Arithmetic of Quadratic Forms

Kitaoka, Yoshiyuki

doi:10.1017/cbo9780511666155

Cited by 294 publications

(224 citation statements)

References 20 publications

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“…At follow-up, all of the patients including the control group received the Ankle-Hindfoot scale of the American Orthopaedic Foot and Ankle Society (AOFAS score) 4 and 3DCT. The pre-and postoperative AOFAS and follow-up scores were compared by means of the Student t test.…”

Section: Methodsmentioning

confidence: 99%

Arthroscopic Treatment for Anterior Impingement Exostosis of the Ankle: Application of Three-Dimensional Computed Tomography

et al. 2004

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The purpose of this study was to evaluate the operative results of excision of anterior impingement exostoses of the ankle. Preoperative three-dimensional computed tomography (3DCT) was used to make the diagnoses. The authors evaluated 16 ankles of 16 patients who underwent arthroscopic resection of the osteophytes of their anterior distal tibia or dorsal talus. They were followed up for 24-51 months. All 16 patients had 3DCT preoperatively, which allowed the authors to determine the exact location, shape, size, and number of the osteophytes. All of the osteophytes were resected using arthroscopic techniques. At the time of the most recent follow-up, the mean AOFAS score was 80.5 ± 4.9 points at preoperation, and 97.0 ± 3.7 points at the most recent follow-up. There were significant differences between the pre-and postoperative AOFAS scores and those of the most recent follow-up period for each group (p < .0001). It is necessary to clarify the location, size, shape, and number of all of the osteophytes preoperatively using 3DCT, and to then resect them all.

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Section: Methodsmentioning

confidence: 99%

Arthroscopic Treatment for Anterior Impingement Exostosis of the Ankle: Application of Three-Dimensional Computed Tomography

et al. 2004

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“…(1, 2, 2), (1, 2, 3), (1, 2, 4), (1,2,6), (1,2,8), (1,3,3), (1,3,4), (1,3,6), (1,3,7), (1,3,8), (1,3,9).…”

Section: Introductionmentioning

confidence: 99%

Ternary Universal Sums of Generalized Pentagonal Numbers

Oh¹

2011

Journal of the Korean Mathematical Society

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with x ∈ Z is said to be a generalized m-gonal number. Let a ≤ b ≤ c and k be positive integers. The quadruple (k, a, b, c) is said to be universal if for every nonnegative integer n there exist integers x, y, z such that n = ap k (x)+bp k (y)+cp k (z). Sun proved in [16] that, when k = 5 or k ≥ 7, there are only 20 candidates for universal quadruples, which he listed explicitly and which all involve only the case of pentagonal numbers (k = 5). He verified that six of the candidates are in fact universal and conjectured that the remaining ones are as well. In a subsequent paper [3], Ge and Sun established universality for all but seven of the remaining candidates, leaving only (5, 1, 1, t) for t = 6, 8, 9, 10, (5, 1, 2, 8) and (5, 1, 3, s) for s = 7, 8 as candidates. In this article, we prove that the remaining seven quadruples given above are, in fact, universal.

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“…Using the geometric language of quadratic spaces and lattices (see [9,11]), the representation problem of quadratic forms can be rephrased as follows. The equivalence class of the quadratic form f (x) corresponds to the isometry class of a Z-lattice M with a basis {v 1 , .…”

Section: Introductionmentioning

confidence: 99%

“…Any unexplained terminologies and notations can be found in [9] and [11]. The term lattice always refers to a Z-lattice on a non-degenerate quadratic space with a bilinear form B and its associated quadratic map Q.…”

Section: Introductionmentioning

confidence: 99%

Extensions of representations of integral quadratic forms

Chan

Kim

et al. 2007

Ramanujan J

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Let N and M be quadratic Z-lattices, and K be a sublattice of N. A representation σ : K → M is said to be extensible to N if there exists a representation ρ : N → M such that ρ| K = σ . We prove in this paper a local-global principle for extensibility of representation, which is a generalization of the main theorems on representations by positive definite Z-lattices by Hsia, Kitaoka and Kneser (J. Reine Angew. Math. 301:132-141, 1978) and Jöchner and Kitaoka (J. Number Theory 48:88-101, 1994). Applications to almost n-universal lattices and systems of quadratic equations with linear conditions are discussed.

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Arithmetic of Quadratic Forms

Cited by 294 publications

References 20 publications

Arthroscopic Treatment for Anterior Impingement Exostosis of the Ankle: Application of Three-Dimensional Computed Tomography

Arthroscopic Treatment for Anterior Impingement Exostosis of the Ankle: Application of Three-Dimensional Computed Tomography

Ternary Universal Sums of Generalized Pentagonal Numbers

Extensions of representations of integral quadratic forms

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