An existence domain of the sum of exponential monomials series

The thermodynamic characteristics of processes in the liquid metal system Fe–Y–Cr–C–O are considered as applied to low-carbon and low-alloy metal. The critical parameters for the state diagram of the oxide system Y2O3–Cr2O3 were established based on the values quoted in literature. The temperature dependence of the melting reaction constant Y2O3·Cr2O3 was determined. The coordinates of eutectic transformation points for the system Y2O3–Cr2O3 were calculated. In accordance with subregular solution theory, the energetic parameters which are necessary to calculate the activities Cr2O3 and Y2O3 of oxide melts in the system Y2O3–Cr2O3 were determined. The energetic parameters of subregular solution theories for the oxide system FeO–Cr2O3–Y2O3 were determined based on the values for the binary systems FeO–Y2O3, FeO–Cr2O3 and Y2O3–Cr2O3. The view of this diagram, as coupled with the existence domain of liquid metal within the framework of the quaternary system Fe–Y–Cr–O–С, suggests that low-carbon chromic liquid metal when injected with yttrium can form the following non-metallic inclusions: |Cr2O3|, |Y2O3|, |FeO·Cr2O3|, |Y2O3·Cr2O3| or oxide melt (FeO, Y2O3, Cr2O3). Oxide melt may contain up to 2 % of divalent chrome (Cr2+). The equilibrium constants for the main reactions of steel deoxidation with the formation of liquid, solid and gas products of chemical reactions were also established. The activity of components dissolved in metal was calculated using interaction parameters. The set of derived expressions for the activity of components and the dependences of equilibrium constants of chemical reactions and phase transformations allowed us to diagram the surface of component solubility in liquid metal (SCSM). SCSM diagrams show the compositions of liquid metal and indicate oxide phases which are in equilibrium with liquid metal.

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Dynamics of two-dimensional multi-peak solitons based on the fractional Schrödinger equation

Journal of Nonlinear Optical Physics & Materials ◽

10.1142/s0218863522500047 ◽

2021 ◽

Author(s):

Xiaoping Ren ◽

Fang Deng

Keyword(s):

Schrödinger Equation ◽

Schrodinger Equation ◽

Optical Lattices ◽

Two Dimensional ◽

Fractional Schrödinger Equation ◽

Existence Domain ◽

Maximal Peak ◽

Band Gap Structure ◽

Fractional Schrodinger Equation ◽

Peak Value

We address the propagation dynamics of two-dimensional multi-peak solitons in the optical lattices based on the fractional Schrödinger equation. The effect of Lévy index and lattice depth on the band-gap structure of optical lattices are presented. Two-, three-, four-, six- and eight-peak solitons all can exist in the first gap and be stable in a wide region of their existence domain. The effective width, maximal peak value and the power of soliton are also studied. It indicates that the Lévy index plays a significant role on the properties of solitons.

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Structure feature of ternary state diagrams of Cr-Ti-V and Cr-Mn-V systems

MATEC Web of Conferences ◽

10.1051/matecconf/201824300014 ◽

2018 ◽

Vol 243 ◽

pp. 00014 ◽

Cited By ~ 1

Author(s):

Anatoliy Klopotov ◽

Irina Kurzina ◽

Alexander Potekaev ◽

Artem Ustinov ◽

Taras Dement ◽

...

Keyword(s):

Solid Solution ◽

Intermetallic Compounds ◽

Solid Solutions ◽

Crystal Chemical ◽

Electron Number ◽

Isothermal Sections ◽

State Diagrams ◽

Bcc Lattice ◽

Existence Domain ◽

Chemical Factors

This paper presents the research results of features of structural and phase states in Cr-Ti-V and Cr-Mn-V systems based on analysis of crystal-geometric and crystal-chemical factors. The diagrams of isothermal sections of state diagrams of Cr-Ti-V and Cr-Mn-V systems were built in coordinates of the electron number (s+d) per atom with homogeneity regions of solid solutions and intermetallic compounds. It was shown that in the Cr-Ti-V system, addition of Mn atoms leads to substantial extension of the existence domain of the disordered solid solution based on the BCC lattice.

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Investigation of the CaO–La2O3–SiO2–P2O5 quaternary diagram. Synthesis, existence domain, and characterization of apatitic phosphosilicates

Solid State Ionics ◽

10.1016/s0167-2738(02)00398-3 ◽

2003 ◽

Vol 156 (1-2) ◽

pp. 209-222 ◽

Cited By ~ 21

Author(s):

R El Ouenzerfi

Keyword(s):

Existence Domain

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ON POSITIVE EIGENFUNCTIONS OF STURM-LIOUVILLE PROBLEM WITH NONLOCAL TWO‐POINT BOUNDARY CONDITION

Mathematical Modelling and Analysis ◽

10.3846/1392-6292.2007.12.215-226 ◽

2007 ◽

Vol 12 (2) ◽

pp. 215-226 ◽

Cited By ~ 8

Author(s):

Sigita Pečiulytė ◽

Artūras Štikonas

Keyword(s):

Boundary Condition ◽

Nonlocal Boundary ◽

Sufficient Conditions ◽

Liouville Problem ◽

Point Boundary ◽

Existence Domain ◽

Classical Boundary ◽

Sturm Liouville ◽

Necessary And Sufficient ◽

Nonlocal Boundary Problem

Positive eigenvalues and corresponding eigenfunctions of the linear Sturm‐Liouville problem with one classical boundary condition and another nonlocal two‐point boundary condition are considered in this paper. Four cases of nonlocal two‐point boundary conditions are analysed. We get positive eigenfunctions existence domain for each case of these problems. This domain depends on the parameters of the nonlocal boundary problem and it gives necessary and sufficient conditions for existing positive eigenvalues with positive eigenfunctions.

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Surface solitons supported by 𝒫𝒯-symmetric lattice with spatially modulated nonlinearity

Journal of Nonlinear Optical Physics & Materials ◽

10.1142/s0218863517500011 ◽

2017 ◽

Vol 26 (01) ◽

pp. 1750001 ◽

Cited By ~ 2

Author(s):

Xin Li ◽

Rangang Yu ◽

Neng Zhang

Keyword(s):

Stability Analysis ◽

Low Power ◽

Periodic Systems ◽

Stable Domain ◽

Power Domain ◽

Existence Domain ◽

Existence And Stability

We report on the formation and stability of induced surface solitons in parity–time ([Formula: see text]) symmetric periodic systems with spatially modulated nonlinearity. We discover that the spatially modulation of the nonlinearity can affect the existence and stability of surface solitons. These surface solitons can be formed in the semi-infinite and first bandgap. Stability analysis shows that odd surface solitons belonging to semi-infinite bandgap are linearly stably in low power domain, and the stable domain becomes narrow with increasing the strength of spatially modulated nonlinearity, both even surface solitons in semi-infinite bandgap and surface solitons in first bandgap are unstable in their existence domain.

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Three-dimensional dissipative optical solitons

Open Physics ◽

10.2478/s11534-008-0083-4 ◽

2008 ◽

Vol 6 (3) ◽

Cited By ~ 3

Author(s):

Dumitru Mihalache

Keyword(s):

Landau Equation ◽

Three Dimensional ◽

Longitudinal Direction ◽

Optical Solitons ◽

Ginzburg Landau ◽

Systematic Analysis ◽

Dissipative Solitons ◽

Existence Domain ◽

Existence And Stability ◽

The Stability

AbstractA brief overview of recent theoretical results in the area of three-dimensional dissipative optical solitons is given. A systematic analysis demonstrates the existence and stability of both fundamental (spinless) and spinning three-dimensional dissipative solitons in both normal and anomalous group-velocity regimes. Direct numerical simulations of the evolution of stationary solitons of the three-dimensional cubic-quintic Ginzburg-Landau equation show full agreement with the predictions based on computation of the instability eigenvalues from the linearized equations for small perturbations. It is shown that the diffusivity in the transverse plane is necessary for the stability of vortex solitons against azimuthal perturbations, while fundamental (zero-vorticity) solitons may be stable in the absence of diffusivity. It has also been found that, at values of the nonlinear gain above the upper border of the soliton existence domain, the three-dimensional dissipative solitons either develop intrinsic pulsations or start to expand in the temporal (longitudinal) direction keeping their structure in the transverse spatial plane.

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Experimental and theoretical research of continuous motion of rotor on anisotropic gapped elastic support

VESTNIK of the Samara State Aerospace University ◽

10.18287/2541-7533-2019-18-2-156-168 ◽

2019 ◽

Vol 18 (2) ◽

pp. 156-168

Author(s):

A. N. Nikiforov ◽

G. Ya. Panovko ◽

A. Ye. Shokhin

Keyword(s):

Contact Forces ◽

Theoretical Research ◽

Complex Conjugate ◽

Flexible Rotor ◽

Dynamic Features ◽

Motion Equations ◽

Displacement Sensors ◽

Existence Domain ◽

Direct Tracking ◽

Full Scale Tests

Kinematic and dynamic features of flexible rotor rolling of an unlubricated uneven-stiffness gapped support are shown with the help of a theoretical model and full-scale tests. A combination of original approaches and well-known analytical and experimental methods is used. In particular, the motion equations are Lagrangian, derived in complex and complex-conjugate coordinates, and their solutions are sought in the exponential form, by excluding contact forces from consideration and by introducing a small parameter. Vibration measurements are not made on the rig frame using accelerometers, but at the source of vibration, i.e. by direct tracking of the rotor axle by a pair of eddy-current contactless displacement sensors installed in a XY configuration. Hence, the preciseness of the experimental data does not depend on the mechanical conductivity of the parts between the source rotor and the receiver sensors. As a result, the frequency, amplitude and existence domain of retrograde precession are reliably determined, and a conclusion is drawn that rolling under a certain combination of parameters is impossible in principle.

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Electron-Acoustic (Un)Modulated Structures in a Plasma Having (r, q)-Distributed Electrons: Solitons, Super Rogue Waves, and Breathers

Symmetry ◽

10.3390/sym13112029 ◽

2021 ◽

Vol 13 (11) ◽

pp. 2029

Author(s):

Wedad Albalawi ◽

Rabia Jahangir ◽

Waqas Masood ◽

Sadah A. Alkhateeb ◽

Samir A. El-Tantawy

Keyword(s):

Acoustic Waves ◽

Rogue Waves ◽

Distribution Functions ◽

Sagdeev Potential ◽

Plasma Parameters ◽

Positive Ions ◽

Modulated Structures ◽

Existence Domain ◽

Unmagnetized Plasma ◽

Electron Acoustic

The propagation of electron-acoustic waves (EAWs) in an unmagnetized plasma, comprising (r,q)-distributed hot electrons, cold inertial electrons, and stationary positive ions, is investigated. Both the unmodulated and modulated EAWs, such as solitary waves, rogue waves (RWs), and breathers are discussed. The Sagdeev potential approach is employed to determine the existence domain of electron acoustic solitary structures and study the perfectly symmetric planar nonlinear unmodulated structures. Moreover, the nonlinear Schrödinger equation (NLSE) is derived and its modulated solutions, including first order RWs (Peregrine soliton), higher-order RWs (super RWs), and breathers (Akhmediev breathers and Kuznetsov–Ma soliton) are presented. The effects of plasma parameters and, in particular, the effects of spectral indices r and q, of distribution functions on the characteristics of both unmodulated and modulated EAWs, are examined in detail. In a limited cases, the (r,q) distribution is compared with Maxwellian and kappa distributions. The present investigation may be beneficial to comprehend and predict the modulated and unmodulated electron acoustic structures in laboratory and space plasmas.

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