• Korean Journal of Mathematics
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• v.26 no.3
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• pp.439-458
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• 2018

In this study, we interpret the notion of homotopy of morphisms in the category of crossed modules in a category C of groups with operations using the categorical equivalence between the categories of crossed modules and of internal categories in C. Further, we characterize the derivations of crossed modules in a category C and obtain new crossed modules using regular derivations of old one.

• Journal of the Korean Mathematical Society
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• v.36 no.1
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• pp.97-107
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• 1999

Given a C-dynamical system (A, G, $\alpha$) with a locally compact group G, two kinds of C-algebras are made from it, called the full C-crossed product and the reduced C-crossed product. In this paper, we extend the theory of the classical C-crossed product to the C-dynamical system (A, G, $\alpha$) with a left-cancellative semigroup M with unit. We construct a new C-algebra A $\alpha$rM, the reduced crossed product of A by the semigroup M under the action $\alpha$ and investigate some properties of A $\alpha$rM.

• The Pure and Applied Mathematics
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• v.25 no.3
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• pp.203-218
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• 2018

An old result of Whitehead says that the set of derivations of a group with values in a crossed G-module has a natural monoid structure. In this paper we introduce derivation of crossed polymodule and actor crossed polymodules by using Lue's and Norrie's constructions. We prove that the set of derivations of a crossed polygroup has a semihypergroup structure with identity. Then, we consider the polygroup of invertible and reversible elements of it and we obtain actor crossed polymodule.

• Bulletin of the Korean Mathematical Society
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• v.52 no.4
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• pp.1077-1095
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• 2015

In this paper we study equivariant crossed modules in its link with strict graded categorical groups. The resulting Schreier theory for equivariant group extensions of the type of an equivariant crossed module generalizes both the theory of group extensions of the type of a crossed module and the one of equivariant group extensions.

• The Pure and Applied Mathematics
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• v.10 no.2
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• pp.103-118
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• 2003

We estimate the stable rank of twisted crossed products of $C^{*}-algebras$ by topological Abelian groups. As an application we estimate the stable rank of twisted crossed products of $C^{*}-algebras$ by solvable Lie groups. In particular, we obtain the stable rank estimate of twisted group $C^{*}-algebras$ of solvable Lie groups by the (reduced) dimension and (generalized) rank of groups.

• Korean Journal of Mathematics
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• v.27 no.2
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• pp.535-545
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• 2019

The main idea of this paper is to introduce the notion of $cat^1$-monoids and to prove that the category of crossed semimodules ${\mathcal{C}}=(A,B,{\partial})$ where A is a group is equivalent to the category of $cat^1$-monoids. This is a generalization of the well known equivalence between category of $cat^1$-groups and that of crossed modules over groups.

• Bulletin of the Korean Mathematical Society
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• v.60 no.2
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• pp.529-539
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• 2023

In this paper, we study nuclearity of semigroup crossed products for quasi-lattice ordered groups. We show the relationships among nuclearity of the semigroup crossed product, amenability of the quasi-lattice ordered group and nuclearity of the underlying C^*-algebra.

• Korean Journal of Mathematics
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• v.32 no.1
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• pp.27-42
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• 2024

In this paper we define generalized double group-groupoids and crossed modules over generalized group-groupoids. Also we prove that these algebraic structures are categorically equivalent.

• Childhood Kidney Diseases
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• v.12 no.2
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• pp.262-266
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• 2008

Crossed renal ectopia is a congenital malformation in which both kidneys lie on the same side of the spine, usually side by side longitudinally. More often on the right side. Fusion of the two renal units is eight times more common than nonfusion. Although crossed renal ectopia is uncommon, this unusual entity must be considered in an infant when cystic mass in the abdomen or pelvis paticularly if no kidney can be found on the opposite side. In many cases of crossed fused ectopia with multicystic dysplastic kidney(MCDK), the diagnosis can be strongly suspected from the sonogram, and no other studies may be necessary. However, both intravenous urography and isotope renography is useful to assess the function of the crossed kidney. Crossed renal ectopia and MCDKs are associated with a greater incidence of ureteropelvic junction obstruction and reflux. So, screening voiding cystourethrography should be performed. Very few studies of MCDK in the setting of crossed fused ectopia have been reported. We have experienced a 3-year-old boy with crossed fused renal ectopia with multicystic dysplasia.

• Journal of the Korean Society of Physical Medicine
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• v.14 no.3
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• pp.13-20
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• 2019

PURPOSE: Sitting with crossed legs may have an effect on maintaining a healthy body posture and proper functioning of the respiratory system. Thus, this study's objective was to identify whether or not sitting with crossed legs affects the vertebral angle, chest wall mobility, the pulmonary function, and the activity of the respiratory muscles. METHODS: Thirty healthy subjects were recruited for this study (16 males and 14 females). The vertebral angle, chest wall mobility, pulmonary function, and the activity of the respiratory muscle were measured while the subjects sat in the correct posture and these factors were again measured with the subjects seated with their legs crossed. Three-dimensional motion analysis was used to determine the trunk and lumbar vertebral angles. Surface electromyography was employed to measure the sternocleidomastoid, the rectus abdominis, and the external and internal oblique abdominis muscles. A tapeline was utilized to evaluate the subjects' chest wall mobility. Spirometry was assessed to determine the forced vital capacity and forced expiratory volume in one second. Paired t-tests were then performed (p<.05). RESULTS: There were significant differences in the trunk and lumbar flexion angles, the chest wall mobility, the activity of the right external oblique muscle, and the left internal oblique abdominis muscle. However, the difference in pulmonary function did not reach statistical significance. CONCLUSION: A crossed leg posture caused slight thoracic extension and lumbar flexion, which may lead to a decrease of the chest wall mobility and also to an imbalance of the abdominal muscles. Therefore, sitting with a crossed leg posture should be avoided. Yet a crossed leg posture did not have any clinical effect on the pulmonary function of healthy people. It may be necessary to study the effects of sitting with crossed legs over an extended period of time for patients suffering with impaired respiratory function.