154 research outputs found
Integrating factors for second order ODEs
A systematic algorithm for building integrating factors of the form mu(x,y),
mu(x,y') or mu(y,y') for second order ODEs is presented. The algorithm can
determine the existence and explicit form of the integrating factors themselves
without solving any differential equations, except for a linear ODE in one
subcase of the mu(x,y) problem. Examples of ODEs not having point symmetries
are shown to be solvable using this algorithm. The scheme was implemented in
Maple, in the framework of the "ODEtools" package and its ODE-solver. A
comparison between this implementation and other computer algebra ODE-solvers
in tackling non-linear examples from Kamke's book is shown.Comment: 21 pages - original version submitted Nov/1997. Related Maple
programs for finding integrating factors together with the ODEtools package
(versions for MapleV R4 and MapleV R5) are available at
http://lie.uwaterloo.ca/odetools.ht
Solutions for the General, Confluent and Biconfluent Heun equations and their connection with Abel equations
In a recent paper, the canonical forms of a new multi-parameter class of Abel
differential equations, so-called AIR, all of whose members can be mapped into
Riccati equations, were shown to be related to the differential equations for
the hypergeometric 2F1, 1F1 and 0F1 functions. In this paper, a connection
between the AIR canonical forms and the Heun General (GHE), Confluent (CHE) and
Biconfluent (BHE) equations is presented. This connection fixes the value of
one of the Heun parameters, expresses another one in terms of those remaining,
and provides closed form solutions in terms of pFq functions for the resulting
GHE, CHE and BHE, respectively depending on four, three and two irreducible
parameters. This connection also turns evident what is the relation between the
Heun parameters such that the solutions admit Liouvillian form, and suggests a
mechanism for relating linear equations with N and N-1 singularities through
the canonical forms of a non-linear equation of one order less.Comment: Original version submitted to Journal of Physics A: 16 pages, related
to math.GM/0002059 and math-ph/0402040. Revised version according to
referee's comments: 23 pages. Sign corrected (June/17) in formula (79).
Second revised version (July/25): 25 pages. See also
http://lie.uwaterloo.ca/odetools.ht
Incomplete beta-function expansions of the solutions to the confluent Heun equation
Several expansions of the solutions to the confluent Heun equation in terms
of incomplete Beta functions are constructed. A new type of expansion involving
certain combinations of the incomplete Beta functions as expansion functions is
introduced. The necessary and sufficient conditions when the derived expansions
are terminated, thus generating closed-form solutions, are discussed. It is
shown that termination of a Beta-function series solution always leads to a
solution that is necessarily an elementary function
Caracterización fisicoquímica de mieles monoflorales de Euphorbia resinifera
The physicochemical characteristics of Euphorbia resinifera honey were studied. Considering the low
water content, the majority of the honeys presented a proper maturity. The values of acidity
revealed the absence of inappropriate fermentation, while the low values of hydroxymethylfurfural
(0.4–16.48 mg/kg) were suitable for of unprocessed honeys. The average values for electrical
conductivity and ashes were 451 µS/cm and 1.6 g/kg, respectively. As for the mineral content, the
K was the most abundant element; Ca, Na, Mg, P, S, and Si are all present in differing quantities in
the honeys. On the other hand, Principal Components Analysis (PCA) and Stepwise Discriminant
Analysis (SDA) were applied to distinguish between three related Euphorbia honey types. PCA
showed that the cumulative variance of the two first factors explained approximately 53%. The
results of SDA showed that variables with a higher discriminant power were K, C*ab and a*, and
100% of the samples were properly classified in their corresponding class.Los parámetros fisicoquímicos de 29 de mieles monoflorales de Euphorbia resinifera fueron estudiadas.
24 parámetros, incluyendo humedad, pH, acidez (libre, lactónica y total), HMF, cenizas, conductividad
eléctrica, monosacáridos (glucosa y fructosa), contenido mineral y parámetros cromáticos fueron
analizados. Desde el punto de vista de su calidad las mieles fueron acordes con la legislación
Europea en cuanto a contenido en agua, acidez y HMF. Los valores de cenizas y conductividad
eléctrica fueron 1,6 g/kg y 451 μS/cm, respectivamente. El contenido en minerales mostró que el K es
el elemento más abundante; mientras que Ca, Na, Mg, P, S y Si se presentaron en contenidos
intermedios. En cuanto a los valores de los parámetros del color fueron típicos de mieles ámbar
claras. Se ha realizado un análisis estadístico multivariante a los datos obtenidos para diferenciar tres
especies de mieles de Euphorbia. El análisis discriminante permite diferencia las mieles por su origen
botánico siendo el contenido en K, C*ab y la variable cromática a* las variables con mayor poder
discriminate, siendo el 100% de las muestras clasificadas correctamente en su grupo
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