• Journal of applied mathematics & informatics
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• v.25 no.1_2
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• pp.329-337
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• 2007

In the paper, we obtain the existence of positive solutions and establish a corresponding iterative scheme for BVPs $$\{^{\;(\phi_p(u'))'\;+\;q(t)f(t,u)=0,\;0\;<\;t\;<\;1,}_{\;u(0)\;-\;B(u'({\eta}))\;=\;0,\;u'(1)\;=\;0}$$ and $$\{^{\;(\phi_p(u'))'\;+\;q(t)f(t,u)=0,\;0\;<\;t\;<\;1,}_{\;u'(0)\;=\;0,\;u(1)+B(u'(\eta))\;=\;0.}$$. The main tool is the monotone iterative technique. Here, the coefficient q(t) may be singular at t = 0, 1.

• Journal of the Korean Mathematical Society
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• v.46 no.3
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• pp.493-511
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• 2009

Let $q\;{\geq}\;3$ be an odd integer and a be an integer coprime to q. Denote by N(a, q) the number of pairs of integers b, c with $bc\;{\equiv}\;a$ (mod q), $1\;{\leq}\;b$, $c\;{\leq}\;{\frac{q-1}{2}}$ and with b, c having different parity. The main purpose of this paper is to study the sum ${\sum}^{'q}_{a=1}\;\(N(a,\;q)\;-\;\frac{{\phi}(q)}{8}\)^2$ and obtain a sharp asymptotic formula.

• Journal of applied mathematics & informatics
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• v.18 no.1_2
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• pp.25-43
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• 2005

Let F be a finite field of prime power order q(odd) and the multiplicative order of q modulo $2^{n}\;(n>1)\;be\; {\phi}(2^{n})/2$. If n > 3, then q is odd number(prime or prime power) of the form $8m{\pm}3$. If q = 8m - 3, then the ring $R_{2^n} = F[x]/ < x^{2^n}-1 >$ has 2n primitive idempotents. The explicit expressions for these primitive idempotents are obtained and the minimal QR cyclic codes of length $2^{n}$ generated by these idempotents are completely described. If q = 8m + 3 then the expressions for the 2n - 1 primitive idempotents of $R_{2^n}$ are obtained. The generating polynomials and the upper bounds of the minimum distance of minimal QR cyclic codes generated by these 2n-1 idempotents are also obtained. The case n = 2,3 is dealt separately.

• Journal of the Korean Institute of Electrical and Electronic Material Engineers
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• v.20 no.9
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• pp.780-786
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• 2007

This paper describes the influence on partial discharge characteristics of defects at the model XLPE/EPDM interfaces of power cable joints. The defects of void and copper which could inadvertently be present at the joint interface. We measured ${\Phi}-n,\;{\Phi}-q$ patterns by a computer-aided partial discharge measuring system. Several parameters i, e, maximum discharge $q_{max}$ [pC], average discharge q [pC/cycle], and average angle of discharge ${\Theta}g$ [deg] were found to depend upon the defect type varying applied voltage. As the result of time evaluation, partial discharges are small different at copper defects, but is decreasing obviously about 20 minutes at void defect. It considered that difference of magnitude of total positive discharge of Q+ [pC] and total negative discharge of Q- [pC] is SPMD(swarming pulsive micro discharges).

• Transactions of the Korean Society of Mechanical Engineers B
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• v.26 no.11
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• pp.1531-1541
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• 2002

The NOx emission characteristics of C$H_4$-Air coflow jet flames were numerically and experimentally investigated. NOx was measured using a chemiluminescent detection and calculated by the parabolic -type equation solver with a detatiled NOx chemistry. The fuel flow rate( $Q_{F}$), the diameter of mixture nozzle and the equivalence rate(Ф) were varied to discuss the EINOx of each flames at the various combustion conditions. The NOx emission index(EINOx) was introduced to quantify the NOx emission from the parametrically varied flames. The results show that Prompt EINOx increases on a logarithmic profile with increasing ${\Phi}$ and keeps nearly constant for the variation of $Q_{F}$. Thermal EINOx reaches the maximum value at around ${\Phi}$ =1.5 and then slowly decrease for ${\Phi}$ >1.5. In addition, Thermal EINOx increases with increasing $Q_{F}$, but nearly indifferent to the variation of the mixture nozzle diameter. Total EINOx also shows a peak at around ${\Phi}$ =1.5, followed by a relatively sharp decrease for 1.5< ${\Phi}$ <2.5 and increase slowly for 2.5 < ${\Phi}$ < $\infty$ The present Total EINOx trend is well explained by a combination of above Thermal and Prompt EINOx trend with the variation of ${\Phi}$ n of ${\Phi}$.

• Kyungpook Mathematical Journal
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• v.55 no.4
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• pp.983-996
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• 2015

In the paper A new parameter for Ramanujan's theta-functions and explicit values, Arab J. Math. Sc., 18 (2012), 105-119, Saikia studied the parameter $A_{k,n}$ involving Ramanujan's theta-functions ${\phi}(q)$ and ${\psi}(q)$ for any positive real numbers k and n and applied it to find explicit values of ${\psi}(q)$. As more application to the parameter $A_{k,n}$, in this paper we prove a new general theorem for explicit evaluation of Ramanujan-$G{\ddot{o}}llnitz$-Gordon continued fraction K(q) in terms of the parameter $A_{k,n}$ and give examples. We also find some new explicit values of the parameter $A_{k,n}$ and offer reciprocity theorems for the continued fraction K(q).

• Journal of the Korean Mathematical Society
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• v.59 no.5
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• pp.843-867
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• 2022

Maynard proved that there exists an effectively computable constant q1 such that if q ≥ q1, then $\frac{{\log}\;q}{\sqrt{q}{\phi}(q)}Li(x){\ll}{\pi}(x;\;q,\;m)<\frac{2}{{\phi}(q)}Li(x)$ for x ≥ q8. In this paper, we will show the following. Let 𝛿1 and 𝛿2 be positive constants with 0 < 𝛿1, 𝛿2 < 1 and 𝛿1 + 𝛿2 > 1. Assume that L ≠ ℚ is a number field. Then there exist effectively computable constants c0 and d1 such that for dL ≥ d1 and x ≥ exp (326n𝛿1L(log dL)1+𝛿2), we have $$\|{\pi}_C(x)-\frac{{\mid}C{\mid}}{{\mid}G{\mid}}Li(x)\|\;{\leq}\;\(1-c_0\frac{1og\;d_L}{d^{7.072}_L}\)\;\frac{{\mid}C{\mid}}{{\mid}G{\mid}}Li(x)$$.

• Bulletin of the Korean Mathematical Society
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• v.52 no.1
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• pp.35-43
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• 2015

Let q > 1 be an odd integer and c be a fixed integer with (c, q) = 1. For each integer a with $1{\leq}a{\leq}q-1$, it is clear that the exists one and only one b with $0{\leq}b{\leq}q-1$ such that $ab{\equiv}c$ (mod q). Let N(c, q) denote the number of all solutions of the congruence equation $ab{\equiv}c$ (mod q) for $1{\leq}a$, $b{\leq}q-1$ in which a and $\bar{b}$ are of opposite parity, where $\bar{b}$ is defined by the congruence equation $b\bar{b}{\equiv}1$ (modq). The main purpose of this paper is using the mean value theorem of Dirichlet L-functions to study the mean value properties of a summation involving $(N(c,q)-\frac{1}{2}{\phi}(q))$ and Kloosterman sums, and give a sharper asymptotic formula for it.

• Proceedings of the KIEE Conference
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• 2000.07c
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• pp.1930-1932
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• 2000

PD in defects of solid insulation system is very harmful since it leads to deteriorate insulation system by. the discharge electrons and ions bombarding the insulation surface and the action of chemical products that are formed by discharges. PD is used to detect and recognize defects and degradation of insulation system However, there are still marked difficulties to recognize defects by PD methode. In this paper, we investigated properties of PD of defects in solid insulation by using statical methods and classified PD patterns of surface discharge, electrical tree and void discharge with source of discharge, we used specific distributions such as $H_n(q)$, $H_{an}(\phi)$, $H_n(\phi)$, $H_a(\phi)$ to recognize defects of solid insulation system.

• Proceedings of the KIEE Conference
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• 1998.07d
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• pp.1478-1480
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• 1998

Inception and propagation of electrical tree and properties of partial discharge(PD) pulses accompanying with tree in low density polyethylene were discussed. We observed the characteristics of process of electrical tree by using optical microscope and investigated the statistical characteristics of the PD pulses by analyzing PD quantities and distribution patterns. The PD pulses were analyzed by q-n, $\phi$-n and $\phi$-q distribution. The statistical operators used were skewness(s), kurtosis(k) and average phase angle. The skewness and average discharge phase angle of PD pulses increased as the Propagation of tree. The kurtosis was about 1.8 at the Inception of tree, but It increased as the propagation of tree.