Strictly Hermitian Positive Definite Functions

Let H be any complex inner product space with inner product . We say that f : C -->C is Hermitian positive definite on H if the matrix (f())_{r,s=1}^n \eqno(*) is Hermitian positive definite for all choice of z^1,...,z^n in H, all n. It is strictly Hermitian positive definite if the matrix (*) is also non-singular for any choice of distinct z^1,...,z^n in H. In this article we prove that if dim H >= 3, then f is Hermitian positive definite on H if and only if f(z) = \sum_{k,m =0}^\infty b_{k,m} z^k \oz^m \eqno(**) where \oz is the conjugate of z, b_{k,m}>= 0, all k,m in Z_+, and the series converges for all z in C. We also prove that f of the form (**) is strictly Hermitian positive definite on any H if and only if the set $$J={(k, m) : b_{k,m}> 0}$$ is such that (0,0) is in J, and every arithmetic sequence in Z intersects the values {k-m : (k,m)\in J} an infinite number of times.Comment: 25 page

Meaningful Measurement: The Role of Assessments in Improving High School Education in the Twenty-First Century

Compiles analyses and policy recommendations on developing common standards and assessments, supporting timely use of data to inform decisions and to improve teaching and learning, and investing in research to prepare students for the global economy

Performance of Oil Pumping Rings: An Analytical and Experimental Study

A steady-state design computer program was developed to predict the performance of pumping rings as functions of geometry, applied loading, speed, ring modulus, and fluid viscosity. Additional analyses were developed to predict transient behavior of the ring and the effects of temperature rises occurring in the hydrodynamic film between the ring and shaft. The analysis was initially compared with previous experimental data and then used to design additional rings for further testing. Tests were performed with Rulon, carbon-graphite, and babbit rings. The design analysis was used to size all of the rings and to select the ranges of clearances, thickness, and loading. Although full quantitative agreement was lacking, relative agreement existed in that rings that were predicted to perform well theoretically, generally performed well experimentally. Some causes for discrepanices between theory and experiment are believed to be due to starvation, leakage past the secondary seal at high pressures, and uncertainties in the small clearances and local inlet temperatures to the pumping ring. A separate preliminary analysis was performed for a pumping Leningrader seal. This anlaysis can be used to predict the film thickness and flow rate thr ough the seal as a function of pressure, speed, loading, and geometry

Necessary conditions for uniqueness in L1-approximation

AbstractLet K be a compact subset of Rm with K = int K. Necessary conditions on an n-dimensional subspace Un of C(K) are given so that for each ƒ ϵ C(K) there exists a unique best L1(w)-approximation Un, for every fixed positive weight function w

Robert I. Grody, Kerry Kaplan vs. State of Indiana (Brief of Appellants) In the Supreme Court of Indiana

Appeal from the Monroe Circuit Court The Honorable Nat U. Hill, Judge No. 1270-S294https://www.repository.law.indiana.edu/histdocs/1028/thumbnail.jp

An Electron Microscopic Study of Human Epidermis After Acetone and Kerosene Administration

An electron microscopic study has been done on normal human skin exposed either 30 or 90 minutes to two lipid solvents (acetone and kerosene) commonly encountered in occupational situations. After topical corneum and stratum spinosum. Keratinized cells, keratohyalin cells, spinous cells, and their nuclear and cell envelopes were affected in somewhat different ways by the two solvents. Marked intracellular edema of keratinized cells and vacuolation of spinous cells were seen after exposure to acetone, while large lacunae and disappearance of the keratin pattern in keratinized cells and evidence of cytolysis of spinous cells were found only after kerosene exposure. These cytoplasmic and nuclear degenerative changes were more pronounced after 90 minutes' than after 30 minutes' exposure. Many cell membranes were ruptured and ruthenium red localization showed that desmosomes and other functional complexes were often disrupted. Seventy-two hours after discontinuation of the solvent application, a high degree of restoration of the normal ultrastructural pattern was observed in superficial layers, while residual evidence of cell degeneration and evidence of early reactive changes were found in the viable portions of the epidermis