Partial metric spaces with negative distances and fixed point theorems

In this paper we consider partial metric spaces in the sense of O'Neill. We introduce the notions of strong partial metric spaces and Cauchy functions. We prove a fixed point theorem for such spaces and functions that improves Matthews' contraction mapping theorem in two ways. First, the existence of fixed points now holds for a wider class of functions and spaces. Second, our theorem also allows for fixed points with nonzero self-distances. We also prove fixed point theorems for orbitally $r$-contractive and orbitally $\phi_r$-contractive maps. We then apply our results to give alternative proofs of some of the other known fixed point theorems in the context of partial metric spaces.Comment: 19 page

Generalized Metrics

The distance on a set is a comparative function. The smaller the distance between two elements of that set, the closer, or more similar, those elements are. Fr\'echet axiomatized the distance into what is today known as a metric. In this thesis we study the generalization of Fr\'echet's axioms in various ways including a partial metric, strong partial metric, partial $n-\mathfrak{M}$etric and strong partial $n-\mathfrak{M}$etric. Those generalizations allow for negative distances, non-zero distances between a point and itself and even the comparison of $n-$tuples. We then present the scoring of a DNA sequence, a comparative function that is not a metric but can be modeled as a strong partial metric. Using the generalized metrics mentioned above we create topological spaces and investigate convergence, limits and continuity in them. As an application, we discuss contractiveness in the language of our generalized metrics and present Banach-like fixed, common fixed and coincidence point theorems.Comment: Thesi

Gastric Bezoar

History of present illness: A 12-year-old female with no past medical history presented with abdominal pain for 3 months. The pain was intermittent, located at the epigastric region, non-radiating, fluctuating intensity up to 8/10, and had worsened over the past month. She did not have fever, nausea, vomiting, diarrhea, constipation, or blood in her stool. The patient also endorsed hair loss over the same time period and noted that her previously long hair was now short and thin. On exam, patient was noted to have shoulder-length hair, a soft, non-distended abdomen with mild tenderness to the epigastric region, and a 5cm hard mass palpated at the epigastrium. Significant findings: In the abdominal radiograph, a nonspecific and non-obstructive bowel gas pattern with no air-fluid level was noted, however the stomach was distended with soft tissue. The CT abdomen/pelvis revealed a distended stomach with undigested heterogeneous contents (presumed bezoar). Discussion: A bezoar is a mass of incompletely digested material typically originating in the stomach and consisting of vegetable fibers, hair, or drugs.1 Bezoars develop after ingested foreign material accumulates in the gastrointestinal tract due to indigestibility, gastric outlet obstruction, or intestinal stasis. Trichobezoars are comprised of hair and classically form in young females with an underlying psychiatric disorder resulting in the urge to pull one’s hair out (trichotillomania) and swallow it (trichophagia).2,3 Gastric bezoars are rare with an approximate incidence of 0.3 percent of patients undergoing upper endoscopy.4 Patients tend to remain asymptomatic for long periods, but may develop abdominal pain, nausea/vomiting, early satiety, anorexia, and weight loss.5 Complications may include gastrointestinal ulcerations, perforations, intussusception, pancreatitis, obstructive jaundice, and death.6-8 The diagnosis of a gastric bezoar can be made using plain films, ultrasound, or CT, and is confirmed via upper endoscopy.2,9 Management of gastric bezoars should be dictated by its composition and may include chemical dissolution, endoscopic removal, or surgical removal.6,9 Our patient underwent urgent surgical removal of the gastric trichobezoar

Binary Metrics

We define a binary metric as a symmetric, distributive lattice ordered magma-valued function of two variables, satisfying a “triangle inequality . Using the notion of a Kuratowski topology, in which topologies are specified by closed sets rather than open sets, we prove that every topology is induced by a binary metric. We conclude with a discussion on the relation between binary metrics and some separation axioms