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

We obtain asymptotics for sums of the formSigma(p)(n=1) e(alpha(k) n(k) + alpha(1)n),involving lower order main terms. As an application, we show that for almost all alpha(2) is an element of [0, 1) one hassup(alpha 1 is an element of[0,1)) | Sigma(1 \u3c= n \u3c= P) e(alpha(1)(n(3) + n) + alpha(2)n(3))| \u3c\u3c P3/4+epsilon,and that in a suitable sense this is best possible. This allows us to improve bounds for the fractal dimension of solutions to the Schrodinger and Airy equations

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

Unilateral congenital elongation of the cervical part of the internal carotid artery with kinking and looping: two case reports and review of the literature

Unilateral and bilateral variation in the course and elongation of the cervical (extracranial) part of the internal carotid artery (ICA) leading to its tortuosity, kinking and coiling or looping is not a rare condition, which could be caused by both embryological and acquired factors. Patients with such variations may be asymptomatic in some cases; in others, they can develop cerebrovascular symptoms due to carotid stenosis affecting cerebral circulation. The risk of transient ischemic attacks in patients with carotid stenosis is high and its surgical correction is indicated for the prevention of ischemic stroke. Detection of developmental variations of the ICA and evaluation of its stenotic areas is very important for surgical interventions and involves specific diagnostic imaging techniques for vascular lesions including contrast arteriography, duplex ultrasonography and magnetic resonance angiography. Examination of obtained images in cases of unusual and complicated variations of vascular pattern of the ICA may lead to confusion in interpretation of data. Awareness about details and topographic anatomy of variations of the ICA may serve as a useful guide for both radiologists and vascular surgeons. It may help to prevent diagnostic errors, influence surgical tactics and interventional procedures and avoid complications during the head and neck surgery. Our present study was conducted with a purpose of updating data about developmental variations of the ICA. Dissections of the main neurovascular bundle of the head and neck were performed on a total 14 human adult cadavers (10 – Africans: 7 males & 3 females and 4 – East Indians: all males). Two cases of unilateral congenital elongation of the cervical part of the ICA with kinking and looping and carotid stenoses were found only in African males. Here we present their detailed case reports with review of the literature

Differential immunohistochemical expression of CD44s, E-cadherin and β-catenin among hyperplastic and neoplastic lesions of the prostate gland

Introduction: CD44s, E-cadherin and β-catenin are cell adhesion molecules (CAMs) and appear to influence organ development, inflammation, cancer invasion and metastasis. We studied the expression of these CAMs in prostate cancer (PCa), high-grade prostatic intraepithelial neoplasia (HGPIN) and nodular adenomatous hyperplasia (NH). Materials and Methods: 135 paraffin blocks of radical prostatectomy specimens were assessed. CAMs were determined by immunohistochemistry. All sections included PCa, HGPIN and NH. The expression was semiquantitatively evaluated in three scores (1+, 2+, 3+). The markers&apos; immunopositivity was statistically investigated with Gleason score and TNM stage. Results and Conclusions: CD44s had score 3+ in 41.5, 46.7 and 37.8% of areas with NH, HGPIN and PCa, respectively. E-cadherin immunostaining was highly detected in 71.1, 78.5 and 63.0% of NH, HGPIN and PCa areas while β-catenin score 3+ was exclusively membranous in 80.7% of NH and nuclear/cytoplasmic in 70.4 and 48.9% of HGPIN and PCa areas. No marker related to the Gleason score (p = 0.352). CD44s and E-cadherin expression was inversely associated with TNM stage (p = 0.021 and p = 0.042, respectively); no such association was observed for β-catenin (p = 0.556). The decreased expression of CD44s and E-cadherin is probably associated with the invasive potential of PCa. The β-catenin staining pattern in neoplastic lesions, either preinvasive or invasive, differs from that in non-neoplastic prostate lesions. Copyright © 2012 S. Karger AG, Basel