Abstract:
In our paper we consider the notion of determinant of Clifford algebra elements. We present some new formulas for determinant of Clifford algebra elements for the cases of dimension 4 and 5. Also we consider the notion of trace of Clifford algebra elements. We use the generalization of the Pauli's theorem for 2 sets of elements that satisfy the main anticommutation conditions of Clifford algebra.

Abstract:
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.

Abstract:
In this note an improvement of the Katz's bound on the number of elements in a finite field with given trace and norm is given. The improvement is obtained by reducing the problem to estimating the number of rational points on certain toric Calabi-Yau hypersurface, and then to use detailed cohomological calculations by Rojas-Leon and the second author for such toric hypersurfaces.

Abstract:
A study was undertaken to determine the major and trace elements in Donghu lake. The total concentrations of a large suite of elements (Al, As, B, Ba, Ca, Cr, Cu, Fe, K, Mg, Mn, Na, P, S, Si, Sr, V, Zn) were determined using spectroscopic techniques. The concentrations of trace and major elements were measured in water samples collected from five locations of Donghu lake from May to October 2007. Since, the surface water is polluted by the discharge of municipal and industrial wastewaters, it is suffering lack of clean water in Donghu, although there is adequacy of total water resource. With the rapid growth of economy in the area, the water pollution seems getting more and more serious. Normally, self cleaning effect is one of the important functions of freshwater system, which some harms posed by pollutants can be alleviated in water. But the human activities decrease it greatly and as a result, the water quality declined, freshwater system eutrophicated and degraded. The variability observed within the water samples is closely connected to the sampling locations; hence, it is primarily a consequence of discharge influent. The trace element levels, in particular those of heavy metals are moderate suggesting an origin from anthropogenic contamination rather than from natural sources.

Abstract:
First data for concentrations and distribution of trace and toxic elements in tissues and organs of brown skua (Catharacta lonnbergi), Gentoo penguin (Pygoscelis papua), crabeater seal (Lobodon carcinophagus), notothenioid fish (Notothenia coriiceps) and in krill (Euphausia superba), collected on Livingston Island (South Shetlands), Antarctica are presented. Lead, cadmium, copper, zinc, manganese and cobalt were studied in liver, kidney, bone, spleen, muscle, heart and feathers. Generally, higher concentrations of toxic elements were found in liver and kidneys than in other investigated tissues. The same elements in krill and notothenioid fish were also determined. The obtained data could serve as a starting point for broader investigation of the possible changes in the ecosystem of Antarctic Peninsula under climate and environmental changes.

Abstract:
Under water and some other conditions, trace and minor elements in coal or waste rocks will migrate and separate out from coal or waste rocks into the soil environment with water. They will move and concentrate in the soil environment,but their capacity of migration is controlled by conditions of soil environment. In this paper, the migration capacity of trace and minor elements is analyzed on the basis of leaching experiments.According to the contents of trace and minor elements in horizontal vertical directions around heaps of waste rocks, the effect on soil and the migration capacity of trace and minor elements are studied. This study will provide data for evaluating environmental effects in the soil environment.

Abstract:
We derive a trace formula for $\sum_n A_{nn}B_{nn}...\delta(E-E_n)$, where $A_{nn}$ is the diagonal matrix element of the operator $A$ in the energy basis of a chaotic system. The result takes the form of a smooth term plus periodic-orbit corrections; each orbit is weighted by the usual Gutzwiller factor times $A_p B_p ...$, where $A_p$ is the average of the classical observable $A$ along the periodic orbit $p$. This structure for the orbit corrections was previously proposed by Main and Wunner (chao-dyn/9904040) on the basis of numerical evidence.

Abstract:
Fresh fruits, vegetables, herbs, and processed foods continue to be the major sources of essential trace elements in humans’ diet required for proper body development. However, food products can potentially be contaminated by toxic heavy metals (HMs) from environmental contamination or industrial food processing. The deleterious health implications of essential trace and macro elements’ deficiency and toxic consequences of HMs in humans necessitate proactive monitoring of the essential trace elements and HMs concentrations in the humans diet to ensure public health safety. Accordingly, this study investigated a comparative analysis of essential elements and potential toxic HMs concentration in food products in the Greensboro metropolis, North Carolina, USA. A total of 49 food samples comprising of 16 difference fresh fruits, 17 fresh vegetables, 4 herbs, and 12 processed foods were purchased from local grocery stores and analyzed for iron (Fe), calcium (Ca), magnesium (Mg), nickel (Ni), zinc (Zn), copper (Cu), lead (Pb), cadmium (Cd), and chromium (Cr) by the use of flame atomic absorption spectrometry (FAAS). The concentrations of elements were subjected to a regression analysis to further gain insight of the inter-element association in the food samples. The results of the study showed high variability in the concentrations of elements in the fresh fruits, vegetables, herbs, and processed foods. The overall average concentrations of Ca (1501 μg/g), Mg (186.5 μg/g), Fe (55.8 μg/g), Zn (22.2 μg/g), Pb (10.2 μg/g), Cu (5.8 μg/g), Cr (<0.1 μg/g), Cd (<0.1 μg/g), and Ni (<0.04 μg/g) were obtained in all food samples categories. The elements concentrations were generally poorly correlated in the food samples. However, a strong inter-element association between Cu and Fe concentration (R2 = 1.000) and a weak association between Ca and Fe (R2 = 0.5609) were found in the food samples. A survey questionnaire was administered to 396 participants in the Greensboro metropolis to evaluate the food consumption pattern and a daily/weekly dietary estimate intake of vegetables, fruits, herbs and processed foods. The results of the food survey analysis showed that the amount of vegetables, fruits, herbs, and processed foods dietary intake varied widely. In general, the participants consumed more processed foods than vegetables, fruits, and herb foods. The low dietary intake of vegetables, fruits, herbs suggests that most participants may be obtaining insufficient essential trace elements and other vital nutrients necessary for normal growth and body development

Abstract:
Four different ashes from three district heating plants firing biomass were studied with the respect to their potential application as soil fertilizers. Major and trace elements content and some important characteristics of the studied ashes are also presented. Five stage sequential extraction procedure was used for the determination of distribution and speciation of As, Ca, Cd, Cr, Cu, K, Mg, Na, Ni, Pb and Zn in studied ash samples.

Abstract:
This study was conducted to investigate the occurrence and extent of potentially trace and major elements in kaolin, widely used in Ghana, using the Instrumental Neutron Activation Analysis (INAA) technique. Soil plays a vital role in human sustenance on earth. Different forms of soil have been used over the years to solve pertinent requirements of man. Kaolin, commonly referred to as “hyire” in Ghana, has been used by women during the delicate periods of their pregnancy and also by lactating mothers. This study has sought to conduct an analysis of the toxic elements that may be ingested by these pregnant women when they eat” hyire” using Instrumental Neutron Activation Analysis (INAA) at the Ghana Research Reactor-1 facility. The study has shown that even though there are useful (major) elements present in the samples used, some toxic elements were also found to be beyond the Recommended Dietary Allowance for those elements. Recommendation has been made to request policy makers to regulate the production and use of “hyire” in order that pregnant women and their foetus are not exposed to effects of the toxic components detected. Industries have been encouraged to invest in this viable sector and to use modern technology to make the manufacturing of “hyire” more refined.