Kurdachenko, L. A., J. Otal, I. Ya. Subbotin.
On permutable fuzzy subgroups
(pp. 83−102)
A B S T R A C T S
COMPARISON AND OSCILLATION THEOREMS FOR SECOND ORDER NONLINEAR NEUTRAL DIFFERENTIAL EQUATIONS OF MIXED TYPE
Ethiraju Thandapani
ethandapani@yahoo.co.in,
Renu Rama
renurama68@gmail.com
2000 Mathematics Subject Classification:
34C15.
Key words:
oscillation theorems, nonlinear, neutral differential equations.
In this paper we establish some comparison theorems for the oscillation of second order neutral differential equations of mixed type
(a(t) ([x(t)+b(t)x(tσ_{1})+c(t)x(t+σ_{2})]
^{α})′)′+q(t) x^{β} (tτ_{1})+p(t)x^{γ}(t+τ_{2}) = 0 

where α, β and γ are ratios of odd positive integers,
σ_{1}, σ_{2}, τ_{1} and τ_{2} are non negative integers. Our results are new even if p(t) = 0 or q(t) = 0. Examples
are provided to illustrate the main results.
SOLUTION STATIONNAIRE DU SYSTÈME
D'ÉQUATIONS DE LA RADIATION ET DE LA TEMPÉRATURE DANS
L'AIR
Nawel Messaadia,
Hisao Fujita Yashima
hisao.fujitayashima@unito.it
2000 Mathematics Subject Classification:
35Q79 (35J61, 86A10).
Key words:
Radiation, air température, integrodifferential equation system.
We consider the integrodifferential equation
system which describes the intensity of the radiation and the
temperature of the air in a domain of R^{3}. Supposing
that the coefficients of absorbation and emission of radiation are
small even if they depend on the position, we prove the existence
of a stationary solution of the equation system in a bounded
domain.
PELL FORM AND PELL EQUATION VIA OBLONG
NUMBERS
Ahmet Tekcan
tekcan@uludag.edu.tr,
Arzu Özkoç
aozkoc@uludag.edu.tr
2000 Mathematics Subject Classification:
11D09, 11D25, 11D41, 11D72, 11E16, 11E18.
Key words:
Binary quadratic form, cycle, proper cycle, Pell form, Pell
equation, oblong number, simple finite continued fraction expansion.
Let k ≥ be an integer. Then it is known that oblong (pronic)
numbers are numbers of the form k(k+1)
which is denoted by O_{k}. In this work, we derive some
algebraic relations on the Pell form
F_{Δk}(x,y) = x^{2}O_{k}y^{2} of discriminant Δ_{k} = 4O_{k}
including cycle, proper cycle, reduction and proper automorphism
of it. Also we determine the integer solutions of the Pell
equation F_{Δk}(x,y) = 1 via oblong numbers O_{k}.
LARGE AND MODERATE DEVIATION PRINCIPLES FOR RECURSIVE KERNEL DENSITY ESTIMATORS DEFINED BY STOCHASTIC APPROXIMATION METHOD
Yousri Slaoui
Yousri.Slaoui@math.univpoitiers.fr
2000 Mathematics Subject Classification:
62G07, 62L20, 60F10.
Key words:
Density estimation, stochastic approximation algorithm, large and moderate deviations principles.
In this paper we prove large and moderate deviations principles
for the recursive kernel estimators of a probability density function defined by the stochastic approximation algorithm introduced by Mokkadem et al. [The stochastic approximation method for the estimation of a probability density. J. Statist. Plann. Inference 139 (2009), 2459−2478]. We show that the estimator constructed using the stepsize which minimize the variance of the class of the recursive estimators defined in [The stochastic approximation method for the estimation of a probability density. J. Statist. Plann. Inference 139 (2009), 2459−2478] gives the same pointwise LDP and MDP as the Rosenblatt kernel estimator. We provide results both for the pointwise and the uniform deviations.
ON PERMUTABLE FUZZY SUBGROUPS
Leonid A. Kurdachenko
lkurdachenko@i.ua,
Javier Otal
otal@unizar.es,
Igor Ya. Subbotin
isubboti@nu.edu,
2000 Mathematics Subject Classification:
Primary 20N25; Secondary 08A72, 03E72.
Key words:
Fuzzy group, fuzzy subgroup,
permutable fuzzy subgroup, level subgroup, support subgroup, characteristic
function.
A fuzzy subgroup κ of a fuzzy group γ on a group G is said
to be permutable in γ if λ⊙κ=κ⊙
λ for every fuzzy subgroup λ of γ. Here μ⊙ν
stands for the product of two fuzzy groups μ and ν on G, that is
(μ⊙ν)(x) = ∨{μ(y)∧ν(z)  y,z∈ G and x=yz}. In this paper, largely extending some previous results, we characterize
the permutability of fuzzy subgroups in terms of the level subgroups and the
support subgroups. We obtain these results emphasizing the role of the
characteristic functions of elements of the group. We also show the
remarkable fact that the (abstract) subgroups of a group having a fuzzy
group whose fuzzy subgroups are permutable, are permutable as well.
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