• 대한수학회지
• /
• 제47권4호
• /
• pp.861-877
• /
• 2010

The works of Erd$\ddot{o}$s et al. about expansions of 1 with respect to a non-integer base q, referred to as q-expansions, are investigated to determine how far they continue to hold when the number 1 is replaced by a positive number x. It is found that most results about q-expansions for real numbers greater than or equal to 1 are in somewhat opposite direction to those for real numbers less than or equal to 1. The situation when a real number has a unique q-expansion, and when it has exactly two q-expansions are studied. The smallest base number q yielding a unique q-expansion is determined and a particular sequence is shown, in certain sense, to be the smallest sequence whose corresponding base number q yields exactly two q-expansions.

• 제어로봇시스템학회:학술대회논문집
• /
• 제어로봇시스템학회 2003년도 ICCAS
• /
• pp.473-478
• /
• 2003

To detect motions of bodies, we have discussed them with two viewpoints; one is a detection algorithm, and another is the hardware implementation. The former is to find small terms expansions for sine/cosine functions. We researched Maclaurin and optimum expansions, and moreover to reduce hardware amounts, revised the expansions. The expansions don't include divide calculations, and the error is within 0.01%. As for the former problem, there is another approach also; that is the cordic method. The method is based on the rotation of a vector on the complex plain. It is simple iterations and don't require large logic. We examined the precision and convergence of the method on C-simulations, and implemented on HDL. The later problem is to make FPGA within small gates. We considered approaches to eliminate a divider and to reduce the bit number of arithmetic. We researched Newton-Raphson's method to get reciprocal numbers. The higher-order expression shows rapid convergence and doesn't be affected by the initial guess. It is an excellent algorithm. Using them, we wish to design a detector, and are developing it on a FPGA.

• 대한수학회논문집
• /
• 제29권2호
• /
• pp.269-283
• /
• 2014

The aim of the paper is to present several new relationships involving the hyperharmonic function introduced by Mez$\ddot{o}$ in (I. Mez$\ddot{o}$, Analytic extension of hyperharmonic numbers, Online J. Anal. Comb. 4, 2009) which is an analytic extension of the hyperharmonic numbers. These relations are obtained by using some fractional calculus theorems as Leibniz rules and Taylor like series expansions.

• 대한수학회보
• /
• 제43권2호
• /
• pp.353-375
• /
• 2006

In this paper we establish a counting method for the Green classes of the Birget-rhodes expansion of finite groups. As an application of the results, we derive explicit enumeration formulas for the Green classes for finite groups of order pq and a finite cyclic group of order $p^m$, where p and q are arbitrary given distinct prime numbers.

• 대한수학회지
• /
• 제55권5호
• /
• pp.1179-1192
• /
• 2018

Several addition formulas for a general class of q-Appell sequences are proved. The q-addition formulas, which are derived, involved not only the generalized q-Bernoulli, the generalized q-Euler and the generalized q-Genocchi polynomials, but also the q-Stirling numbers of the second kind and several general families of hypergeometric polynomials. Some q-umbral calculus generalizations of the addition formulas are also investigated.

• 호남수학학술지
• /
• 제30권4호
• /
• pp.733-741
• /
• 2008

Let T be Gauss transformation on the unit interval defined by T (x) = ${\frac{1}{x}}$ where {x} is the fractional part of x. Gauss transformation is closely related to the continued fraction expansions of real numbers. We show that almost every x is mod M normal number of Gauss transformation with respect to intervals whose endpoints are rational or quadratic irrational. Its connection to Central Limit Theorem is also shown.

• 호남수학학술지
• /
• 제43권2호
• /
• pp.259-267
• /
• 2021

Given 𝛽 > 1, we consider real numbers whose 𝛽-expansions are Sturmian words. When the slope of Sturmian words varies, their behaviors have been well studied from analytical point of view. The regularity enables us to find the Fourier series expansion, while the singularity at rational slopes yields a new kind of trigonometric series representing 𝜋.

• Korean Journal of Mathematics
• /
• 제32권2호
• /
• pp.285-295
• /
• 2024

In this work, we consider the function $${\Psi}(z)=\frac{z}{\ln(1+z)}=1+\sum\limits_{n=1}^{\infty}\,G_nz^n$$ whose coefficients G_n are the Gregory coefficients related to Stirling numbers of the first kind and introduce a new subclass ${\mathcal{G}}^{{\lambda},{\mu}}_{\Sigma}(\Psi)$ of analytic bi-univalent functions subordinate to the function Ψ. For functions belong to this class, we investigate the estimates for the general Taylor-Maclaurin coefficients by using the Faber polynomial expansions. In certain cases, our estimates improve some of those existing coefficient bounds.

• Ocean Systems Engineering
• /
• 제6권4호
• /
• pp.363-375
• /
• 2016

Large numbers of submarine pipelines are laid as the world now is attaching great importance to offshore oil exploitation. Free spanning of submarine pipelines may be caused by seabed unevenness, change of topology, artificial supports, etc. By combining Iwan's wake oscillator model with the differential equation which describes the vibration behavior of free-span submarine pipelines, the pipe-fluid coupling equation is developed and solved in order to study the effect of both internal and external fluid on the vibration behavior of free-span submarine pipelines. Through generalized integral transform technique (GITT), the governing equation describing the transverse displacement is transformed into a system of second-order ordinary differential equations (ODEs) in temporal variable, eliminating the spatial variable. The MATHEMATICA built-in function NDSolve is then used to numerically solve the transformed ODE system. The good convergence of the eigenfunction expansions proved that this method is applicable for predicting the dynamic response of free-span pipelines subjected to both internal flow and external current.

• Journal of Genetic Medicine
• /
• 제10권1호
• /
• pp.33-37
• /
• 2013

Purpose: Myotonic dystrophy 1 (DM1, OMIM 160900) is an autosomal-dominant muscular disorder caused by an expansion of CTG repeats in the 3' UTR of the DMPK gene. Variable expansions of CTG repeats preclude the accurate determination of repeat size. We tried to show the clinical and analytical validity of the application of Southern blotting after long-range PCR was demonstrated in Korean DM1 patients. Materials and Methods: The Southern blotting of long-range PCR was applied to 1,231 cases with clinical suspicion of DM1, between 2000 and 2011. PCR was performed using genomic DNA with forward 5'-CAGTTCACAACCGCTCCGAGC-3' and reverse 5'-CGTGGAGGATGGAACACGGAC-3' primers. Subsequently, the PCR fragments were subjected to gel electrophoresis, capillary transfer to a nylon membrane, hybridization with a labeled (CAG)10 probe. The correlation between clinical manifestations and the CTG repeat expansions were analyzed. Results: Among a total of 1,231 tested cases, 642 individuals were diagnosed with DM1 and the range of the detected expansion was 50 to 2,500 repeats; fourteen cases with mild DM1 ($75{\pm}14$ repeats), 602 cases with classical DM1 ($314{\pm}143$ repeats), and 26 cases with congenital DM1 ($1,219{\pm}402$ repeats). The positive and negative predictive values were 100%. The age at test requested and the CTG repeat numbers were inversely correlated (R=-0.444, P<0.01). Conclusion: This study indicates that Southern blotting after long-range PCR is a reliable diagnostic method DM1.