• Title/Summary/Keyword: $R-{\theta}$

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SOME NEW IDENTITIES CONCERNING THE HORADAM SEQUENCE AND ITS COMPANION SEQUENCE

  • Keskin, Refik;Siar, Zafer
    • Communications of the Korean Mathematical Society
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    • v.34 no.1
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    • pp.1-16
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    • 2019
  • Let a, b, P, and Q be real numbers with $PQ{\neq}0$ and $(a,b){\neq}(0,0)$. The Horadam sequence $\{W_n\}$ is defined by $W_0=a$, $W_1=b$ and $W_n=PW_{n-1}+QW_{n-2}$ for $n{\geq}2$. Let the sequence $\{X_n\}$ be defined by $X_n=W_{n+1}+QW_{n-1}$. In this study, we obtain some new identities between the Horadam sequence $\{W_n\}$ and the sequence $\{X_n\}$. By the help of these identities, we show that Diophantine equations such as $$x^2-Pxy-y^2={\pm}(b^2-Pab-a^2)(P^2+4),\\x^2-Pxy+y^2=-(b^2-Pab+a^2)(P^2-4),\\x^2-(P^2+4)y^2={\pm}4(b^2-Pab-a^2),$$ and $$x^2-(P^2-4)y^2=4(b^2-Pab+a^2)$$ have infinitely many integer solutions x and y, where a, b, and P are integers. Lastly, we make an application of the sequences $\{W_n\}$ and $\{X_n\}$ to trigonometric functions and get some new angle addition formulas such as $${\sin}\;r{\theta}\;{\sin}(m+n+r){\theta}={\sin}(m+r){\theta}\;{\sin}(n+r){\theta}-{\sin}\;m{\theta}\;{\sin}\;n{\theta},\\{\cos}\;r{\theta}\;{\cos}(m+n+r){\theta}={\cos}(m+r){\theta}\;{\cos}(n+r){\theta}-{\sin}\;m{\theta}\;{\sin}\;n{\theta},$$ and $${\cos}\;r{\theta}\;{\sin}(m+n){\theta}={\cos}(n+r){\theta}\;{\sin}\;m{\theta}+{\cos}(m-r){\theta}\;{\sin}\;n{\theta}$$.

CHARACTERIZATIONS OF ELEMENTS IN PRIME RADICALS OF SKEW POLYNOMIAL RINGS AND SKEW LAURENT POLYNOMIAL RINGS

  • Cheon, Jeoung-Soo;Kim, Eun-Jeong;Lee, Chang-Ik;Shin, Yun-Ho
    • Bulletin of the Korean Mathematical Society
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    • v.48 no.2
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    • pp.277-290
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    • 2011
  • We show that the ${\theta}$-prime radical of a ring R is the set of all strongly ${\theta}$-nilpotent elements in R, where ${\theta}$ is an automorphism of R. We observe some conditions under which the ${\theta}$-prime radical of coincides with the prime radical of R. Moreover we characterize elements in prime radicals of skew Laurent polynomial rings, studying (${\theta}$, ${\theta}^{-1}$)-(semi)primeness of ideals of R.

A CLASS OF MAPPINGS BETWEEN Rz-SUPERCONTINUOUS FUNCTIONS AND Rδ-SUPERCONTINUOUS FUNCTIONS

  • Prasannan, A.R.;Aggarwal, Jeetendra;Das, A.K.;Biswas, Jayanta
    • Honam Mathematical Journal
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    • v.39 no.4
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    • pp.575-590
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    • 2017
  • A new class of functions called $R_{\theta}$-supercontinuous functions is introduced. Their basic properties are studied and their place in the hierarchy of strong variants of continuity, which already exist in the literature, is elaborated. The class of $R_{\theta}$-supercontinuous functions properly contains the class of $R_z$-supercontinuous functions [39] which in turn properly contains the class of $R_{cl}$-supercontinuous functions [43] and so includes all cl-supercontinuous (clopen continuous) functions ([38], [34]) and is properly contained in the class of $R_{\delta}$-supercontinuous functions [24].

Ptr,s)-CLOSED SPACES AND PRE-(ωr,s)t-θf-CLUSTER SETS

  • Afsan, Bin Mostakim Uzzal;Basu, Chanchal Kumar
    • Communications of the Korean Mathematical Society
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    • v.26 no.1
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    • pp.135-149
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    • 2011
  • Using (r, s)-preopen sets [14] and pre-${\omega}_t$-closures [6], a new kind of covering property $P^t_{({\omega}_r,s)}$-closedness is introduced in a bitopological space and several characterizations via filter bases, nets and grills [30] along with various properties of such concept are investigated. Two new types of cluster sets, namely pre-(${\omega}_r$, s)t-${\theta}_f$-cluster sets and (r, s)t-${\theta}_f$-precluster sets of functions and multifunctions between two bitopological spaces are introduced. Several properties of pre-(${\omega}_r$, s)t-${\theta}_f$-cluster sets are investigated and using the degeneracy of such cluster sets, some new characterizations of some separation axioms in topological spaces or in bitopological spaces are obtained. A sufficient condition for $P^t_{({\omega}_r,s)}$-closedness has also been established in terms of pre-(${\omega}_r$, s)t-${\theta}_f$-cluster sets.

Edge-Maximal 𝜃k+1-Edge Disjoint Free Graphs

  • Jaradat, Mohammed M.M.;Bataineh, Mohammed S.A.
    • Kyungpook Mathematical Journal
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    • v.54 no.1
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    • pp.23-30
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    • 2014
  • For two positive integers r and s, $\mathcal{G}$(n; r; ${\theta}_s$) denotes to the class of graphs on n vertices containing no r of edge disjoint ${\theta}_s$-graphs and f(n; r; ${\theta}_s$) = max{${\varepsilon}(G)$ : G ${\in}$ $\mathcal{G}$(n; r; ${\theta}_s$)}. In this paper, for integers r, $k{\geq}2$, we determine f(n; r; ${\theta}_{2k+1}$) and characterize the edge maximal members in G(n; r; ${\theta}_{2k+1}$).

Properties of a Surface Curvature in Toric Lens (토릭렌즈의 표면 곡률 특성 연구)

  • Park, Sang-An;Kim, Yong-Geun
    • Journal of Korean Ophthalmic Optics Society
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    • v.6 no.2
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    • pp.65-70
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    • 2001
  • We obtained the sum of two curvature ($C_x+C_y$) in toric lens which two toroidal surface is the right angle each other. $$C_x+C_y=\frac{x^2+y^2}{2r_1}+\frac{x^2}{2}(\frac{1}{r_2}-\frac{1}{r_1})$$ and the sum of two curvature ($C_a+C_b$) in toric lens about the cross angle. $$(C_a+C_b)=\frac{x^2cos^2{\alpha}_1}{2r_1}+\frac{x^2cos^2{\alpha}_2}{2r_2}+\frac{y^2sin^2{\alpha}_1}{2r_1}+\frac{y^2sin^2{\alpha}_2}{2r_2}$$ and claculated the parameter S, C, ${\theta}$ of a combination power in toric lens of the cross angle including surface curvature ($C_x$, $C_y$) values. $$S=(n-1)\[\frac{C_x}{x^2}+\frac{C_y}{y^2}\]-\frac{C}{2},\;C=-\frac{2(n-1)}{sin2{\theta}}\[\frac{C_x}{x^2}+\frac{C_y}{y^2}\]$$ $${\theta}=\frac{1}{2}tan^{-1}\[-\frac{{C_xy^2sin2{\theta}_1}+{C_yx^2sin2{\theta}_2}}{{C_xy^2cos2{\theta}_1}+{C_yx^2cos2{\theta}_2}}\]$$.

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SOME RESULTS CONCERNING ($\theta,\;\varphi$)-DERIVATIONS ON PRIME RINGS

  • Park, Kyoo-Hong;Jung Yong-Soo
    • The Pure and Applied Mathematics
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    • v.10 no.4
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    • pp.207-215
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    • 2003
  • Let R be a prime ring with characteristic different from two and let $\theta,\varphi,\sigma,\tau$ be the automorphisms of R. Let d : $R{\rightarrow}R$ be a nonzero ($\theta,\varphi$)-derivation. We prove the following results: (i) if $a{\in}R$ and [d(R), a]$_{{\theta}o{\sigma},{\varphi}o{\tau}}$=0, then $\sigma(a)\;+\;\tau(a)\;\in\;Z$, the center of R, (ii) if $d([R,a]_{\sigma,\;\tau)\;=\;0,\;then\;\sigma(a)\;+\;\tau(a)\;\in\;Z$, (iii) if $[ad(x),\;x]_{\sigma,\;\tau}\;=\;0;for\;all\;x\;\in\;RE$, then a = 0 or R is commutative.

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Model Equations to Estimate the Soil Water Characteristics Curve Using Scaling Factor (Scaling Factor를 이용한 토양수분특성곡선 추정모형)

  • Eom, Ki-Cheol;Song, Kwan-Cheol;Ryu, Kwan-Shig;Sonn, Yeon-Kyu;Lee, Sang-Eun
    • Korean Journal of Soil Science and Fertilizer
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    • v.28 no.3
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    • pp.227-232
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    • 1995
  • The model equations including scaling factors to estimate the soil water characteristics curve(SWCC) without direct measurement of soil water tension were developed. Scaling were applied to a data set of soil water content, soil water tension, particle size distribution, and OM contents of the 134 soil samples with the 10 soil textural classes. The capability of the model equations was tested on another 205 soil samples. The parameter, ${\theta}^*$, of soil water contents was used by scale transformation as follows : ${\theta}^*=[{\theta}i-{\theta}(1.5MPa)]$/$[{\theta}(10KPa)-{\theta}(1.5MPa)]$ Using ${\theta}^*$ a model equation to estimate SWCC, which was applicable to all textural classes, was developed as follows: $H(0.1MPa)=0.13{\cdot}({\theta}^*)^{-2.04}$. Other model equations to estimate the water content at the soil water tension of 10KPa [${\theta}(10KPa)$] and 1.5MPa [${\theta}(1.5MPa)$], which are required to ${\theta}^*$ were developed by using scale factors of sand(S) and silt(Si) content and organic matter content(OM) as foilows : ${\theta}(10KPa)=26.80-3.99ln[S]+2.36{\sqrt{[Si]}}+2.88[OM]$ ($R=0.81^{**}$) ${\theta}(1.5KPa)=15.75-2.86ln[S]+0.55{\sqrt{[Si]}}+0.70[OM]$ ($R=0.76^{**}$) The measured and estimated values of ${\theta}(1/30MPa)$ on the 205 soil samples were highly correlated on 1 : 1 corresponding line with $R=0.85^{**}$.

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Sequential Confidence Set of the Mean Vector of a Multivariate Distribution

  • Kim, Sung Lai
    • Journal of the Chungcheong Mathematical Society
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    • v.5 no.1
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    • pp.87-97
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    • 1992
  • Sequential procedure with ${\beta}$-protection for the mean vector ${\mu}(\theta)$ of a p(> 1)-variate multivariate distribution $P_{\theta}$, ${\theta}{\in}{\Theta}$, with covariance matrix ${\sum}(\theta)$ is considered when the only nuisance parameters is ${\sum}(\theta)$. We obtain a confidence set for ${\mu}(\theta)$ with coverage probability condition and ${\beta}$-protection at ${\mu}-{\delta}(\mu)$ for some imprecision function ${\delta}:\mathbb{R}^p{\rightarrow}\mathbb{R}^p$.

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