Codimension one spheres in Rn with double tangent balls

In contrast to the situation in $\text{R}$³, where a 2-sphere with double tangent balls at each point must be tamely embedded in $\text{R}$³, there exist wild (n?1)-spheres in $\text{R}$ⁿ for n>3 with this same geometric property. However, if the sphere Σ is tame moduio a subset X that lies in a polyhedron P that is tame in Σ, the dimension of P is less than n?2, n>4, and Σ has double tangent balls over X, then Σ must be tame in $\text{R}$ⁿ. Also if the tangent balls extend over P and are pairwise congruent, the dimensional restriction on P can be dropped. Examples are given to support the necessity of the hypotheses of the included theorems.     

Polypseudologarithms revisited

Lee, in a series of papers, described a unified formulation of the statistical thermodynamics of ideal quantum gases in terms of the polylogarithm functions, . It is aimed here to investigate the functions , for s=0,−1,−2,…, which are, following Lee, referred to as the polypseudologarithms (or polypseudologs) of order n≡−s. Various known results regarding polypseudologs, mainly obtained in widely differing contexts and currently scattered throughout the literature, have been brought together along with many new results and insights and they all have been proved in a simple and unified manner. In addition, a new general explicit closed-form formula for these functions involving the Carlitz-Scoville higher tangent numbers has been established.     

The participation puzzle with reference-dependent expected utility preferences

Expected utility theory with a smooth utility function predicts that, when allocating wealth between a risky and a riskless asset, investors allocate a positive amount to the risky asset whenever its expected return exceeds the riskless rate of return. A large number of people invest none of their wealth in risky assets, though, leading to the ”participation puzzle.” This paper explores whether the participation puzzle can be addressed when the utility function has a kink at the reference wealth level. It shows that when the reference wealth level is initial wealth increased by the riskless rate of return, there exists a range of expected excess returns for the risky asset for which the investor takes no position. Moreover, this range of expected excess returns is described by comparing a common performance measure of stock returns, the Omega Function, to a function of preference parameters. However, if the reference wealth level is any other constant, the usual expected utility prediction holds and investors allocate at least some of their wealth to the risky asset whenever it has a positive expected excess return.     

Optimization designs and performance comparison of two CUSUM schemes for monitoring process shifts in mean and variance

In statistical process control (SPC), when dealing with a quality characteristic x that is a variable, it is usually necessary to monitor both the mean value and variability. This article proposes an optimization algorithm (called the holistic algorithm) to design the CUSUM charts for this purpose. It facilitates the determination of the charting parameters of the CUSUM charts and considerably or significantly increases their overall detection effectiveness. A single CUSUM chart (called the ABS CUSUM chart) has been developed by the holistic algorithm and fully investigated. This chart is able to detect two-sided mean shifts and increasing variance shifts by inspecting the absolute value of sample mean shift. The results of performance studies show that the overall performance of the ABS CUSUM chart is nearly as good as an optimal 3-CUSUM scheme (a scheme incorporating three individual CUSUM charts). However, since the ABS CUSUM chart is easier for implementation and design, it may be more suitable for many SPC applications in which both mean and variance of a variable have to be monitored.     

Units of measurement

It is shown that the symbols for the fundamental units of mechanics, namely length, time and mass, are capable of a meaningful interpretation as positive real parameters. Then a suitable parameter domain allows one to take the derived units into account. The formal manipulations usually carried out with symbols of physical quantities, involving real coefficients and units of different kinds affected by rational exponents, are consistent with that interpretation. The note is expected to serve as a well-motivated introduction to a deeper study of dimensional analysis.     

Corrigendum to “reduced tangent cones and conductor at multiplanar isolated singularities”

We explicitly explain how to fix a wrong statement in a preliminary result of our previous paper on the conductor at a multiplanar singularity.     

Glass Transition (Tg) and Dielectric Loss in Na2O-Bi2O3-TiO2-P2O5 GLASSES

Thermal and dielectric loss properties of Na₂O-Bi₂O₃-TiO₂-P₂O₅ phosphate glasses, have been studied by the differential scanning calorimetry (DSC) and electrical factor loss (tgi) measurements. The variation of tgi has been followed along two lines inside the quaternary systems Na₂O-Bi₂O₃-5TiO₂-P₂O₅ and Na₂O-5Bi₂O₃-TiO₂-P₂O₅. Experiments have been carried out from ambient temperature to 450°C and show a strong influence of sodium, bismuth and titanium ions on T_g and tgi.     

Bayesian analysis of loss reserving using dynamic models with generalized beta distribution

A Bayesian approach is presented in order to model long tail loss reserving data using the generalized beta distribution of the second kind (GB2) with dynamic mean functions and mixture model representation. The proposed GB2 distribution provides a flexible probability density function, which nests various distributions with light and heavy tails, to facilitate accurate loss reserving in insurance applications. Extending the mean functions to include the state space and threshold models provides a dynamic approach to allow for irregular claims behaviors and legislative change which may occur during the claims settlement period. The mixture of GB2 distributions is proposed as a mean of modeling the unobserved heterogeneity which arises from the incidence of very large claims in the loss reserving data. It is shown through both simulation study and forecasting that model parameters are estimated with high accuracy.