Extension of the two-variable Pierce-Birkhoff conjecture to generalized polynomials

Let R denote the reals, and let h: R^n --> R be a continuous, piecewise-polynomial function. The Pierce-Birkhoff conjecture (1956) is that any such h is representable in the form sup_i inf_j f_{ij}, for some finite collection of polynomials f_{ij} in R[x_1,...,x_n]. (A simple example is h(x_1) = |x_1| = sup{x_1, -x_1}.) In 1984, L. Mahe and, independently, G. Efroymson, proved this for n 2. In this paper we prove an analogous result for "generalized polynomials" (also known as signomials), i.e., where the exponents are allowed to be arbitrary real numbers, and not just natural numbers; in this version, we restrict to the positive orthant, where each x_i > 0. As before, our methods work only for n < 3.Comment: 16 pages, 4 figure

Continuous Pythagoras numbers for rational quadratic forms

We introduce the concept of the continuous Pythagoras number Pc(S) of a subset S of a commutative topological ring to be, roughly, the least number m ≤ ∞ such that the set of sums of squares of elements of S can be represented as sums of m squares of elements of S, by means of m continuous functions. Heilbronn had already shown that Pc(Q) = 4. Letting Ln(F) be the set of linear n-ary forms over the field F, we show that Pc(Ln(R)) = n. We then allow continuously varying nonnegative rational weights on the m square summands. If these continuous weight functions and the continuous functions giving the coefficients of the m linear forms, are required to be Q-rational functions of the coefficients of the given positive semidefinite quadratic forms, then we show that Pc(L1(R)) = 1 and Pc(Ln(R)) = ∞ for n \u3e 1. However, if only the product of the weight functions and the coefficient functions is required to be continuous, then n ≤ Pc(Ln(R)) \u3c [n!e] (where e is the base of the natural logarithms) and 2 \u3c Pc(L2(R)); we conjecture that n \u3c Pc(Ln(R)) also for n \u3e 2. On the other hand, if these weight functions and coefficient functions are required only to be rational in the weaker sense of taking rational values at rational arguments, then Pc(L2(Q)) = 2, and we conjecture that Pc(Ln(Q)) = n also for n \u3e 2. © 1987

Relationships Between Vocational Identity, Substance Use and Criminal Thinking Among Emerging Adults

Successful formation of a self-chosen, purposeful identity in personal, social, educational and vocational areas is a primary task for emerging adults, with failure to do so often resulting in cycles of substance use, unemployment, and delinquent/criminal behavior. The purpose of this quantitative study was to examine relationships between vocational identity, substance use, and criminal thinking within the population of emerging adults. The expectancy value theory of motivation, which states that identity may be a motivational construct between self-efficacy and subjective self-values, provided the foundation for the study. The online inventory platform PsychData was used to garner data from a sample of 78 emerging adults measuring vocational identity (using the Vocational Identity Status Assessment [VISA]), substance use (using the CAGE-AID questionnaire), and criminal thinking (using the General Criminal Thinking-GCT scale of the Psychological Inventory of Criminal Thinking Styles). A bivariate correlational analysis using SPSS allowed for comparison of the 6 vocational identity statuses of Achieved, Searching Moratorium, Moratorium, Foreclosed, Diffused, and Undifferentiated, against the CAGE-AID scores and the GCT scores for possible relationships. The study did not result in significant correlations between variables; however, poststudy analysis revealed that the Diffused level of vocational identity, which is generally associated with the most negative life patterns, was strongly reflected in the responses of 25-year-old participants. Further research on the significance of vocational identity among older emerging adults may serve both the individual and society through encouraging successful transition to stable and healthy adult roles

A Biochemical Approach to Characterize a Divergent Trypanosoma brucei Mitochondrial DNA Polymerase, POLIB

Trypanosoma brucei is a single-celled parasitic protist that causes African sleeping sickness in people and nagana in cattle in sub-Saharan Africa. T. brucei and related trypanosomatid parasites contain an unusual catenated mitochondrial genome known as kinetoplast DNA (kDNA) composed of dozens of 23 kb maxicircles and thousands of 1 kb minicircles. The kDNA structure and replication mechanism are divergent from other eukaryotes and essential for parasite survival. POLIB is one of three Family A DNA polymerases that are independently essential to maintain the kDNA network, and has been implicated in minicircle replication. However, the division of labor among the paralogs, particularly which might be a replicative, proofreading enzyme remains enigmatic. De novo modelling of POLIB suggested a structure that is divergent from all other Family A polymerases in which the thumb subdomain contains a 369 amino acid insertion with homology to DEDDh DnaQ family 3\u27-5\u27 exonucleases. In chapter 2, we explore the polymerase and exonuclease activity of POLIB using purified his-tagged v recombinant variants that have been truncated and codon optimized for expression in E. coli. Using this recombinant protein variants we demonstrated that the 3\u27-5\u27 exonuclease activity of recombinant POLIB prefers DNA vs. RNA substrates and prefers singlestranded vs. double-stranded substrates. POLIB exonuclease activity prevails over polymerase activity on DNA substrates at pH 8.0, while DNA primer extension is favored at pH 6.0. Mutations that ablate POLIB polymerase activity slow the exonuclease rate suggesting crosstalk between the domains. We show that POLIB extends an RNA primer more efficiently than a DNA primer in the presence of dNTPs but does not incorporate rNTPs efficiently using either RNA or DNA primers. Immunoprecipitation of Pol I-like paralogs from T. brucei corroborate the pH selectivity and RNA primer preferences of POLIB and revealed that the other paralogs efficiently extend a DNA primer. We also show that overexpression of the exonuclease-ablated variant of POLIB in T. brucei results in a loss of fitness and impacts kDNA replication. We postulate that this unique enzyme and the machinery associated with it in the process of kDNA replication could be excellent drug targets worthy of further study