• Honam Mathematical Journal
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• v.45 no.3
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• pp.542-554
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• 2023

Given a real number α, the Lagrange number of α is the supremum of all real numbers L > 0 for which the inequality |α - p/q| < (Lq²)^-1 holds for infinitely many rational numbers p/q. All Lagrange numbers less than 3 can be arranged as a set {l_p/q : p/q ∈ ℚ ∩ [0, 1]} using the Farey index. The present paper considers a function C(α) devised from Sturmian words. We demonstrate that the function C(α) contains all information on Lagrange numbers less than 3. More precisely, we prove that for any real number α ∈ (0, 1], the value C(α) - C(0) is equal to the sum of all numbers 3 - l_p/q where the Farey index p/q is less than α.

• Structural Engineering and Mechanics
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• v.26 no.6
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• pp.673-685
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• 2007

Reinforced concrete beams can be strengthened in a structural retrofit process by attaching steel plates to their sides by bolting. Whilst bolting produces a confident degree of shear connection under conditions of either static or seismic overload, the plates are susceptible to local buckling. The aim of this paper is to investigate the local buckling of unilaterally-restrained plates with point supports in a generic fashion, but with particular emphasis on the provision of the restraints by bolts, and on the geometric configuration of these bolts on the buckling loads. A numerical procedure, which is based on the Rayleigh-Ritz method in conjunction with the technique of Lagrange multipliers, is developed to study the unilateral local buckling of rectangular plates bolted to the concrete with various arrangements of the pattern of bolting. A sufficient number of separable polynomials are used to define the flexural buckling displacements, while the restraint condition is modelled as a tensionless foundation using a penalty function approach to this form of mathematical contact problem. The additional constraint provided by the bolts is also modelled using Lagrange multipliers, providing an efficacious method of numerical analysis. Local buckling coefficients are determined for a range of bolting configurations, and these are compared with those developed elsewhere with simplifying assumptions. The interaction of the actions in bolted plates during buckling is also considered.

• 제어로봇시스템학회:학술대회논문집
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• 2003.10a
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• pp.2246-2251
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• 2003

This paper proposes shape design of frame structures for vibration suppression and weight reduction. The $H_{\infty}$ norm of the transfer function from disturbance sources to the output points where vibration should be suppressed, is adopted as the performance index to represent the magnitude of vibration transfer. The design parameters are the node positions of the frame structure, on which constraints are imposed so that the structure achieves given tasks. For computation of Pareto optimal solutions to the two-objective design problem, a number of linear combinations of the $H_{\infty}$ norm and the total weight of the structure are considered and minimized. For minimization of the scalared objective function, a Lagrange function is defined by the objective function and the imposed constraints on the design parameters. The solution for which the Lagrange function satisfies the Karush-Kuhn-Tucker condition, is searched by the sequential quadratic programming (SQP) method. Numerical examples are presented to demonstrate the effectiveness of the proposed design method.

• Structural Engineering and Mechanics
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• v.23 no.3
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• pp.263-278
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• 2006

This work presents a time-truncation scheme, based on the Lagrange interpolation polynomial, for the solution of the two-dimensional scalar wave problem by the time-domain boundary element method. The aim is to reduce the number of stored matrices, due to the convolution integral performed from the initial time to the current time, and to keep a compromise between computational economy and efficiency and the numerical accuracy. In order to verify the accuracy of the proposed formulation, three examples are presented and discussed at the end of the article.

• 한국연소학회:학술대회논문집
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• 1999.10a
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• pp.71-79
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• 1999

Numerical analysis on the characteristics of nitrogen oxides (NOx) formation in turbulent nonpremixed hydrogen-air flames was carried out. Lagrange IEM model and Assumed PDF model were applied to consider turbulence-chemistry interaction known to affect the production of NOx. Partial equilibrium assumption was used to predict nonequilibrium effect to which one-half power dependence between EINOx normalized by flame residence time and global strain rate is attributed. As a result. such one-half power dependence could be reproduced only by reaction model including $HO_{2}$and $H_{2}O_{2}$, which means its dependence on Damkohler number; nonequilibrium effect. This dependence was shown better in the region of higher global strain. Besides, the improvement of turbulence model is required to predict mean flow properties quantitatively in the radial direction.

• Steel and Composite Structures
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• v.17 no.6
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• pp.867-890
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• 2014

This paper deals with the geometric nonlinear bending analysis of laminated composite stiffened plates subjected to uniform transverse loading. The eight-noded degenerated shell element and three-noded degenerated curved beam element with five degrees of freedom per node are adopted in the present analysis to model the plate and stiffeners respectively. The Green-Lagrange strain displacement relationship is adopted and the total Lagrangian approach is taken in the formulation. The convergence study of the present formulation is carried out first and the results are compared with the results published in the literature. The stiffener element is reformulated taking the torsional rigidity in an efficient manner. The effects of lamination angle, depth of stiffener and number of layers, on the bending response of the composite stiffened plates are considered and the results are discussed.

• The Mathematical Education
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• v.15 no.1
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• pp.18-21
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• 1976

Lagrange proved that any positive integer is the sum of at most four squares. We consider a elliptic function f$_{\alpha}$(v│$\tau$) of periods 1. $\tau$ derived from $\theta$-functions. From the important number-theoretical interpretation (equation omitted) we obtain $A_4$(n) the number of representations entations of n as a sum of 4-squares.m of 4-squares.

• Journal of the military operations research society of Korea
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• v.16 no.2
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• pp.83-95
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• 1990

This paper addresses the problem of determining the optimal number of spares for a system consisting of multi-item parts. In commercial sector, the cost minimization is mainly considered as an objective functions in most inventory models. However, in the military inventory systems, it is more stressed on maximizing the system availability than minimizing the system cost because the field commander always wants the system to be in perfect working condition to prepare against an emergence case. In this point of view, this paper develops an inventory model which decides the optimal number of spares by minimizing units short and simultaneously achieving a certain level of system availability. Solution algorithms are derived using the generalized Lagrange multiplier approach and marginal analysis approach. Sample data and output results are provided and sensitivity analysis is performed as the level of system availability changes in order to decide the optimal number of spares and availability in terms of economic sense.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.13 no.4
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• pp.315-321
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• 2009

In this paper, a level set based energy functional is proposed, the minimization of which results in simultaneous reference background image modeling and foreground segmentation. Due to the mutual constraint of the two processes, a good estimate of the background can be obtained with a small number of frames, and due to the use of the level set, an Euler-Lagrange equation that directly solves the problem can be derived.

• Journal of the Korean Mathematical Society
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• v.33 no.3
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• pp.651-655
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• 1996

Lagrange's famous Four Square Theorem [L] says that every positive integer can be represented by the sum of four squares. This marvelous theorem was generalized by Mordell [M1] and Ko [K1] as follows : every positive definite integral quadratic form of two, three, four, and five variables is represented by the sum of five, six, seven, and eight squares, respectively. And they tried to extend this to positive definite integral quadratic forms of six or more variables.