We introduce the notion of length for non-associative finite-dimensional unitary algebras and obtain a sharp upper bound for the lengths of algebras belonging to this class. We also put forward a new ...method of characteristic sequences based on linear algebra technique, which provides an efficient tool for computing the length function in non-associative case. Then we apply the introduced method to obtain an upper bound for the length of an arbitrary locally complex algebra. In the last case the length is bounded in terms of the Fibonacci sequence. We present concrete examples demonstrating the sharpness of our bounds.
Epitaxial layers in a system of InAs
1–
x
–
y
Sb
y
P
x
solid solutions in the composition range of 0 <
x
< 0.72 were obtained on an InAs(001) substrate by metalorganic vapor phase epitaxy (MOVPE). ...The layer-by-layer analysis of obtained structures by secondary ion mass spectrometry showed a gradient change in the composition along the growth direction. A dramatic change in the composition at the layer/substrate heteroboundary was observed for the quaternary InAsSbP solid solutions due to the presence of radicals of arsenic compounds in the gas phase. Upon MOVPE deposition on the InAs substrate in a system of InAsSbP solid solutions, the decrease in the solid-phase content of arsenium by less than (1–
x
–
y
) < 0.3 resulted in a suppression of the deposited layer gradientness, as well as suppressed fluctuations in the composition in the initial transition layer.
A lower and an upper bounds for the length of a direct sum of nonassociative algebras are obtained, and their sharpness is established. Note that while the lower bound for the length of a direct sum ...in the associative and nonassociative cases turns out to be the same, the upper bound in the nonassociative case significantly exceeds its associative counterpart.
In this paper we study the relations between numerical characteristics of finite dimensional algebras and such classical combinatorial objects as additive chains. We study the behavior of the length ...function via so-called characteristic sequences of quadratic algebras. As one of our main results, we prove the sharp upper bound for the length of quadratic algebras in terms of the Fibonacci numbers depending on the dimension of the algebra. Moreover, we show that quadratic algebras have the extremal behavior with respect to this bound. In addition, we obtain the description of the set of characteristic sequences for quadratic algebras. Namely, we completely determine the set of combinatorial properties which are satisfied for characteristic sequences of quadratic algebras and show that they belong to the family of additive chains known in combinatorics. Conversely, for a given integer sequence being an additive chain and satisfying these combinatorial properties, we construct a quadratic algebra with a characteristic sequence equal to this sequence.
The obtained information on characteristic sequences is then applied to investigate the problem of realizability for the length function. In particular, we determine certain subsets of integers which are not realizable as values of the length function on quadratic algebras.
The radioactive element astatine exists only in trace amounts in nature. Its properties can therefore only be explored by study of the minute quantities of artificially produced isotopes or by ...performing theoretical calculations. One of the most important properties influencing the chemical behaviour is the energy required to remove one electron from the valence shell, referred to as the ionization potential. Here we use laser spectroscopy to probe the optical spectrum of astatine near the ionization threshold. The observed series of Rydberg states enabled the first determination of the ionization potential of the astatine atom, 9.31751(8) eV. New ab initio calculations are performed to support the experimental result. The measured value serves as a benchmark for quantum chemistry calculations of the properties of astatine as well as for the theoretical prediction of the ionization potential of superheavy element 117, the heaviest homologue of astatine.
Characteristic effects of magnetic ordering and conduction in semiconductor heterostructures with a GaAs : Be/Ga
0.84
In
0.16
As/GaAs quantum well and manganese δ-layers of different thickness (from ...0.4 to 2 monolayers) were studied based on analysis of magnetic field and temperature dependences, galvanomagnetic effects, and magnetization. An anomalous dependence of the conductivity on the manganese atoms concentration in the δ-layer was observed, which was due to a strong scattering of charge carriers in the structures with the low content of magnetic impurities. Magnetic properties of the heterostructures clearly indicated the magnetic ordering of the impurity system (saturation and hysteresis of the magnetization and fulfillment of the Curie–Weiss law at increasing temperature). Parameters of the magnetic subsystem allowed revealing different types of ordering in the systems with different concentrations of the magnetic impurity. Changing the concentration of the Mn admixture in the δ-layer was shown to influence significantly the conductivity and magnetism in the studied structures.
The classical Hurwitz theorem claims that there are exactly four normed algebras with division: the real numbers (ℝ), complex numbers (ℂ), quaternions (ℍ), and octonions (𝕆). The length of ℝ as an ...algebra over itself is zero; the length of ℂ as an ℝ-algebra equals one. The purpose of the present paper is to prove that the lengths of the ℝ-algebras of quaternions and octonions equal two and three, respectively.
The temperature and field dependences of the specific magnetization and magnetoresistance in heterostructures with a GaAs/Ga
0.84
In
0.16
As/GaAs quantum well and a δ-layer of atomic Mn in the ...barrier layer near the quantum well filled with holes are studied. A change in the resistance and magnetization behavior upon ordering of localized magnetic moments in the cap layer due to a change in the manganese ion distribution topology is detected.
Purpose of this work is to use analytical and numerical methods to consider the problem of the structure and dynamics of the kinks in the sine-Gordon model with “impurities” (or spatial inhomogeneity ...of the periodic potential). Methods. Using the method of collective variables for the case of three identical point impurities located at the same distance from each other, a system of differential equations is obtained. Resulting system of equations makes it possible to describe the dynamics of the kink taking into account the excitation of localized waves on impurities. To analyze the dynamics of the kink in the case of extended impurities, a numerical finite difference method with an explicit integration scheme was applied. Frequency analysis of kink oscillations and localized waves calculated numerically was performed using a discrete Fourier transform. Results. For the kink dynamics, taking into account the excitation of oscillations in modes, a system of equations for the coordinate of the kink center and the amplitudes of waves localized on impurities is obtained and investigated. Significant differences are observed in the dynamics of the kink when interacting with a repulsive and attractive impurity. The dynamics of the kink in a model with three identical extended impurities, taking into account possible resonant effects, was solved numerically. It is established that the found scenarios of kink dynamics for an extended rectangular impurity are qualitatively similar to the scenarios obtained for a point impurity described using a delta function. All possible scenarios of kink dynamics were determined and described taking into account resonant effects. Conclusion. The analysis of the influence of system parameters and initial conditions on possible scenarios of kink dynamics is carried out. Critical and resonant kink velocities are found as functions of the impurity parameters.
Kinetic simulations of a direct-current positive-column Ar plasma revealed that electron fluxes are sensitive to energy dependence of an elastic collision cross section. Paradoxical behavior of the ...electron flux in a coordinate-energy phase space is presented. The direction of the electron flux at the "elastic" energy region (5-10 eV) in whole volume, except the region near the wall, turned out to be oppositely directed to that conventionally assumed.
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