Elements A1 Research Articles

A finiteness theorem for a class of exponential congruences

For given elements a1,... , ak and 3 belonging to the ring of integers A of a number field we consider the set of all k-tuples (al,... , ak) in Nk for which Ek l caiai divides Ekcaiai for any z E A, and prove under some reasonable assumptions that the set of solutions is finite. The original motivation for this work comes from a problem raised by J. L. Selfridge (see Guy [1], problem B47) who asks for what pairs (a, b) does 2a-2b divide nanb for all integers n. A related (but more difficult) problem proposed by H. Ruderman asks to show that if 2a 2b divides 3a -3b, then 2a 2b divides n nb for all integers n. This was investigated by B. Velez in [6]. While Ruderman's problem is still open, Selfridge's problem was solved by Pomerance [2], who combined results of Schinzel [4] with Velez's work. It turns out that there are exactly 14 solutions. The problem was also solved by Sun Qi and Zhang Ming Zhi [5]. In this paper we show that the above finiteness result is a particular case of a more general phenomenon. Let IC be a number field, A = AKC its ring of integers and U = UKc its group of units. Let c1,... , ak and p be nonzero elements of A. We consider the set of all k-tuples (al,... , ak) in Nk for which

The multiplicity of a set of homogeneous polynomials over a commutative ring

Some thirty years ago Buchsbaum and Rim [1] extended the notion of multiplicity e(a1, …, an; E) for elements a1, …, an of a commutative ring R with identity and a Noetherian R-module E (≠0) with lengthR (E/[sum ]nt=1aiE) finite to give a multiplicity e((aij); E) associated with E and an m×n matrix (aij) over R satisfying a certain extended finiteness condition. One of their results states that for each of a set of m complexes depending on E, (aij) the Euler–Poincaré characteristic is a certain integer multiple of e((aij); E), at least when R is a local ring.Some of these ideas were taken up in [4] where it is shown that when n[ges ]m−1, each of the complexes K((aij); E; t) with t∈ℤ introduced in [3] also have e((aij); E) as their Euler–Poincaré characteristic. With a slight change in viewpoint (aij) can be replaced by linear forms aj=[sum ]mi=1aijxi (j=1, …, n) of the graded polynomial ring R[x1, …, xm]; the complex K((aij); E; t) then becomes the component of degree t in a certain graded double complexformula herewhere K(a1, …, an; F) is the standard Koszul complex (see [4; section 2]). From this point of view the construction can be extended to allow the homogeneous polynomials a1, …, an to have any (possibly unequal) positive degrees [6]. The main aim of the present note is to extend similarly the results of [4] and to strengthen those results to give information on the vanishing of the multiplicity.

Sorting in Parallel

Although more than a century passed before Menabrea's vision became a reality, today parallel computers with hundreds and even thousands of processors are used in a broad range of applications. One of the great challenges of parallel computing is the design of algorithms that make efficient use of the large number of processors in the machine. In this paper we explore parallel algorithms for one of the most fundamental problems in computer science: the problem of sorting a list of numbers. Specifically, given a list a1, a2, ... , an of integers, the sorting problem is that of finding a permutation of this list, 7T, such that a.(l) < a.(2) < ...< a(n) Before considering parallel sorting algorithms, we examine a sequential sorting algorithm; an algorithm for a computer with a single processor. One sequential algorithm for sorting is known as selection sort. The selection sort algorithm first examines each of the n integers a1, ... , an to find an integer ai with minimum value. Elements a1 and ai are then exchanged so that the element with minimum value is now in the first position in the list. Next, the algorithm examines each of the n 1 integers a2, ... ., an to find an integer with minimum value in this portion of the list. This integer is exchanged with a2 so that now the second smallest integer is in the second position in the list. In general, in the k th iteration the algorithm examines integers ak, .. , an, finds a minimum element in this list, and exchanges it with ak. At the end of nth iteration, the list is sorted. Observe that the first iteration of the algorithm requires n steps to find an element with minimum value and then some constant additional number of steps, c, to exchange this minimum value with a1. The next iteration requires n 1 steps to locate a minimum value in the remaining list and c steps to perform the exchange. In general, the kth iteration requires n k + 1 steps to locate a minimum value in the remaining list and c steps to perform the exchange. Since there are n iterations, the total number of steps or running time of the algorithm is

Determination of trace elements in lichens by instrumental neutron activation analysis

Instrumental neutron activation analysis has been applied to analyze epiphytic specimens of the lichen Canoparmelia texana. Samples collected from the barks of the trees were previously cleaned, lyophilized and ground to be irradiated under a thermal neutron flux in the IEA-RI nuclear reactor. Elements A1, As, Br, Ca, Cd, C1, Co, Cr, Cs, Fe, Hf, K, La, Ce, Nd, Sin, Eu, rib, Yb, Lu, Mg, Mn, Na, Rb, Sb, Sc, Se, Th, Ti, U, V and Zn were determined quantitatively by using short arid long irradiations. Comparisons were made between the results for washed and unwashed samples as well as those collected in two different years, from individual palm trees and also from differelat sites of Sgo Paulo State. The precision and the accuracy of the method were evaluated by analyzing NIST 1572a Citrus I_eaves, IAEA 336 Lichen and USGS W-1 Rock reference materials.

Representations of -algebras and dynamical systems

Here Pk(·) are polynomials in the non-commuting variables a1, . . . , an over C such that P ∗ k (·) = Pk(·). In other words, A is a quotient of the free ∗-algebra C〈a1, . . . , an〉 generated by the self-adjoint elements a1, . . . , an with respect to the two-sided ideal generated by the relations (1). Representations ot A (∗-homomorphisms π:A → L(H) of the ∗-algebra A into a ∗algebra L(H) of bounded operators on a separable Hilbert space H or into a ∗-algebra of unbounded operators) are of interest both from mathematical point of view and for their physical applications. A choice of a representation π(·) corresponds to a choice of a model with observables Ak = π(ak) (k = 1, . . . , n), which are connected by the relations Pk(A1, . . . , An) = 0 (k = 1, . . . ,m). (2)

Cryptic variations of minor elements Al, Cr, Ti and Mn in Lower and Critical Zone orthopyroxenes of the Western Bushveld Complex

AbstractCompositional variations with respect to minor elements A1, Cr, Ti and Mn, and major elements Fe and Mg, in orthopyroxenes along ca. 160 km of strike of the Lower (LZ), Lower Critical (LCZ) and Upper Critical (UCZ) Zones are reviewed on the basis of 1900 analyses by electron microprobe. AI increases with stratigraphic height and declining Mg/(Mg + Fe2) ratios (hereafter MMF ratios) through the LZ and LCZ, reaching peak values close to the base of the UCZ, where the first cumulus plagioclase appears in the succession. Above this, Al contents decline as MMF ratios decline. Through the same interval, subdued increase in Ti occurs through &gt;1000 m of ultramafic cumulates, but this increase accelerates within the ca. 450 m UCZ sequence. Mn increases linearly with declining MMF ratios through the entire succession, Cr levels are highest in orthopyroxenes of the ultramafic LZ and LCZ, and olivine norites of the UCZ, but decline in more evolved norites and associated anorthosites of the UCZ.This pattern of cryptic variations, displayed by a thick succession of cumulates, is consistent with the model of Bence and Papike (1972) and Grove and Bence (1977) for basaltic rocks, which links the levels of minor elements in pyroxenes with entry of plagioclase into the paragenesis.

Hydrothermal fluids responsible for the formation of precious minerals in the Nigerian Younger Granite Province

Preliminary investigations in the Younger Granite Province of Nigeria have revealed that precious and semi-precious minerals like rubies, sapphires, emeralds, aquamarine, zircon and fluorite can be found in the region. The gem minerals are shown to have been produced either by direct deposition along fissures, veins and greisens by hydrothermal fluids or as a result of hydrothermal fluids reacting with wall-rocks. These wall rocks are either biotite granites from which the hydrothermal fluids originated or basement rocks or any other rocks which the biotite granites intrude and their residual hydrothermal fluids have invaded. The hydrothermal fluids appear to have been rich in alkalis (Na+, K+, etc.), rare elements (Be, Zr, F, REE, etc.) and siliceous. As these fluids rose through fractures and channel ways through the rocks, they either deposited the gem minerals in the fractures at the appropriate stability conditions or reacted with the wall-rocks producing the gem minerals at the expense of elements like Ca and A1 in the minerals of these rocks.

Four major sequence elements of simian virus 40 large T antigen coordinate its specific and nonspecific DNA binding

By mutational analysis, we have identified a motif critical to the proper recognition and binding of simian virus 40 large tumor antigen (T antigen) to virus DNA sequences at the origin of DNA replication. This motif is tripartite and consists of two elements (termed A1 and B2) that are necessary for sequence-specific binding of the origin and a central element (B1) which is required for nonspecific DNA-binding activity. Certain amino acids in elements A1 (residues 152 to 155) and B2 (203 to 207) may make direct contact with the GAGGC pentanucleotide sequences in binding sites I and II on the DNA. Alternatively, these two elements could determine the proper structure of the DNA-binding domain, although for a number of reasons we favor the first possibility. In contrast, element B1 (183 to 187) is most likely important for recognizing a general structural feature of DNA. Elements A1 and B2 are nearly identical in all known papovavirus T antigens, whereas B1 is identical only in the closely related papovaviruses simian virus 40, BK virus, and JC virus. In addition to these three elements, a fourth (B3; residues 215 to 219) is necessary for the binding of T antigen to site II but not to site I. We propose that additional contact sites on T antigen are involved in the interaction with site II to initiate the replication of the viral DNA.

Some Peculiarities of Space-Time Distribution of the Atoms of the Elements in the Plasma of A D.C. Arc in a Regime of Anode Evaporation

Abstract The space-time distribution of the atoms of the elements A1, (T1), Mg, Ag, (Sn), (Si), Sb and Be in d.c. arc plasma during anode evaporation of samples containing a readily ionized additive, Li (as Li2CO3), was studied by spectral monochromatic equidensitometry. It was established that the distribution of the atoms in the interelectrode gap depends on the concentration of the readily ionized additive, the moment of burning of the arc and the physicochemical properties and ionization potential of the microadditive. The results obtained may be used to determine the optimum conditions for analysis of microelements in powdery samples.

Free non-associative principal train algebras

The definitions of finite dimensional baric, train, and special train algebras, and of genetic algebras in the senses of Schafer and Gonshor (which coincide when the ground field is algebraically closed, and which I call special triangular) are given in Worz-Busekros's monograph [8]. In [6] I introduced applications requiring infinite dimensional generalisations. The elements of these algebras were infinite linear forms in basis elements a0, a1,… and complex coefficients such that In this paper I consider only algebras whose elements are forms which only a finite number of the xi are non zero.

Local Stability and Saturation in Spaces of Orderings

If k is a f.r. (= formally real) field which is partially ordered with positive cone, P, XP denotes the space of total orders T of k with P ⊂ T. Suppose you have a subset A ⊂ XP and an element T ∈ XP, T ∉ A. Then the main question investigated in this paper is the following: How can T be separated from A by using elements of k? To be more specific, this is split up into two different questions.Question 1. Suppose A is closed. Then there is an n ∈ N and elements a1, …, an ∈ k such that the basic open set H = H(a1, …, an) is a neighborhood of T and has empty intersection with A. Now, if T is given, what is the least n ∊ N (if it exists) such that T has a neighborhood basis consisting of basic open sets of the form H(a1, …, an)?

On-Line Analysis of Iron Ore for Pellet Production

The paper describes a method of continuous analyzing of iron ore concentrate and of the mixture iron ore and additives in a pellet-processing plant being commissioned at present in steelworks in the USSR.These are two X-ray emission gauges with the pertinent sampling and sample preparing facilities which have been adapted to this specific application and used for that duty for the first time.The system is meant to detect continuously the quality of the iron ore concentrate and, after admixing of additives, to determine the mixture composition and, hence, the basicity of the pellets in the two production lines.Special importance has been attached to the design of the sampling and sample preparing system. A representative sample is taken from the continuous flow of material directly from the belt conveyor and then dried. Since the moist material shows a strong tendency to stick, that portion of the system has been laid out with special care. After the sample has been dried in a temperature-controlled rotary dryer and heated in vibratory trough conveyors, it is split and fed into the continuously analyzing X-ray emission gauge.That system has also been set up locally and simultaneously analyzes the elements Fe, Ca, Si and A1 while the flow of sample material passes the point of measuring.A microprocessor provides for internal controlling, correction to the original reference values and output of the information suitable for computer processing. There are parallel analog outputs which enable the chemical composition being plotted by recorders as a function of time.The process computer that has been connected in series can evaluate the analytical data, check them for limit values, record them and take over controlling for metering the additives.

Remarks on generalized analytic independence

This paper is concerned with the notion of independence relating sets of elements in a ring A to a proper ideal a of A. A set of elements a1, …, an ∈A is called a-independent if every form in A[X1, …, Xn] vanishing at a1,…, an has all its coefficients in a. This notion leads to many questions (cf. (2) and (12)), which are of some interest in their own right, several of which are considered here. On the other hand, this independence is related to the structure of the graded ring associated to the ideal generated by the set of elements, hence is often relevant to some problems concerning regular sequences and complex.

The determination of the elemental composition of glasses for forensic application using a combination of proton induced X-ray emission and inelastic proton scattering

Proton induced X-ray emission (PIXE) is used in an attempt to identify glass samples with the aid to determine, trace element concentrations. The elements Na and A1 are simultaneously determined using the (p, p' γ) nuclear reaction. The preparation of very thin and homogeneous glass targets is discussed and a procedure is described to obtain an accurate equipment calinration, checked afterwards with NBS bovine liver. Results are given for 11 different glass samples.

Remarks on regular sequences

In this note we exhibit a condition under which a sequence of elements a1, …, an in a commutative noetherian local ring A form an A-sequence, and derive a number of corollaries.

Fine Microstructure of a Mechanically Alloyed Oxide Dispersion Strengthened Nickel-Base Superalloy

The previous paper briefly described the fine microstructure of a mechanically alloyed oxide dispersion strengthened nickel-base solid solution. This note examines the fine microstructure of another mechanically alloyed system. This alloy differs from the one described previously in that it is more generously endowed with coherent precipitate γ forming elements A1 and Ti and it contains a higher volume fraction of the finely dispersed Y2O3 oxide. An interesting question to answer in the comparative study of the creep and stress rupture of these two ODS systems is the role of the precipitate γ' in the mechanisms of creep and stress rupture in alloys already containing oxide dispersoids.The nominal chemical composition of this alloy is Ni - 20%Cr - 2.5%Ti - 1.5% A1 - 1.3%Y203 by weight. The system receives a three stage heat treatment-- the first designed to produce a coarse grain structure similar to the solid solution alloy but with a smaller grain aspect ratio of about ten.

A multiplicative property of R-sequences and H1-sets

Let R be a commutative ring which may not contain a multiplicative identity. A set of elements a1,…,ak in R will be called an H1-set (this notation is explained in section 1) if for each relation r1a1 + … +rkak = 0 (ri ∈ R) there exist elements sij ∈ R such thatwhere Xl,…,Xk are indeterminates. Any R-sequence is an H1-set, but there do exist H1-sets which are not R-sequences (see section 1). Throughout this note we consider an H1-set a1,…,ak which we suppose to be partitioned into two non-empty sets bl…, br and cl,…, cs. Our main purpose is to show that the ideals B = Rb1 + … + Rbr and C = Rc1 + … + Rcs satisfy Bm ∩ Cn = BmCn for all positive integers m and n (Corollary 1). This generalizes Lemma 2 of Caruth(2) where the result is proved when a1,…, ak is a permutable R-sequence. Our proof involves more detail than is necessary just for this, and we obtain various other properties of H1-sets. In particular we extend the main results of Corsini(3) concerning the symmetric and Rees algebras of a power of the ideal Ra1 +… + Rak (Corollary 3).

Quantitative determination of mineral inclusions in polycaproamide by a spectral method

The analysis consists-of the calcination of the specimens, recording the spectra on photographic plates with a spectrograph and comparing the blackening of the analytical lines on the spectrum of the individual elements with the blackening of the corresponding lines on the spectra of reference mixtures. The spectral analysis makes possible the quantitative determination of the elements Cu, Ca, A1, Fe, Mg~ and Si. The results of determinations of inclusions in specimens of several batches of polycaproamide (Capron fibre) are set out in Table 1.

A note on a theorem of Vaught

In [2] Vaught showed that if T is a complete theory formalized in the first-order predicate calculus, then it is not possible for T to have exactly (up to isomorphism) two countable models. In this note we extend his methods to obtain a theorem which implies the above.First some definitions. We define Fn(T) to be the set of well-formed formulas (wffs) in the language of T whose free variables are among x1 x2, …, xn. An n-type of T is a maximal consistent set of wffs of Fn(T); equivalently, a subset P of Fn(T) is an n-type of T if there is a model M of T and elements a1, a2, …, an of M such that M ⊧ ϕ(a1, a2, …, an) for every ϕ ∈ P. In the latter case we say that 〈a1, a2, …, an〉 ony realizes P in M. Every set of wffs of Fn(T) which is consistent with T can be extended to an n-type of T.

Comparative Elemental Analyses of Plant Tissue by Spark Emission and Atomic Absorption Spectroscopy

AbstractThe analyses of 91 plant tissues for their A1, Ca, Cu, Fe, K, Mg, Mn, and Zn content by direct reading emission and atomic absorption spectroscopy were comparable for most elements. Statistically significant differences were obtained for the elements K and Mg although the numerical differences were small. The variances within each laboratory were sufficiently large for A1, Ca, Cu, Fe, Mn, and Zn to mask significant differences between laboratories. Discrepancies between laboratories were observed in several plant specie samples for the elements AI, Cu, and Fe. Reproducibility by atomic absorption was better than that obtained by emission spectroscopy except for the elements A1 and Cu. Although some differences were found for certain elements and plant tissue samples, the analytical results compared favorably between the methods and laboratories.