On the Hausdorff dimension of ultrametric subsets in R^n

For every e>0, any subset of R^n with Hausdorff dimension larger than (1-e)n must have ultrametric distortion larger than 1/(4e).Comment: 4 pages, improved layou

A node-capacitated Okamura-Seymour theorem

The classical Okamura-Seymour theorem states that for an edge-capacitated, multi-commodity flow instance in which all terminals lie on a single face of a planar graph, there exists a feasible concurrent flow if and only if the cut conditions are satisfied. Simple examples show that a similar theorem is impossible in the node-capacitated setting. Nevertheless, we prove that an approximate flow/cut theorem does hold: For some universal c > 0, if the node cut conditions are satisfied, then one can simultaneously route a c-fraction of all the demands. This answers an open question of Chekuri and Kawarabayashi. More generally, we show that this holds in the setting of multi-commodity polymatroid networks introduced by Chekuri, et. al. Our approach employs a new type of random metric embedding in order to round the convex programs corresponding to these more general flow problems.Comment: 30 pages, 5 figure

Measured descent: A new embedding method for finite metrics

We devise a new embedding technique, which we call measured descent, based on decomposing a metric space locally, at varying speeds, according to the density of some probability measure. This provides a refined and unified framework for the two primary methods of constructing Frechet embeddings for finite metrics, due to [Bourgain, 1985] and [Rao, 1999]. We prove that any n-point metric space (X,d) embeds in Hilbert space with distortion O(sqrt{alpha_X log n}), where alpha_X is a geometric estimate on the decomposability of X. As an immediate corollary, we obtain an O(sqrt{(log lambda_X) \log n}) distortion embedding, where \lambda_X is the doubling constant of X. Since \lambda_X\le n, this result recovers Bourgain's theorem, but when the metric X is, in a sense, ``low-dimensional,'' improved bounds are achieved. Our embeddings are volume-respecting for subsets of arbitrary size. One consequence is the existence of (k, O(log n)) volume-respecting embeddings for all 1 \leq k \leq n, which is the best possible, and answers positively a question posed by U. Feige. Our techniques are also used to answer positively a question of Y. Rabinovich, showing that any weighted n-point planar graph embeds in l_\infty^{O(log n)} with O(1) distortion. The O(log n) bound on the dimension is optimal, and improves upon the previously known bound of O((log n)^2).Comment: 17 pages. No figures. Appeared in FOCS '04. To appeaer in Geometric & Functional Analysis. This version fixes a subtle error in Section 2.

Gastrointestinal Stromal Tumor (GIST) in Long Standing Crohn’s disease on Anti-TNF Therapy

Introduction Patients suffering from inflammatory bowel disease (IBD) are at increased risk for developing cancer. Adenocarcinomas are the most commonly observed tumors of the gastrointestinal tract whereas data on gastrointestinal stromal tumor (GIST) in IBD patients is limited. GIST is a neoplasm that originates from the interstitial cells of Cajal in the smooth muscle layers of the gastrointestinal tract. [1] The association between GIST and Crohn’s disease (CD) is debated, as the tumor inconsistently present in areas of inflammatory activity. We report an interesting case of CD maintained on Infliximab, who presented with a flare that revealed GIST in the stomach. To our knowledge, this is the first reported occurrence of GIST in stomach in a patient with CD maintained on anti-TNF therapy. Case Report A 40-year-old Caucasian man with a history of small bowel Crohn’s disease on infliximab therapy presented with a two-day history of abdominal pain, hematochezia, and diffuse joint pain. Upon admission, the patient was hemodynamically stable and afebrile, with a blood pressure of 140/70 mmHg, heart rate of 90 beats per minute, and respiratory rate of 14 per minute. Physical exam was remarkable for abdominal distension and diffuse abdominal tenderness. Complete blood count, comprehensive metabolic panel, and C-reactive protein were within normal range. The patient reported no history of alcohol abuse, smoking, recent abdominal procedures, or trauma. The patient had computed tomography (CT) of the abdomen done that revealed a 2.5-centimeter exophytic mass in the stomach with possible liver metastases (Fig. 1). Endoscopic ultrasound (EUS) guided biopsies of the exophytic mass confirmed gastrointestinal stromal tumor (GIST) on fine needle aspiration and flow cytometry results (Fig. 2,3). The patient underwent surgical resection without complication and is back to his usual state of health. Discussion GIST is the most common mesenchymal neoplasm in the gastrointestinal tract [1,2]. The annual incidence of GIST has been reported as 11-19.6 per million [3,4], however a more recent analysis in 2015 estimates the annual incidence to be 6.8 per million with a 53% predominance in males and 73% predominance in Caucasians [5]. Individuals are typically diagnosed with GIST in their seventh decade of life [5]. Immunologically, it is reported that 70-80% of GIST have a mutation in the KIT gene, leading to a continuously active KIT receptor, independent of its activating ligand [1]. KIT activation leads to overexpression of the protein CD117. In KIT-negative GIST, a small number are observed to have a mutation in platelet-derived growth factor receptor-a (PDGFRA). Dysregulated activation of either of these genes results in uncontrolled cell growth and survival. It is estimated that 10-15% of GIST do not have mutations in either KIT or PDGRFA, and while they are considered wild-type, they are shown to express high levels of KIT [1]. More recently, Novelli et al. found that the presence of proteins CD117 and DOG1 had the highest sensitivity and specificity for GIST [6]. The majority of GIST develop in the stomach (60%), with the jejunum and ileum representing the next most common site of involvement (30%) [7]. Several prognostic factors have been researched, most notably tumor location and mitotic index. Emory et al. found that GIST originating from the esophagus had the highest survival rate, followed by those that arose from the stomach, small bowel, colon/rectum, and omentum/mesentery in decreasing order [8]. Additionally, mitotic index, defined as the number of mitotic figures per high-power field (HPF), is reported an independent prognostic factor, with greater than 10 mitotic figures per 50 HPF showing the largest difference in survival in gastric GIST [8]. Small bowel GIST exhibited minimally different survival curves with respect to mitotic index. Age was also found to be an independent prognostic factor of survival in GIST [8]. Later research by Miettinen demonstrated that larger gastric GIST with a diameter of 10cm and 5 mitotic figures per 50 HPF carried a lower metastatic risk in comparison to gastric GIST with diameter of \u3e 5cm but with \u3e 5 mitotic figures per 50 HPF [9]. This may suggest that in gastric GIST, mitotic index carries the most prognostic value. Miettinen found that in intestinal GIST, a diameter of \u3e 5cm and \u3e 5 mitotic figures per HPF each independently carried a moderate or high risk of metastasis, respectively. Intestinal GIST carried a 39% tumor-related mortality rate, compared to 17% for gastric GIST [10,11]. Currently, surgery is the primary treatment modality for nonmetastatic GIST that is technically amenable to resection. Imatinib, a tyrosine kinase inhibitor (TKI), may be used as neoadjuvant therapy or as initial therapy for nonresectable disease [12]. Imatinib directly binds to the KIT protein and prevents further signaling [1]. This medication first demonstrated favorable treatment effects in 2002, with over 50% of the 147 patients showing at least a partial response to therapy [13]. Some patients develop resistance to Imatinib, prompting the development of alternative TKI therapy. Currently, Sunitinib is FDA approved for Imatinib-resistant GIST [14], with a host of other TKI’s and alternative therapies under investigation [1]. In 2012, Körner examined glucagon-like peptide-2 receptor (GLP-2) expression in a variety of neoplasm and found that 68% of the GISTs expressed this receptor in the intestinal myenteric plexus [15]. Additionally, this receptor was expressed in high density in patients with Crohn’s disease. Interestingly, this expression was absent in active or inactive ulcerative colitis as well as Hirschsprung’s disease [15]. Table 1: GIST with concurrent IBD. Author (ref) Age, Sex IBD Symptoms Location of GIST Imaging or operative findings Pfeffela, 1999 [16] 51, M CD Weight loss, Abdominal pain, Fever, Fatigue Ileum Large tumorous lesions in the right lower abdomen (terminal ileum) measuring 8 × 5 × 6 cm Grieco, 2002 [17] 57, F UC Melena, progressive anemia Ileum Solid mass in the left pelvic cavity with a diameter of 7 cm Mijandrusić Sincić, 2005 [18] 81, M CD Ileus Meckel’s diverticulum Dilated loops of intestine with large packets of gas and anti-peristalsis Kaiser, 2006 [19] 64, M UC Severe bleeding, abdominal distension Omentum 8 cm mass attached to greater omentum Ruffolo, 2010 [20] 59, M UC Rectal bleeding Rectum 0.5 cm GIST located 20 cm from anal adenocarcinoma Theodoropoulos, 2009 [21] 45, M CD Abdominal pain, vomiting, constipation, bloating Jejunum and Ileum 6 mm GIST within jejunoileal intussusception Bocker U, 2008 [22] 26, F CD Abdominal cramping, gastrointestinal bleeding Duodenum Ulcerated lesion noted 140 cm past proximal duodenum on enteroscopy Gianluca, 2016 [7] 38, M CD Asymptomatic Small bowel A mass found along the small bowel Gianluca, 2016 [7] 53, M UC Abrupt postoperative bleeding Stomach No evidences of masses at surgery. Gastric bleeding at endoscopy Present paper 40, M CD Abdominal pain, hematochezia Stomach 2.5 cm exophytic mass in the stomach with possible liver metastases CONCLUSION Our case of Crohn’s disease diagnosed with gastric GIST sheds light on a rare link between two separate disease entities native to the gastrointestinal system. While there exists a well-known association between inflammatory bowel disease and colon cancer, other malignancies are described much less frequently in the literature. The development of gastric GIST with underlying Crohn’s disease is a rare occurrence, but is one that should be kept in mind when evaluating patients with inflammatory bowel disease found to have new masses on imaging. References: 1. Corless CL, Barnett CM, Heinrich MC. Gastrointestinal stromal tumours: Origin and molecular oncology. Nat Rev Cancer. 2011;11(12):865-878. doi:10.1038/nrc3143 2. Katzka DA, Loftus E V., Camilleri M. Evolving molecular targets in the treatment of nonmalignant gastrointestinal diseases. Clin Pharmacol Ther. 2012;92(3):306-320. doi:10.1038/clpt.2012.77 3. Nilsson B, Bümming P, Meis-Kindblom JM, et al. Gastrointestinal stromal tumors: The incidence, prevalence, clinical course, and prognostication in the preimatinib mesylate era - A population-based study in western Sweden. Cancer. 2005;103(4):821-829. doi:10.1002/cncr.20862 4. Goettsch WG, Bos SD, Breekveldt-Postma N, Casparie M, Herings RMC, Hogendoorn PCW. Incidence of gastrointestinal stromal tumours is underestimated: Results of a nation-wide study. Eur J Cancer. 2005;41(18):2868-2872. doi:10.1016/j.ejca.2005.09.009 5. Ma GL, Murphy JD, Martinez ME, Sicklick JK. Epidemiology of gastrointestinal stromal tumors in the era of histology codes: Results of a population-based study. Cancer Epidemiol Biomarkers Prev. 2015;24(1):298-302. doi:10.1158/1055-9965.EPI-14-1002 6. Novelli M, Rossi S, Rodriguez-Justo M, et al. DOG1 and CD117 are the antibodies of choice in the diagnosis of gastrointestinal stromal tumours. Histopathology. 2010;57(2):259-270. doi:10.1111/j.1365-2559.2010.03624.x 7. Pellino G, Marcellinaro R, Candilio G, et al. The experience of a referral centre and literature overview of GIST and carcinoid tumours in inflammatory bowel diseases. Int J Surg. 2016;28:S133-S141. doi:10.1016/j.ijsu.2015.12.051 8. Emory TS, Sobin LH, Lukes L, Lee DH, O’Leary TJ. Prognosis of gastrointestinal smooth-muscle (stromal) tumors: Dependence on anatomic site. Am J Surg Pathol. 1999;23(1):82-87. doi:10.1097/00000478-199901000-00009 9. Miettinen M, Lasota J. Gastrointestinal stromal tumors: Pathology and prognosis at different sites. Semin Diagn Pathol. 2006. doi:10.1053/j.semdp.2006.09.001 10. Miettinen M, Sobin LH, Lasota J. Gastrointestinal stromal tumors of the stomach: A clinicopathologic, immunohistochemical, and molecular genetic study of 1765 cases with long-term follow-up. Am J Surg Pathol. 2005;29(1):52-68. doi:10.1097/01.pas.0000146010.92933.de 11. Miettinen M, Makhlouf H, Sobin LH, Lasota J. Gastrointestinal stromal tumors of the jejunum and ileum: A clinicopathologic, immunohistochemical, and molecular genetic study of 906 cases before imatinib with long-term follow-up. Am J Surg Pathol. 2006;30(4):477-489. doi:10.1097/00000478-200604000-00008 12. Demetri GD, Benjamin R, Blanke CD, et al. NCCN Task Force Report: Optimal Management of Patients with Gastrointestinal Stromal Tumor (GIST)--Expansion and Update of NCCN Clinical Practice Guidelines. Vol 2 Suppl 1.; 2004. 13. Eorge D Emetri GD, Argaret Von Ehren MM, Harles B Lanke CD, et al. The New Eng Land Jour Nal of Medicine EFFICACY AND SAFETY OF IMATINIB MESYLATE IN ADVANCED GASTROINTESTINAL STROMAL TUMORS A BSTRACT Background Constitutive Activation of KIT Receptor. Vol 347.; 2002. www.nejm.org. Accessed January 29, 2020. 14. Demetri GD, van Oosterom AT, Garrett CR, et al. Efficacy and safety of sunitinib in patients with advanced gastrointestinal stromal tumour after failure of imatinib: a randomised controlled trial. Lancet. 2006;368(9544):1329-1338. doi:10.1016/S0140-6736(06)69446-4 15. Körner M, Rehmann R, Reubi JC. GLP-2 receptors in human disease: High expression in gastrointestinal stromal tumors and Crohn’s disease. Mol Cell Endocrinol. 2012;364(1-2):46-53. doi:10.1016/j.mce.2012.08.008 16. Pfeffel F, Stiglbauer W, Depisch D, Oberhuber G, Raderer M, Scheithauer W. Coincidence of Crohn’s disease and a high-risk gastrointestinal stromal tumor of the terminal ileum. Digestion. 1999. doi:10.1159/000007684 17. Grieco A, Cavallaro A, Potenza AE, et al. Gastrointestinal stromal tumor (GIST) and ulcerative colitis. J Exp Clin Cancer Res. 2002. 18. Mijandrusic Sincic BM, Kovać D, Jašić M, Grbas H, Uravić M, Depolo A. Crohn’s disease and a gastrointestinal stromal tumor in an 81-year-old man - A rare coincidence. Zentralbl Chir. 2005. doi:10.1055/s-2005-918206 19. Kaiser AM, Kang JC, Tolazzi AR, Sherrod AE, Beart RW. Primary solitary extragastrointestinal stromal tumor of the greater omentum coexisting with ulcerative colitis. Dig Dis Sci. 2006;51(10):1850-1852. doi:10.1007/s10620-006-9217-y 20. Ruffolo C, Massani M, Rossi S, Caratozzolo E, Antoniutti M, Bassi N. Adenocarcinoma and GIST in ulcerative colitis. Int J Colorectal Dis. 2010;25(8):1027-1028. doi:10.1007/s00384-010-0905-x 21. Theodoropoulos GE, Linardoutsos D, Tsamis D, et al. Gastrointestinal stromal tumor causing small bowel intussusception in a patient with Crohn’s disease. World J Gastroenterol. 2009;15(41):5224-5227. doi:10.3748/wjg.15.5224 22. Böcker U, Löhr JM, Marx A. Twenty-six-year-old female with assumed Crohn’s disease and a gastrointestinal stromal tumor: Response. Inflamm Bowel Dis. 2009;15(4):489-490. doi:10.1002/ibd.2065

Negative-Energy Spinors and the Fock Space of Lattice Fermions at Finite Chemical Potential

Recently it was suggested that the problem of species doubling with Kogut-Susskind lattice fermions entails, at finite chemical potential, a confusion of particles with antiparticles. What happens instead is that the familiar correspondence of positive-energy spinors to particles, and of negative-energy spinors to antiparticles, ceases to hold for the Kogut-Susskind time derivative. To show this we highlight the role of the spinorial ``energy'' in the Osterwalder-Schrader reconstruction of the Fock space of non-interacting lattice fermions at zero temperature and nonzero chemical potential. We consider Kogut-Susskind fermions and, for comparison, fermions with an asymmetric one-step time derivative.Comment: 14p