On generating functions in additive number theory, II: Lower-order terms and applications to PDEs

We obtain asymptotics for sums of the form $$\sum_{n=1}^P e(\alpha_kn^k + \alpha_1n),$$ involving lower order main terms. As an application, we show that for almost all $\alpha_2 \in [0,1)$ one has $$\sup_{\alpha_1 \in [0,1)} \Big| \sum_{1 \le n \le P} e(\alpha_1(n^3+n) + \alpha_2 n^3) \Big| \ll P^{3/4 + \varepsilon},$$ and that in a suitable sense this is best possible. This allows us to improve bounds for the fractal dimension of solutions to the Schr\"odinger and Airy equations

Metastatic breast carcinoma in the mandible presenting as a periodontal abscess: a case report

<p>Abstract</p> <p>Introduction</p> <p>Tumors can metastasize to the oral cavity and affect the jaws, soft tissue and salivary glands. Oral cavity metastases are considered rare and represent approximately 1% of all oral malignancies. Because of their rarity and atypical clinical and radiographic appearance, metastatic lesions are considered a diagnostic challenge. The purpose of this report is to present a rare case of a metastatic breast carcinoma mimicking a periodontal abscess in the mandible.</p> <p>Case presentation</p> <p>A 55-year-old Caucasian woman was referred to our clinic for evaluation of bisphosphonate-induced jaw osteonecrosis. She had undergone modified radical mastectomy with axillary lymph node dissection for invasive ductal carcinoma of the left breast. Her clinical examination showed diffuse swelling and a periodontal pocket of 6 mm exhibiting suppuration in the posterior right mandible. Moreover, paresthesia of the lower right lip and chin was noted. There were no significant radiographic findings other than alveolar bone loss due to her periodontal disease. Although the lesion resembled a periodontal abscess, metastatic carcinoma of the breast was suspected on the basis of the patient's medical history. The area was biopsied, and histological analysis confirmed the final diagnosis of metastatic breast carcinoma.</p> <p>Conclusion</p> <p>The general dentist or dental specialist should maintain a high level of suspicion while evaluating patients with a history of cancer. Paresthesias of the lower lip and the chin should be considered ominous signs of metastatic disease. This case highlights the importance of the value of a detailed medical history and thorough clinical examination for the early detection of metastatic tumors in the oral cavity.</p