Mutual Dimension

We define the lower and upper mutual dimensions $mdim(x:y)$ and $Mdim(x:y)$ between any two points $x$ and $y$ in Euclidean space. Intuitively these are the lower and upper densities of the algorithmic information shared by $x$ and $y$. We show that these quantities satisfy the main desiderata for a satisfactory measure of mutual algorithmic information. Our main theorem, the data processing inequality for mutual dimension, says that, if $f:\mathbb{R}^m \rightarrow \mathbb{R}^n$ is computable and Lipschitz, then the inequalities $mdim(f(x):y) \leq mdim(x:y)$ and $Mdim(f(x):y) \leq Mdim(x:y)$ hold for all $x \in \mathbb{R}^m$ and $y \in \mathbb{R}^t$. We use this inequality and related inequalities that we prove in like fashion to establish conditions under which various classes of computable functions on Euclidean space preserve or otherwise transform mutual dimensions between points.Comment: This article is 29 pages and has been submitted to ACM Transactions on Computation Theory. A preliminary version of part of this material was reported at the 2013 Symposium on Theoretical Aspects of Computer Science in Kiel, German

Discovering Successful Pastoral Mentors Based on their Spiritual Gifts Cluster and Engaging Them in Mentorship in the Wisconsin Conference of the Seventh-day Adventist Church

Problem Many pastors are placed into pastoral ministry with passion and education but lack sufficient mentoring to help them succeed in ministry. These pastors would benefit from a mentor who could support them and/or grow them through intentional mentorship in specific agreed upon areas. The concept of mentoring is biblical, Jesus being a prime example, and one that is in line with the principles of the Seventh-day Adventist Church. There is no conflict with the concept of mentoring, but nothing practical has been put into place in the Wisconsin Conference to discover good pastoral mentors and then engage them in the mentoring of other pastors. A pastor’s ministry is greatly enhanced by mentoring. Method Research was done to identify an appropriate spiritual gifts survey. The identified survey was used to discover whether or not pastors, who are good mentors, have a consistent spiritual gifts cluster. This was achieved by interviewing over sixty pastors in the North American Division of Seventh-day Adventists who were considered by their peers to be good pastoral mentors. The interview, combined with the results from their spiritual gifts survey, gave the necessary data for this project. Following this research, pastors in the Wisconsin Conference with the identified spiritual gifts cluster were sought out and engaged in the mentoring of other pastors within the conference. Results The study of pastoral mentors in the North American Division showed that there is indeed a common spiritual gifts cluster in pastors who are considered by their peers to be good mentors. This data was used as a baseline to identify other pastors who might be good mentors. The pastors in the Wisconsin Conference of Seventh-day Adventists were used as the test group. A spiritual gifts survey was administered to the pastors who were willing to participate, and their data was compared to the baseline information. Four pastors were identified as having the spiritual gifts cluster of a good pastoral mentor. These pastors were then trained and engaged in mentoring other pastors over a period of six months. The feedback from the mentors and the mentees at the end of the process validated that it was indeed possible to identify good pastoral mentors based on their spiritual gifts. Conclusion Based on the data retrieved from the mentees and the pastors identified as potential mentors, it is apparent that there is a consistent spiritual gifts cluster among pastors who are good mentors. Our specific test was to see if pastor mentors who are not currently involved in mentoring could be identified based on their spiritual gifts, trained, and successfully engaged in mentoring. Of the four pastors in the Wisconsin Conference who fit the mentor profile, all four agreed to enter into a mentoring relationship. The results of their mentoring relationships indicate that each one of them proved themselves to be a competent and successful mentor. Based on this project, it has been found that using the spiritual gifts survey is a valid tool to discover pastors who can successfully mentor other pastor

Mutual dimension, data processing inequalities, and randomness

This dissertation makes progress in the area of constructive dimension, an effectivization of classical Hausdorff dimension. Using constructive dimension, one may assign a non-zero number to the dimension of individual sequences and individual points in Euclidean space. The primary objective of this dissertation is to develop a framework for mutual dimension, i.e., the density of algorithmic mutual information between two infinite objects, that has similar properties as those of classical Shannon mutual information. Chapter 1 presents a brief history of the development of constructive dimension along with its relationships to algorithmic information theory, algorithmic randomness, and classical Hausdorff dimension. Some applications of this field are discussed and an overview of each subsequent chapter is provided. Chapter 2 defines and analyzes the mutual algorithmic information between two points x and y at a given precision r. In fact, we describe two plausible definitions for this quantity, I_r(x:y) and J_r(x:y), and show that they are closely related. In order to do this, we prove and make use of a generalization of Levin\u27s coding theorem. Chapter 3 defines the lower and upper mutual dimensions between two points in Euclidean space and presents results on its basic properties. A large portion of this chapter is dedicated to studying data processing inequalities for points in Euclidean space. Generally speaking, a data processing inequality says that the amount of information between two objects cannot be significantly increased when one of the objects is processed by a particular type of transformation. We show that it is possible to derive several kinds of data processing inequalities for points in Euclidean space depending on the continuity properties of the computable transformation that is used. Chapter 4 focuses on extending mutual dimension to sequences over an arbitrary alphabet. First, we prove that the mutual dimension between two sequences is equal to the mutual dimension between the sequences\u27 real representations. Using this result, we show that the lower and upper mutual dimensions between sequences have nice properties. We also provide an analysis of data processing inequalities for sequences where transformations are represented by Turing functionals whose use and yield are bounded by computable functions. Chapter 5 relates mutual dimension to the study of algorithmic randomness. Specifically, we show that a particular class of coupled random sequences, i.e., sequences generated by independent tosses of coins whose biases may or may not be correlated, can be characterized by classical Shannon mutual information. We also prove that any two sequences that are independently random with respect to computable probability measures have zero mutual dimension and that the converse of this statement is not true. We conclude this chapter with some initial investigations on Billingsley mutual dimension, i.e., mutual dimension with respect to probability measures, and prove the existence of a mutual divergence formula

On the Origin of Superoxide Dismutase: An Evolutionary Perspective of Superoxide-Mediated Redox Signaling.

The field of free radical biology originated with the discovery of superoxide dismutase (SOD) in 1969. Over the last 5 decades, a plethora of research has been performed in species ranging from bacteria to mammals that has elucidated the molecular reaction, subcellular location, and specific isoforms of SOD. However, while humans have only begun to study this class of enzymes over the past 50 years, it has been estimated that these enzymes have existed for billions of years, and may be some of the original enzymes found in primitive life. As life evolved over this expanse of time, these enzymes have taken on new and different functional roles potentially in contrast to how they were originally derived. Herein, examination of the evolutionary history of these enzymes provides both an explanation and further inquiries into the modern-day role of SOD in physiology and disease

Defining the Nuanced Nature of Redox Biology in Post-Traumatic Stress Disorder

Post-traumatic stress disorder (PTSD) is a mental health disorder that arises after experiencing or witnessing a traumatic event. Despite affecting around 7% of the population, there are currently no definitive biological signatures or biomarkers used in the diagnosis of PTSD. Thus, the search for clinically relevant and reproducible biomarkers has been a major focus of the field. With significant advances of large-scale multi-omic studies that include genomic, proteomic, and metabolomic data, promising findings have been made, but the field still has fallen short. Amongst the possible biomarkers examined, one area is often overlooked, understudied, or inappropriately investigated: the field of redox biology. Redox molecules are free radical and/or reactive species that are generated as a consequence of the necessity of electron movement for life. These reactive molecules, too, are essential for life, but in excess are denoted as oxidative stress and often associated with many diseases. The few studies that have examined redox biology parameters have often utilized outdated and nonspecific methods, as well as have reported confounding results, which has made it difficult to conclude the role for redox in PTSD. Herein, we provide a foundation of how redox biology may underlie diseases like PTSD, critically examine redox studies of PTSD, and provide future directions the field can implement to enhance standardization, reproducibility, and accuracy of redox assessments for the use of diagnosis, prognosis, and therapy of this debilitating mental health disorder

Absence of manganese superoxide dismutase delays p53-induced tumor formation.

BACKGROUND: Manganese superoxide dismutase (MnSOD) is a mitochondrial antioxidant enzyme that is down-regulated in a majority of cancers. Due to this observation, as well as MnSOD\u27s potent antioxidant enzymatic activity, MnSOD has been suggested as a tumor suppressor for over 30 years. However, testing this postulate has proven difficult due to the early post-natal lethality of the MnSOD constitutive knock-out mouse. We have previously used a conditional tissue-specific MnSOD knock-out mouse to study the effects of MnSOD loss on the development of various cell types, but long-term cancer development studies have not been performed. We hypothesized the complete loss of MnSOD would significantly increase the rate of tumor formation in a tissue-specific manner. RESULTS: Utilizing a hematopoietic stem cell specific Cre-recombinase mouse model, we created pan-hematopoietic cell MnSOD knock-out mice. Additionally, we combined this MnSOD knock-out with two well established models of lymphoma development: B-lymphocyte specific Myc over-expression and conditional pan-hematopoietic cell p53 knock-out. Mice were allowed to age unchallenged until illness or death had occurred. Contrary to our initial hypothesis, the loss of MnSOD alone was insufficient in causing an increase in tumor formation, but did cause significant life-shortening skin pathology in a strain-dependent manner. Moreover, the loss of MnSOD in conjunction with either Myc overexpression or p53 knock-out did not accelerate tumor formation, and in fact delayed lymphomagenesis in the p53 knock-out model. CONCLUSIONS: Our findings strongly suggest that MnSOD does not act as a classical tumor suppressor in hematological tissues. Additionally, the complete loss of MnSOD may actually protect from tumor development by the creation of an unfavorable redox environment for tumor progression. In summary, these results in combination with our previous work suggest that MnSOD needs to be tightly regulated for proper cellular homeostasis, and altering the activity in either direction may lead to cellular dysfunction, oncogenesis, or death