On the invertibility of mappings arising in 2D grid generation problems

In adapting a grid for a Computational Fluid Dynamics problem one uses a mapping from the unit square onto itself that is the solution of an elliptic partial differential equation with rapidly varying coefficients. For a regular discretization this mapping has to be invertible. We will show that such result holds for general elliptic operators (in two dimensions). The Carleman-Hartman-Wintner Theorem will be fundamental in our proof. We will also explain why such a general result cannot be expected to hold for the (three-dimensional) cube

On a Dirichlet problem related to the invertibility of mappings arising in 2D grid generation problems

this paper depends strongly on a theorem of Carleman-HartmanWintner. This theorem is only true in two dimensional domains. In fact a straightforward generalization to more than two dimensional domains cannot be true. A counterexample to the proof of [15]forthe three dimensional case can be found by using a special harmonic function due to Kellogg [12]. This function is shown in [2]. A direct counterexample can be found in [13]. 2 Main result on smooth domain

Optimal estimates from below for biharmonic Green functions

Optimal pointwise estimates are derived for the biharmonic Green function under Dirichlet boundary conditions in arbitrary $C^{4,\gamma}$-smooth domains. Maximum principles do not exist for fourth order elliptic equations and the Green function may change sign. It prevents using a Harnack inequality as for second order problems and hence complicates the derivation of optimal estimates. The present estimate is obtained by an asymptotic analysis. The estimate shows that this Green function is positive near the singularity and that a possible negative part is small in the sense that it is bounded by the product of the squared distances to the boundary.Comment: 11 pages. To appear in "Proceedings of the AMS

A sufficient condition for a discrete spectrum of the Kirchhoff plate with an infinite peak

Sufficient conditions for a discrete spectrum of the biharmonic equation in a two-dimensional peak-shaped domain are established. Different boundary conditions from Kirchhoff's plate theory are imposed on the boundary and the results depend both on the type of boundary conditions and the sharpness exponent of the peak.Comment: 12 pages, 1 figure, submitted to Math. Mech. Compl. Sy

IMPROVING SEPTIC PATIENT OUTCOMES THROUGH A NURSE-LED QUALITY IMPROVEMENT PROJECT IN A RURAL, MIDWESTERN HOSPITAL

Sepsis is a complex and costly problem that continues to afflict hospitals across the globe (Napolitano, 2018). If it is not identified quickly and treated promptly, sepsis will progress to septic shock and create an increased risk of mortality (Napolitano, 2018). The best way to improve patient outcomes is through early detection and precise treatment (Rello, Valenzuela-Sánchez, Ruiz-Rodriguez, & Moyano, 2017). Through staff education and clinical decision tools, such as checklists, septic patients are more likely to have a favorable outcome (Storozuk, MacLeod, Freeman, & Banner, 2019). The purpose of this DNP project was to decrease fluid resuscitation fallouts and sepsis-related deaths by enhancing the education provided to nursing staff regarding the early warning signs of sepsis and providing a comprehensive checklist to help standardize care for septic patients admitted to the acute care unit at a rural midwestern hospital. Education regarding sepsis early warning signs and current sepsis protocols was provided to the acute care unit nursing staff through an in-person PowerPoint presentation. A comprehensive sepsis checklist was also implemented to standardize care for patients with sepsis. Using a quasi-experimental design, sepsis case fatality rates and fluid resuscitation pre-and post-intervention were examined. There was not a significant difference in sepsis case fatality rates (p = .98) or fluid resuscitation fallout rates pre-and post-intervention (p = .68). This project was implemented within a year of the COVID-19 pandemic, which likely impacted the results. While the results of this DNP project were not statistically significant, previous research does support the use of a sepsis checklist and nursing education

The phenomenological approach to religious experience in the theology of Paul Tillich

Thesis (M.A.)--Boston UniversityThe purpose of this thesis, as the title indicates, is to examine Tillich's concept of religious experience in the light of the phenomenological method. Tillich follows St. Augustine in his approach to the problem of the knowledge of God, believing that one finds clues to the basic nature of religious experience in man himself rather than in nature or by rational deduction. For Tillich the key question is not, "Does God exist?" but what does the experience of God mean in terms of finite human experience. To carry out his analysis of the ontological nature of human existence as finite and as raising the question of God (that which can account for the fact that despite all the threats to the contingent order, this order persists and therefore must be grounded on something non-contingent and unconditional) Tillich finds the phenomenological method a valuable tool. He does not accept this method uncritically though he is greatly indebted to Husserl and Heidegger. Tillich feels that if phenomenology is to be an adequate tool for theology, it must be "corrected" by reference to final revelation (the New Being in Christ) as the criterion of the type of experience that needs to be subjected to a phenomenological analysis. What Tillich finds missing in Husserl's approach is any recognition of the importance of the experience selected as normative. Not only must phenomenology describe meanings, it must also indicate where and to whom the experience has occurred. Nonetheless, before we can understand the particular approach Tillich takes, we must have a general knowledge of what the phenomenological movement is, especially as found in the writings of the two phenomenologists who directly influenced Tillich: Edmund Husserl and Martin Heidegger. Accordingly, the main body of the thesis begins with a chapter entitled "The Phenomenological Movement". This chapter begins with a general discussion of the principles and techniques of the phenomenological method. Here it is pointed out that one of the major tenets of the phenomenological method is to avoid pre-suppositional prejudices which would distort one's approach to the subject matter. The most common prejudice cited by the phenomenologists is that of the natural standpoint. This is discussed at some length by Husserl in Ideas as he wishes to point out that we do not primarily and basically experience the world as independent and "out there", somehow making impressions on a largely passive consciousness. Rather, our world is always permeated by certain valuations and perspective variations. This leads us to think that consciousness has a more active and formative role in the knowing process than common sense assumes. Therefore, to study objectively the world and thus make it possible for philosophy to be a "rigorous science" (Husserl), we must turn our gaze away from the natural objects to the structures of consciousness and concentrate on what it is that makes it possible for us to have the experiences that we do in the way that we do. In order to do this we must "bracket" (shut off) any references which assume the world as objectively given, and reduce what is presented to us until we are aware of their essences (what makes them what they are). One of the things we discover when we bracket the transcendent referents of consciousness (Husserl) is that we become aware that consciousness is intentional, i.e. it always has an object. In Heidegger, this concept is signifi• cant because it opens the possibility of discovering or becoming aware of Being, the goal of human existence and knowing. What intentionality reveals, for Heidegger, is the fundamental fact of the self's relatedness to the world. Man is Dasein, Being-in-the-world and an analysis of Being as manifest in this structure is our best means of access to the nature of Being. In Heidegger, then, phenomenology is changed from an abstract methodology (Husserl) to an existential hermeneutic. Tillich is indebted to both approaches. He is indebted to Husserl for his emphasis on the need to focus on what concepts mean, quite apart from their existential references, and he is indebted to Heidegger for his concept of phenomenology as the means by which one can discover Being (God in religious terms) as manifest in the human self. Therefore, the relation between Tillich and the phenomenologists is very close at key points. For example, Tillich's concern for the meaning of religious statements involves a bracketing of such questions as ''Does God exist?" which involve the "natural standpoint" assumption that there is a being "God", 'out there' somewhere and to whom one may choose to be related or not, or about whose existence one may argue. Tillich points out that unless what is meant by the word "God" is somehow present to man even in the very asking of the question and unless the question deals with a matter of threatening or saving one's being, we are not really dealing with a religious question at all. Granted that the experience of God is a reality to man before he becomes aware of it and makes explicit the question of God, how should we proceed to discover this fact? How may the "hidden" God of our experience be brought to awareness? Here Tillich relies on Heidegger and the use of phenomenology as an existential hermeneutic. If God is present to man even in his pre-reflective experience, and if what is meant by "God" is the Being or Reality which sustains the finite order and gives meaning to human life, then the kind of approach that must be taken is to analyze human experience in the attempt to see what is really given in that experience. What Tillich finds explicitly as the result of his ontological investigation by means of the phenomenological method is the amazing fact that despite the contingency and threat of nonbeing, experienced as anxiety, man still exists and can affirm his being. How is this possible? Only because, says Tillich, man experiences an unconditional power which gives him the courage to be and this power of being is what is meant by and pointed to by the religious symbol "God"