• Kyungpook Mathematical Journal
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• v.46 no.2
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• pp.255-260
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• 2006

Let $m$ and $k$ be any positive integers, let $\mathbb{Z}_k$ the ring of integers of modulo $k$, let $G_m(\mathbb{Z}_k)$ the group of all $m$ by $m$ nonsingular matrices over $\mathbb{Z}_k$ and let ${\phi}_m(k)$ the order of $G_m(\mathbb{Z}_k)$. In this paper, ${\phi}_m(k)$ can be computed by the following investigation: First, for any relatively prime positive integers $s$ and $t$, $G_m(\mathbb{Z}_{st})$ is isomorphic to $G_m(\mathbb{Z}_s){\times}G_m(\mathbb{Z}_t)$. Secondly, for any positive integer $n$ and any prime $p$, ${\phi}_m(p^n)=p^{m^2}{\cdot}{\phi}_m(p^{n-1})=p{^{2m}}^2{\cdot}{\phi}_m(p^{n-2})={\cdots}=p^{{(n-1)m}^2}{\cdot}{\phi}_m(p)$, and so ${\phi}_m(k)={\phi}_m(p_1^n1){\cdot}{\phi}_m(p_2^{n2}){\cdots}{\phi}_m(p_s^{ns})$ for the prime factorization of $k$, $k=p_1^{n1}{\cdot}p_2^{n2}{\cdots}p_s^{ns}$.

• Bulletin of the Korean Mathematical Society
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• v.60 no.6
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• pp.1651-1672
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• 2023

In 2010, Li-Ye [13, Theorem 0.1] proved that P(ζ(z), ζ'(z), . . . , ζ^(m)(z), Γ(z), Γ'(z), Γ"(z)) ≢ 0 in ℂ, where m is a non-negative integer, and P(u₀, u₁, . . . , u_m, v₀, v₁, v₂) is any non-trivial polynomial in its arguments with coefficients in the field ℂ. Later on, Li-Ye [15, Theorem 1] proved that P(z, Γ(z), Γ'(z), . . . , Γ⁽ⁿ⁾(z), ζ(z)) ≢ 0 in z ∈ ℂ for any non-trivial distinguished polynomial P(z, u₀, u₁, . . ., u_n, v) with coefficients in a set L_δ of the zero function and a class of nonzero functions f from ℂ to ℂ ∪ {∞} (cf. [15, Definition 1]). In this paper, we prove that P(z, ζ(z), ζ'(z), . . . , ζ^(m)(z), Γ(z), Γ'(z), . . . , Γ⁽ⁿ⁾(z)) ≢ 0 in z ∈ ℂ, where m and n are two non-negative integers, and P(z, u₀, u₁, . . . , u_m, v₀, v₁, . . . , v_n) is any non-trivial polynomial in the m + n + 2 variables u₀, u₁, . . . , u_m, v₀, v₁, . . . , v_n with coefficients being meromorphic functions of order less than one, and the polynomial P(z, u₀, u₁, . . . , u_m, v₀, v₁, . . . , v_n) is a distinguished polynomial in the n + 1 variables v₀, v₁, . . . , v_n. The question studied in this paper is concerning the conjecture of Markus from [16]. The main results obtained in this paper also extend the corresponding results from Li-Ye [12] and improve the corresponding results from Chen-Wang [5] and Wang-Li-Liu-Li [23], respectively.

• Journal of the Korean Mathematical Society
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• v.38 no.3
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• pp.623-631
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• 2001

Let $\kappa$ be a real abelian field of conductor f and $\kappa$(sub)$\infty$ = ∪(sub)n$\geq$0$\kappa$(sub)n be its Z(sub)p-extension for an odd prime p such that płf$\phi$(f). he aim of this paper is ot compute the cohomology groups of circular units. For m>n$\geq$0, let G(sub)m,n be the Galois group Gal($\kappa$(sub)m/$\kappa$(sub)n) and C(sub)m be the group of circular units of $\kappa$(sub)m. Let l be the number of prime ideals of $\kappa$ above p. Then, for mm>n$\geq$0, we have (1) C(sub)m(sup)G(sub)m,n = C(sub)n, (2) H(sup)i(G(sub)m,n, C(sub)m) = (Z/p(sup)m-n Z)(sup)l-1 if i is even, (3) H(sup)i(G(sub)m,n, C(sub)m) = (Z/P(sup)m-n Z)(sup l) if i is odd (※Equations, See Full-text).

• Bulletin of the Korean Mathematical Society
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• v.58 no.4
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• pp.983-1002
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• 2021

We investigate the non-existence of finite order transcendental entire solutions of Fermat-type differential-difference equations [f(z)f'(z)]ⁿ + P²(z)f^m(z + 𝜂) = Q(z) and [f(z)f'(z)]ⁿ + P(z)[∆_𝜂f(z)]^m = Q(z), where P(z) and Q(z) are non-zero polynomials, m and n are positive integers, and 𝜂 ∈ ℂ \ {0}. In addition, we discuss transcendental entire solutions of finite order of the following Fermat-type differential-difference equation P²(z) [f^(k)(z)]² + [αf(z + 𝜂) - 𝛽f(z)]² = e^r(z), where $P(z){\not\equiv}0$ is a polynomial, r(z) is a non-constant polynomial, α ≠ 0 and 𝛽 are constants, k is a positive integer, and 𝜂 ∈ ℂ \ {0}. Our results generalize some previous results.

• The Pure and Applied Mathematics
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• v.12 no.1
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• pp.57-63
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• 2005

Let ∑(p)(p ∈ N) be the class of functions f(z) = z/sup -p/ + α/sub 1-p/ z/sup 1-p/ + α/sub 2-p/z/sup 2-p/ + ... analytic in 0 < ｜z｜ < 1 and let M(p, λ, μ)(0 < λ≤ 2 and 2λ(λ - 1) ≤ μ ≤ λ²) denote the class of functions f(z) ∈ ∑(p) which satisfy (equation omitted). The object of the present paper is to derive some properties of functions in the class M(p, λ, μ).

• Bulletin of the Korean Mathematical Society
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• v.50 no.3
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• pp.983-991
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• 2013

For each real number $n$ > 6, we prove that there is a sequence $\{pk(n,z)\}^{\infty}_{k=1}$ of fourth degree self-reciprocal polynomials such that the zeros of $p_k(n,z)$ are all simple and real, and every $p_{k+1}(n,z)$ has the largest (in modulus) zero ${\alpha}{\beta}$ where ${\alpha}$ and ${\beta}$ are the first and the second largest (in modulus) zeros of $p_k(n,z)$, respectively. One such sequence is given by $p_k(n,z)$ so that $$p_k(n,z)=z^4-q_{k-1}(n)z^3+(q_k(n)+2)z^2-q_{k-1}(n)z+1$$, where $q_0(n)=1$ and other $q_k(n)^{\prime}s$ are polynomials in n defined by the severely nonlinear recurrence $$4q_{2m-1}(n)=q^2_{2m-2}(n)-(4n+1)\prod_{j=0}^{m-2}\;q^2_{2j}(n),\\4q_{2m}(n)=q^2_{2m-1}(n)-(n-2)(n-6)\prod_{j=0}^{m-2}\;q^2_{2j+1}(n)$$ for $m{\geq}1$, with the usual empty product conventions, i.e., ${\prod}_{j=0}^{-1}\;b_j=1$.

• Communications of the Korean Mathematical Society
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• v.38 no.3
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• pp.755-772
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• 2023

Our aim is to establish certain image formulas of the (p, q)-extended modified Bessel function of the second kind M_ν,p,q(z) by employing the Marichev-Saigo-Maeda fractional calculus (integral and differential) operators including their composition formulas and using certain integral transforms involving (p, q)-extended modified Bessel function of the second kind M_ν,p,q(z). Corresponding assertions for the Saigo's, Riemann-Liouville (R-L) and Erdélyi-Kober (E-K) fractional integral and differential operators are deduced. All the results are represented in terms of the Hadamard product of the (p, q)-extended modified Bessel function of the second kind M_ν,p,q(z) and Fox-Wright function _rΨ_s(z).

• Bulletin of the Korean Mathematical Society
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• v.57 no.4
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• pp.991-1002
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• 2020

We show that when n > m, the following delay differential equation fⁿ(z)f'(z) + p(z)(f(z + c) - f(z))^m = r(z)e^q(z) of rational coefficients p, r doesn't admit any transcendental entire solutions f(z) of finite order. Furthermore, we study the conditions of α₁, α₂ that ensure existence of transcendental meromorphic solutions of the equation fⁿ(z) + f^n-2(z)f'(z) + P_d(z, f) = p₁(z)e^α₁(z) + p₂(z)e^α₂(z). These results have improved some known theorems obtained most recently by other authors.

• Journal of the Chungcheong Mathematical Society
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• v.22 no.1
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• pp.89-99
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• 2009

Let $\mu$ be a finite positive Borel measure on the unit ball $B{\subset}\mathbb{C}^n$ and $\nu$ be the Euclidean volume measure such that ${\nu}(B)=1$. For the unit sphere $S=\{z:{\mid}z{\mid}=1\}$, $\sigma$ is the rotation-invariant measure on S such that ${\sigma}(S)=1$. Let $\mathcal{P}[f]$ be the invariant Poisson integral of f. We will show that there is a constant M > 0 such that $\int_B{\mid}{\mathcal{P}}[f](z){\mid}^{p}d{\mu}(z){\leq}M\;{\int}_B{\mid}{\mathcal{P}}[f](z)^pd{\nu}(z)$ for all $f{\in}L^p({\sigma})$ if and only if ${\parallel}{\mu}{\parallel_r}\;=\;sup_{z{\in}B}\;\frac{\mu(E(z,r))}{\nu(E(z,r))}\;<\;\infty$.

• Magazine of the Korean Society of Agricultural Engineers
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• v.20 no.1
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• pp.4592-4598
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• 1978

The purpose of this research is to find out mathemetically an economical intake water depth in the design of head works through the derivation of some formulas. For the performance of the purpose the following formulas were found out for the design intake water depth in each flow type of intake sluice, such as overflow type and orifice type. (1) The conditional equations of !he economical intake water depth in .case that weir body is placed on permeable soil layer ; (a) in the overflow type of intake sluice, {{{{ { zp}_{1 } { Lh}_{1 }+ { 1} over {2 } { Cp}_{3 }L(0.67 SQRT { q} -0.61) { ( { d}_{0 }+ { h}_{1 }+ { h}_{0 } )}^{- { 1} over {2 } }- { { { 3Q}_{1 } { p}_{5 } { h}_{1 } }^{- { 5} over {2 } } } over { { 2m}_{1 }(1-s) SQRT { 2gs} }+[ LEFT { b+ { 4C TIMES { 0.61}^{2 } } over {3(r-1) }+z( { d}_{0 }+ { h}_{0 } ) RIGHT } { p}_{1 }L+(1+ SQRT { 1+ { z}^{2 } } ) { p}_{2 }L+ { dcp}_{3 }L+ { nkp}_{5 }+( { 2z}_{0 }+m )(1-s) { L}_{d } { p}_{7 } ] =0}}}} (b) in the orifice type of intake sluice, {{{{ { zp}_{1 } { Lh}_{1 }+ { 1} over {2 } C { p}_{3 }L(0.67 SQRT { q} -0.61)}}}} {{{{ { ({d }_{0 }+ { h}_{1 }+ { h}_{0 } )}^{ - { 1} over {2 } }- { { 3Q}_{1 } { p}_{ 6} { { h}_{1 } }^{- { 5} over {2 } } } over { { 2m}_{ 2}m' SQRT { 2gs} }+[ LEFT { b+ { 4C TIMES { 0.61}^{2 } } over {3(r-1) }+z( { d}_{0 }+ { h}_{0 } ) RIGHT } { p}_{1 }L }}}} {{{{+(1+ SQRT { 1+ { z}^{2 } } ) { p}_{2 } L+dC { p}_{4 }L+(2 { z}_{0 }+m )(1-s) { L}_{d } { p}_{7 }]=0 }}}} where, z=outer slope of weir body (value of cotangent), h1=intake water depth (m), L=total length of weir (m), C=Bligh's creep ratio, q=flood discharge overflowing weir crest per unit length of weir (m3/sec/m), d0=average height to intake sill elevation in weir (m), h0=freeboard of weir (m), Q1=design irrigation requirements (m3/sec), m1=coefficient of head loss (0.9∼0.95) s=(h1-h2)/h1, h2=flow water depth outside intake sluice gate (m), b=width of weir crest (m), r=specific weight of weir materials, d=depth of cutting along seepage length under the weir (m), n=number of side contraction, k=coefficient of side contraction loss (0.02∼0.04), m2=coefficient of discharge (0.7∼0.9) m'=h0/h1, h0=open height of gate (m), p1 and p4=unit price of weir body and of excavation of weir site, respectively (won/㎥), p2 and p3=unit price of construction form and of revetment for protection of downstream riverbed, respectively (won/㎡), p5 and p6=average cost per unit width of intake sluice including cost of intake canal having the same one as width of the sluice in case of overflow type and orifice type respectively (won/m), zo : inner slope of section area in intake canal from its beginning point to its changing point to ordinary flow section, m: coefficient concerning the mean width of intak canal site,a : freeboard of intake canal. (2) The conditional equations of the economical intake water depth in case that weir body is built on the foundation of rock bed ; (a) in the overflow type of intake sluice, {{{{ { zp}_{1 } { Lh}_{1 }- { { { 3Q}_{1 } { p}_{5 } { h}_{1 } }^{- {5 } over {2 } } } over { { 2m}_{1 }(1-s) SQRT { 2gs} }+[ LEFT { b+z( { d}_{0 }+ { h}_{0 } )RIGHT } { p}_{1 }L+(1+ SQRT { 1+ { z}^{2 } } ) { p}_{2 }L+ { nkp}_{5 }}}}} {{{{+( { 2z}_{0 }+m )(1-s) { L}_{d } { p}_{7 } ]=0 }}}} (b) in the orifice type of intake sluice, {{{{ { zp}_{1 } { Lh}_{1 }- { { { 3Q}_{1 } { p}_{6 } { h}_{1 } }^{- {5 } over {2 } } } over { { 2m}_{2 }m' SQRT { 2gs} }+[ LEFT { b+z( { d}_{0 }+ { h}_{0 } )RIGHT } { p}_{1 }L+(1+ SQRT { 1+ { z}^{2 } } ) { p}_{2 }L}}}} {{{{+( { 2z}_{0 }+m )(1-s) { L}_{d } { p}_{7 } ]=0}}}} The construction cost of weir cut-off and revetment on outside slope of leeve, and the damages suffered from inundation in upstream area were not included in the process of deriving the above conditional equations, but it is true that magnitude of intake water depth influences somewhat on the cost and damages. Therefore, in applying the above equations the fact that should not be over looked is that the design value of intake water depth to be adopted should not be more largely determined than the value of h1 satisfying the above formulas.