The Word Problem for Omega-Terms over the Trotter-Weil Hierarchy

For two given $\omega$-terms $\alpha$ and $\beta$, the word problem for $\omega$-terms over a variety $\boldsymbol{\mathrm{V}}$ asks whether $\alpha=\beta$ in all monoids in $\boldsymbol{\mathrm{V}}$. We show that the word problem for $\omega$-terms over each level of the Trotter-Weil Hierarchy is decidable. More precisely, for every fixed variety in the Trotter-Weil Hierarchy, our approach yields an algorithm in nondeterministic logarithmic space (NL). In addition, we provide deterministic polynomial time algorithms which are more efficient than straightforward translations of the NL-algorithms. As an application of our results, we show that separability by the so-called corners of the Trotter-Weil Hierarchy is witnessed by $\omega$-terms (this property is also known as $\omega$-reducibility). In particular, the separation problem for the corners of the Trotter-Weil Hierarchy is decidable

Serum Calcium Levels Are Associated with Novel Cardiometabolic Risk Factors in the Population-Based CoLaus Study

BACKGROUND: Associations of serum calcium levels with the metabolic syndrome and other novel cardio-metabolic risk factors not classically included in the metabolic syndrome, such as those involved in oxidative stress, are largely unexplored. We analyzed the association of albumin-corrected serum calcium levels with conventional and non-conventional cardio-metabolic risk factors in a general adult population. METHODOLOGY/PRINCIPAL FINDINGS: The CoLaus study is a population-based study including Caucasians from Lausanne, Switzerland. The metabolic syndrome was defined using the Adult Treatment Panel III criteria. Non-conventional cardio-metabolic risk factors considered included: fat mass, leptin, LDL particle size, apolipoprotein B, fasting insulin, adiponectin, ultrasensitive CRP, serum uric acid, homocysteine, and gamma-glutamyltransferase. We used adjusted standardized multivariable regression to compare the association of each cardio-metabolic risk factor with albumin-corrected serum calcium. We assessed associations of albumin-corrected serum calcium with the cumulative number of non-conventional cardio-metabolic risk factors. We analyzed 4,231 subjects aged 35 to 75 years. Corrected serum calcium increased with both the number of the metabolic syndrome components and the number of non-conventional cardio-metabolic risk factors, independently of the metabolic syndrome and BMI. Among conventional and non-conventional cardio-metabolic risk factors, the strongest positive associations were found for factors related to oxidative stress (uric acid, homocysteine and gamma-glutamyltransferase). Adiponectin had the strongest negative association with corrected serum calcium. CONCLUSIONS/SIGNIFICANCE: Serum calcium was associated with the metabolic syndrome and with non-conventional cardio-metabolic risk factors independently of the metabolic syndrome. Associations with uric acid, homocysteine and gamma-glutamyltransferase were the strongest. These novel findings suggest that serum calcium levels may be associated with cardiovascular risk via oxidative stress