493,806 research outputs found
EQUATIONS ON PARTIAL WORDS
It is well-known that some of the most basic properties of words, like the commutativity (xy = yx) and the conjugacy (xz = zy), can be expressed as solutions of word equations. An important problem is to decide whether or not a given equation on words has a solution. For instance, the equation xMyN = zP has only periodic solutions in a free monoid, that is, if xMyN = zP holds with integers m,n,p = 2, then there exists a word w such that x, y, z are powers of w. This result, which received a lot of attention, was first proved by Lyndon and SchĂŒtzenberger for free groups. In this paper, we investigate equations on partial words. Partial words are sequences over a finite alphabet that may contain a number of âdo not knowâ symbols. When we speak about equations on partial words, we replace the notion of equality (=) with compatibility (?). Among other equations, we solve xy ? yx, xz ? zy, and special cases of xmyn ? zp for integers m,n,p = 2.
Compatible flat metrics
We solve the problem of description for nonsingular pairs of compatible flat
metrics in the general N-component case. The integrable nonlinear partial
differential equations describing all nonsingular pairs of compatible flat
metrics (or, in other words, nonsingular flat pencils of metrics) are found and
integrated. The integrating of these equations is based on reducing to a
special nonlinear differential reduction of the Lame equations and using the
Zakharov method of differential reductions in the dressing method (a version of
the inverse scattering method).Comment: 30 page
On some stochastic singular integro-partial differential equations and the parabolic transform
Some stochastic singular integro-partial differential equations are studied without any restrictions on the characteristic forms of the partial differential operators. Linear and nonlinear cases are studied. Using the parabolic transform, existence and stability results are obtained. The Cauchy problem of fractional stochastic partial differential equations can be considered as a special case from the obtained results. Key words: Singular integral equations, Stochastic partial differential equations, Existence and stability of solutions, Fractional stochastic partial differential equations
Ergodic BSDEs under weak dissipative assumptions
In this paper we study ergodic backward stochastic differential equations
(EBSDEs) dropping the strong dissipativity assumption needed in the previous
work. In other words we do not need to require the uniform exponential decay of
the difference of two solutions of the underlying forward equation, which, on
the contrary, is assumed to be non degenerate. We show existence of solutions
by use of coupling estimates for a non-degenerate forward stochastic
differential equations with bounded measurable non-linearity. Moreover we prove
uniqueness of "Markovian" solutions exploiting the recurrence of the same class
of forward equations. Applications are then given to the optimal ergodic
control of stochastic partial differential equations and to the associated
ergodic Hamilton-Jacobi-Bellman equations
Singular limit of an integrodifferential system related to the entropy balance
A thermodynamic model describing phase transitions with thermal memory, in
terms of an entropy equation and a momentum balance for the microforces, is
adressed. Convergence results and error estimates are proved for the related
integrodifferential system of PDE as the sequence of memory kernels converges
to a multiple of a Dirac delta, in a suitable sense.Comment: Key words: entropy equation, thermal memory, phase field model,
nonlinear partial differential equations, asymptotics on the memory ter
Global existence for a hydrogen storage model with full energy balance
A thermo-mechanical model describing hydrogen storage by use of metal
hydrides has been recently proposed in a paper by Bonetti, Fr\'emond and
Lexcellent. It describes the formation of hydrides using the phase transition
approach. By virtue of the laws of continuum thermo-mechanics, the model leads
to a phase transition problem in terms of three state variables: the
temperature, the phase parameter representing the fraction of one solid phase,
and the pressure, and is derived within a generalization of the principle of
virtual powers proposed by Fr\'emond accounting for micro-forces, responsible
for the phase transition, in the whole energy balance of the system. Three
coupled nonlinear partial differential equations combined with initial and
boundary conditions have to be solved. The main difficulty in investigating the
resulting system of partial differential equations relies on the presence of
the squared time derivative of the order parameter in the energy balance
equation. Here, the global existence of a solution to the full problem is
proved by exploiting known and sharp estimates on parabolic equations with
right hand side in L^1. Some complementary results on stability and steady
state solutions are also given.Comment: Key-words: phase transition model; hydrogen storage; nonlinear
parabolic system; existenc
On The Component Analysis and Transformation of An Explicit Fourth â Stage Fourth â Order Runge â Kutta Methods
This work is designed to transform the fourth stage â fourth order explicit Runge-Kutta method with the aim of projecting a new method of implementing it through tree diagram analysis. Efforts will be made to represent the equations derived from the y derivatives and x,y derivatives separately on Butcherâs rooted trees. This idea is derivable from general graphs and combinatorics. Key words: Rooted tree diagram, Transformation, Vertex, explicit, y partial derivatives, x,y partial derivatives, Runge-Kutta Methods, Linear and non- linear equations, Taylor series, Graphs. Keywords: key words, Runge - Kutta, Combinatorics, Derivatives, Transformation, Analysi
- âŠ