• Kyungpook Mathematical Journal
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• v.46 no.2
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• pp.285-295
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• 2006

We present a concept of negative definite functions on a commutative normal hypercomplex system $L_1$(Q, $m$) with basis unity. Negative definite functions were studied in [5] and [4] for commutative groups and semigroups respectively. The definition of such functions on Q is a natural generalization of that defined on a commutative hypergroups.

• Kyungpook Mathematical Journal
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• v.48 no.4
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• pp.559-575
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• 2008

A commutative hypercomplex system $L_1$(Q,m) is, roughly speaking, a space which is defined by a structure measure (c(A,B, r), (A,$B{\in}{\beta}$(Q)). Such space has bee studied by Berezanskii and Krein. Our main purpose is to establish a generalization of convolution semigroups and to discuss the role of the L$\'{e}$vy measure in the L$\'{e}$vy-Khinchin representation in terms of continuous negative definite functions on the dual hypercomplex system.

• The Korean Journal of Nuclear Medicine
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• v.13 no.1_2
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• pp.61-71
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• 1979

The authors investigated the incidence of antithyroglobulin antibodies and antibodies and antimicrosomal antibodies measured by tanned red cell hemagglutination method in subjects suffering from various thyroid disorders. 1) In 15 normal patients, neither suffering from any thyroid diseases nor from any other autoimmune disorders, the anti thyroglobulin antibodies were all negative, but the antimicrosomal antibody was positive only in one patient (6.7%). 2) The antithyroglobulin antibodies were positive in 31.5% (34 patients) of 108 patients with various thyroid diseases, and the antimicrosomal antibodies were positive in 37.0% (40 patients). 3) of the 25 patients with Graves' diseases, 7 patients (28.0%) showed positive for the antithyroglobulin antibodies, and 9(36.0%) for the antimicrosomal antibodies. There was no definite differences in clinical and thyroid functions between the groups with positive and negative results. 4) Both antibodies were positive in 16(88.9%) and 17(94.4%) patients respectively among 18 patients with Hashimoto's thyroiditis, all of them were diagnosed histologically. 5) Three out of 35 patients with thyroid adenoma showed positive antibodies, and 3 of 16 patients with thyroid carcinoma revealed positive antibodies. 6) TRCH antibodies demonstrated negative results in 2 patients with subacute thyroiditis, but positive in one patient with idiopathic primary myxedema. 7) The number of patients with high titers ($>1:80^2$) was 16 for antithyroglobulin antibody and 62.5%(10 patients) of which was Hashimoto's thyroiditis. Thirteen(65.0) of 20 patients with high titers($>1:80^2$) for antimicrosomal antibody was Hashimoto's thyroiditis. TRCH test is a simple, sensitive method, and has high reliablity and reproducibility. The incidences and titers of antihyroglobulin antibody and antimicrosomal antibody are especially high in Hashimoto's thyroiditis.