825 research outputs found
Theory and Design of Flight-Vehicle Engines
Papers are presented on such topics as the testing of aircraft engines, errors in the experimental determination of the parameters of scramjet engines, the effect of the nonuniformity of supersonic flow with shocks on friction and heat transfer in the channel of a hypersonic ramjet engine, and the selection of the basic parameters of cooled GTE turbines. Consideration is also given to the choice of optimal total wedge angle for the acceleration of aerospace vehicles, the theory of an electromagnetic-resonator engine, the dynamic characteristics of the pumps and turbines of liquid propellant rocket engines in transition regimes, and a hierarchy of mathematical models for spacecraft control engines
Conditional symmetry and spectrum of the one-dimensional Schr\"odinger equation
We develop an algebraic approach to studying the spectral properties of the
stationary Schr\"odinger equation in one dimension based on its high order
conditional symmetries. This approach makes it possible to obtain in explicit
form representations of the Schr\"odinger operator by matrices for
any and, thus, to reduce a spectral problem to a purely
algebraic one of finding eigenvalues of constant matrices. The
connection to so called quasi exactly solvable models is discussed. It is
established, in particular, that the case, when conditional symmetries reduce
to high order Lie symmetries, corresponds to exactly solvable Schr\"odinger
equations. A symmetry classification of Sch\"odinger equation admitting
non-trivial high order Lie symmetries is carried out, which yields a hierarchy
of exactly solvable Schr\"odinger equations. Exact solutions of these are
constructed in explicit form. Possible applications of the technique developed
to multi-dimensional linear and one-dimensional nonlinear Schr\"odinger
equations is briefly discussed.Comment: LaTeX-file, 31 pages, to appear in J.Math.Phys., v.37, N7, 199
Comment on "Control landscapes are almost always trap free: a geometric assessment"
We analyze a recent claim that almost all closed, finite dimensional quantum
systems have trap-free (i.e., free from local optima) landscapes (B. Russell
et.al. J. Phys. A: Math. Theor. 50, 205302 (2017)). We point out several errors
in the proof which compromise the authors' conclusion.
Interested readers are highly encouraged to take a look at the "rebuttal"
(see Ref. [1]) of this comment published by the authors of the criticized work.
This "rebuttal" is a showcase of the way the erroneous and misleading
statements under discussion will be wrapped up and injected in their future
works, such as R. L. Kosut et.al, arXiv:1810.04362 [quant-ph] (2018).Comment: 6 pages, 1 figur
Group classification of heat conductivity equations with a nonlinear source
We suggest a systematic procedure for classifying partial differential
equations invariant with respect to low dimensional Lie algebras. This
procedure is a proper synthesis of the infinitesimal Lie's method, technique of
equivalence transformations and theory of classification of abstract low
dimensional Lie algebras. As an application, we consider the problem of
classifying heat conductivity equations in one variable with nonlinear
convection and source terms. We have derived a complete classification of
nonlinear equations of this type admitting nontrivial symmetry. It is shown
that there are three, seven, twenty eight and twelve inequivalent classes of
partial differential equations of the considered type that are invariant under
the one-, two-, three- and four-dimensional Lie algebras, correspondingly.
Furthermore, we prove that any partial differential equation belonging to the
class under study and admitting symmetry group of the dimension higher than
four is locally equivalent to a linear equation. This classification is
compared to existing group classifications of nonlinear heat conductivity
equations and one of the conclusions is that all of them can be obtained within
the framework of our approach. Furthermore, a number of new invariant equations
are constructed which have rich symmetry properties and, therefore, may be used
for mathematical modeling of, say, nonlinear heat transfer processes.Comment: LaTeX, 51 page
Conditional Lie-B\"acklund symmetry and reduction of evolution equations.
We suggest a generalization of the notion of invariance of a given partial
differential equation with respect to Lie-B\"acklund vector field. Such
generalization proves to be effective and enables us to construct principally
new Ans\"atze reducing evolution-type equations to several ordinary
differential equations. In the framework of the said generalization we obtain
principally new reductions of a number of nonlinear heat conductivity equations
with poor Lie symmetry and obtain their exact solutions.
It is shown that these solutions can not be constructed by means of the
symmetry reduction procedure.Comment: 12 pages, latex, needs amssymb., to appear in the "Journal of Physics
A: Mathematical and General" (1995
On the new approach to variable separation in the time-dependent Schr\"odinger equation with two space dimensions
We suggest an effective approach to separation of variables in the
Schr\"odinger equation with two space variables. Using it we classify
inequivalent potentials such that the corresponding Schr\" odinger
equations admit separation of variables. Besides that, we carry out separation
of variables in the Schr\" odinger equation with the anisotropic harmonic
oscillator potential and obtain a complete list of
coordinate systems providing its separability. Most of these coordinate systems
depend essentially on the form of the potential and do not provide separation
of variables in the free Schr\" odinger equation ().Comment: 21 pages, latex, to appear in the "Journal of Mathematical Physics"
(1995
Group classification of (1+1)-Dimensional Schr\"odinger Equations with Potentials and Power Nonlinearities
We perform the complete group classification in the class of nonlinear
Schr\"odinger equations of the form
where is an arbitrary
complex-valued potential depending on and is a real non-zero
constant. We construct all the possible inequivalent potentials for which these
equations have non-trivial Lie symmetries using a combination of algebraic and
compatibility methods. The proposed approach can be applied to solving group
classification problems for a number of important classes of differential
equations arising in mathematical physics.Comment: 10 page
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