On the number of NN-free elements with prescribed trace

In this paper we derive a formula for the number of $N$-free elements over a finite field $\mathbb{F}_q$ with prescribed trace, in particular trace zero, in terms of Gaussian periods. As a consequence, we derive a simple explicit formula for the number of primitive elements, in quartic extensions of Mersenne prime fields, having absolute trace zero. We also give a simple formula in the case when $Q = (q^m-1)/(q-1)$ is prime. More generally, for a positive integer $N$ whose prime factors divide $Q$ and satisfy the so called semi-primitive condition, we give an explicit formula for the number of $N$-free elements with arbitrary trace. In addition we show that if all the prime factors of $q-1$ divide $m$, then the number of primitive elements in $\mathbb{F}_{q^m}$, with prescribed non-zero trace, is uniformly distributed. Finally we explore the related number, $P_{q, m, N}(c)$, of elements in $\mathbb{F}_{q^m}$ with multiplicative order $N$ and having trace $c \in \mathbb{F}_q$. Let $N \mid q^m-1$ such that $L_Q \mid N$, where $L_Q$ is the largest factor of $q^m-1$ with the same radical as that of $Q$. We show there exists an element in $\mathbb{F}_{q^m}^*$ of (large) order $N$ with trace $0$ if and only if $m \neq 2$ and $(q,m) \neq (4,3)$. Moreover we derive an explicit formula for the number of elements in $\mathbb{F}_{p^4}$ with the corresponding large order $L_Q = 2(p+1)(p^2+1)$ and having absolute trace zero, where $p$ is a Mersenne prime

Trace element nutrition in the tropical zebu cattle type : lessons from the Gilgel Gibe catchment, Ethiopia

In the tropics, cattle are majorly dependent on natural pastures growing on poor soils and as a result prone to trace element deficiencies. This work investigated several aspects of trace element nutrition in zebu (Bos indicus) cattle, the most widely used cattle type in the tropics. Through several trials conducted at the Gilgel Gibe catchment, Ethiopia, we aimed to study factors influencing the flow of trace elements from soil to plant and grazing zebu cattle, and to investigate the impact of trace element status on health, production and trace elements in animal products. Overall, trace element, e.g., Cu, deficiencies were widely prevalent. Environment and certain grazing strategies were intimately related and affected trace element supply whereas cattle type affected trace element status. The effect of trace element status on trace elements in animal products was obvious, whereas the impact on health and production was less clear-cut. Key words: trace elements, minerals, cattle, zebu, Bos indicus, Ethiopia, tropic

Monotone trace functions of several variables

We investigate monotone operator functions of several variables under a trace or a trace-like functional. In particular, we prove the inequality \tau(x_1... x_n)\le\tau(y_1... y_n) for a trace \tau on a C^*-algebra and abelian n-tuples (x_1,...,x_n)\le (y_1,...,y_n) of positive elements. We formulate and prove Jensen's inequality for expectation values, and we study matrix functions of several variables which are convex or monotone with respect to the weak majorization for matrices

Metals in the shell of Bathymodiolus azoricus from a hydrothermal vent site on the Mid-Atlantic Ridge

Specimens of the mussel Bathymodiolus azoricus were collected from Menez Gwen, a relatively shallow (850 m) hydrothermal vent field on the Mid-Atlantic Ridge. Each bivalve shell (n = 21) was individually cleaned by selective chemical. The residual crystal matrix of each shell was individually analysed for the concentrations of the minor elements magnesium and strontium and the trace elements iron, manganese, copper and zinc. The chemical composition of the crystal matrix is unusual. B. azoricus is identified as a species having one of the most strontium impoverished shells amongst the marine molluscs. For a bimineral species the magnesium concentration is also extraordinary low. Despite originating from a trace metal rich environment; the metal concentrations in the shells were exceptionally low. Mean concentrations of iron, manganese, copper and zinc were 20.6, 3.7, 0.6 and 9.4 microg g(-1) respectively. Minor and trace element concentrations exhibited a marked intra-population variability. Copper concentrations increased and iron and zinc concentrations decreased with increasing shell weight. Due to its insensitivity to the high environmental levels of trace elements and the variability in intra-population concentrations induced by shell weight the crystal matrix of the shell of B. azoricus has little potential for use in environmental trace metal monitoring in areas contiguous to deep-sea hydrothermal vents.info:eu-repo/semantics/publishedVersio

Geochemical-geophysical investigations, Fairbanks district

Trace element distribution in a subarctic valley in the Cleary Hill area of the Fairbanks gold district has been studied. Zinc and arsenic have been found excellent pathfinder elements for auriferous deposits. Methods of analysis for copper, lead, zinc, molybdenum, silver and arsenic as well as heavy metals are discussed. The University of Alaska method #2 has been improved, Terrain, slope, and frozen ground have little effect upon the distribution of trace elements associated with the Cleary H i l l vein. A new method for the determination of zinc using dilute acid is proposed. Analysis of geochemical data by trend surface procedures proved effective for localization of anomalies

Trace Element Estimation – Methods & Clinical Context

Understanding the effects of trace metals on human health is as complex as it is fascinating. As mentioned earlier, the high concentrations may prove toxic, as also, depletion in the concentration of the essential trace elements may cause various metabolic instabilities due to enzyme dysfunction. In the era of rapid industrialization and technological advances, it is imperative to watch keenly for contamination of the environment and its vital composition from heavy metal wastes emanating out of industries. Many metabolic disorders in man are accompanied by alterations in the concentration of one or more trace elements in some body fluid, especially blood serum or plasma It is thus important to update ourselves with various techniques available for such determinations, their operational aspects, advantages / disadvantages etc. More recently, element analysis from hair and nail has been stated as the best indices of such contamination and has also been discussed in this article

Spectral characterization of sums of commutators II

For countably generated ideals, \Jc, of B(\Hil), geometric stability is necessary for the canonical spectral characterization of sums of (\Jc,B(\Hil))--commutators to hold. This answers a question raised by Dykema, Figiel, Weiss and Wodzicki. There are some ideals, \Jc, having quasi--nilpotent elements that are not sums of (\Jc,B(\Hil))--commutators. Also, every trace on every geometrically stable ideal is a spectral trace

Scalar multiplication in compressed coordinates in the trace-zero subgroup

We consider trace-zero subgroups of elliptic curves over a degree three field extension. The elements of these groups can be represented in compressed coordinates, i.e. via the two coefficients of the line that passes through the point and its two Frobenius conjugates. In this paper we give the first algorithm to compute scalar multiplication in the degree three trace-zero subgroup using these coordinates.Comment: 23 page