• Journal of applied mathematics & informatics
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• v.40 no.3_4
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• pp.709-717
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• 2022

The objective of this research is to prove that an additive mapping T : R → R is a left as well as right centralizer on R if it satisfies any one of the following identities: (i) T(xⁿyⁿ + yⁿxⁿ) = T(xⁿ)yⁿ + yⁿT(xⁿ) (ii) 2T(xⁿyⁿ) = T(xⁿ)yⁿ + yⁿT(xⁿ) for each x, y ∈ R, where n ≥ 1 is a fixed integer and R is any n!-torsion free semiprime ring. In addition, we talk over above identities in the setting of *-ring(ring with involution).

• Korean Journal of Veterinary Research
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• v.39 no.3
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• pp.653-664
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• 1999

This study was undertaken to determine the normal appearance of the postpartum uterine involution. Postpartum changes in uterine shape, architecture, echogenicity and diameter were monitored with ultrasonography in 11 Korea Jin-do dogs. Serial ultrasonographic examination was done daily during the first week, 3 days interval from 8 to 30 days, and weekly from 31 to 100 days postpartum, respectively. Of 11 postpartum bitches, 10 bitches (90.9%) had normal involution and 1 bitch (9.1%) had subinvolution of the placental sites (SIPS) by gross findings, vaginal discharges, and by ultrasographic findings, uterine shape and echogenicity. The excretory period of vaginal discharges in 10 normal bitches of uterine involution was finished completely at $20.2{\pm}4.6$ days (Mean${\pm}$SD) postpartum, but in 1 SIPS at 50 days postpartum. The short axis shape of the uterus was initially often flaccid-appearing. It varied from circular to crescent shaped to polygonal. This lasted until $16.5{\pm}3.7$ days postpartum, during which time the short axis uterine shape gradually changed to circular. Also, the long axis shape of the uterus was created a beaded appearance of the horns until $29.9{\pm}3.2$ days postpartum. After 30 days, it was appeared as tubular shape without distinguished between placental and interplacental sites. The ultrasonographic image of the postpartum uterus consisted of four echogenicity distinct layers. Uterine wall was represented the very hyperechoic serosa, hypoechoic myometrium, hyper echoic endometrium and anechoic structures of fluid in the uterine cavity until 7 days postpartum. The individual uterine layers were most prominent during the first week postpartum, and they became progressively less distinct throughout the course of uterine involution. Anechoic structures of fluid in the uterine cavity was detected until $25.0{\pm}6.4$ days postpartum, after which time it was not reliably detected. The uterine diameter was decreased not only in the placental sites frorm $24.1{\pm}2.5mm$ at 1 day to $15.4{\pm}1.4mm$ at 7 days postpartum, but also in the interplacental sites $14.9{\pm}1.5mm$ at 1 day, $10.6{\pm}0.8mm$ at 7 days postpartum. There was a general trend of decreasing uterine diameter, which occurred more rapidly at the placental sites. At 31 days pastpartum, these diameter reached almost same size, after that time, they could be not distinguished between placental and interplacental sites. At 87 days postpartum, the uterine diameter was $5.6{\pm}0.6mm$ both placental and interplacental sites, and the uterine horns were uniform hypoechoic, tubular structures without enlargement. Therefore, complete involution of the uterus occurred at that time. It was concluded that normal postpartum uterine involution in Korea Jin-do bitches appeared to be completed around 87 days postpartum by gross findings such as vaginal discharges, and by ultrasonographic findings, uterine shape and echogenicity. Also ultrasonographic characteristics of the postpartum uterine involution were described. Therefore, these result suggest that ultrasonographic assessment is a reliable method for diagnosing the SIPS and uterine dysfunction, such as pyometra in bitch.

• Journal of Embryo Transfer
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• v.25 no.1
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• pp.77-84
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• 2010

This study was carried out the normal serial ultrasonographic appearance of the postpartum uterine involution with small pet bitches. Postpartum changes in uterine shape, architecture, echogenicity and diameter were monitored with ultrasonography in small pet bitches (Miniature Schnauzer, Cocker Spaniel and Shih-tzu). In the Miniature Schnauzer bitches, uterine diameter of the placental sites and interplacental sites were decreased from $24.31{\pm}1.65\;mm$ at 1 day to $13.56{\pm}0.76\;mm$at 7 day, and $15.19{\pm}1.28\;mm$ at 1 day to $10.41{\pm}0.76\;mm$ at 7 day postpartum. In the Cocker Spaniel, the placental sites and interplacental sites were decreased from $24.90{\pm}1.62\;mm$ at 1 day to $14.14{\pm}1.21\;mm$at 7 day, and $15.33{\pm}1.47\;mm$ at 1 day to $10.19{\pm}0.83\;mm$ at 7 day postpartum. In the Shih-tzu, the placental sites and interplacental sites were decreased from $23.96{\pm}1.55\;mm$at 1 day to $13.84{\pm}0.94\;mm$ at 7 day, and $14.93{\pm}1.29\;mm$ at 1 day to $9.98{\pm}0.84\;mm$ at 7 day postpartum. And uterine diameter was no statistically significant difference among bitches (p>0.05). At 67 days, the uterine diameter in Miniature Schnauzer bitches were 6~7 mm both placental and interplacental sites, and the uterine horns showed uniform hypoechoic, tubular structures without enlargement. In present study, the involution of the uterus was completed at 67 days after parturition in Miniature Schnauzer, and 65 days in Cocker Spaniel and the Shih-tzu. There were no significant differences of normal postpartum uterine involution between small pet bitches (p>0.05). In conclusion, the postpartum involution of small pet bitches appeared to be completed normally at 65~67 days after parturition and could be identified by gross findings such as vaginal discharges and ultrasonographic findings of uterine shape and echogenicity.

• Journal of the Chungcheong Mathematical Society
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• v.27 no.2
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• pp.261-269
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• 2014

In this paper, using fixed point method, we prove the Hyers-Ulam stability of the following functional equation $$(k+1)f(x+y)+f(x+{\sigma}(y))+kf({\sigma}(x)+y)-2(k+1)f(x)-2(k+1)f(y)=0$$ with an involution ${\sigma}$ for a fixed non-zero real number k with $k{\neq}-1$.

• East Asian mathematical journal
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• v.34 no.5
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• pp.693-702
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• 2018

In this paper, we consider the generalized Hyers-Ulam stability for the following additive-quadratic functional equation with involution $f(x+2y)-f(2x+y)+f(x+y)+f({\sigma}(x)+y)+f(x)-4f(y)-f({\sigma}(y))=0$ in non-Archimedean spaces.

• Journal of applied mathematics & informatics
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• v.34 no.5_6
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• pp.509-517
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• 2016

In this paper, we consider the following functional equation with involution f(x + y) + f(x + σ(y)) = 2f(x) + f(y) + f(σ(y)) and prove the generalized Hyers-Ulam stability for it when the target space is a non-Archimedean Banach space.

• Communications of the Korean Mathematical Society
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• v.29 no.4
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• pp.497-503
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• 2014

In this paper, we consider characters of $SL_2(\mathbb{Z})$ and then apply them to newforms of integral weight, level 4 and of trivial character. More precisely, we prove that all of them are actually level 1 forms of some nontrivial character. As a byproduct, we prove that they all are eigenfunctions of the Fricke involution with eigenvalue -1.

• Communications of the Korean Mathematical Society
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• v.11 no.3
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• pp.595-600
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• 1996

We give a characterization of a deductive system. We introduce the concept of maximal deductive systems and show that every bounded Hilbert algebra with at least two elements contains at least one maximal deductive system. Moreover, we introduce the notion of radical and semisimple in a Hilbert algebra and prove that if H is a bounded Hilbert algebra in which every element is an involution, then H is semisimple.

• Korean Journal of Mathematics
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• v.16 no.3
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• pp.425-437
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• 2008

Let P be a poset and G = G(P) be the incomparability graph of P. Stanley [7] defined the chromatic symmetric function $X_{G(P)}$ which generalizes the chromatic polynomial ${\chi}_G$ of G, and showed all coefficients are nonnegative in the e-expansion of $X_{G(P)}$ for a poset P of rank 1. In this paper, we construct a sign reversing involution on the set of special rim hook P-tableaux with some conditions. It gives a combinatorial proof for (3+1)-free conjecture of a poset P of rank 1.

• Kyungpook Mathematical Journal
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• v.56 no.2
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• pp.343-347
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• 2016

Posner's first theorem states that if R is a prime ring of characteristic different from two, $d_1$ and $d_2$ are derivations on R such that the iterate $d_1d_2$ is also a derivation of R, then at least one of $d_1$, $d_2$ is zero. In the present paper we extend this result to *-prime rings of characteristic different from two.