead theorem which says that a homology equivalence implies a homotopy equivalence for nilpotent spaces. We make some theorems on a homotopy equivalence of non-nilpotent spaces, e.g., the solvable space or space satisfying the condition (T**) or space X with $\pi$1(X) Engel, or locally nilpotent space with some properties. Furthermore we find some conditions that the Wall invariant will be trivial.

• Bulletin of the Korean Mathematical Society
• /
• v.57 no.6
• /
• pp.1475-1479
• /
• 2020

Let G be a finite group and ${\sigma}_1(G)={\frac{1}{{\mid}G{\mid}}}\;{\sum}_{H{\leq}G}\;{\mid}H{\mid}$. Under some restrictions on the number of conjugacy classes of (non-normal) maximal subgroups of G, we prove that if ${\sigma}_1(G)<{\frac{117}{20}}$, then G is solvable. This partially solves an open problem posed in [9].